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Inmathematics,finite field arithmeticisarithmeticin afinite field(afieldcontaining a finite number ofelements) contrary to arithmetic in a field with an infinite number of elements, like the field ofrational numbers.
There are infinitely many different finite fields. Theirnumber of elementsis necessarily of the formpnwherepis aprime numberandnis apositive integer, and two finite fields of the same size areisomorphic. The primepis called thecharacteristicof the field, and the positive integernis called thedimensionof the field over itsprime field.
Finite fields are used in a variety of applications, including in classicalcoding theoryinlinear block codessuch asBCH codesandReed–Solomon error correction, incryptographyalgorithms such as theRijndael(AES) encryption algorithm, in tournament scheduling, and in thedesign of experiments.
The finite field withpnelements is denoted GF(pn) and is also called theGalois fieldof orderpn, in honor of the founder of finite field theory,Évariste Galois. GF(p), wherepis a prime number, is simply theringof integersmodulop. That is, one can perform operations (addition, subtraction, multiplication) using the usual operation on integers, followed by reduction modulop. For instance, in GF(5),4 + 3 = 7is reduced to 2 modulo 5. Division is multiplication by the inverse modulop, which may be computed using theextended Euclidean algorithm.
A particular case isGF(2), where addition isexclusive OR(XOR) and multiplication isAND. Since the only invertible element is 1, division is theidentity function.
Elements of GF(pn) may be represented aspolynomialsof degree strictly less thannover GF(p). Operations are then performed modulom(x)wherem(x)is anirreducible polynomialof degreenover GF(p), for instance usingpolynomial long division. Addition is the usual addition of polynomials, but the coefficients are reduced modulop. Multiplication is also the usual multiplication of polynomials, but with coefficients multiplied modulopand polynomials multiplied modulo the polynomialm(x).[1]This representation in terms of polynomial coefficients is called amonomial basis(a.k.a. 'polynomial basis').
There are other representations of the elements of GF(pn); some are isomorphic to the polynomial representation above and others look quite different (for instance, using matrices). Using anormal basismay have advantages in some contexts.
When the prime is 2, it is conventional to express elements of GF(pn) asbinary numbers, with the coefficient of each term in a polynomial represented by one bit in the corresponding element's binary expression. Braces ( "{" and "}" ) or similar delimiters are commonly added to binary numbers, or to their hexadecimal equivalents, to indicate that the value gives the coefficients of a basis of a field, thus representing an element of the field. For example, the following are equivalent representations of the same value in a characteristic 2 finite field:
There are many irreducible polynomials (sometimes calledreducing polynomials) that can be used to generate a finite field, but they do not all give rise to the same representation of the field.
Amonicirreducible polynomialof degreenhaving coefficients in the finite field GF(q), whereq=ptfor some primepand positive integert, is called aprimitive polynomialif all of its roots areprimitive elementsof GF(qn).[2][3]In the polynomial representation of the finite field, this implies thatxis a primitive element. There is at least one irreducible polynomial for whichxis a primitive element.[4]In other words, for a primitive polynomial, the powers ofxgenerate every nonzero value in the field.
In the following examples it is best not to use the polynomial representation, as the meaning ofxchanges between the examples. The monic irreducible polynomialx8+x4+x3+x+ 1overGF(2)is not primitive. Letλbe a root of this polynomial (in the polynomial representation this would bex), that is,λ8+λ4+λ3+λ+ 1 = 0. Nowλ51= 1, soλis not a primitive element of GF(28) and generates a multiplicative subgroup of order 51.[5]The monic irreducible polynomialx8+x4+x3+x2+ 1overGF(2)is primitive, and all 8 roots are generators ofGF(28).
All GF(28) have a total of 128 generators (seeNumber of primitive elements), and for a primitive polynomial, 8 of them are roots of the reducing polynomial. Havingxas a generator for a finite field is beneficial for many computational mathematical operations.
Addition and subtraction are performed by adding or subtracting two of these polynomials together, and reducing the result modulo the characteristic.
In a finite field with characteristic 2, addition modulo 2, subtraction modulo 2, and XOR are identical. Thus,
Under regular addition of polynomials, the sum would contain a term 2x6. This term becomes 0x6and is dropped when the answer is reduced modulo 2.
Here is a table with both the normal algebraic sum and the characteristic 2 finite field sum of a few polynomials:
In computer science applications, the operations are simplified for finite fields of characteristic 2, also called GF(2n)Galois fields, making these fields especially popular choices for applications.
Multiplication in a finite field is multiplicationmoduloanirreduciblereducing polynomial used to define the finite field. (I.e., it is multiplication followed by division using the reducing polynomial as the divisor—the remainder is the product.) The symbol "•" may be used to denote multiplication in a finite field.
Rijndael(standardised as AES) uses the characteristic 2 finite field with 256 elements, which can also be called the Galois field GF(28). It employs the following reducing polynomial for multiplication:
For example, {53} • {CA} = {01} in Rijndael's field because
and
The latter can be demonstrated throughlong division(shown using binary notation, since it lends itself well to the task. Notice thatexclusive ORis applied in the example and not arithmetic subtraction, as one might use in grade-school long division.):
(The elements {53} and {CA} aremultiplicative inversesof one another since their product is1.)
Multiplication in this particular finite field can also be done using a modified version of the "peasant's algorithm". Each polynomial is represented using the same binary notation as above. Eight bits is sufficient because only degrees 0 to 7 are possible in the terms of each (reduced) polynomial.
This algorithm uses threevariables(in the computer programming sense), each holding an eight-bit representation.aandbare initialized with the multiplicands;paccumulates the product and must be initialized to 0.
At the start and end of the algorithm, and the start and end of each iteration, thisinvariantis true:ab+pis the product. This is obviously true when the algorithm starts. When the algorithm terminates,aorbwill be zero sopwill contain the product.
This algorithm generalizes easily to multiplication over other fields of characteristic 2, changing the lengths ofa,b, andpand the value0x1bappropriately.
Themultiplicative inversefor an elementaof a finite field can be calculated a number of different ways:
When developing algorithms for Galois field computation on small Galois fields, a common performance optimization approach is to find ageneratorgand use the identity:
to implement multiplication as a sequence of table look ups for the logg(a) andgyfunctions and an integer addition operation. This exploits the property that every finite field contains generators. In the Rijndael field example, the polynomialx+ 1(or {03}) is one such generator. A necessary but not sufficient condition for a polynomial to be a generator is to beirreducible.
An implementation must test for the special case ofaorbbeing zero, as the product will also be zero.
This same strategy can be used to determine the multiplicative inverse with the identity:
Here, theorderof the generator, |g|, is the number of non-zero elements of the field. In the case of GF(28) this is28− 1 = 255. That is to say, for the Rijndael example:(x+ 1)255= 1. So this can be performed with two look up tables and an integer subtract. Using this idea for exponentiation also derives benefit:
This requires two table look ups, an integer multiplication and an integer modulo operation. Again a test for the special casea= 0must be performed.
However, in cryptographic implementations, one has to be careful with such implementations since thecache architectureof many microprocessors leads to variable timing for memory access. This can lead to implementations that are vulnerable to atiming attack.
For binary fields GF(2n), field multiplication can be implemented using a carryless multiply such asCLMUL instruction set, which is good forn≤ 64. A multiplication uses one carryless multiply to produce a product (up to 2n− 1 bits), another carryless multiply of a pre-computed inverse of the field polynomial to produce a quotient = ⌊product / (field polynomial)⌋, a multiply of the quotient by the field polynomial, then an xor: result = product ⊕ ((field polynomial) ⌊product / (field polynomial)⌋). The last 3 steps (pclmulqdq, pclmulqdq, xor) are used in theBarrett reductionstep for fast computation of CRC using thex86pclmulqdq instruction.[8]
Whenkis acomposite number, there will existisomorphismsfrom a binary field GF(2k) to an extension field of one of its subfields, that is, GF((2m)n) wherek=mn. Utilizing one of these isomorphisms can simplify the mathematical considerations as the degree of the extension is smaller with the trade off that the elements are now represented over a larger subfield.[9]To reduce gate count for hardware implementations, the process may involve multiple nesting, such as mapping from GF(28) to GF(((22)2)2).[10]
Here is someCcode which will add and multiply numbers in the characteristic 2 finite field of order 28, used for example by Rijndael algorithm or Reed–Solomon, using theRussian peasant multiplication algorithm:
This example hascache, timing, and branch prediction side-channelleaks, and is not suitable for use in cryptography.
ThisDprogram will multiply numbers in Rijndael's finite field and generate aPGMimage:
This example does not use any branches or table lookups in order to avoid side channels and is therefore suitable for use in cryptography.
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Inmathematics, more specificallyabstract algebra, afinite ringis aringthat has a finite number of elements.
Everyfinite fieldis an example of a finite ring, and the additive part of every finite ring is an example of anabelianfinite group, but the concept of finite rings in their own right has a more recent history.
Although rings have more structure than groups do, the theory of finite rings is simpler than that of finite groups. For instance, theclassification of finite simple groupswas one of the major breakthroughs of 20th century mathematics, its proof spanning thousands of journal pages. On the other hand, it has been known since 1907 that any finitesimple ringis isomorphic to the ringMn(Fq){\displaystyle \mathrm {M} _{n}(\mathbb {F} _{q})}– then-by-nmatrices over a finite field of orderq(as a consequence of Wedderburn's theorems, described below).
The number of rings withmelements, forma natural number, is listed underOEIS:A027623in theOn-Line Encyclopedia of Integer Sequences.
The theory offinite fieldsis perhaps the most important aspect of finite ring theory due to its intimate connections withalgebraic geometry,Galois theoryandnumber theory. An important, but fairly old aspect of the theory is the classification of finite fields:[1]
Despite the classification, finite fields are still an active area of research, including recent results on theKakeya conjectureand open problems regarding the size of smallestprimitive roots(in number theory).
A finite fieldFmay be used to build avector spaceof dimensionnoverF. Thematrix ringAofn-by-nmatrices with elements fromFis used inGalois geometry, with theprojective linear groupserving as themultiplicative groupofA.
Wedderburn's little theoremasserts that any finitedivision ringis necessarily commutative:
Nathan Jacobsonlater discovered yet another condition that guarantees commutativity of a ring: if for every elementrofRthere exists an integern> 1such thatrn=r, thenRis commutative.[2]More general conditions that imply commutativity of a ring are also known.[3]
Yet another theorem by Wedderburn has, as its consequence, a result demonstrating that the theory of finitesimple ringsis relatively straightforward in nature. More specifically, any finite simple ring is isomorphic to the ringMn(Fq){\displaystyle \mathrm {M} _{n}(\mathbb {F} _{q})}, then-by-nmatrices over a finite field of orderq. This follows from two theorems ofJoseph Wedderburnestablished in 1905 and 1907 (one of which is Wedderburn's little theorem).
In 1964David Singmasterproposed the following problem in theAmerican Mathematical Monthly: "(1) What is the order of the smallest non-trivial ring with identity which is not a field? Find two such rings with this minimal order. Are there more? (2) How many rings of order four are there?"
One can find the solution by D.M. Bloom in a two-page proof[4]that there are eleven rings (meaningrngs) of order 4, four of which have a multiplicative identity. Indeed, four-element rings introduce the complexity of the subject. There are three rings over thecyclic groupC4and eight rings over theKlein four-group. There is an interesting display of the discriminatory tools (nilpotents,zero-divisors,idempotents, and left- and right-identities) in Gregory Dresden's lecture notes.[5]
The occurrence ofnon-commutativityin finite rings was described in (Eldridge 1968) in two theorems: If the ordermof a finite ring with 1 has acube-freefactorization, then it iscommutative. And if anon-commutativefinite ring with 1 has the order of a prime cubed, then the ring is isomorphic to the upper triangular 2 × 2 matrix ring over the Galois field of the prime.
The study of rings of order the cube of a prime was further developed in (Raghavendran 1969) and (Gilmer & Mott 1973). Next Flor and Wessenbauer (1975) made improvements on the cube-of-a-prime case. Definitive work on the isomorphism classes came with (Antipkin & Elizarov 1982) proving that forp> 2, the number of classes is 3p+ 50.
There are earlier references in the topic of finite rings, such as Robert Ballieu[6]and Scorza.[7]
These are a few of the facts that are known about the number of finite rings (not necessarily with unity) of a given order (supposepandqrepresent distinct prime numbers):
The number of rings (meaningrngs) withnelements are (witha(0) = 1)
For rings with identity (rings with 1), starting fromn= 1, we get
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In mathematics,Galois ringsare a type offinitecommutative ringswhich generalize both thefinite fieldsand therings of integers moduloaprime power. A Galois ring is constructed from the ringZ/pnZ{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }similar to how a finite fieldFpr{\displaystyle \mathbb {F} _{p^{r}}}is constructed fromFp{\displaystyle \mathbb {F} _{p}}. It is aGalois extensionofZ/pnZ{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }, when the concept of a Galois extension is generalized beyond the context offields.
Galois rings were studied byKrull(1924),[1]and independently by Janusz (1966)[2]and by Raghavendran (1969),[3]who both introduced the nameGalois ring. They are named afterÉvariste Galois, similar toGalois fields, which is another name for finite fields. Galois rings have found applications incoding theory, where certain codes are best understood aslinear codesoverZ/4Z{\displaystyle \mathbb {Z} /4\mathbb {Z} }using Galois rings GR(4,r).[4][5]
A Galois ring is a commutative ring ofcharacteristicpnwhich haspnrelements, wherepis prime andnandrare positive integers. It is usually denoted GR(pn,r). It can be defined as aquotient ring
wheref(x)∈Z[x]{\displaystyle f(x)\in \mathbb {Z} [x]}is amonic polynomialof degreerwhich isirreduciblemodulop.[6][7]Up to isomorphism, the ring depends only onp,n, andrand not on the choice offused in the construction.[8]
The simplest examples of Galois rings are important special cases:
A less trivial example is the Galois ring GR(4, 3). It is of characteristic 4 and has 43= 64 elements. One way to construct it isZ[x]/(4,x3+2x2+x−1){\displaystyle \mathbb {Z} [x]/(4,x^{3}+2x^{2}+x-1)}, or equivalently,(Z/4Z)[ξ]{\displaystyle (\mathbb {Z} /4\mathbb {Z} )[\xi ]}whereξ{\displaystyle \xi }is a root of the polynomialf(x)=x3+2x2+x−1{\displaystyle f(x)=x^{3}+2x^{2}+x-1}. Although any monic polynomial of degree 3 which is irreducible modulo 2 could have been used, this choice offturns out to be convenient because
in(Z/4Z)[x]{\displaystyle (\mathbb {Z} /4\mathbb {Z} )[x]}, which makesξ{\displaystyle \xi }a 7throot of unityin GR(4, 3). The elements of GR(4, 3) can all be written in the forma2ξ2+a1ξ+a0{\displaystyle a_{2}\xi ^{2}+a_{1}\xi +a_{0}}where each ofa0,a1, anda2is inZ/4Z{\displaystyle \mathbb {Z} /4\mathbb {Z} }. For example,ξ3=2ξ2−ξ+1{\displaystyle \xi ^{3}=2\xi ^{2}-\xi +1}andξ4=2ξ3−ξ2+ξ=−ξ2−ξ+2{\displaystyle \xi ^{4}=2\xi ^{3}-\xi ^{2}+\xi =-\xi ^{2}-\xi +2}.[4]
Every Galois ring GR(pn,r) has aprimitive (pr– 1)-th root of unity. It is the equivalence class ofxin the quotientZ[x]/(pn,f(x)){\displaystyle \mathbb {Z} [x]/(p^{n},f(x))}whenfis chosen to be aprimitive polynomial. This means that, in(Z/pnZ)[x]{\displaystyle (\mathbb {Z} /p^{n}\mathbb {Z} )[x]}, the polynomialf(x){\displaystyle f(x)}dividesxpr−1−1{\displaystyle x^{p^{r}-1}-1}and does not dividexm−1{\displaystyle x^{m}-1}for allm<pr– 1. Such anfcan be computed by starting with aprimitive polynomialof degreerover the finite fieldFp{\displaystyle \mathbb {F} _{p}}and usingHensel lifting.[9]
A primitive (pr– 1)-th root of unityξ{\displaystyle \xi }can be used to express elements of the Galois ring in a useful form called thep-adic representation. Every element of the Galois ring can be written uniquely as
where eachαi{\displaystyle \alpha _{i}}is in the set{0,1,ξ,ξ2,...,ξpr−2}{\displaystyle \{0,1,\xi ,\xi ^{2},...,\xi ^{p^{r}-2}\}}.[7][9]
Every Galois ring is alocal ring. The uniquemaximal idealis theprincipal ideal(p)=pGR(pn,r){\displaystyle (p)=p\operatorname {GR} (p^{n},r)}, consisting of all elements which are multiples ofp. Theresidue fieldGR(pn,r)/(p){\displaystyle \operatorname {GR} (p^{n},r)/(p)}is isomorphic to the finite field of orderpr. Furthermore,(0),(pn−1),...,(p),(1){\displaystyle (0),(p^{n-1}),...,(p),(1)}are all the ideals.[6]
The Galois ring GR(pn,r) contains a uniquesubringisomorphic to GR(pn,s) for everyswhich dividesr. These are the only subrings of GR(pn,r).[10]
Theunitsof a Galois ringRare all the elements which are not multiples ofp. The group of units,R×, can be decomposed as adirect productG1×G2, as follows. The subgroupG1is the group of (pr− 1)-th roots of unity. It is acyclic groupof orderpr− 1. The subgroupG2is 1+pR, consisting of all elements congruent to 1 modulop. It is a group of orderpr(n−1), with the following structure:
This description generalizes the structure of themultiplicative group of integers modulopn, which is the caser= 1.[11]
Analogous to the automorphisms of the finite fieldFpr{\displaystyle \mathbb {F} _{p^{r}}}, theautomorphism groupof the Galois ring GR(pn,r) is a cyclic group of orderr.[12]The automorphisms can be described explicitly using thep-adic representation. Specifically, the map
(where eachαi{\displaystyle \alpha _{i}}is in the set{0,1,ξ,ξ2,...,ξpr−2}{\displaystyle \{0,1,\xi ,\xi ^{2},...,\xi ^{p^{r}-2}\}}) is an automorphism, which is called the generalizedFrobenius automorphism. Thefixed pointsof the generalized Frobenius automorphism are the elements of the subringZ/pnZ{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }. Iterating the generalized Frobenius automorphism gives all the automorphisms of the Galois ring.[13]
The automorphism group can be thought of as theGalois groupof GR(pn,r) overZ/pnZ{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }, and the ring GR(pn,r) is aGalois extensionofZ/pnZ{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }. More generally, wheneverris a multiple ofs, GR(pn,r) is a Galois extension of GR(pn,s), with Galois group isomorphic toGal(Fpr/Fps){\displaystyle \operatorname {Gal} (\mathbb {F} _{p^{r}}/\mathbb {F} _{p^{s}})}.[14][13]
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Instatisticsandcoding theory, aHamming spaceis usually the set of all2N{\displaystyle 2^{N}}binary stringsof lengthN, where different binary strings are considered to beadjacentwhen they differ only in one position. The total distance between any two binary strings is then the total number of positions at which the corresponding bits are different, called theHamming distance.[1][2]Hamming spaces are named after American mathematicianRichard Hamming, who introduced the concept in 1950.[3]They are used in the theory of coding signals and transmission.
More generally, a Hamming space can be defined over anyalphabet(set)Qas the set ofwordsof a fixed lengthNwith letters fromQ.[4][5]IfQis afinite field, then a Hamming space overQis anN-dimensionalvector spaceoverQ. In the typical, binary case, the field is thusGF(2)(also denoted byZ2).[4]
In coding theory, ifQhasqelements, then anysubsetC(usually assumed ofcardinalityat least two) of theN-dimensional Hamming space overQis called aq-arycodeof length N; the elements ofCare calledcodewords.[4][5]In the case whereCis alinear subspaceof its Hamming space, it is called alinear code.[4]A typical example of linear code is theHamming code. Codes defined via a Hamming space necessarily have the same length for every codeword, so they are calledblock codeswhen it is necessary to distinguish them fromvariable-length codesthat are defined by unique factorization on a monoid.
TheHamming distanceendows a Hamming space with ametric, which is essential in defining basic notions of coding theory such aserror detecting and error correcting codes.[4]
Hamming spaces over non-field alphabets have also been considered, especially overfinite rings(most notably overZ4) giving rise tomodulesinstead of vector spaces andring-linear codes(identified withsubmodules) instead of linear codes. The typical metric used in this case theLee distance. There exist aGray isometrybetweenZ22m{\displaystyle \mathbb {Z} _{2}^{2m}}(i.e. GF(22m)) with the Hamming distance andZ4m{\displaystyle \mathbb {Z} _{4}^{m}}(also denoted as GR(4,m)) with the Lee distance.[6][7][8]
Thislinear algebra-related article is astub. You can help Wikipedia byexpanding it.
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Inmathematics, aquasi-finite field[1]is a generalisation of afinite field. Standardlocal class field theoryusually deals withcomplete valued fieldswhose residue field isfinite(i.e.non-archimedean local fields), but the theory applies equally well when the residue field is only assumed quasi-finite.[2]
Aquasi-finite fieldis aperfect fieldKtogether with anisomorphismoftopological groups
whereKsis analgebraic closureofK(necessarily separable becauseKis perfect). Thefield extensionKs/Kis infinite, and theGalois groupis accordingly given theKrull topology. The groupZ^{\displaystyle {\widehat {\mathbb {Z} }}}is theprofinite completionofintegerswith respect to its subgroups of finite index.
This definition is equivalent to saying thatKhas a unique (necessarilycyclic) extensionKnof degreenfor each integern≥ 1, and that the union of these extensions is equal toKs.[3]Moreover, as part of the structure of the quasi-finite field, there is a generatorFnfor each Gal(Kn/K), and the generators must becoherent, in the sense that ifndividesm, the restriction ofFmtoKnis equal toFn.
The most basic example, which motivates the definition, is the finite fieldK=GF(q). It has a unique cyclic extension of degreen, namelyKn=GF(qn). The union of theKnis the algebraic closureKs. We takeFnto be theFrobenius element; that is,Fn(x) =xq.
Another example isK=C((T)), the ring offormal Laurent seriesinTover the fieldCofcomplex numbers. (These are simplyformal power seriesin which we also allow finitely many terms of negative degree.) ThenKhas a unique cyclic extension
of degreenfor eachn≥ 1, whose union is an algebraic closure ofKcalled the field ofPuiseux series, and that a generator of Gal(Kn/K) is given by
This construction works ifCis replaced by any algebraically closed fieldCofcharacteristiczero.[4]
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Inalgebra, adivision ring, also called askew field(or, occasionally, asfield[1][2]), is anontrivialringin whichdivisionby nonzero elements is defined. Specifically, it is a nontrivial ring[3]in which every nonzero elementahas amultiplicative inverse, that is, an element usually denoteda–1, such thataa–1=a–1a= 1. So, (right)divisionmay be defined asa/b=ab–1, but this notation is avoided, as one may haveab–1≠b–1a.
A commutative division ring is afield.Wedderburn's little theoremasserts that all finite division rings are commutative and thereforefinite fields.
Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".[7]In some languages, such asFrench, the word equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" (commutative field) or "corps gauche" (skew field).
All division rings aresimple. That is, they have no two-sidedidealbesides thezero idealand itself.
All fields are division rings, and every non-field division ring is noncommutative. The best known example is the ring ofquaternions. If one allows onlyrationalinstead ofrealcoefficients in the constructions of the quaternions, one obtains another division ring. In general, ifRis a ring andSis asimple moduleoverR, then, bySchur's lemma, theendomorphism ringofSis a division ring;[8]every division ring arises in this fashion from some simple module.
Much oflinear algebramay be formulated, and remains correct, formodulesover a division ringDinstead ofvector spacesover a field. Doing so, one must specify whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. In particular, every module has abasis, andGaussian eliminationcan be used. So, everything that can be defined with these tools works on division algebras.Matricesand their products are defined similarly.[citation needed]However, a matrix that is leftinvertibleneed not to be right invertible, and if it is, its right inverse can differ from its left inverse. (SeeGeneralized inverse § One-sided inverse.)
Determinantsare not defined over noncommutative division algebras. Most things that require this concept cannot be generalized to noncommutative division algebras, although generalizations such asquasideterminantsallow some results[clarification needed]to be recovered.
Working in coordinates, elements of a finite-dimensional right module can be represented by column vectors, which can be multiplied on the right by scalars, and on the left by matrices (representing linear maps); for elements of a finite-dimensional left module, row vectors must be used, which can be multiplied on the left by scalars, and on the right by matrices. The dual of a right module is a left module, and vice versa. The transpose of a matrix must be viewed as a matrix over the opposite division ringDopin order for the rule(AB)T=BTATto remain valid.
Every module over a division ring isfree; that is, it has a basis, and all bases of a modulehave the same number of elements. Linear maps between finite-dimensional modules over a division ring can be described bymatrices; the fact that linear maps by definition commute with scalar multiplication is most conveniently represented in notation by writing them on theoppositeside of vectors as scalars are. TheGaussian eliminationalgorithm remains applicable. The column rank of a matrix is the dimension of the right module generated by the columns, and the row rank is the dimension of the left module generated by the rows; the same proof as for the vector space case can be used to show that these ranks are the same and define the rank of a matrix.
Division rings are the onlyringsover which every module is free: a ringRis a division ring if and only if everyR-module isfree.[9]
Thecenterof a division ring is commutative and therefore a field.[10]Every division ring is therefore adivision algebraover its center. Division rings can be roughly classified according to whether or not they are finite dimensional or infinite dimensional over their centers. The former are calledcentrally finiteand the lattercentrally infinite. Every field is one dimensional over its center. The ring ofHamiltonian quaternionsforms a four-dimensional algebra over its center, which is isomorphic to the real numbers.
Wedderburn's little theorem: All finite division rings are commutative and thereforefinite fields. (Ernst Wittgave a simple proof.)
Frobenius theorem: The only finite-dimensional associative division algebras over the reals are the reals themselves, thecomplex numbers, and thequaternions.
Division ringsused to becalled "fields" in an older usage. In many languages, a word meaning "body" is used for division rings, in some languages designating either commutative or noncommutative division rings, while in others specifically designating commutative division rings (what we now call fields in English). A more complete comparison is found in the article onfields.
The name "skew field" has an interestingsemanticfeature: a modifier (here "skew")widensthe scope of the base term (here "field"). Thus a field is a particular type of skew field, and not all skew fields are fields.
While division rings and algebras as discussed here are assumed to have associative multiplication,nonassociative division algebrassuch as theoctonionsare also of interest.
Anear-fieldis an algebraic structure similar to a division ring, except that it has only one of the twodistributive laws.
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Inmathematics, especially inabstract algebra, aquasigroupis analgebraic structurethat resembles agroupin the sense that "division" is always possible. Quasigroups differ from groups mainly in that theassociativeandidentity elementproperties are optional. In fact, a nonempty associative quasigroup is a group.[1][2]
A quasigroup that has an identity element is called aloop.
There are at least two structurally equivalent formal definitions of quasigroup:
Thehomomorphicimageof a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations.[3]We begin with the first definition.
Aquasigroup(Q, ∗)is a non-emptysetQwith a binary operation∗(that is, amagma, indicating that a quasigroup has to satisfy the closure property), obeying theLatin square property. This states that, for eachaandbinQ, there exist unique elementsxandyinQsuch that botha∗x=b{\displaystyle a\ast x=b}y∗a=b{\displaystyle y\ast a=b}hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, orCayley table. This property ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is aLatin square.) The requirement thatxandybe unique can be replaced by the requirement that the magma becancellative.[4][a]
The unique solutions to these equations are writtenx=a\bandy=b/a. The operations '\' and '/' are called, respectively,left divisionandright division. With regard to the Cayley table, the first equation (left division) means that thebentry in thearow is in thexcolumn while the second equation (right division) means that thebentry in theacolumn is in theyrow.
Theempty setequipped with theempty binary operationsatisfies this definition of a quasigroup. Some authors accept the empty quasigroup but others explicitly exclude it.[5][6]
Given somealgebraic structure, anidentityis an equation in which all variables are tacitlyuniversally quantified, and in which alloperationsare among the primitive operations proper to the structure. Algebraic structures that satisfy axioms that are given solely by identities are called avariety. Many standard results inuniversal algebrahold only for varieties. Quasigroups form a variety if left and right division are taken as primitive.
Aright-quasigroup(Q, ∗, /)is a type(2, 2)algebra that satisfy both identities:y=(y/x)∗x{\displaystyle y=(y/x)\ast x}y=(y∗x)/x{\displaystyle y=(y\ast x)/x}
Aleft-quasigroup(Q, ∗, \)is a type(2, 2)algebra that satisfy both identities:y=x∗(x∖y){\displaystyle y=x\ast (x\backslash y)}y=x∖(x∗y){\displaystyle y=x\backslash (x\ast y)}
Aquasigroup(Q, ∗, \, /)is a type(2, 2, 2)algebra (i.e., equipped with three binary operations) that satisfy the identities:[b]y=(y/x)∗x{\displaystyle y=(y/x)\ast x}y=(y∗x)/x{\displaystyle y=(y\ast x)/x}y=x∗(x∖y){\displaystyle y=x\ast (x\backslash y)}y=x∖(x∗y){\displaystyle y=x\backslash (x\ast y)}
In other words: Multiplication and division in either order, one after the other, on the same side by the same element, have no net effect.
Hence if(Q, ∗)is a quasigroup according to the definition of the previous section, then(Q, ∗, \, /)is the same quasigroup in the sense of universal algebra. And vice versa: if(Q, ∗, \, /)is a quasigroup according to the sense of universal algebra, then(Q, ∗)is a quasigroup according to the first definition.
Aloopis a quasigroup with anidentity element; that is, an element,e, such that
It follows that the identity element,e, is unique, and that every element ofQhas uniqueleftandright inverses(which need not be the same).
A quasigroup with anidempotent elementis called apique("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given anabelian group,(A, +), taking its subtraction operation as quasigroup multiplication yields a pique(A, −)with the group identity (zero) turned into a "pointed idempotent". (That is, there is aprincipal isotopy(x,y,z) ↦ (x, −y,z).)
A loop that is associative is a group. A group can have a strictly nonassociative pique isotope, but it cannot have a strictly nonassociative loop isotope.
There are weaker associativity properties that have been given special names.
For instance, aBol loopis a loop that satisfies either:
or else
A loop that is both a left and right Bol loop is aMoufang loop. This is equivalent to any one of the following single Moufang identities holding for allx,y,z:
According to Jonathan D. H. Smith, "loops" were named after theChicago Loop, as their originators were studying quasigroups in Chicago at the time.[9]
Smith (2007)names the following important properties and subclasses:
A quasigroup issemisymmetricif any of the following equivalent identities hold for allx,y:[c]
Although this class may seem special, every quasigroupQinduces a semisymmetric quasigroupQΔ on the direct product cubeQ3via the following operation:
where "//" and "\\" are theconjugate division operationsgiven byy//x=x/yandy\\x=x\y.
A quasigroup may exhibit semisymmetrictriality.[10]
A narrower class is atotally symmetric quasigroup(sometimes abbreviatedTS-quasigroup) in which allconjugatescoincide as one operation:x∗y=x/y=x\y. Another way to define (the same notion of) totally symmetric quasigroup is as a semisymmetric quasigroup that is commutative, i.e.x∗y=y∗x.
Idempotent total symmetric quasigroups are precisely (i.e. in a bijection with)Steiner triples, so such a quasigroup is also called aSteiner quasigroup, and sometimes the latter is even abbreviated assquag. The termslooprefers to an analogue for loops, namely, totally symmetric loops that satisfyx∗x= 1instead ofx∗x=x. Without idempotency, total symmetric quasigroups correspond to the geometric notion ofextended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC).
A quasigroup(Q, ∗)is calledweakly totally anti-symmetricif for allc,x,y∈Q, the following implication holds.[11]
A quasigroup(Q, ∗)is calledtotally anti-symmetricif, in addition, for allx,y∈Q, the following implication holds:[11]
This property is required, for example, in theDamm algorithm.
Quasigroups have thecancellation property: ifab=ac, thenb=c. This follows from the uniqueness of left division ofaboracbya. Similarly, ifba=ca, thenb=c.
The Latin square property of quasigroups implies that, given any two of the three variables inxy=z, the third variable is uniquely determined.
The definition of a quasigroup can be treated as conditions on the left and rightmultiplication operatorsLx,Rx:Q→Q, defined by
The definition says that both mappings arebijectionsfromQto itself. A magmaQis a quasigroup precisely when all these operators, for everyxinQ, are bijective. The inverse mappings are left and right division, that is,
In this notation the identities among the quasigroup's multiplication and division operations (stated in the section onuniversal algebra) are
whereiddenotes the identity mapping onQ.
The multiplication table of a finite quasigroup is aLatin square: ann×ntable filled withndifferent symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
Conversely, every Latin square can be taken as the multiplication table of a quasigroup in many ways: the border row (containing the column headers) and the border column (containing the row headers) can each be any permutation of the elements. SeeSmall Latin squares and quasigroups.
For acountably infinitequasigroupQ, it is possible to imagine an infinite array in which every row and every column corresponds to some elementqofQ, and where the elementa∗bis in the row corresponding toaand the column responding tob. In this situation too, the Latin square property says that each row and each column of the infinite array will contain every possible value precisely once.
For anuncountably infinitequasigroup, such as the group of non-zeroreal numbersunder multiplication, the Latin square property still holds, although the name is somewhat unsatisfactory, as it is not possible to produce the array of combinations to which the above idea of an infinite array extends since the real numbers cannot all be written in asequence. (This is somewhat misleading however, as the reals can be written in a sequence of lengthc{\displaystyle {\mathfrak {c}}}, assuming thewell-ordering theorem.)
The binary operation of a quasigroup isinvertiblein the sense that bothLxandRx, theleft and right multiplication operators, are bijective, and henceinvertible.
Every loop element has a unique left and right inverse given by
A loop is said to have (two-sided)inversesifxλ=xρfor allx. In this case the inverse element is usually denoted byx−1.
There are some stronger notions of inverses in loops that are often useful:
A loop has theinverse propertyif it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties. In fact, any loop that satisfies any two of the above four identities has the inverse property and therefore satisfies all four.
Any loop that satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.
A quasigroup or loophomomorphismis amapf:Q→Pbetween two quasigroups such thatf(xy) =f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).
LetQandPbe quasigroups. Aquasigroup homotopyfromQtoPis a triple(α,β,γ)of maps fromQtoPsuch that
for allx,yinQ. A quasigroup homomorphism is just a homotopy for which the three maps are equal.
Anisotopyis a homotopy for which each of the three maps(α,β,γ)is abijection. Two quasigroups areisotopicif there is an isotopy between them. In terms of Latin squares, an isotopy(α,β,γ)is given by a permutation of rowsα, a permutation of columnsβ, and a permutation on the underlying element setγ.
Anautotopyis an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup forms a group with theautomorphism groupas a subgroup.
Every quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup that is isotopic to a group need not be a group. For example, the quasigroup onRwith multiplication given by(x,y) ↦ (x+y)/2is isotopic to the additive group(R, +), but is not itself a group as it has no identity element. Everymedialquasigroup is isotopic to anabelian groupby theBruck–Toyoda theorem.
Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation ∗ (i.e.,x∗y=z) we can form five new operations:xoy:=y∗x(theoppositeoperation),/and\, and their opposites. That makes a total of six quasigroup operations, which are called theconjugatesorparastrophesof ∗. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).
If the setQhas two quasigroup operations, ∗ and ·, and one of them is isotopic to a conjugate of the other, the operations are said to beisostrophicto each other. There are also many other names for this relation of "isostrophe", e.g.,paratopy.
Ann-ary quasigroupis a set with ann-ary operation,(Q,f)withf:Qn→Q, such that the equationf(x1, ...,xn) =yhas a unique solution for any one variable if all the othernvariables are specified arbitrarily.Polyadicormultiarymeansn-ary for some nonnegative integern.
A 0-ary, ornullary, quasigroup is just a constant element ofQ. A 1-ary, orunary, quasigroup is a bijection ofQto itself. Abinary, or 2-ary, quasigroup is an ordinary quasigroup.
An example of a multiary quasigroup is an iterated group operation,y=x1·x2· ··· ·xn; it is not necessary to use parentheses to specify the order of operations because the group is associative. One can also form a multiary quasigroup by carrying out any sequence of the same or different group or quasigroup operations, if the order of operations is specified.
There exist multiary quasigroups that cannot be represented in any of these ways. Ann-ary quasigroup isirreducibleif its operation cannot be factored into the composition of two operations in the following way:
where1 ≤i<j≤nand(i,j) ≠ (1,n). Finite irreduciblen-ary quasigroups exist for alln> 2; seeAkivis & Goldberg (2001)for details.
Ann-ary quasigroup with ann-ary version ofassociativityis called ann-ary group.
The number of isomorphism classes of small quasigroups (sequenceA057991in theOEIS) and loops (sequenceA057771in theOEIS) is given here:[14]
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https://en.wikipedia.org/wiki/Quasigroup#Algebra
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Inmathematics, especially inabstract algebra, aquasigroupis analgebraic structurethat resembles agroupin the sense that "division" is always possible. Quasigroups differ from groups mainly in that theassociativeandidentity elementproperties are optional. In fact, a nonempty associative quasigroup is a group.[1][2]
A quasigroup that has an identity element is called aloop.
There are at least two structurally equivalent formal definitions of quasigroup:
Thehomomorphicimageof a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations.[3]We begin with the first definition.
Aquasigroup(Q, ∗)is a non-emptysetQwith a binary operation∗(that is, amagma, indicating that a quasigroup has to satisfy the closure property), obeying theLatin square property. This states that, for eachaandbinQ, there exist unique elementsxandyinQsuch that botha∗x=b{\displaystyle a\ast x=b}y∗a=b{\displaystyle y\ast a=b}hold. (In other words: Each element of the set occurs exactly once in each row and exactly once in each column of the quasigroup's multiplication table, orCayley table. This property ensures that the Cayley table of a finite quasigroup, and, in particular, a finite group, is aLatin square.) The requirement thatxandybe unique can be replaced by the requirement that the magma becancellative.[4][a]
The unique solutions to these equations are writtenx=a\bandy=b/a. The operations '\' and '/' are called, respectively,left divisionandright division. With regard to the Cayley table, the first equation (left division) means that thebentry in thearow is in thexcolumn while the second equation (right division) means that thebentry in theacolumn is in theyrow.
Theempty setequipped with theempty binary operationsatisfies this definition of a quasigroup. Some authors accept the empty quasigroup but others explicitly exclude it.[5][6]
Given somealgebraic structure, anidentityis an equation in which all variables are tacitlyuniversally quantified, and in which alloperationsare among the primitive operations proper to the structure. Algebraic structures that satisfy axioms that are given solely by identities are called avariety. Many standard results inuniversal algebrahold only for varieties. Quasigroups form a variety if left and right division are taken as primitive.
Aright-quasigroup(Q, ∗, /)is a type(2, 2)algebra that satisfy both identities:y=(y/x)∗x{\displaystyle y=(y/x)\ast x}y=(y∗x)/x{\displaystyle y=(y\ast x)/x}
Aleft-quasigroup(Q, ∗, \)is a type(2, 2)algebra that satisfy both identities:y=x∗(x∖y){\displaystyle y=x\ast (x\backslash y)}y=x∖(x∗y){\displaystyle y=x\backslash (x\ast y)}
Aquasigroup(Q, ∗, \, /)is a type(2, 2, 2)algebra (i.e., equipped with three binary operations) that satisfy the identities:[b]y=(y/x)∗x{\displaystyle y=(y/x)\ast x}y=(y∗x)/x{\displaystyle y=(y\ast x)/x}y=x∗(x∖y){\displaystyle y=x\ast (x\backslash y)}y=x∖(x∗y){\displaystyle y=x\backslash (x\ast y)}
In other words: Multiplication and division in either order, one after the other, on the same side by the same element, have no net effect.
Hence if(Q, ∗)is a quasigroup according to the definition of the previous section, then(Q, ∗, \, /)is the same quasigroup in the sense of universal algebra. And vice versa: if(Q, ∗, \, /)is a quasigroup according to the sense of universal algebra, then(Q, ∗)is a quasigroup according to the first definition.
Aloopis a quasigroup with anidentity element; that is, an element,e, such that
It follows that the identity element,e, is unique, and that every element ofQhas uniqueleftandright inverses(which need not be the same).
A quasigroup with anidempotent elementis called apique("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given anabelian group,(A, +), taking its subtraction operation as quasigroup multiplication yields a pique(A, −)with the group identity (zero) turned into a "pointed idempotent". (That is, there is aprincipal isotopy(x,y,z) ↦ (x, −y,z).)
A loop that is associative is a group. A group can have a strictly nonassociative pique isotope, but it cannot have a strictly nonassociative loop isotope.
There are weaker associativity properties that have been given special names.
For instance, aBol loopis a loop that satisfies either:
or else
A loop that is both a left and right Bol loop is aMoufang loop. This is equivalent to any one of the following single Moufang identities holding for allx,y,z:
According to Jonathan D. H. Smith, "loops" were named after theChicago Loop, as their originators were studying quasigroups in Chicago at the time.[9]
Smith (2007)names the following important properties and subclasses:
A quasigroup issemisymmetricif any of the following equivalent identities hold for allx,y:[c]
Although this class may seem special, every quasigroupQinduces a semisymmetric quasigroupQΔ on the direct product cubeQ3via the following operation:
where "//" and "\\" are theconjugate division operationsgiven byy//x=x/yandy\\x=x\y.
A quasigroup may exhibit semisymmetrictriality.[10]
A narrower class is atotally symmetric quasigroup(sometimes abbreviatedTS-quasigroup) in which allconjugatescoincide as one operation:x∗y=x/y=x\y. Another way to define (the same notion of) totally symmetric quasigroup is as a semisymmetric quasigroup that is commutative, i.e.x∗y=y∗x.
Idempotent total symmetric quasigroups are precisely (i.e. in a bijection with)Steiner triples, so such a quasigroup is also called aSteiner quasigroup, and sometimes the latter is even abbreviated assquag. The termslooprefers to an analogue for loops, namely, totally symmetric loops that satisfyx∗x= 1instead ofx∗x=x. Without idempotency, total symmetric quasigroups correspond to the geometric notion ofextended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC).
A quasigroup(Q, ∗)is calledweakly totally anti-symmetricif for allc,x,y∈Q, the following implication holds.[11]
A quasigroup(Q, ∗)is calledtotally anti-symmetricif, in addition, for allx,y∈Q, the following implication holds:[11]
This property is required, for example, in theDamm algorithm.
Quasigroups have thecancellation property: ifab=ac, thenb=c. This follows from the uniqueness of left division ofaboracbya. Similarly, ifba=ca, thenb=c.
The Latin square property of quasigroups implies that, given any two of the three variables inxy=z, the third variable is uniquely determined.
The definition of a quasigroup can be treated as conditions on the left and rightmultiplication operatorsLx,Rx:Q→Q, defined by
The definition says that both mappings arebijectionsfromQto itself. A magmaQis a quasigroup precisely when all these operators, for everyxinQ, are bijective. The inverse mappings are left and right division, that is,
In this notation the identities among the quasigroup's multiplication and division operations (stated in the section onuniversal algebra) are
whereiddenotes the identity mapping onQ.
The multiplication table of a finite quasigroup is aLatin square: ann×ntable filled withndifferent symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
Conversely, every Latin square can be taken as the multiplication table of a quasigroup in many ways: the border row (containing the column headers) and the border column (containing the row headers) can each be any permutation of the elements. SeeSmall Latin squares and quasigroups.
For acountably infinitequasigroupQ, it is possible to imagine an infinite array in which every row and every column corresponds to some elementqofQ, and where the elementa∗bis in the row corresponding toaand the column responding tob. In this situation too, the Latin square property says that each row and each column of the infinite array will contain every possible value precisely once.
For anuncountably infinitequasigroup, such as the group of non-zeroreal numbersunder multiplication, the Latin square property still holds, although the name is somewhat unsatisfactory, as it is not possible to produce the array of combinations to which the above idea of an infinite array extends since the real numbers cannot all be written in asequence. (This is somewhat misleading however, as the reals can be written in a sequence of lengthc{\displaystyle {\mathfrak {c}}}, assuming thewell-ordering theorem.)
The binary operation of a quasigroup isinvertiblein the sense that bothLxandRx, theleft and right multiplication operators, are bijective, and henceinvertible.
Every loop element has a unique left and right inverse given by
A loop is said to have (two-sided)inversesifxλ=xρfor allx. In this case the inverse element is usually denoted byx−1.
There are some stronger notions of inverses in loops that are often useful:
A loop has theinverse propertyif it has both the left and right inverse properties. Inverse property loops also have the antiautomorphic and weak inverse properties. In fact, any loop that satisfies any two of the above four identities has the inverse property and therefore satisfies all four.
Any loop that satisfies the left, right, or antiautomorphic inverse properties automatically has two-sided inverses.
A quasigroup or loophomomorphismis amapf:Q→Pbetween two quasigroups such thatf(xy) =f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).
LetQandPbe quasigroups. Aquasigroup homotopyfromQtoPis a triple(α,β,γ)of maps fromQtoPsuch that
for allx,yinQ. A quasigroup homomorphism is just a homotopy for which the three maps are equal.
Anisotopyis a homotopy for which each of the three maps(α,β,γ)is abijection. Two quasigroups areisotopicif there is an isotopy between them. In terms of Latin squares, an isotopy(α,β,γ)is given by a permutation of rowsα, a permutation of columnsβ, and a permutation on the underlying element setγ.
Anautotopyis an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup forms a group with theautomorphism groupas a subgroup.
Every quasigroup is isotopic to a loop. If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup that is isotopic to a group need not be a group. For example, the quasigroup onRwith multiplication given by(x,y) ↦ (x+y)/2is isotopic to the additive group(R, +), but is not itself a group as it has no identity element. Everymedialquasigroup is isotopic to anabelian groupby theBruck–Toyoda theorem.
Left and right division are examples of forming a quasigroup by permuting the variables in the defining equation. From the original operation ∗ (i.e.,x∗y=z) we can form five new operations:xoy:=y∗x(theoppositeoperation),/and\, and their opposites. That makes a total of six quasigroup operations, which are called theconjugatesorparastrophesof ∗. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).
If the setQhas two quasigroup operations, ∗ and ·, and one of them is isotopic to a conjugate of the other, the operations are said to beisostrophicto each other. There are also many other names for this relation of "isostrophe", e.g.,paratopy.
Ann-ary quasigroupis a set with ann-ary operation,(Q,f)withf:Qn→Q, such that the equationf(x1, ...,xn) =yhas a unique solution for any one variable if all the othernvariables are specified arbitrarily.Polyadicormultiarymeansn-ary for some nonnegative integern.
A 0-ary, ornullary, quasigroup is just a constant element ofQ. A 1-ary, orunary, quasigroup is a bijection ofQto itself. Abinary, or 2-ary, quasigroup is an ordinary quasigroup.
An example of a multiary quasigroup is an iterated group operation,y=x1·x2· ··· ·xn; it is not necessary to use parentheses to specify the order of operations because the group is associative. One can also form a multiary quasigroup by carrying out any sequence of the same or different group or quasigroup operations, if the order of operations is specified.
There exist multiary quasigroups that cannot be represented in any of these ways. Ann-ary quasigroup isirreducibleif its operation cannot be factored into the composition of two operations in the following way:
where1 ≤i<j≤nand(i,j) ≠ (1,n). Finite irreduciblen-ary quasigroups exist for alln> 2; seeAkivis & Goldberg (2001)for details.
Ann-ary quasigroup with ann-ary version ofassociativityis called ann-ary group.
The number of isomorphism classes of small quasigroups (sequenceA057991in theOEIS) and loops (sequenceA057771in theOEIS) is given here:[14]
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https://en.wikipedia.org/wiki/Loop_(algebra)
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Anauthentication protocolis a type of computercommunications protocolorcryptographic protocolspecifically designed for transfer ofauthenticationdata between two entities. It allows the receiving entity to authenticate the connecting entity (e.g. Client connecting to a Server) as well as authenticate itself to the connecting entity (Server to a client) by declaring the type of information needed for authentication as well as syntax.[1]It is the most important layer of protection needed for secure communication within computer networks.
With the increasing amount of trustworthy information being accessible over the network, the need for keeping unauthorized persons from access to this data emerged. Stealing someone's identity is easy in the computing world - special verification methods had to be invented to find out whether the person/computer requesting data is really who he says he is.[2]The task of the authentication protocol is to specify the exact series of steps needed for execution of the authentication. It has to comply with the main protocol principles:
An illustration of password-based authentication using simple authentication protocol:
Alice (an entity wishing to be verified) and Bob (an entity verifying Alice's identity) are both aware of the protocol they agreed on using. Bob has Alice's password stored in a database for comparison.
This is an example of a very basic authentication protocol vulnerable to many threats such aseavesdropping,replay attack,man-in-the-middleattacks,dictionary attacksorbrute-force attacks. Most authentication protocols are more complicated in order to be resilient against these attacks.[4]
Protocols are used mainly byPoint-to-Point Protocol(PPP) servers to validate the identity of remote clients before granting them access to server data. Most of them use a password as the cornerstone of the authentication. In most cases, the password has to be shared between the communicating entities in advance.[5]
Password Authentication Protocolis one of the oldest authentication protocols. Authentication is initialized by the client sending a packet withcredentials(username and password) at the beginning of the connection, with the client repeating the authentication request until acknowledgement is received.[6]It is highly insecure because credentials are sent "in the clear" and repeatedly, making it vulnerable even to the most simple attacks likeeavesdroppingandman-in-the-middlebased attacks. Although widely supported, it is specified that if an implementation offers a stronger authentication method, that methodmustbe offered before PAP. Mixed authentication (e.g. the same client alternately using both PAP and CHAP) is also not expected, as the CHAP authentication would be compromised by PAP sending the password in plain-text.
The authentication process in this protocol is always initiated by the server/host and can be performed anytime during the session, even repeatedly. The server sends a random string (usually 128B long). The client uses the password and the string received as input to a hash function and then sends the result together with username in plain text. The server uses the username to apply the same function and compares the calculated and received hash. An authentication is successful when the calculated and received hashes match.
EAP was originally developed for PPP(Point-to-Point Protocol) but today is widely used inIEEE 802.3,IEEE 802.11(WiFi) orIEEE 802.16as a part ofIEEE 802.1xauthentication framework. The latest version is standardized in RFC 5247. The advantage of EAP is that it is only a general authentication framework for client-server authentication - the specific way of authentication is defined in its many versions called EAP-methods. More than 40 EAP-methods exist, the most common are:
Complex protocols used in larger networks for verifying the user (Authentication), controlling access to server data (Authorization) and monitoring network resources and information needed for billing of services (Accounting).
The oldest AAA protocol using IP based authentication without any encryption (usernames and passwords were transported as plain text). Later version XTACACS (Extended TACACS) added authorization and accounting. Both of these protocols were later replaced by TACACS+. TACACS+ separates the AAA components thus they can be segregated and handled on separate servers (It can even use another protocol for e.g. Authorization). It usesTCP(Transmission Control Protocol) for transport and encrypts the whole packet. TACACS+ is Cisco proprietary.
Remote Authentication Dial-In User Service(RADIUS) is a fullAAA protocolcommonly used byISPs. Credentials are mostly username-password combination based, and it usesNASandUDPprotocol for transport.[7]
Diameter (protocol)evolved from RADIUS and involves many improvements such as usage of more reliable TCP orSCTPtransport protocol and higher security thanks toTLS.[8]
Kerberos is a centralized network authentication system developed atMITand available as a free implementation from MIT but also in many commercial products. It is the default authentication method inWindows 2000and later. The authentication process itself is much more complicated than in the previous protocols - Kerberos usessymmetric key cryptography, requires atrusted third partyand can usepublic-key cryptographyduring certain phases of authentication if need be.[9][10][11]
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https://en.wikipedia.org/wiki/Authentication_protocol
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Anelectronic signature, ore-signature, isdatathat is logically associated with other data and which is used by thesignatoryto sign the associated data.[1][2][3]This type of signature has the same legal standing as a handwritten signature as long as it adheres to the requirements of the specific regulation under which it was created (e.g.,eIDASin theEuropean Union,NIST-DSSin theUSAorZertESinSwitzerland).[4][5]
Electronic signatures are a legal concept distinct fromdigital signatures, a cryptographic mechanism often used to implement electronic signatures. While an electronic signature can be as simple as a name entered in an electronic document,digital signaturesare increasingly used ine-commerceand in regulatory filings to implement electronic signatures in acryptographically protectedway. Standardization agencies likeNISTorETSIprovide standards for their implementation (e.g.,NIST-DSS,XAdESorPAdES).[4][6]The concept itself is not new, withcommon lawjurisdictions having recognizedtelegraphsignatures as far back as the mid-19th century andfaxedsignatures since the 1980s.
The USA'sE-Sign Act,[7][8]signed June 30, 2000 byPresident Clintonwas described months later as "more like a seal than a signature."[9]
An electronic signature is intended to provide a secure and accurate identification method for the signatory during a transaction.
Definitions of electronic signatures vary depending on the applicablejurisdiction. A common denominator in most countries is the level of anadvanced electronic signaturerequiring that:
Electronic signatures may be created with increasing levels of security, with each having its own set of requirements and means of creation on various levels that prove the validity of the signature. To provide an even strongerprobative valuethan the above described advanced electronic signature, some countries like member states of the European Union or Switzerland introduced the qualified electronic signature. It is difficult to challenge the authorship of a statement signed with aqualified electronic signature- the statement isnon-repudiable.[11]Technically, a qualified electronic signature is implemented through an advanced electronic signature that utilizes a digital certificate, which has been encrypted through a security signature-creating device[12]and which has been authenticated by aqualified trust service provider.[13]
Since well before theAmerican Civil Warbegan in 1861,morse codewas used to send messages electrically via the telegraph. Some of these messages were agreements to terms that were intended as enforceablecontracts. An early acceptance of the enforceability of telegraphic messages as electronic signatures came from aNew Hampshire Supreme Courtcase, Howley v. Whipple, in 1869.[14][15]
In the 1980s, many companies and even some individuals began using fax machines for high-priority or time-sensitive delivery of documents. Although the original signature on the original document was on paper, the image of the signature and its transmission was electronic.[16]
Courts in various jurisdictions have decided that enforceable legality of electronic signatures can include agreements made by email, entering apersonal identification number(PIN) into a bankATM, signing a credit or debit slip with a digital pen pad device (an application ofgraphics tablettechnology) at apoint of sale, installing software with aclickwrapsoftware license agreementon the package, and signing electronic documents online.
The first agreement signed electronically by two sovereign nations was a Joint Communiqué recognizing the growing importance of the promotion of electronic commerce, signed by the United States and Ireland in 1998.[17]
In 1996 theUnited Nationspublished the UNCITRAL Model Law on Electronic Commerce.[18]Article 7 of the UNCITRAL Model Law on Electronic Commerce was highly influential in the development of electronic signature laws around the world, including in the US.[19]In 2001, UNCITRAL concluded work on a dedicated text, the UNCITRAL Model Law on Electronic Signatures,[20]which has been adopted in some 30 jurisdictions.[21]Article 9, paragraph 3 of theUnited Nations Convention on the Use of Electronic Communications in International Contracts, 2005, which establishes a mechanism for functional equivalence between electronic and handwritten signatures at the international level as well as for the cross-border recognition. The latest UNCITRAL text dealing with electronic signatures is article 16 of the UNCITRAL Model Law on the Use and Cross-border Recognition of Identity Management and Trust Services (2022).
Canadian law (PIPEDA) attempts to clarify the situation by first defining a generic electronic signature as "a signature that consists of one or more letters, characters, numbers or other symbols in digital form incorporated in, attached to or associated with an electronic document," then defining a secure electronic signature as an electronic signature with specific properties. PIPEDA's secure electronic signature regulations refine the definition as being a digital signature applied and verified in a specific manner.[22]
In theEuropean Union, EURegulationNo 910/2014 on electronic identification and trust services for electronic transactions in the Europeaninternal market(eIDAS) sets the legal frame for electronic signatures. It repealsDirective1999/93/EC.[2]The current and applicable version of eIDAS was published by theEuropean Parliamentand theEuropean Councilon July 23, 2014. Following Article 25 (1) of the eIDAS regulation, anadvanced electronic signatureshall “not be denied legal effect and admissibility as evidence in legal proceedings". However it will reach a higherprobative valuewhen enhanced to the level of aqualified electronic signature. By requiring the use of aqualified electronic signature creation device[23]and being based on a certificate that has been issued by a qualified trust service provider, the upgraded advanced signature then carries according to Article 25 (2) of the eIDAS Regulation the same legal value as a handwritten signature.[2][10]However, this is only regulated in the European Union and similarly throughZertESinSwitzerland. A qualified electronic signature is not defined in the United States.[24][25]
The U.S. Code defines an electronic signature for the purpose of US law as "an electronic sound, symbol, or process, attached to or logically associated with a contract or other record and executed or adopted by a person with the intent to sign the record."[26]It may be an electronic transmission of the document which contains the signature, as in the case offacsimiletransmissions, or it may be encoded message, such astelegraphyusingMorse code.
In the United States, the definition of what qualifies as an electronic signature is wide and is set out in theUniform Electronic Transactions Act("UETA") released by the National Conference of Commissioners on Uniform State Laws (NCCUSL) in 1999.[27]It was influenced byABAcommittee white papers and the uniform law promulgated by NCCUSL. Under UETA, the term means "an electronic sound, symbol, or process, attached to or logically associated with a record and executed or adopted by a person with the intent to sign the record." This definition and many other core concepts of UETA are echoed in the U.S.ESign Actof 2000.[26]48 US states, the District of Columbia, and the US Virgin Islands have enacted UETA.[28]Only New York and Illinois have not enacted UETA,[28]but each of those states has adopted its own electronic signatures statute.[29][30][31]As of June 11, 2020, Washington State Office of CIO adopted UETA.[32]
In Australia, an electronic signature is recognised as "not necessarily the writing in of a name, but maybe any mark which identifies it as the act of the party.”[33]Under the Electronic Transactions Acts in each Federal, State and Territory jurisdiction, an electronic signature may be considered enforceable if (a) there was a method used to identify the person and to indicate that person’s intention in respect of the information communicated and the method was either: (i) as reliable as appropriate for the purpose for which the electronic communication was generated or communicated, in light of all the circumstances, including the relevant agreement; or (ii) proven in fact to have fulfilled the functions above by itself or together with further evidence and the person to whom the signature is required to be given consents to that method.[34]
Various laws have been passed internationally to facilitate commerce by using electronic records and signatures in interstate and foreign commerce. The intent is to ensure the validity and legal effect of contracts entered electronically. For instance,
In 2016,Aberdeen Strategy and Researchreported that 73% of "best-in-class" and 34% of all other respondents surveyed made use of electronic signature processes insupply chainandprocurement, delivering benefits in the speed and efficiency of key procurement activities. The percentages of their survey respondents using electronic signatures inaccounts payableandaccounts receivableprocesses were a little lower, 53% of "best-in-class" respondents in each case.[40]
Digital signatures arecryptographicimplementations ofelectronic signaturesused as a proof ofauthenticity,data integrityandnon-repudiationof communications conducted over theInternet. When implemented in compliance to digital signature standards, digital signing should offer end-to-end privacy with the signing process being user-friendly and secure. Digital signatures are generated and verified through standardized frameworks such as theDigital Signature Algorithm(DSA)[6][41]byNISTor in compliance to theXAdES,PAdESorCAdESstandards, specified by theETSI.[42]
There are typically three algorithms involved with the digital signature process:
The process of digital signing requires that its accompanying public key can then authenticate the signature generated by both the fixed message and private key. Using these cryptographic algorithms, the user's signature cannot be replicated without having access to their private key.[43]Asecure channelis not typically required. By applying asymmetric cryptography methods, the digital signature process prevents several common attacks where the attacker attempts to gain access through the following attack methods.[1]
The most relevant standards on digital signatures with respect to size of domestic markets are theDigital Signature Standard (DSS)[41]by theNational Institute of Standards and Technology(NIST) and theeIDASRegulation[2]enacted by theEuropean Parliament.[4]OpenPGPis a non-proprietary protocol for email encryption throughpublic key cryptography. It is supported byPGPandGnuPG, and some of theS/MIMEIETFstandards and has evolved into the most popular email encryption standard in the world.[44]
An electronic signature may also refer to electronic forms of processing or verifying identity through the use of biometric "signatures" or biologically identifying qualities of an individual. Such signatures use the approach of attaching some biometric measurement to a document as evidence. Biometric signatures include fingerprints,hand geometry(finger lengths and palm size),iris patterns,voice characteristics, retinal patterns, or any other human body property. All of these are collected using electronic sensors of some kind.
Biometric measurements of this type are useless aspasswordsbecause they can't be changed if compromised. However, they might be serviceable, except that to date, they have been so easily deceived that they can carry little assurance that the person who purportedly signed a document was actually the person who did. For example, a replay of the electronic signal produced and submitted to the computer system responsible for 'affixing' a signature to a document can be collected via wiretapping techniques.[citation needed]Many commercially available fingerprint sensors have low resolution and can be deceived with inexpensive household items (for example,gummy bearcandy gel).[45]In the case of a user's face image, researchers in Vietnam successfully demonstrated in late 2017 how a specially crafted mask could beatApple'sFace IDoniPhone X.[46]
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Authorizationorauthorisation(seespelling differences), ininformation security,computer securityandIAM(Identity and Access Management),[1]is the function of specifying rights/privileges for accessing resources, in most cases through an access policy, and then deciding whether a particularsubjecthas privilege to access a particularresource. Examples ofsubjectsinclude human users, computersoftwareand otherhardwareon the computer. Examples ofresourcesinclude individual files or an item'sdata,computer programs, computerdevicesand functionality provided bycomputer applications. For example, user accounts forhuman resourcesstaff are typically configured with authorization for accessing employee records.
Authorization is closely related toaccess control, which is what enforces the authorization policy by deciding whether access requests to resources from (authenticated) consumers shall be approved (granted) or disapproved (rejected).[2]
Authorization should not be confused withauthentication, which is the process of verifying someone's identity.
IAMconsists the following two phases: the configuration phase where a user account is created and its corresponding access authorization policy is defined, and the usage phase where user authentication takes place followed by access control to ensure that the user/consumer only gets access to resources for which they are authorized. Hence, access control incomputersystems andnetworksrelies on access authorization specified during configuration.
Authorization is the responsibility of anauthority, such as a department manager, within the application domain, but is often delegated to a custodian such as a system administrator. Authorizations are expressed as access policies in some types of "policy definition application", e.g. in the form of anaccess control listor acapability, or a policy administration point e.g.XACML.
Broken authorization is often listed as the number one risk in web applications.[3]On the basis of the "principle of least privilege", consumers should only be authorized to access whatever they need to do their jobs, and nothing more.[4]
"Anonymous consumers" or "guests", are consumers that have not been required to authenticate. They often have limited authorization. On a distributed system, it is often desirable to grant access without requiring a unique identity. Familiar examples ofaccess tokensinclude keys, certificates and tickets: they grant access without proving identity.
A widely used framework for authorizing applications isOAuth 2. It provides a standardized way for third-party applications to obtain limited access to a user's resources without exposing their credentials.[5]
In modern systems, a widely used model for authorization isrole-based access control(RBAC) where authorization is defined by granting subjects one or more roles, and then checking that the resource being accessed has been assigned at least one of those roles.[5]However, with the rise of social media,Relationship-based access controlis gaining more prominence.[6]
Even when access is controlled through a combination of authentication andaccess control lists, the problems of maintaining the authorization data is not trivial, and often represents as much administrative burden as managing authentication credentials. It is often necessary to change or remove a user's authorization: this is done by changing or deleting the corresponding access rules on the system. Usingatomic authorizationis an alternative to per-system authorization management, where atrusted third partysecurely distributes authorization information.
Inpublic policy, authorization is a feature of trusted systems used forsecurityorsocial control.
Inbanking, anauthorizationis a hold placed on a customer's account when a purchase is made using adebit cardorcredit card.
Inpublishing, sometimes public lectures and other freely available texts are published without the approval of theauthor. These are called unauthorized texts. An example is the 2002'The Theory of Everything: The Origin and Fate of the Universe', which was collected fromStephen Hawking's lectures and published without his permission as per copyright law.[citation needed]
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OpenIDis anopen standardanddecentralizedauthenticationprotocolpromoted by the non-profitOpenID Foundation. It allows users to be authenticated by co-operating sites (known asrelying parties, or RP) using a third-party identity provider (IDP) service, eliminating the need forwebmastersto provide their ownad hoclogin systems, and allowing users to log in to multiple unrelated websites without having to have a separate identity and password for each.[1]Users create accounts by selecting an OpenIDidentity provider,[1]and then use those accounts to sign on to any website that accepts OpenID authentication. Several large organizations either issue or accept OpenIDs on their websites.[2]
The OpenID standard provides a framework for the communication that must take place between the identity provider and the OpenID acceptor (the "relying party").[3]An extension to the standard (the OpenID Attribute Exchange) facilitates the transfer of user attributes, such as name and gender, from the OpenID identity provider to the relying party (each relying party may request a different set of attributes, depending on its requirements).[4]The OpenID protocol does not rely on a central authority to authenticate a user's identity. Moreover, neither services nor the OpenID standard may mandate a specific means by which to authenticate users, allowing for approaches ranging from the common (such as passwords) to the novel (such assmart cardsor biometrics).
The final version of OpenID is OpenID 2.0, finalized and published in December 2007.[5]The termOpenIDmay also refer to an identifier as specified in the OpenID standard; these identifiers take the form of a uniqueUniform Resource Identifier(URI), and are managed by some "OpenID provider" that handles authentication.[1]
As of March 2016[update], there are over 1 billion OpenID-enabled accounts on the Internet (see below) and approximately 1,100,934 sites have integrated OpenID consumer support:[6]AOL,Flickr,Google,Amazon.com,Canonical(provider nameUbuntu One),LiveJournal,Microsoft(provider nameMicrosoft account),Mixi,Myspace,Novell,OpenStreetMap,Orange,Sears,Sun,Telecom Italia,Universal Music Group,VeriSign,WordPress,Yahoo!, theBBC,[7]IBM,[8]PayPal,[9]andSteam,[10]although some of those organizations also have their own authentication management.
Many if not all of the larger organizations require users to provide authentication in the form of an existing email account or mobile phone number in order to sign up for an account (which then can be used as an OpenID identity). There are several smaller entities that accept sign-ups with no extra identity details required.
Facebook did use OpenID in the past, but moved toFacebook Connect.[11]Blogger also used OpenID, but since May 2018 no longer supports it.[12]
OpenID is a decentralized authentication protocol that allows users to authenticate with multiple websites using a single set of credentials, eliminating the need for separate usernames and passwords for each website. OpenID authenticates a user with an identity provider (IDP), who then provides the user with a unique identifier (called an OpenID). This identifier can then be used to authenticate the user with any website that supports OpenID.
When a user visits a website that supports OpenID authentication, the website will redirect the user to their chosen IDP. The IDP will then prompt the user to authenticate themselves (e.g., by entering a username and password). Once the user is authenticated, the IDP will generate an OpenID and send it back to the website. The website can then use this OpenID to authenticate the user without needing to know their actual credentials.
OpenID is built on top of several existing standards, including HTTP, HTML, and XML. OpenID relies on a number of technologies, including a discovery mechanism that allows websites to find the IDP associated with a particular OpenID, as well as security mechanisms to protect against phishing and other attacks.[13]
One of the key benefits of OpenID is that it allows users to control their own identity information, rather than relying on individual websites to store and manage their login credentials. This can be particularly important in cases where websites are vulnerable to security breaches or where users are concerned about the privacy of their personal information.
OpenID has been widely adopted by a number of large websites and service providers, including Google, Yahoo!, and PayPal. The protocol is also used by a number of open source projects and frameworks, including Ruby on Rails and Django.
The end user interacts with a relying party (such as a website) that provides an option to specify an OpenID for the purposes of authentication; an end user typically has previously registered an OpenID (e.g.alice.openid.example.org) with an OpenID provider (e.g.openid.example.org).[1]
The relying party typically transforms the OpenID into a canonical URL form (e.g.http://alice.openid.example.org/).
There are two modes in which the relying party may communicate with the OpenID provider:
Thecheckid_immediatemode can fall back to thecheckid_setupmode if the operation cannot be automated.
First, the relying party and the OpenID provider (optionally) establish ashared secret, referenced by anassociate handle, which the relying party then stores. If using thecheckid_setupmode, the relying party redirects the end user's user-agent to the OpenID provider so the end user can authenticate directly with the OpenID provider.
The method of authentication may vary, but typically, an OpenID provider prompts the end user for a password or some cryptographic token, and then asks whether the end user trusts the relying party to receive the necessary identity details.
If the end user declines the OpenID provider's request to trust the relying party, then the user-agent is redirected back to the relying party with a message indicating that authentication was rejected; the relying party in turn refuses to authenticate the end user.
If the end user accepts the OpenID provider's request to trust the relying party, then the user-agent is redirected back to the relying party along with the end user's credentials. That relying party must then confirm that the credentials really came from the OpenID provider. If the relying party and OpenID provider had previously established a shared secret, then the relying party can validate the identity of the OpenID provider by comparing its copy of the shared secret against the one received along with the end user's credentials; such a relying party is calledstatefulbecause it stores the shared secret between sessions. In contrast, astatelessordumbrelying party must make one more background request (check_authentication) to ensure that the data indeed came from the OpenID provider.
After the OpenID has been verified, authentication is considered successful and the end user is considered logged into the relying party under the identity specified by the given OpenID (e.g.alice.openid.example.org). The relying party typically then stores the end user's OpenID along with the end user's other session information.
To obtain an OpenID-enabledURLthat can be used to log into OpenID-enabled websites, a user registers an OpenID identifier with an identity provider. Identity providers offer the ability to register a URL (typically a third-level domain, e.g. username.example.com) that will automatically be configured with OpenID authentication service.
Once they have registered an OpenID, a user can also use an existing URL under their own control (such as a blog or home page) as an alias or "delegated identity". They simply insert the appropriate OpenID tags in theHTML[14]or serve aYadisdocument.[15]
Starting with OpenID Authentication 2.0 (and some 1.1 implementations), there are two types of identifiers that can be used with OpenID: URLs and XRIs.
XRIsare a new form ofInternetidentifierdesigned specifically for cross-domain digital identity. For example, XRIs come in two forms—i-namesandi-numbers—that are usually registered simultaneously assynonyms. I-names are reassignable (like domain names), while i-numbers are never reassigned. When an XRI i-name is used as an OpenID identifier, it is immediately resolved to the synonymous i-number (the CanonicalID element of the XRDS document). This i-number is the OpenID identifier stored by the relying party. In this way, both the user and the relying party are protected from the end user's OpenID identity ever being taken over by another party as can happen with a URL based on a reassignable DNS name.
The OpenID Foundation (OIDF) promotes and enhances the OpenID community and technologies. The OIDF is a non-profit international standards development organization of individual developers, government agencies and companies who wish to promote and protect OpenID. The OpenID Foundation was formed in June 2007 and serves as a public trust organization representing an open community of developers, vendors and users. OIDF assists the community by providing needed infrastructure and help in promoting and supporting adoption of OpenID. This includes managing intellectual property and trade marks as well a fostering viral growth and global participation in OpenID.
The OpenID Foundation's board of directors has six community board members and eight corporate board members:[16]
Community board members
Corporate board members
OIDF is a global organization to promote digital identity and to encourage the further adoption of OpenID, the OIDF has encouraged the creation of member chapters. Member chapters are officially part of the Foundation and work within their own constituency to support the development and adoption of OpenID as a framework for user-centric identity on the internet.
The OIDF ensures that OpenID specifications are freely implementable therefore the OIDF requires all contributors to sign a contribution agreement. This agreement both grants a copyright license to the Foundation to publish the collective specifications and includes a patent non-assertion agreement. The non-assertion agreement states that the contributor will not sue someone for implementing OpenID specifications.
The OpenID trademark in the United States was assigned to the OpenID Foundation in March 2008.[17]It had been registered by NetMesh Inc. before the OpenID Foundation was operational.[18][19]In Europe, as of August 31, 2007, the OpenID trademark is registered to the OpenID Europe Foundation.[20]
The OpenID logo was designed by Randy "ydnar" Reddig, who in 2005 had expressed plans to transfer the rights to an OpenID organization.[21]
Since the original announcement of OpenID, the official site has stated:[22]
Nobody should own this. Nobody's planning on making any money from this. The goal is to release every part of this under the most liberal licenses possible, so there's no money or licensing or registering required to play. It benefits the community as a whole if something like this exists, and we're all a part of the community.
Sun Microsystems,VeriSignand a number of smaller companies involved in OpenID have issued patentnon-assertion covenantscovering OpenID 1.1 specifications. The covenants state that the companies will not assert any of their patents against OpenID implementations and will revoke their promises from anyone who threatens, or asserts, patents against OpenID implementors.[23][24]
In March, 2012, a research paper[25]reported two generic security issues in OpenID. Both issues allow an attacker to sign in to a victim's relying party accounts. For the first issue, OpenID and Google (an Identity Provider of OpenID) both published security advisories to address it.[26][27]Google's advisory says "An attacker could forge an OpenID request that doesn't ask for the user's email address, and then insert an unsigned email address into the IDPs response. If the attacker relays this response to a website that doesn't notice that this attribute is unsigned, the website may be tricked into logging the attacker in to any local account." The research paper claims that many popular websites have been confirmed vulnerable, includingYahoo! Mail,smartsheet.com,Zoho,manymoon.com,diigo.com. The researchers have notified the affected parties, who have then fixed their vulnerable code.
For the second issue, the paper called it "Data Type Confusion Logic Flaw", which also allows attackers to sign in to victims' RP accounts.GoogleandPayPalwere initially confirmed vulnerable. OpenID published a vulnerability report[28]on the flaw. The report says Google and PayPal have applied fixes, and suggest other OpenID vendors to check their implementations.
Some observers have suggested that OpenID has security weaknesses and may prove vulnerable tophishingattacks.[29][30][31]For example, a malicious relaying party may forward the end user to a bogus identity provider authentication page asking that end user to input their credentials. On completion of this, the malicious party (who in this case also controls the bogus authentication page) could then have access to the end user's account with the identity provider, and then use that end user's OpenID to log into other services.
In an attempt to combat possible phishing attacks, some OpenID providers mandate that the end user needs to be authenticated with them prior to an attempt to authenticate with the relying party.[32]This relies on the end user knowing the policy of the identity provider. In December 2008, the OpenID Foundation approved version 1.0 of the Provider Authentication Policy Extension (PAPE), which "enables Relying Parties to request that OpenID Providers employ specified authentication policies when authenticating users and for OpenID Providers to inform the Relying Parties which policies were actually used."[33]
Other security issues identified with OpenID involve lack of privacy and failure to address thetrust problem.[34]However, this problem is not unique to OpenID and is simply the state of the Internet as commonly used.[citation needed]
The Identity Provider does, however, get a log of your OpenID logins; they know when you logged into what website, makingcross-site trackingmuch easier. A compromised OpenID account is also likely to be a more serious breach of privacy than a compromised account on a single site.
Another important vulnerability is present in the last step in the authentication scheme when TLS/SSL are not used: the redirect-URL from the identity provider to the relying party. The problem with this redirect is the fact that anyone who can obtain this URL (e.g. by sniffing the wire) can replay it and get logged into the site as the victim user. Some of the identity providers usenonces(a number used just once) to allow a user to log into the site once and fail all the consecutive attempts. The nonce solution works if the user is the first one to use the URL. However, a fast attacker who is sniffing the wire can obtain the URL and immediately reset a user's TCP connection (as an attacker is sniffing the wire and knows the required TCP sequence numbers) and then execute the replay attack as described above. Thus nonces only protect against passive attackers, but cannot prevent active attackers from executing the replay attack.[35]Use of TLS/SSL in the authentication process can significantly reduce this risk.
This can be restated as:
On May 1, 2014, a bug dubbed "Covert Redirectrelated toOAuth2.0 and OpenID" was disclosed.[36][37]It was discovered by mathematics doctoral student Wang Jing at the School of Physical and Mathematical Sciences,Nanyang Technological University, Singapore.[38][39][40]
The announcement of OpenID is:
"'Covert Redirect', publicized in May 2014, is an instance of attackers using open redirectors – a well-known threat, with well-known means of prevention. The OpenID Connect protocol mandates strict measures that preclude open redirectors to prevent this vulnerability."[41]
"The general consensus, so far, is that Covert Redirect is not as bad, but still a threat. Understanding what makes it dangerous requires a basic understanding of Open Redirect, and how it can be exploited."[42]
A patch was not immediately made available. Ori Eisen, founder, chairman and chief innovation officer at 41st Parameter told Sue Marquette Poremba, "In any distributed system, we are counting of the good nature of the participants to do the right thing. In cases like OAuth and OpenID, the distribution is so vast that it is unreasonable to expect each and every website to patch up in the near future".[43]
The original OpenID authentication protocol was developed in May 2005[44]byBrad Fitzpatrick, creator of popular community websiteLiveJournal, while working atSix Apart.[45]Initially referred to as Yadis (an acronym for "Yet another distributed identity system"),[46]it was named OpenID after the openid.netdomain namewas given to Six Apart to use for the project.[47]OpenID support was soon implemented onLiveJournaland fellow LiveJournalenginecommunityDeadJournalfor blog post comments and quickly gained attention in the digital identity community.[48][49]Web developerJanRainwas an early supporter of OpenID, providing OpenIDsoftware librariesand expanding its business around OpenID-based services.
In late June, discussions started between OpenID users and developers fromenterprise softwarecompany NetMesh, leading to collaboration on interoperability between OpenID and NetMesh's similarLight-weight Identity(LID) protocol. The direct result of the collaboration was theYadisdiscovery protocol, adopting the name originally used for OpenID. The new Yadis was announced on October 24, 2005.[50]After a discussion at the 2005Internet Identity Workshopa few days later,XRI/i-namesdevelopers joined the Yadis project,[51]contributing their Extensible Resource Descriptor Sequence (XRDS) format for utilization in the protocol.[52]
In December, developers at Sxip Identity began discussions with the OpenID/Yadis community[53]after announcing a shift in the development of version 2.0 of its Simple Extensible Identity Protocol (SXIP) to URL-based identities like LID and OpenID.[54]In March 2006, JanRain developed a Simple Registration (SREG) extension for OpenID enabling primitive profile-exchange[55]and in April submitted a proposal to formalize extensions to OpenID. The same month, work had also begun on incorporating fullXRIsupport into OpenID.[56]Around early May, key OpenID developerDavid Recordonleft Six Apart, joining VeriSign to focus more on digital identity and guidance for the OpenID spec.[49][57]By early June, the major differences between the SXIP 2.0 and OpenID projects were resolved with the agreement to support multiple personas in OpenID by submission of an identity provider URL rather than a full identity URL. With this, as well as the addition of extensions and XRI support underway, OpenID was evolving into a full-fledged digital identity framework, with Recordon proclaiming "We see OpenID as being an umbrella for the framework that encompasses the layers for identifiers, discovery, authentication and a messaging services layer that sits atop and this entire thing has sort of been dubbed 'OpenID 2.0'.[58]" In late July, Sxip began to merge its Digital Identity Exchange (DIX) protocol into OpenID, submitting initial drafts of the OpenID Attribute Exchange (AX) extension in August. Late in 2006, aZDNetopinion piece made the case for OpenID to users, web site operators and entrepreneurs.[59]
On January 31, 2007,Symantecannounced support for OpenID in its Identity Initiative products and services.[60]A week later, on February 6Microsoftmade a joint announcement with JanRain, Sxip, and VeriSign to collaborate on interoperability between OpenID and Microsoft'sWindows CardSpacedigital identity platform, with particular focus on developing a phishing-resistant authentication solution for OpenID. As part of the collaboration, Microsoft pledged to support OpenID in its future identity server products and JanRain, Sxip, and VeriSign pledged to add support for Microsoft'sInformation Cardprofile to their future identity solutions.[61]In mid-February,AOLannounced that an experimental OpenID provider service was functional for all AOL andAOL Instant Messenger(AIM) accounts.[62]
In May,Sun Microsystemsbegan working with the OpenID community, announcing an OpenID program,[63]as well as entering a non-assertion covenant with the OpenID community, pledging not to assert any of its patents against implementations of OpenID.[23]In June, OpenID leadership formed the OpenID Foundation, an Oregon-basedpublic benefit corporationfor managing the OpenID brand and property.[64]The same month, an independent OpenID Europe Foundation was formed in Belgium[65]by Snorri Giorgetti. By early December, non-assertion agreements were collected by the major contributors to the protocol and the final OpenID Authentication 2.0 and OpenID Attribute Exchange 1.0 specifications were ratified on December 5.[66]
In mid-January 2008,Yahoo!announced initial OpenID 2.0 support, both as a provider and as a relying party, releasing the provider service by the end of the month.[67]In early February, Google, IBM, Microsoft, VeriSign and Yahoo! joined the OpenID Foundation as corporate board members.[68]Around early May,SourceForge, Inc.introduced OpenID provider and relying party support to leading open source software development websiteSourceForge.net.[69]In late July, popularsocial network serviceMySpaceannounced support for OpenID as a provider.[70]In late October, Google launched support as an OpenID provider and Microsoft announced thatWindows Live IDwould support OpenID.[71]In November, JanRain announced a free hosted service, RPX Basic, that allows websites to begin accepting OpenIDs for registration and login without having to install, integrate and configure the OpenID open source libraries.[72]
In January 2009, PayPal joined the OpenID Foundation as a corporate member, followed shortly by Facebook in February. The OpenID Foundation formed an executive committee and appointed Don Thibeau as executive director. In March, MySpace launched their previously announced OpenID provider service, enabling all MySpace users to use their MySpace URL as an OpenID. In May, Facebook launched their relying party functionality,[73][74]letting users use an automatic login-enabled OpenID account (e.g. Google) to log into Facebook.[75]
In September 2013,Janrainannounced that MyOpenID.com would be shut down on February 1, 2014; a pie chart showed Facebook and Google dominate the social login space as of Q2 2013.[76]Facebook has since left OpenID; it is no longer a sponsor, represented on the board, or permitting OpenID logins.[16][77]
In May 2016, Symantec announced that they would be discontinuing their pip.verisignlabs.com OpenID personal identity portal service.[78][79]
In March 2018, Stack Overflow announced an end to OpenID support, citing insufficient usage to justify the cost. In the announcement, it was stated that based on activity, users strongly preferred Facebook, Google, and e-mail/password based account authentication.[80]
OpenID is a way to use a single set of user credentials to access multiple sites, whileOAuthfacilitates the authorization of one site to access and use information related to the user's account on another site. Although OAuth is not anauthenticationprotocol, it can be used as part of one.
Authentication in the context of a user accessing an application tells an application who the current user is and whether or not they're present. [...] Authentication is all about the user and their presence with the application, and an internet-scale authentication protocol needs to be able to do this across network and security boundaries.
However, OAuth tells the application none of that. OAuth says absolutely nothing about the user, nor does it say how the user proved their presence or even if they're still there. As far as an OAuth client is concerned, it asked for a token, got a token, and eventually used that token to access some API. It doesn't know anything about who authorized the application or if there was even a user there at all. In fact, much of the point of OAuth is about giving this delegated access for use in situations where the user is not present on the connection between the client and the resource being accessed. This is great for client authorization, but it's really bad for authentication where the whole point is figuring out if the user is there or not (and who they are).[81]
The following drawing highlights the differences between using OpenID versusOAuthfor authentication. Note that with OpenID, the process starts with the application asking the user for their identity (typically an OpenID URI), whereas in the case of OAuth, the application directly requests a limited access OAuth Token (valet key) to access the APIs (enter the house) on user's behalf. If the user can grant that access, the application can retrieve the unique identifier for establishing the profile (identity) using the APIs.
OpenID provides a cryptographic verification mechanism that prevents the attack below against users who misuse OAuth for authentication.
Note that the valet key does not describe the user in any way, it only provides limited access rights, to some house (which is not even necessarily the user's, they just had a key). Therefore if the key becomes compromised (the user is malicious and managed to steal the key to someone else's house), then the user can impersonate the house owner to the application who requested their authenticity. If the key is compromised by any point in the chain of trust, a malicious user may intercept it and use it to impersonate user X for any application relying on OAuth2 for pseudo authentication against the same OAuth authorization server. Conversely, the notarized letter contains the user's signature, which can be checked by the requesting application against the user, so this attack is not viable.[82]
The letter can usepublic-key cryptographyto be authenticated.
Published in February 2014[83]by the OpenID Foundation, OpenID Connect (OIDC) is the third generation of OpenID technology. It is an authentication layer on top of theOAuth 2.0authorization framework.[84]It allows computing clients to verify the identity of an end user based on the authentication performed by an authorization server, as well as to obtain the basic profile information about the end user in an interoperable and REST-like manner. In technical terms, OpenID Connect specifies a RESTful HTTP API, usingJSONas a data format.
OpenID Connect allows a range of parties, including web-based, mobile and JavaScript clients, to request and receive information about authenticated sessions and end users. The OpenID Connect specification is extensible, supporting optional features such as encryption of identity data, discovery of OpenID providers, and session management.
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https://en.wikipedia.org/wiki/OpenID
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OTPWis aone-time passwordsystem developed forauthenticationinUnix-likeoperating systemsbyMarkus Kuhn.[1]A user's real password is not directly transmitted across thenetwork. Rather, a series of one-time passwords is created from a short set of characters (constant secret) and a set of one-time tokens. As each single-use password can only be used once, passwords intercepted by apassword snifferorkey loggerare not useful to an attacker.
OTPW is supported inUnixandLinux(viapluggable authentication modules),OpenBSD,NetBSD, andFreeBSD, and a generic open source implementation can be used to enable its use on other systems.
OTPW, like the other one-time password systems, is sensitive to aman in the middle attackif used by itself. This could for example be solved by puttingSSL, SPKM or similar security protocol "under it" which authenticates the server and gives point-to-point security between the client and server.
UnlikeS/KEY, OTPW is not based on theLamport's scheme in which every one-time password is the one-wayhash valueof its successor. Password lists based on the Lamport's scheme have the problem that if the attacker can see one of the last passwords on the list, then all previous passwords can be calculated from it. It also does not store the encrypted passwords as suggested by Aviel D. Rubin inIndependent One-Time Passwords, in order to keep the host free of files with secrets.
In OTPW a one-way hash value of every single password is stored in a potentially widely readablefilein the user’shome directory. For instance, hash values of 300 passwords (a typicalA4 page) require only a four kilobyte long.otpwfile, a typically negligible amount of storage space.
The passwords are carefully generated random numbers. Therandom number generatoris based on theRIPEMD-160securehash function, and it is seeded by hashing together the output of variousshellcommands. These provide unpredictability in the form of a system random number seed, access times of important system files, usage history of the host, and more. The random state is the 160-bit output of the hash function. The random state is iterated after each use by concatenating the old state with the current high-resolution timer output and hashing the result again. The first 72 bits of the hash output are encoded with a modifiedbase64scheme to produce readable passwords, while the remaining 88 bits represent the undisclosed internal state of the random number generator.
In many fonts, the characters0andOor1andlandIare difficult to distinguish, therefore the modified base64 encoding replaces the three characters01lby corresponding:,=and%. If for instance a zero is confused with a capital O by the user, the password verification routine will automatically correct for this.S/KEYuses sequences of shortEnglishwords as passwords. OTPW uses by default a base64 encoding instead, because that allows more passwords to be printed on a single page, with the same passwordentropy. In addition, an average human spy needs over 30 seconds to write a 12-character random string into short-term memory, which provides a good protection against brief looks that an attacker might have on a password list. Lists of short words on the other hand are much faster to memorize. OTPW can handle arbitrary password generation algorithms, as long as the length of the password is fixed. In the current version, theotpw-genprogram can generate both base-64 encoded (option -p) and 4-letter-word encoded (option -p1) passwords with a user-specified entropy (option -e).
The prefix passwordensures that neither stealing the password list nor eavesdropping the line alone can provide unauthorized access. Admittedly, the security obtained by OTPW is not comparable with that of achallenge–responsesystem in which the user has a PIN-protected special calculator that generates the response. On the other hand, a piece of paper is much more portable, much more robust, and much cheaper than a special calculator. OTPW was designed for the large user base, for which an extra battery-powered device is inconvenient or not cost effective and who therefore still use normal Unix passwords everywhere.
In contrast to the suggestion made in RFC 1938, OTPW does not lock more than one one-time password at a time. If it did this, an attacker could easily exhaust its list of unlocked passwords and force it to either not login at all or use the normal Unix login password. Therefore, OTPW locks only one single password andfor all further logins a triple-challenge is issued. If more than 100 unused passwords remain available, then there are over a million different challenges and an attacker has very little chance to perform a successful race attack while the authorized user finishes password entry.
One-time password authentication with the OTPW package is accomplished via a file.otpwlocated in the user’s home directory. No state is kept in any system-wide files, therefore OTPW does not introduce any newsetuidroot programs. As long as a user does not have.otpwin his home directory, the one-time-password facility has not been activated for him.
A user who wants to set up the one-time-password capability just executes theotpw-genprogram. The program will ask for aprefix passwordand it will then write a password list to standard output. The chosenprefix passwordshould be memorized and the password list can be formatted and printed.
Where one-time-password authentication is used, the password prompt will be followed by a 3-digit password number. Enter first the prefix password that was given tootpw-gen, followed directly (without hitting return between) by the password with the requested number from the printed password list:
In this example, geHeim wasthe prefix password.
A clever attacker might observe the password being entered and might try to use the fact that computers can send data much faster than users can finish entering passwords. In the several hundred milliseconds that the user needs to press the return key after the last character, an attacker could on a parallel connection to the same machine send the code of the return key faster than the user.
To prevent such arace-for-the-last-key attack, any login attempt that is taking place concurrently with another attempt will require three one-time passwords to be entered, neither of which will ever be the password which is locked by the concurrent authentication attempt.
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OPIEis the initialism of "One time Passwords In Everything".
Opie is amature,Unix-likelogin andpasswordpackage
installed on the server and the client which makes untrusted networks safer against password-sniffingpacket-analysissoftware likedSniffand safe againstshoulder surfing.
It works by circumventing the delayed attack method because the same password is never used twice after installing Opie.
OPIE implements aone-time password(OTP) scheme based onS/KEY, which will require a secretpassphrase(not echoed) to generate a password for the current session, or a list of passwords.
OPIE uses an MD4 or MD5hash functionto generate passwords.
OPIE can restrict its logins based on IP address. It uses its ownpasswdand login modules.
If theEnter key↵ Enteris pressed at the password prompt, it will turn echo on, so what is being typed can be seen when entering an unfamiliar password from a printout.
OPIE can improve security when accessing online banking at conferences, hotels and airports. Some countries require banks to implement OTP.
OPIE shipped withDragonFly BSD,FreeBSDandOpenSUSE. It can be installed on a Unix-like server and clients for improved security.
The commands are
This cryptography-related article is astub. You can help Wikipedia byexpanding it.
ThisUnix-related article is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/OPIE_Authentication_System
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ThePGP Word List("Pretty Good Privacyword list", also called abiometric word listfor reasons explained below) is a list ofwordsfor conveying databytesin a clear unambiguous way via a voice channel. They are analogous in purpose to theNATO phonetic alphabet, except that a longer list of words is used, each word corresponding to one of the 256 distinct numeric byte values.
The PGP Word List was designed in 1995 byPatrick Juola, acomputational linguist, andPhilip Zimmermann, creator ofPGP.[1][2]The words were carefully chosen for theirphoneticdistinctiveness, usinggenetic algorithmsto select lists of words that had optimum separations inphonemespace. The candidate word lists were randomly drawn fromGrady Ward'sMoby Pronunciatorlist as raw material for the search, successively refined by the genetic algorithms. The automated search converged to an optimized solution in about 40 hours on aDEC Alpha, a particularly fast machine in that era.
The Zimmermann–Juola list was originally designed to be used inPGPfone, a secure VoIP application, to allow the two parties to verbally compare a short authentication string to detect aman-in-the-middle attack(MiTM). It was called abiometricword list because the authentication depended on the two human users recognizing each other's distinct voices as they read and compared the words over the voice channel, binding the identity of the speaker with the words, which helped protect against the MiTM attack. The list can be used in many other situations where a biometric binding of identity is not needed, so calling it a biometric word list may be imprecise. Later, it was used inPGPto compare and verify PGPpublic keyfingerprintsover a voice channel. This is known in PGP applications as the "biometric" representation. When it was applied to PGP, the list of words was further refined, with contributions byJon Callas. More recently, it has been used inZfoneand theZRTPprotocol, the successor to PGPfone.
The list is actually composed of two lists, each containing 256phoneticallydistinct words, in which each word represents a different byte value between 0 and 255. Two lists are used because reading aloud long random sequences of human words usually risks three kinds of errors: 1) transposition of two consecutive words, 2) duplicate words, or 3) omitted words. To detect all three kinds of errors, the two lists are used alternately for the even-offset bytes and the odd-offset bytes in the byte sequence. Each byte value is actually represented by two different words, depending on whether that byte appears at an even or an odd offset from the beginning of the byte sequence. The two lists are readily distinguished by the number ofsyllables; the even list has words of two syllables, the odd list has three. The two lists have a maximum word length of 9 and 11 letters, respectively. Using a two-list scheme was suggested by Zhahai Stewart.
Here are the two lists of words as presented in the PGPfone Owner's Manual.[3]
Each byte in a bytestring is encoded as a single word. A sequence of bytes is rendered innetwork byte order, from left to right. For example, the leftmost (i.e. byte 0) is considered "even" and is encoded using the PGP Even Word table. The next byte to the right (i.e. byte 1) is considered "odd" and is encoded using the PGP Odd Word table. This process repeats until all bytes are encoded. Thus, "E582" produces "topmost Istanbul", whereas "82E5" produces "miser travesty".
A PGP public key fingerprint that displayed inhexadecimalas
would display in PGP Words (the "biometric" fingerprint) as
The order of bytes in a bytestring depends onendianness.
There are several other word lists for conveying data in a clear unambiguous way via a voice channel:
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Incryptography, theavalanche effectis the desirable property of cryptographicalgorithms, typically block ciphers[1]andcryptographic hash functions, wherein if an input is changed slightly (for example, flipping a single bit), the output changes significantly (e.g., half the output bits flip). In the case of high-quality block ciphers, such a small change in either thekeyor theplaintextshould cause a drastic change in theciphertext. The actual term was first used byHorst Feistel,[1]although the concept dates back to at leastShannon'sdiffusion.
If a block cipher or cryptographic hash function does not exhibit the avalanche effect to a significant degree, then it has poor randomization, and thus acryptanalystcan make predictions about the input, being given only the output. This may be sufficient to partially or completely break the algorithm. Thus, the avalanche effect is a desirable condition from the point of view of the designer of the cryptographic algorithm or device. Failure to incorporate this characteristic leads to the hash function being exposed to attacks includingcollision attacks,length extension attacks, andpreimage attacks.[2]
Constructing a cipher or hash to exhibit a substantial avalanche effect is one of the primary design objectives, and mathematically the construction takes advantage of thebutterfly effect.[3]This is why most block ciphers areproduct ciphers. It is also why hash functions have large data blocks. Both of these features allow small changes to propagate rapidly through iterations of the algorithm, such that everybitof the output should depend on every bit of the input before the algorithm terminates.[citation needed]
Thestrict avalanche criterion(SAC) is a formalization of the avalanche effect. It is satisfied if, whenever a single input bit iscomplemented, each of the output bits changes with a 50% probability. The SAC builds on the concepts ofcompletenessand avalanche and was introduced by Webster and Tavares in 1985.[4]
Higher-order generalizations of SAC involve multiple input bits.
Boolean functions which satisfy the highest order SAC are alwaysbent functions,
also called maximally nonlinear functions,
also called "perfect nonlinear" functions.[5]
Thebit independence criterion(BIC) states that output bitsjandkshould change independently when any single input bitiis inverted, for alli,jandk.[6]
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The following tables compare general and technical information for a number of cryptographic hash functions. See the individual functions' articles for further information. This article is not all-inclusive or necessarily up-to-date. An overview of hash function security/cryptanalysiscan be found athash function security summary.
Basic general information about thecryptographic hash functions: year, designer, references, etc.
The following tables compare technical information forcompression functionsofcryptographic hash functions. The information comes from the specifications, please refer to them for more details.
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Incryptographic protocoldesign,cryptographic agilityorcrypto-agilityis the ability to switch between multiplecryptographic primitives.
A cryptographically agile system implementing a particular standard can choose which combination of primitives to use. The primary goal of cryptographic agility is to enable rapid adaptations of newcryptographic primitivesandalgorithmswithout making disruptive changes to the system's infrastructure.
Cryptographic agility acts as a safety measure or an incident response mechanism for when a cryptographic primitive of a system is discovered to be vulnerable.[1]A security system is considered crypto-agile if its cryptographic algorithms or parameters can be replaced with ease and is at least partly automated.[2][3]The impending arrival of aquantum computerthat can break existingasymmetric cryptographyis raising awareness of the importance of cryptographic agility.[4][5]
TheX.509public key certificateillustrates crypto-agility. A public key certificate hascryptographic parametersincluding key type, key length, and ahash algorithm. X.509 version v.3, with key type RSA, a 1024-bit key length, and the SHA-1 hash algorithm were found byNISTto have a key length that made it vulnerable to attacks, thus prompting the transition to SHA-2.[6]
With the rise ofsecure transport layer communicationin the end of the 1990s, cryptographic primitives and algorithms have been increasingly popular; for example, by 2019, more than 80% of all websites employed some form of security measures.[7]Furthermore, cryptographic techniques are widely incorporated to protect applications and business transactions.
However, as cryptographic algorithms are deployed, research of their security intensifies, and new attacks against cryptographic primitives (old and new alike) are discovered. Crypto-agility tries to tackle the implied threat to information security by allowing swift deprecation of vulnerable primitives and replacement with new ones.
This threat is not merely theoretical; many algorithms that were once considered secure (DES, 512-bitRSA,RC4) are now known to be vulnerable, some even to amateur attackers. On the other hand, new algorithms (AES,Elliptic curve cryptography) are often both more secure and faster in comparison to old ones. Systems designed to meet crypto-agility criteria are expected to be less affected should current primitives be found vulnerable, and may enjoy betterlatencyor battery usage by using new and improved primitives.
For example,quantum computing, if feasible, is expected to be able to defeat existing public key cryptography algorithms. The overwhelming majority of existing public-key infrastructure relies on the computational hardness of problems such asinteger factorizationanddiscrete logarithms(which includeselliptic-curve cryptographyas a special case). Quantum computers runningShor's algorithmcan solve these problems exponentially faster than the best-known algorithms for conventional computers.[8]Post-quantum cryptographyis the subfield of cryptography that aims to replace quantum-vulnerable algorithms with new ones that are believed hard to break even for a quantum computer. The main families of post-quantum alternatives to factoring and discrete logarithms includelattice-based cryptography,multivariate cryptography,hash-based cryptography, andcode-based cryptography.
System evolution and crypto-agility are not the same. System evolution progresses on the basis of emerging business and technical requirements. Crypto-agility is related instead to computing infrastructure and requires consideration by security experts, system designers, and application developers.[9]
Best practices about dealing with crypto-agility include:[10]
Cryptographic agility typically increases the complexity of the applications that rely on it. Developers need to build support for each of the optional cryptographic primitives, introducing more code and increasing the chance of implementation flaws as well as increasing maintenance and support costs.[14]Users of the systems need to select which primitives they wish to use; for example,OpenSSLusers can select from dozens of ciphersuites when usingTLS.[15]Further, when two parties negotiate the cryptographic primitives for their message exchange, it creates the opportunity for downgrade attacks by intermediaries (such asPOODLE), or for the selection of insecure primitives.[16]
One alternative approach is to dramatically limit the choices available to both developers and users, so that there is less scope for implementation or configuration flaws.[17]In this approach, the designers of the library or system choose the primitives and do not offer a choice of cryptographic primitives (or, if they do, it is a very constrained set of choices). Opinionated encryption is visible in tools likeLibsodium, where high-level APIs explicitly aim to discourage developers from picking primitives, and inWireguard, where single primitives are picked to intentionally eliminate crypto-agility.[18]
If opinionated encryption is used and a vulnerability is discovered in one of the primitives in a protocol, there is no way to substitute better primitives. Instead, the solution is to use versioned protocols. A new version of the protocol will include the fixed primitive. As a consequence of this, two parties running different versions of the protocol will not be able to communicate.
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https://en.wikipedia.org/wiki/Cryptographic_agility
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CRYPTRECis theCryptography Research and Evaluation Committeesset up by theJapanese Governmentto evaluate and recommendcryptographictechniques for government and industrial use. It is comparable in many respects to theEuropean Union'sNESSIEproject and to theAdvanced Encryption Standard processrun byNational Institute of Standards and Technologyin theU.S.
There is some overlap, and some conflict, between the NESSIE selections and the CRYPTREC draft recommendations. Both efforts include some of the best cryptographers in the world[citation needed]therefore conflicts in their selections and recommendations should be examined with care. For instance, CRYPTREC recommends several 64 bit block ciphers while NESSIE selected none, but CRYPTREC was obliged by its terms of reference to take into account existing standards and practices, while NESSIE was not. Similar differences in terms of reference account for CRYPTREC recommending at least onestream cipher,RC4, while the NESSIE report specifically said that it wasnotablethat they had not selected any of those considered. RC4 is widely used in theSSL/TLSprotocols; nevertheless, CRYPTREC recommended that it only be used with 128-bit keys. Essentially the same consideration led to CRYPTREC's inclusion of 160-bit message digest algorithms, despite their suggestion that they be avoided in new system designs. Also, CRYPTREC was unusually careful to examine variants and modifications of the techniques, or at least to discuss their care in doing so; this resulted in particularly detailed recommendations regarding them.
CRYPTREC includes members from Japaneseacademia,industry, andgovernment. It was started in May 2000 by combining efforts from several agencies who were investigating methods and techniques for implementing 'e-Government' in Japan. Presently, it is sponsored by
It is also the organization that provides technical evaluation and recommendations concerning regulations that implement Japanese laws. Examples include the Electronic Signatures and Certification Services (Law 102 of FY2000, taking effect as from April 2001), the Basic Law on the Formulation of an Advanced Information and Telecommunications Network Society of 2000 (Law 144 of FY2000), and the Public Individual Certification Law of December 2002. Furthermore, CRYPTEC has responsibilities with regard to the Japanese contribution to theISO/IECJTC 1/SC27 standardization effort.
In the first release in 2003,[1]many Japanese ciphers were selected for the "e-Government Recommended Ciphers List":CIPHERUNICORN-E(NEC),Hierocrypt-L1(Toshiba), andMISTY1(Mitsubishi Electric) as 64 bit block ciphers,Camellia(Nippon Telegraph and Telephone,Mitsubishi Electric),CIPHERUNICORN-A(NEC),Hierocrypt-3(Toshiba), andSC2000(Fujitsu) as 128 bit block ciphers, and finallyMUGIandMULTI-S01(Hitachi) as stream ciphers.
In the revised release of 2013,[2]the list was divided into three: "e-Government Recommended Ciphers List", "Candidate Recommended Ciphers List", and "Monitored Ciphers List". Most of the Japanese ciphers listed in the previous list (except for Camellia) have moved from the "Recommended Ciphers List" to the "Candidate Recommended Ciphers List". There were several new proposals, such asCLEFIA(Sony) as a 128 bit block cipher as well asKCipher-2(KDDI) andEnocoro-128v2(Hitachi) as stream ciphers. However, only KCipher-2 has been listed on the "e-Government Recommended Ciphers List". The reason why most Japanese ciphers have not been selected as "Recommended Ciphers" is not that these ciphers are necessarily unsafe, but that these ciphers are not widely used in commercial products, open-source projects, governmental systems, or international standards. There is the possibility that ciphers listed on "Candidate Recommended Ciphers List" will be moved to the "e-Government Recommended Ciphers List" when they are utilized more widely.
In addition, 128 bitRC4andSHA-1are listed on "Monitored Ciphers List". These are unsafe and only permitted to remain compatible with old systems.
After the revision in 2013, there are several updates such as addition ofChaCha20-Poly1305,EdDSAandSHA-3, move ofTriple DESto Monitored list, and deletion ofRC4, etc.
As of March 2023[update]
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File fixityis adigital preservationterm referring to the property of a digital file being fixed, or unchanged.[1]Fixity checking is the process of verifying that a digital object has not been altered or corrupted.[2]During transfer, a repository may run a fixity check to ensure a transmitted file has not been altered en route. Within the repository, fixity checking is used to ensure that digital files have not been affected bydata rot, or other digital preservation dangers.[3]By itself, fixity checking does not ensure the preservation of a digital file. Instead, it allows a repository to identify which corrupted files to replace with a clean copy from the producer or from a backup.
In practice, a fixity check is most often accomplished by computingchecksumorcryptographic hash functionvalues for a file and comparing them to a stored value or throughdigital signatures.[4]File fixity figures prominently inPreservation Metadata: Implementation Strategies (PREMIS), theGovernment Printing Office's work on the Authenticity of Electronic Federal Government Publications,[5]and fixity checking practices are used by a range of cultural heritage organizations.[6]
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Ahash chainis the successive application of acryptographic hash functionto a piece of data. Incomputer security, a hash chain is a method used to produce manyone-time keysfrom a singlekeyorpassword. Fornon-repudiation, a hash function can be applied successively to additional pieces of data in order to record the chronology of data's existence.
Ahash chainis a successive application of acryptographic hash functionh{\displaystyle h}to a stringx{\displaystyle x}.
For example,
h(h(h(h(x)))){\displaystyle h(h(h(h(x))))}gives a hash chain of length 4, often denotedh4(x){\displaystyle h^{4}(x)}
Leslie Lamport[1]suggested the use of hash chains as a password protection scheme in an insecure environment. A server which needs to provideauthenticationmay store a hash chain rather than aplain textpassword and prevent theft of the password in transmission or theft from the server. For example, a server begins by storingh1000(password){\displaystyle h^{1000}(\mathrm {password} )}which is provided by the user. When the user wishes to authenticate, they supplyh999(password){\displaystyle h^{999}(\mathrm {password} )}to the server. The server computesh(h999(password))=h1000(password){\displaystyle h(h^{999}(\mathrm {password} ))=h^{1000}(\mathrm {password} )}and verifies this matches the hash chain it has stored. It then storesh999(password){\displaystyle h^{999}(\mathrm {password} )}for the next time the user wishes to authenticate.
An eavesdropper seeingh999(password){\displaystyle h^{999}(\mathrm {password} )}communicated to the server will be unable to re-transmit the same hash chain to the server forauthenticationsince the server now expectsh998(password){\displaystyle h^{998}(\mathrm {password} )}. Due to theone-way propertyofcryptographically secure hash functions, it is infeasible for the eavesdropper to reverse the hash function and obtain an earlier piece of the hash chain. In this example, the user could authenticate 1000 times before the hash chain were exhausted. Each time the hash value is different, and thus cannot be duplicated by an attacker.
Binary hash chains are commonly used in association with ahash tree. A binary hash chain takes two hash values as inputs, concatenates them and applies a hash function to the result, thereby producing a third hash value.
The above diagram shows a hash tree consisting of eight leaf nodes and the hash chain for the third leaf node. In addition to the hash values themselves the order of concatenation (right or left 1,0) or "order bits" are necessary to complete the hash chain.
Winternitz chains(also known asfunction chains[2]) are used inhash-based cryptography. The chain is parameterized by theWinternitz parameterw(number of bits in a "digit"d) andsecurity parametern(number of bits in the hash value, typically double thesecurity strength,[3]256 or 512). The chain consists of2w{\displaystyle 2^{w}}values that are results of repeated application of aone-way "chain" functionFto a secret keysk:sk,F(sk),F(F(sk)),...,F2w−1(sk){\displaystyle sk,F(sk),F(F(sk)),...,F^{2^{w-1}}(sk)}. The chain function is typically based on a standardcryptographic hash, but needs to be parameterized ("randomized"[4]), so it involves few invocations of the underlying hash.[5]In the Winternitz signature scheme a chain is used to encode one digit of them-bit message, so the Winternitz signature uses approximatelymn/w{\displaystyle mn/w}bits, its calculation takes about2wm/w{\displaystyle 2^{w}m/w}applications of the function F.[3]Note that some signature standards (likeExtended Merkle signature scheme, XMSS) definewas the number of possible values in a digit, sow=16{\displaystyle w=16}in XMSS corresponds tow=4{\displaystyle w=4}in standards (likeLeighton-Micali Signature, LMS) that definewin the same way as above - as a number of bits in the digit.[6]
A hash chain is similar to ablockchain, as they both utilize acryptographic hash functionfor creating a link between two nodes. However, a blockchain (as used byBitcoinand related systems) is generally intended to support distributed agreement around a public ledger (data), and incorporates a set of rules for encapsulation of data and associated data permissions.
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Incryptographyandcomputer security, alength extension attackis a type ofattackwhere an attacker can useHash(message1) and the length ofmessage1to calculateHash(message1‖message2) for an attacker-controlledmessage2, without needing to know the content ofmessage1. This is problematic when thehashis used as amessage authentication codewith constructionHash(secret‖message),[1]andmessageand the length ofsecretis known, because an attacker can include extra information at the end of the message and produce a valid hash without knowing the secret. Algorithms likeMD5,SHA-1and most ofSHA-2that are based on theMerkle–Damgård constructionare susceptible to this kind of attack.[1][2][3]Truncated versions of SHA-2, including SHA-384 andSHA-512/256are not susceptible,[4]nor is theSHA-3algorithm.[5]HMACalso uses a different construction and so is not vulnerable to length extension attacks.[6]A secret suffix MAC, which is calculated asHash(message‖secret), isn't vulnerable to a length extension attack, but is vulnerable to another attack based on a hash collision.[7]
The vulnerable hashing functions work by taking the input message, and using it to transform an internal state. After all of the input has been processed, the hash digest is generated by outputting the internal state of the function. It is possible to reconstruct the internal state from the hash digest, which can then be used to process the new data. In this way, one may extend the message and compute the hash that is a valid signature for the new message.
A server for delivering waffles of a specified type to a specific user at a location could be implemented to handle requests of the given format:
The server would perform the request given (to deliver ten waffles of type eggo to the given location for user "1") only if the signature is valid for the user. The signature used here is aMAC, signed with a key not known to the attacker.[note 1]
It is possible for an attacker to modify the request in this example by switching the requested waffle from "eggo" to "liege." This can be done by taking advantage of a flexibility in the message format if duplicate content in the query string gives preference to the latter value. This flexibility does not indicate an exploit in the message format, because the message format was never designed to be cryptographically secure in the first place, without the signature algorithm to help it.
In order to sign this new message, typically the attacker would need to know the key the message was signed with, and generate a new signature by generating a new MAC. However, with a length extension attack, it is possible to feed the hash (the signature given above) into the state of the hashing function, and continue where the original request had left off, so long as the length of the original request is known. In this request, the original key's length was 14 bytes, which could be determined by trying forged requests with various assumed lengths, and checking which length results in a request that the server accepts as valid.
The message as fed into the hashing function is oftenpadded, as many algorithms can only work on input messages whose lengths are a multiple of some given size. The content of this padding is always specified by the hash function used. The attacker must include all of these padding bits in their forged message before the internal states of their message and the original will line up. Thus, the attacker constructs a slightly different message using these padding rules:
This message includes all of the padding that was appended to the original message inside of the hash function before their payload (in this case, a 0x80 followed by a number of 0x00s and a message length, 0x228 = 552 = (14+55)*8, which is the length of the key plus the original message, appended at the end). The attacker knows that the state behind the hashed key/message pair for the original message is identical to that of new message up to the final "&." The attacker also knows the hash digest at this point, which means they know the internal state of the hashing function at that point. It is then trivial to initialize a hashing algorithm at that point, input the last few characters, and generate a new digest which can sign his new message without the original key.
By combining the new signature and new data into a new request, the server will see the forged request as a valid request due to the signature being the same as it would have been generated if the password was known.
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https://en.wikipedia.org/wiki/Length_extension_attack
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Incryptography,MD5CRKwas avolunteer computingeffort (similar todistributed.net) launched by Jean-Luc Cooke and his company, CertainKey Cryptosystems, to demonstrate that theMD5message digestalgorithmis insecure by finding acollision– two messages that produce the same MD5 hash. The project went live on March 1, 2004. The project ended on August 24, 2004, after researchers independently demonstrated a technique for generating collisions in MD5 using analytical methods byXiaoyun Wang, Feng,Xuejia Lai, and Yu.[1]CertainKey awarded a 10,000Canadian Dollarprize to Wang, Feng, Lai and Yu for their discovery.[2]
A technique calledFloyd's cycle-finding algorithmwas used to try to find a collision for MD5. The algorithm can be described by analogy with arandom walk. Using the principle that any function with a finite number of possible outputs placed in a feedback loop will cycle, one can use a relatively small amount of memory to store outputs with particular structures and use them as "markers" to better detect when a marker has been "passed" before. These markers are calleddistinguished points, the point where two inputs produce the same output is called acollision point. MD5CRK considered any point whose first 32bitswere zeroes to be a distinguished point.
The expected time to find a collision is not equal to2N{\displaystyle 2^{N}}whereN{\displaystyle N}is the number of bits in the digest output. It is in fact2N!(2N−K)!×2NK{\displaystyle 2^{N}! \over {(2^{N}-K)!\times {2^{N}}^{K}}}, whereK{\displaystyle K}is the number of function outputs collected.
For this project, the probability of success afterK{\displaystyle K}MD5 computations can be approximated by:11−eK×(1−K)2N+1{\displaystyle 1 \over {1-e^{K\times (1-K) \over 2^{N+1}}}}.
The expected number of computations required to produce a collision in the 128-bit MD5 message digest function is thus:1.17741×2N/2=1.17741×264{\displaystyle {1.17741\times 2^{N/2}}={1.17741\times 2^{64}}}
To give some perspective to this, using Virginia Tech'sSystem Xwith a maximum performance of 12.25 Teraflops, it would take approximately2.17×1019/12.25×1012≈1,770,000{\displaystyle {2.17\times 10^{19}/12.25\times 10^{12}\approx 1,770,000}}seconds or about 3 weeks. Or for commodity processors at 2 gigaflops it would take 6,000 machines approximately the same amount of time.
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https://en.wikipedia.org/wiki/MD5CRK
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NESSIE(New European Schemes for Signatures, Integrity and Encryption) was aEuropeanresearch project funded from 2000 to 2003 to identify securecryptographicprimitives. The project was comparable to theNISTAES processand the Japanese Government-sponsoredCRYPTRECproject, but with notable differences from both. In particular, there is both overlap and disagreement between the selections and recommendations from NESSIE and CRYPTREC (as of the August 2003 draft report). The NESSIE participants include some of the foremost activecryptographersin the world, as does the CRYPTREC project.
NESSIE was intended to identify and evaluate quality cryptographic designs in several categories, and to that end issued a public call for submissions in March 2000. Forty-two were received, and in February 2003 twelve of the submissions were selected. In addition, five algorithms already publicly known, but not explicitly submitted to the project, were chosen as "selectees". The project has publicly announced that "no weaknesses were found in the selected designs".
The selected algorithms and their submitters or developers are listed below. The five already publicly known, but not formally submitted to the project, are marked with a "*". Most may be used by anyone for any purpose without needing to seek a patent license from anyone; a license agreement is needed for those marked with a "#", but the licensors of those have committed to "reasonable non-discriminatory license terms for all interested", according to a NESSIE project press release.
None of the sixstream cipherssubmitted to NESSIE were selected because every one fell tocryptanalysis. This surprising result led to theeSTREAMproject.
Entrants that did not get past the first stage of the contest includeNoekeon,Q,Nimbus,NUSH,Grand Cru,Anubis,Hierocrypt,SC2000, andLILI-128.
The contractors and their representatives in the project were:
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https://en.wikipedia.org/wiki/NESSIE
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Incryptography, arandom oracleis anoracle(a theoreticalblack box) that responds to everyunique querywith a (truly)randomresponse chosenuniformlyfrom its output domain. If a query is repeated, it responds thesame wayevery time that query is submitted.
Stated differently, a random oracle is amathematical functionchosen uniformly at random, that is, a function mapping each possible query to a (fixed) random response from its output domain.
Random oracles first appeared in the context of complexity theory, in which they were used to argue that complexity class separations may face relativization barriers, with the most prominent case being theP vs NP problem, two classes shown in 1981 to be distinct relative to a random oraclealmost surely.[1]They made their way into cryptography by the publication ofMihir BellareandPhillip Rogawayin 1993, which introduced them as a formal cryptographic model to be used in reduction proofs.[2]
They are typically used when the proof cannot be carried out using weaker assumptions on thecryptographic hash function. A system that is proven secure when every hash function is replaced by a random oracle is described as being secure in therandom oracle model, as opposed to secure in thestandard model of cryptography.
Random oracles are typically used as anidealisedreplacement forcryptographic hash functionsin schemes where strong randomness assumptions are needed of the hash function's output. Such a proof often shows that a system or a protocol is secure by showing that an attacker must require impossible behavior from the oracle, or solve some mathematical problem believedhardin order to break it. However, it only proves such properties in the random oracle model, making sure no major design flaws are present. It is in general not true that such a proof implies the same properties in the standard model. Still, a proof in the random oracle model is considered better than no formal security proof at all.[3]
Not all uses of cryptographic hash functions require random oracles: schemes that require only one or more properties having a definition in thestandard model(such ascollision resistance,preimage resistance,second preimage resistance, etc.) can often be proven secure in the standard model (e.g., theCramer–Shoup cryptosystem).
Random oracles have long been considered incomputational complexity theory,[4]and many schemes have been proven secure in the random oracle model, for exampleOptimal Asymmetric Encryption Padding,RSA-FDHandPSS. In 1986,Amos FiatandAdi Shamir[5]showed a major application of random oracles – the removal of interaction from protocols for the creation of signatures.
In 1989,Russell ImpagliazzoandSteven Rudich[6]showed the limitation of random oracles – namely that their existence alone is not sufficient for secret-key exchange.
In 1993,Mihir BellareandPhillip Rogaway[2]were the first to advocate their use in cryptographic constructions. In their definition, the random oracle produces a bit-string ofinfinitelength which can be truncated to the length desired.
When a random oracle is used within a security proof, it is made available to all players, including the adversary or adversaries.
A single oracle may be treated as multiple oracles by pre-pending a fixed bit-string to the beginning of each query (e.g., queries formatted as "1||x" or "0||x" can be considered as calls to two separate random oracles, similarly "00||x", "01||x", "10||x" and "11||x" can be used to represent calls to four separate random oracles). This practice is usually calleddomain separation.Oracle cloningis the re-use of the once-constructed random oracle within the same proof (this in practice corresponds to the multiple uses of the samecryptographic hashwithin one algorithm for different purposes).[7]Oracle cloning with improper domain separation breaks security proofs and can lead to successful attacks.[8]
According to theChurch–Turing thesis, no functioncomputableby a finite algorithm can implement a true random oracle (which by definition requires an infinite description because it has infinitely many possible inputs, and its outputs are all independent from each other and need to be individually specified by any description).
In fact, certain contrived signature and encryption schemes are known which are proven secure in the random oracle model, but which are trivially insecure when any real function is substituted for the random oracle.[9][10]Nonetheless, for any more natural protocol a proof of security in the random oracle model gives very strong evidence of thepracticalsecurity of the protocol.[11]
In general, if a protocol is proven secure, attacks to that protocol must either be outside what was proven, or break one of the assumptions in the proof; for instance if the proof relies on the hardness ofinteger factorization, to break this assumption one must discover a fast integer factorization algorithm. Instead, to break the random oracle assumption, one must discover some unknown and undesirable property of the actual hash function; for good hash functions where such properties are believed unlikely, the considered protocol can be considered secure.
Although the Baker–Gill–Solovay theorem[12]showed that there exists an oracle A such that PA= NPA, subsequent work by Bennett and Gill,[13]showed that for arandom oracleB (a function from {0,1}nto {0,1} such that each input element maps to each of 0 or 1 with probability 1/2, independently of the mapping of all other inputs), PB⊊ NPBwith probability 1. Similar separations, as well as the fact that random oracles separate classes with probability 0 or 1 (as a consequence of theKolmogorov's zero–one law), led to the creation of theRandom Oracle Hypothesis, that two "acceptable" complexity classes C1and C2are equal if and only if they are equal (with probability 1) under a random oracle (the acceptability of a complexity class is defined in BG81[13]). This hypothesis was later shown to be false, as the two acceptable complexity classesIPandPSPACEwere shown to be equal[14]despite IPA⊊ PSPACEAfor a random oracle A with probability 1.[15]
Anidealcipher is arandom permutationoracle that is used to model an idealized block cipher. A random permutation decrypts each ciphertext block into one and only one plaintext block and vice versa, so there is aone-to-one correspondence. Some cryptographic proofs make not only the "forward" permutation available to all players, but also the "reverse" permutation.
Recent works showed that an ideal cipher can be constructed from a random oracle using 10-round[16]or even 8-round[17]Feistel networks.
An ideal permutation is an idealized object sometimes used in cryptography to model the behaviour of a permutation whose outputs are indistinguishable from those of a random permutation. In the ideal permutation model, an additional oracle access is given to the ideal permutation and its inverse. The ideal permutation model can be seen as a special case of the ideal cipher model where access is given to only a single permutation, instead of a family of permutations as in the case of the ideal cipher model.
Post-quantum cryptographystudies quantum attacks on classical cryptographic schemes. As a random oracle is an abstraction of ahash function, it makes sense to assume that a quantum attacker can access the random oracle inquantum superposition.[18]Many of the classical security proofs break down in that quantum random oracle model and need to be revised.
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https://en.wikipedia.org/wiki/Random_oracle
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Incryptography,cryptographic hash functionscan be divided into two main categories. In the first category are those functions whose designs are based on mathematical problems, and whose security thus follows from rigorous mathematical proofs,complexity theoryandformal reduction. These functions are calledprovably secure cryptographic hash functions. To construct these is very difficult, and few examples have been introduced. Their practical use is limited.
In the second category are functions which are not based on mathematical problems, but on an ad-hoc constructions, in which the bits of the message are mixed to produce the hash. These are then believed to be hard to break, but no formal proof is given. Almost all hash functions in widespread use reside in this category. Some of these functions are already broken, and are no longer in use.SeeHash function security summary.
Generally, thebasicsecurity ofcryptographic hash functionscan be seen from different angles: pre-image resistance, second pre-image resistance, collision resistance, and pseudo-randomness.
The basic question is the meaning ofhard. There are two approaches to answer this question. First is the intuitive/practical approach: "hardmeans that it is almost certainly beyond the reach of any adversary who must be prevented from breaking the system for as long as the security of the system is deemed important." The second approach is theoretical and is based on thecomputational complexity theory: if problemAis hard, then there exists a formalsecurity reductionfrom a problem which is widely considered unsolvable inpolynomial time, such asinteger factorizationor thediscrete logarithmproblem.
However, non-existence of a polynomial time algorithm does not automatically ensure that the system is secure. The difficulty of a problem also depends on its size. For example,RSA public-key cryptography(which relies on the difficulty ofinteger factorization) is considered secure only with keys that are at least 2048 bits long, whereas keys for theElGamal cryptosystem(which relies on the difficulty of thediscrete logarithmproblem) are commonly in the range of 256–512 bits.
If the set of inputs to the hash is relatively small or is ordered by likelihood in some way, then a brute force search may be practical, regardless of theoretical security. The likelihood of recovering the preimage depends on the input set size and the speed or cost of computing the hash function. A common example is the use of hashes to storepasswordvalidation data. Rather than store the plaintext of user passwords, an access control system typically stores a hash of the password. When a person requests access, the password they submit is hashed and compared with the stored value. If the stored validation data is stolen, then the thief will only have the hash values, not the passwords. However, most users choose passwords in predictable ways, and passwords are often short enough so that all possible combinations can be tested if fast hashes are used.[1]Special hashes calledkey derivation functionshave been created to slow searches.SeePassword cracking.
Most hash functions are built on an ad-hoc basis, where the bits of the message are nicely mixed to produce the hash. Variousbitwise operations(e.g. rotations),modular additions, andcompression functionsare used in iterative mode to ensure high complexity and pseudo-randomness of the output. In this way, the security is very hard to prove and the proof is usually not done. Only a few years ago[when?], one of the most popular hash functions,SHA-1, was shown to be less secure than its length suggested: collisions could be found in only 251[2]tests, rather than the brute-force number of 280.
In other words, most of the hash functions in use nowadays are not provably collision-resistant. These hashes are not based on purely mathematical functions. This approach results generally in more effective hashing functions, but with the risk that a weakness of such a function will be eventually used to find collisions. One famous case isMD5.
In this approach, the security of a hash function is based on some hard mathematical problem, and it is proved that finding collisions of the hash function is as hard as breaking the underlying problem. This gives a somewhat stronger notion of security than just relying on complex mixing of bits as in the classical approach.
A cryptographic hash function hasprovable security against collision attacksif finding collisions is provablypolynomial-time reduciblefrom a problemPwhich is supposed to be unsolvable in polynomial time. The function is then called provably secure, or just provable.
It means that if finding collisions would be feasible in polynomial time by algorithmA, then one could find and use polynomial time algorithmR(reduction algorithm) that would use algorithmAto solve problemP, which is widely supposed to be unsolvable in polynomial time. That is a contradiction. This means that finding collisions cannot be easier than solvingP.
However, this only indicates that finding collisions is difficult insomecases, as not all instances of a computationally hard problem are typically hard. Indeed, very large instances of NP-hard problems are routinely solved, while only the hardest are practically impossible to solve.
Examples of problems that are assumed to be not solvable in polynomial time include
SWIFFTis an example of a hash function that circumvents these security problems. It can be shown that, for any algorithm that can break SWIFFT with probabilitypwithin an estimated timet, one can find an algorithm that solves theworst-casescenario of a certain difficult mathematical problem within timet′depending ontandp.[citation needed]
Lethash(m) =xmmodn, wherenis a hard-to-factor composite number, andxis some prespecified base value. A collisionxm1≡xm2(modn)reveals a multiplem1−m2of themultiplicative orderofxmodulon. This information can be used to factornin polynomial time, assuming certain properties ofx.
But the algorithm is quite inefficient because it requires on average 1.5 multiplications modulonper message-bit.
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https://en.wikipedia.org/wiki/Security_of_cryptographic_hash_functions
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SHA-3(Secure Hash Algorithm 3) is the latest[4]member of theSecure Hash Algorithmfamily of standards, released byNISTon August 5, 2015.[5][6][7]Although part of the same series of standards, SHA-3 is internally different from theMD5-likestructureofSHA-1andSHA-2.
SHA-3 is a subset of the broader cryptographic primitive familyKeccak(/ˈkɛtʃæk/or/ˈkɛtʃɑːk/),[8][9]designed byGuido Bertoni,Joan Daemen,Michaël Peeters, andGilles Van Assche, building uponRadioGatún. Keccak's authors have proposed additional uses for the function, not (yet) standardized by NIST, including astream cipher, anauthenticated encryptionsystem, a "tree" hashing scheme for faster hashing on certain architectures,[10][11]andAEADciphers Keyak and Ketje.[12][13]
Keccak is based on a novel approach calledsponge construction.[14]Sponge construction is based on a wide random function or randompermutation, and allows inputting ("absorbing" in sponge terminology) any amount of data, and outputting ("squeezing") any amount of data, while acting as a pseudorandom function with regard to all previous inputs. This leads to great flexibility.
As of 2022, NIST does not plan to withdraw SHA-2 or remove it from the revised Secure Hash Standard.[15]The purpose of SHA-3 is that it can be directly substituted for SHA-2 in current applications if necessary, and to significantly improve the robustness of NIST's overall hash algorithm toolkit.[16]
For small message sizes, the creators of the Keccak algorithms and the SHA-3 functions suggest using the faster functionKangarooTwelvewith adjusted parameters and a new tree hashing mode without extra overhead.
The Keccak algorithm is the work of Guido Bertoni,Joan Daemen(who also co-designed theRijndaelcipher withVincent Rijmen), Michaël Peeters, andGilles Van Assche. It is based on earlier hash function designsPANAMAandRadioGatún. PANAMA was designed by Daemen and Craig Clapp in 1998. RadioGatún, a successor of PANAMA, was designed by Daemen, Peeters, and Van Assche, and was presented at the NIST Hash Workshop in 2006.[17]Thereference implementationsource codewas dedicated topublic domainviaCC0waiver.[18]
In 2006,NISTstarted to organize theNIST hash function competitionto create a new hash standard, SHA-3. SHA-3 is not meant to replaceSHA-2, as no significant attack on SHA-2 has been publicly demonstrated[needs update]. Because of the successful attacks onMD5,SHA-0andSHA-1,[19][20]NIST perceived a need for an alternative, dissimilar cryptographic hash, which became SHA-3.
After a setup period, admissions were to be submitted by the end of 2008. Keccak was accepted as one of the 51 candidates. In July 2009, 14 algorithms were selected for the second round. Keccak advanced to the last round in December 2010.[21]
During the competition, entrants were permitted to "tweak" their algorithms to address issues that were discovered. Changes that have been made to Keccak are:[22][23]
On October 2, 2012, Keccak was selected as the winner of the competition.[8]
In 2014, the NIST published a draftFIPS202 "SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions".[24]FIPS 202 was approved on August 5, 2015.[25]
On August 5, 2015, NIST announced that SHA-3 had become a hashing standard.[26]
In early 2013 NIST announced they would select different values for the "capacity", the overall strength vs. speed parameter, for the SHA-3 standard, compared to the submission.[27][28]The changes caused some turmoil.
The hash function competition called for hash functions at least as secure as the SHA-2 instances. It means that ad-bit output should haved/2-bit resistance tocollision attacksandd-bit resistance topreimage attacks, the maximum achievable fordbits of output. Keccak's security proof allows an adjustable level of security based on a "capacity"c, providingc/2-bit resistance to both collision and preimage attacks. To meet the original competition rules, Keccak's authors proposedc= 2d. The announced change was to accept the samed/2-bit security for all forms of attack and standardizec=d. This would have sped up Keccak by allowing an additionaldbits of input to be hashed each iteration. However, the hash functions would not have been drop-in replacements with the same preimage resistance as SHA-2 any more; it would have been cut in half, making it vulnerable to advances in quantum computing, which effectively would cut it in half once more.[29]
In September 2013,Daniel J. Bernsteinsuggested on theNISThash-forum mailing list[30]to strengthen the security to the 576-bit capacity that was originally proposed as the default Keccak, in addition to and not included in the SHA-3 specifications.[31]This would have provided at least a SHA3-224 and SHA3-256 with the same preimage resistance as their SHA-2 predecessors, but SHA3-384 and SHA3-512 would have had significantly less preimage resistance than their SHA-2 predecessors. In late September, the Keccak team responded by stating that they had proposed 128-bit security by settingc= 256as an option already in their SHA-3 proposal.[32]Although the reduced capacity was justifiable in their opinion, in the light of the negative response, they proposed raising the capacity toc= 512bits for all instances. This would be as much as any previous standard up to the 256-bit security level, while providing reasonable efficiency,[33]but not the 384-/512-bit preimage resistance offered by SHA2-384 and SHA2-512. The authors stated that "claiming or relying onsecurity strengthlevels above 256 bits is meaningless".
In early October 2013,Bruce Schneiercriticized NIST's decision on the basis of its possible detrimental effects on the acceptance of the algorithm, saying:
There is too much mistrust in the air. NIST risks publishing an algorithm that no one will trust and no one (except those forced) will use.[34]
He later retracted his earlier statement, saying:
I misspoke when I wrote that NIST made "internal changes" to the algorithm. That was sloppy of me. The Keccak permutation remains unchanged. What NIST proposed was reducing the hash function's capacity in the name of performance. One of Keccak's nice features is that it's highly tunable.[34]
Paul Crowley, a cryptographer and senior developer at an independent software development company, expressed his support of the decision, saying that Keccak is supposed to be tunable and there is no reason for different security levels within one primitive. He also added:
Yes, it's a bit of a shame for the competition that they demanded a certain security level for entrants, then went to publish a standard with a different one. But there's nothing that can be done to fix that now, except re-opening the competition. Demanding that they stick to their mistake doesn't improve things for anyone.[35]
There was some confusion that internal changes may have been made to Keccak, which were cleared up by the original team, stating that NIST's proposal for SHA-3 is a subset of the Keccak family, for which one can generate test vectors using their reference code submitted to the contest, and that this proposal was the result of a series of discussions between them and the NIST hash team.[36]
In response to the controversy, in November 2013 John Kelsey of NIST proposed to go back to the originalc= 2dproposal for all SHA-2 drop-in replacement instances.[37]The reversion was confirmed in subsequent drafts[38]and in the final release.[5]
SHA-3 uses thesponge construction,[14]in which data is "absorbed" into the sponge, then the result is "squeezed" out. In the absorbing phase, message blocks areXORedinto a subset of the state, which is then transformed as a whole using apermutation function(ortransformation)f{\displaystyle f}. In the "squeeze" phase, output blocks are read from the same subset of the state, alternated with the state transformation functionf{\displaystyle f}. The size of the part of the state that is written and read is called the "rate" (denotedr{\displaystyle r}), and the size of the part that is untouched by input/output is called the "capacity" (denotedc{\displaystyle c}). The capacity determines the security of the scheme. The maximumsecurity levelis half the capacity.
Given an input bit stringN{\displaystyle N}, a padding functionpad{\displaystyle pad}, a permutation functionf{\displaystyle f}that operates on bit blocks of widthb{\displaystyle b}, a rater{\displaystyle r}and an output lengthd{\displaystyle d}, we have capacityc=b−r{\displaystyle c=b-r}and the sponge constructionZ=sponge[f,pad,r](N,d){\displaystyle Z={\text{sponge}}[f,pad,r](N,d)}, yielding a bit stringZ{\displaystyle Z}of lengthd{\displaystyle d}, works as follows:[6]: 18
The fact that the internal stateScontainscadditional bits of information in addition to what is output toZprevents thelength extension attacksthat SHA-2, SHA-1, MD5 and other hashes based on theMerkle–Damgård constructionare susceptible to.
In SHA-3, the stateSconsists of a5 × 5array ofw-bit words (withw= 64),b= 5 × 5 ×w= 5 × 5 × 64 = 1600 bits total. Keccak is also defined for smaller power-of-2 word sizeswdown to 1 bit (total state of 25 bits). Small state sizes can be used to test cryptanalytic attacks, and intermediate state sizes (fromw= 8, 200 bits, tow= 32, 800 bits) can be used in practical, lightweight applications.[12][13]
For SHA3-224, SHA3-256, SHA3-384, and SHA3-512 instances,ris greater thand, so there is no need for additional block permutations in the squeezing phase; the leadingdbits of the state are the desired hash. However, SHAKE128 and SHAKE256 allow an arbitrary output length, which is useful in applications such asoptimal asymmetric encryption padding.
To ensure the message can be evenly divided intor-bit blocks, padding is required. SHA-3 uses the pattern 10...01 in its padding function: a 1 bit, followed by zero or more 0 bits (maximumr− 1) and a final 1 bit.
The maximum ofr− 1zero bits occurs when the last message block isr− 1bits long. Then another block is added after the initial 1 bit, containingr− 1zero bits before the final 1 bit.
The two 1 bits will be added even if the length of the message is already divisible byr.[6]: 5.1In this case, another block is added to the message, containing a 1 bit, followed by a block ofr− 2zero bits and another 1 bit. This is necessary so that a message with length divisible byrending in something that looks like padding does not produce the same hash as the message with those bits removed.
The initial 1 bit is required so messages differing only in a few additional 0 bits at the end do not produce the same hash.
The position of the final 1 bit indicates which raterwas used (multi-rate padding), which is required for the security proof to work for different hash variants. Without it, different hash variants of the same short message would be the same up to truncation.
The block transformationf, which is Keccak-f[1600] for SHA-3, is a permutation that usesXOR,ANDandNOToperations, and is designed for easy implementation in both software and hardware.
It is defined for any power-of-twowordsize,w= 2ℓbits. The main SHA-3 submission uses 64-bit words,ℓ= 6.
The state can be considered to be a5 × 5 ×warray of bits. Leta[i][j][k]be bit(5i+j) ×w+kof the input, using alittle-endianbit numbering convention androw-majorindexing. I.e.iselects the row,jthe column, andkthe bit.
Index arithmetic is performed modulo 5 for the first two dimensions and modulowfor the third.
The basic block permutation function consists of12 + 2ℓrounds of five steps:
The speed of SHA-3 hashing of long messages is dominated by the computation off= Keccak-f[1600] and XORingSwith the extendedPi, an operation onb= 1600 bits. However, since the lastcbits of the extendedPiare 0 anyway, and XOR with 0 is a NOP, it is sufficient to perform XOR operations only forrbits (r= 1600 − 2 × 224 = 1152 bits for SHA3-224, 1088 bits for SHA3-256, 832 bits for SHA3-384 and 576 bits for SHA3-512). The lowerris (and, conversely, the higherc=b−r= 1600 −r), the less efficient but more secure the hashing becomes since fewer bits of the message can be XORed into the state (a quick operation) before each application of the computationally expensivef.
The authors report the following speeds for software implementations of Keccak-f[1600] plus XORing 1024 bits,[1]which roughly corresponds to SHA3-256:
For the exact SHA3-256 on x86-64, Bernstein measures 11.7–12.25 cpb depending on the CPU.[40]: 7SHA-3 has been criticized for being slow on instruction set architectures (CPUs) which do not have instructions meant specially for computing Keccak functions faster – SHA2-512 is more than twice as fast as SHA3-512, and SHA-1 is more than three times as fast on an Intel Skylake processor clocked at 3.2 GHz.[41]The authors have reacted to this criticism by suggesting to use SHAKE128 and SHAKE256 instead of SHA3-256 and SHA3-512, at the expense of cutting the preimage resistance in half (but while keeping the collision resistance). With this, performance is on par with SHA2-256 and SHA2-512.
However, inhardware implementations, SHA-3 is notably faster than all other finalists,[42]and also faster than SHA-2 and SHA-1.[41]
As of 2018, ARM's ARMv8[43]architecture includes special instructions which enable Keccak algorithms to execute faster and IBM'sz/Architecture[44]includes a complete implementation of SHA-3 and SHAKE in a single instruction. There have also been extension proposals forRISC-Vto add Keccak-specific instructions.[45]
The NIST standard defines the following instances, for messageMand output lengthd:[6]: 20, 23
With the following definitions
SHA-3 instances are drop-in replacements for SHA-2, intended to have identical security properties.
SHAKE will generate as many bits from its sponge as requested, thus beingextendable-output functions(XOFs). For example, SHAKE128(M, 256) can be used as a hash function with a 256 character bitstream with 128-bit security strength. Arbitrarily large lengths can be used as pseudo-random number generators. Alternately, SHAKE256(M, 128) can be used as a hash function with a 128-bit length and 128-bit resistance.[6]
All instances append some bits to the message, the rightmost of which represent thedomain separationsuffix. The purpose of this is to ensure that it is not possible to construct messages that produce the same hash output for different applications of the Keccak hash function. The following domain separation suffixes exist:[6][46]
In December 2016NISTpublished a new document, NIST SP.800-185,[47]describing additional SHA-3-derived functions:
• X is the main input bit string. It may be of any length, including zero.
• L is an integer representing the requested output length in bits.
• N is a function-name bit string, used by NIST to define functions based on cSHAKE. When no function other than cSHAKE is desired, N is set to the empty string.
• S is a customization bit string. The user selects this string to define a variant of the function. When no customization is desired, S is set to the empty string.
• K is a key bit string of any length, including zero.
• B is the block size in bytes for parallel hashing. It may be any integer such that 0 < B < 22040.
In 2016 the same team that made the SHA-3 functions and the Keccak algorithm introduced faster reduced-rounds (reduced to 12 and 14 rounds, from the 24 in SHA-3) alternatives which can exploit the availability of parallel execution because of usingtree hashing: KangarooTwelve and MarsupilamiFourteen.[49]
These functions differ from ParallelHash, the FIPS standardized Keccak-based parallelizable hash function, with regard to the parallelism, in that they are faster than ParallelHash for small message sizes.
The reduced number of rounds is justified by the huge cryptanalytic effort focused on Keccak which did not produce practical attacks on anything close to twelve-round Keccak. These higher-speed algorithms are not part of SHA-3 (as they are a later development), and thus are not FIPS compliant; but because they use the same Keccak permutation they are secure for as long as there are no attacks on SHA-3 reduced to 12 rounds.[49]
KangarooTwelve is a higher-performance reduced-round (from 24 to 12 rounds) version of Keccak which claims to have 128 bits of security[50]while having performance as high as 0.55cycles per byteon aSkylakeCPU.[51]This algorithm is anIETFRFCdraft.[52]
MarsupilamiFourteen, a slight variation on KangarooTwelve, uses 14 rounds of the Keccak permutation and claims 256 bits of security. Note that 256-bit security is not more useful in practice than 128-bit security, but may be required by some standards.[50]128 bits are already sufficient to defeat brute-force attacks on current hardware, so having 256-bit security does not add practical value, unless the user is worried about significant advancements in the speed ofclassicalcomputers. For resistance againstquantumcomputers, see below.
KangarooTwelve and MarsupilamiFourteen are Extendable-Output Functions, similar to SHAKE, therefore they generate closely related output for a common message with different output length (the longer output is an extension of the shorter output). Such property is not exhibited by hash functions such as SHA-3 or ParallelHash (except of XOF variants).[6]
In 2016, the Keccak team released a different construction calledFarfalle construction, and Kravatte, an instance of Farfalle using the Keccak-p permutation,[53]as well as two authenticated encryption algorithms Kravatte-SANE and Kravatte-SANSE[54]
RawSHAKE is the basis for the Sakura coding for tree hashing, which has not been standardized yet. Sakura uses a suffix of 1111 for single nodes, equivalent to SHAKE, and other generated suffixes depending on the shape of the tree.[46]: 16
There is a general result (Grover's algorithm) that quantum computers can perform a structuredpreimage attackin2d=2d/2{\displaystyle {\sqrt {2^{d}}}=2^{d/2}}, while a classical brute-force attack needs 2d. A structured preimage attack implies a second preimage attack[29]and thus acollision attack. A quantum computer can also perform abirthday attack, thus break collision resistance, in2d3=2d/3{\displaystyle {\sqrt[{3}]{2^{d}}}=2^{d/3}}[55](although that is disputed).[56]Noting that the maximum strength can bec/2{\displaystyle c/2}, this gives the following upper[57]bounds on the quantum security of SHA-3:
It has been shown that theMerkle–Damgård construction, as used by SHA-2, is collapsing and, by consequence, quantum collision-resistant,[58]but for the sponge construction used by SHA-3, the authors provide proofs only for the case when the block functionfis not efficiently invertible; Keccak-f[1600], however, is efficiently invertible, and so their proof does not apply.[59][original research]
The following hash values are from NIST.gov:[60]
Changing a single bit causes each bit in the output to change with 50% probability, demonstrating anavalanche effect:
In the table below,internal statemeans the number of bits that are carried over to the next block.
Optimized implementation usingAVX-512VL(i.e. fromOpenSSL, running onSkylake-XCPUs) of SHA3-256 do achieve about 6.4 cycles per byte for large messages,[66]and about 7.8 cycles per byte when usingAVX2onSkylakeCPUs.[67]Performance on other x86, Power and ARM CPUs depending on instructions used, and exact CPU model varies from about 8 to 15 cycles per byte,[68][69][70]with some older x86 CPUs up to 25–40 cycles per byte.[71]
Below is a list of cryptography libraries that support SHA-3:
Apple A13ARMv8 six-coreSoCCPU cores have support[72]for accelerating SHA-3 (and SHA-512) using specialized instructions (EOR3, RAX1, XAR, BCAX) from ARMv8.2-SHA crypto extension set.[73]
Some software libraries usevectorizationfacilities of CPUs to accelerate usage of SHA-3. For example, Crypto++ can useSSE2on x86 for accelerating SHA3,[74]andOpenSSLcan useMMX,AVX-512orAVX-512VLon many x86 systems too.[75]AlsoPOWER8CPUs implement 2x64-bit vector rotate, defined in PowerISA 2.07, which can accelerate SHA-3 implementations.[76]Most implementations for ARM do not useNeonvector instructions asscalar codeis faster. ARM implementations can however be accelerated usingSVEand SVE2 vector instructions; these are available in theFujitsu A64FXCPU for instance.[77]
The IBMz/Architecturesupports SHA-3 since 2017 as part of the Message-Security-Assist Extension 6.[78]The processors support a complete implementation of the entire SHA-3 and SHAKE algorithms via the KIMD and KLMD instructions using a hardware assist engine built into each core.
Ethereumuses the Keccak-256 hash function (as per version 3 of the winning entry to the SHA-3 contest by Bertoni et al., which is different from the final SHA-3 specification).[79]
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Incryptographyauniversal one-way hash function(UOWHF, often pronounced "woof") is a type ofuniversal hash functionof particular importance tocryptography. UOWHFs are proposed as an alternative tocollision-resistant hash functions(CRHFs). CRHFs have a strong collision-resistance property: that it is hard, given randomly chosen hash function parameters, to find any collision of the hash function. In contrast, UOWHFs require that it be hard to find a collision where onepreimageis chosen independently of the hash function parameters. The primitive was suggested byMoni NaorandMoti Yungand is also known as "target collision resistant" hash functions; it was employed to construct general digital signature schemes without trapdoor functions, and also within chosen-ciphertext secure public key encryption schemes.
The UOWHF family contains a finite number of hash functions with each
having the same probability of being used.
The security property of a UOWHF is as follows. LetA{\displaystyle A}be an algorithm that operates in two phases:
Then for all polynomial-timeA{\displaystyle A}the probability thatA{\displaystyle A}succeeds is negligible.
UOWHFs are thought to be less computationally expensive than CRHFs, and are most often used for efficiency purposes in schemes where the choice of the hash function happens at some stage of execution, rather than beforehand. For instance, theCramer–Shoup cryptosystemuses a UOWHF as part of the validity check in its ciphertexts.
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Google Authenticatoris a software-basedauthenticatorbyGoogle. It implementsmulti-factor authenticationservices using thetime-based one-time password(TOTP; specified in RFC 6238) andHMAC-based one-time password(HOTP; specified in RFC 4226), for authenticating users of software applications.[5]
When logging into a site supporting Authenticator (including Google services) or using Authenticator-supporting third-party applications such aspassword managersorfile hosting services, Authenticator generates a six- to eight-digitone-time passwordwhich users must enter in addition to their usual login details.
Google providesAndroid,[6]Wear OS,[7]BlackBerry, andiOS[8]versions of Authenticator.
An officialopen sourcefork of the Android app is available onGitHub.[9]However, this fork was archived in Apr 6, 2021 and is now read only.[10]
Current software releases areproprietaryfreeware.[11]
Theappis first installed on a smartphone to use Authenticator. It must be set up for each site with which it is to be used: the site provides ashared secretkey to the user over a secure channel, to be stored in the Authenticator app. This secret key will be used for all future logins to the site.
To log into a site or service that usestwo-factor authenticationand supports Authenticator, the user provides a username and password to the site. The site then computes (but does not display) the required six- to eight-digitone-time passwordand asks the user to enter it. The user runs the Authenticator app, which independently computes and displays the same password, which the user types in, authenticating their identity.[citation needed]
With this kind of two-factor authentication, mere knowledge of username and password is insufficient to break into a user's account - the attacker also needs knowledge of the shared secret key or physical access to the device running the Authenticator app. An alternative route of attack is aman-in-the-middle attack: if the device used for the login process is compromised bymalware, the credentials and one-time password can be intercepted by the malware, which then can initiate its login session to the site, or monitor and modify the communication between the user and the site.[12]
During setup, the service provider generates an 80-bit secret key for each user (whereas RFC 4226 §4 requires 128 bits and recommends 160 bits).[13]This is transferred to the Authenticator app as a 16, 26, or 32-characterbase32string, or as aQR code.
Subsequently, when the user opens the Authenticator app, it calculates anHMAC-SHA1hash value using this secret key. The message can be:
A portion of the HMAC is extracted and displayed to the user as a six- to eight-digit code; The lastnibble(4 bits) of the result is used as a pointer, to a 32-bit integer, in the result byte array, and masks out the 31st bit.
The Google Authenticator app forAndroidwas originally open source, but later became proprietary.[11]Google made earlier source for their Authenticator app available on itsGitHubrepository; the associated development page stated:
"This open source project allows you to download the code that powered version 2.21 of the application. Subsequent versions contain Google-specific workflows that are not part of the project."[14]
The latest open-source release was in 2020.[9]
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FreeOTPis afree and open-sourceauthenticatorbyRedHat. It implementsmulti-factor authenticationusingHOTPandTOTP.Tokenscan be added by scanning aQR codeor by manually entering the token configuration. It is licensed under theApache 2.0 license, and supportsAndroidandiOS.[4][5][6]
This mobile software article is astub. You can help Wikipedia byexpanding it.
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Initiative for Open Authentication(OATH) is an industry-wide collaboration to develop an openreference architectureusingopen standardsto promote the adoption of strong authentication. It has close to thirty coordinating and contributing members and is proposing standards for a variety of authentication technologies, with the aim of lowering costs and simplifying their functions.
The nameOATHis an acronym from the phrase "open authentication", and is pronounced as the English word "oath".[1]
OATH is not related toOAuth, an open standard forauthorization, however, most logging systems employ a mixture of both.
Thiscomputer securityarticle is astub. You can help Wikipedia byexpanding it.
This cryptography-related article is astub. You can help Wikipedia byexpanding it.
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KYPS(Keep Your Password Secret) is a free web-basedservicethat enables users to log into websites, which usually require a username/password combination, usingone-time passwords. The main difference between KYPS and similarpassword managementtechnologies is that the password is never disclosed to the local computer. This makes KYPS effective against password theft byspywareorkeyloggers, particularly when using public computers such as in anInternet cafe.[1]
More details about the internal workings of KYPS were published at the CSIE 2009 conference.[1]KYPS is also featured onMakeuseof,[2]andheise.de[3]Some of the internal workings of its predecessor, anopen-sourceproject called "Impostor",[4]were published at the peer-reviewedGLOBECOMconference in 2004.[5]
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Incryptology, acodeis a method used toencryptamessagethat operates at the level of meaning; that is, words or phrases are converted into something else. A code might transform "change" into "CVGDK" or "cocktail lounge". The U.S.National Security Agencydefined a code as "A substitution cryptosystem in which the plaintext elements are primarily words, phrases, or sentences, and the code equivalents (called "code groups") typically consist of letters or digits (or both) in otherwise meaningless combinations of identical length."[1]: Vol I, p. 12Acodebookis needed to encrypt, and decrypt the phrases or words.
By contrast,ciphersencrypt messages at the level of individual letters, or small groups of letters, or even, in modern ciphers, individualbits. Messages can be transformed first by a code, and then by a cipher.[2]Suchmultiple encryption, or "superencryption" aims to makecryptanalysismore difficult.
Another comparison between codes and ciphers is that a code typically represents a letter or groups of letters directly without the use of mathematics. As such the numbers are configured to represent these three values: 1001 = A, 1002 = B, 1003 = C, ... . The resulting message, then would be 1001 1002 1003 to communicate ABC. Ciphers, however, utilize a mathematical formula to represent letters or groups of letters. For example, A = 1, B = 2, C = 3, ... . Thus the message ABC results by multiplying each letter's value by 13. The message ABC, then would be 13 26 39.
Codes have a variety of drawbacks, including susceptibility tocryptanalysisand the difficulty of managing the cumbersomecodebooks, so ciphers are now the dominant technique in modern cryptography.
In contrast, because codes are representational, they are not susceptible to mathematical analysis of the individual codebook elements. In the example, the message 13 26 39 can be cracked by dividing each number by 13 and then ranking them alphabetically. However, the focus of codebook cryptanalysis is the comparative frequency of the individual code elements matching the same frequency of letters within the plaintext messages usingfrequency analysis. In the above example, the code group, 1001, 1002, 1003, might occur more than once and that frequency might match the number of times that ABC occurs in plain text messages.
(In the past, or in non-technical contexts,codeandcipherare often used to refer to any form ofencryption).
Codes are defined by "codebooks" (physical or notional), which are dictionaries of codegroups listed with their corresponding plaintext. Codes originally had the codegroups assigned in 'plaintext order' for convenience of the code designed, or the encoder. For example, in a code using numeric code groups, a plaintext word starting with "a" would have a low-value group, while one starting with "z" would have a high-value group. The same codebook could be used to "encode" a plaintext message into a coded message or "codetext", and "decode" a codetext back into plaintext message.
In order to make life more difficult for codebreakers, codemakers designed codes with no predictable relationship between the codegroups and the ordering of the matching plaintext. In practice, this meant that two codebooks were now required, one to find codegroups for encoding, the other to look up codegroups to find plaintext for decoding. Such "two-part" codes required more effort to develop, and twice as much effort to distribute (and discard safely when replaced), but they were harder to break. TheZimmermann Telegramin January 1917 used the German diplomatic "0075" two-part code system which contained upwards of 10,000 phrases and individual words.[3]
A one-time code is a prearranged word, phrase or symbol that is intended to be used only once to convey a simple message, often the signal to execute or abort some plan or confirm that it has succeeded or failed. One-time codes are often designed to be included in what would appear to be an innocent conversation. Done properly they are almost impossible to detect, though a trained analyst monitoring the communications of someone who has already aroused suspicion might be able to recognize a comment like "Aunt Bertha has gone into labor" as having an ominous meaning. Famous example of one time codes include:
Sometimes messages are not prearranged and rely on shared knowledge hopefully known only to the recipients. An example is the telegram sent to U.S. PresidentHarry Truman, then at thePotsdam Conferenceto meet with Soviet premierJoseph Stalin, informing Truman of thefirst successful testof anatomic bomb.
See alsoone-time pad, an unrelated cypher algorithm
Anidiot codeis a code that is created by the parties using it. This type of communication is akin to the hand signals used by armies in the field.
Example: Any sentence where 'day' and 'night' are used means 'attack'. The location mentioned in the following sentence specifies the location to be attacked.
An early use of the term appears to be by George Perrault, a character in the science fiction bookFriday[5]byRobert A. Heinlein:
Terrorism expert Magnus Ranstorp said that the men who carried out theSeptember 11 attackson the United States used basic e-mail and what he calls "idiot code" to discuss their plans.[6]
While solving amonoalphabetic substitution cipheris easy, solving even a simple code is difficult. Decrypting a coded message is a little like trying to translate a document written in a foreign language, with the task basically amounting to building up a "dictionary" of the codegroups and the plaintext words they represent.
One fingerhold on a simple code is the fact that some words are more common than others, such as "the" or "a" in English. In telegraphic messages, the codegroup for "STOP" (i.e., end of sentence or paragraph) is usually very common. This helps define the structure of the message in terms of sentences, if not their meaning, and this is cryptanalytically useful.
Further progress can be made against a code by collecting many codetexts encrypted with the same code and then using information from other sources
For example, a particular codegroup found almost exclusively in messages from a particular army and nowhere else might very well indicate the commander of that army. A codegroup that appears in messages preceding an attack on a particular location may very well stand for that location.
Cribscan be an immediate giveaway to the definitions of codegroups. As codegroups are determined, they can gradually build up a critical mass, with more and more codegroups revealed from context and educated guesswork. One-part codes are more vulnerable to such educated guesswork than two-part codes, since if the codenumber "26839" of a one-part code is determined to stand for "bulldozer", then the lower codenumber "17598" will likely stand for a plaintext word that starts with "a" or "b". At least, for simple one part codes.
Various tricks can be used to "plant" or "sow" information into a coded message, for example by executing a raid at a particular time and location against an enemy, and then examining code messages sent after the raid. Coding errors are a particularly useful fingerhold into a code; people reliably make errors, sometimes disastrous ones. Planting data and exploiting errors works against ciphers as well.
Constructing a new code is like building a new language and writing a dictionary for it; it was an especially big job before computers. If a code is compromised, the entire task must be done all over again, and that means a lot of work for both cryptographers and the code users. In practice, when codes were in widespread use, they were usually changed on a periodic basis to frustrate codebreakers, and to limit the useful life of stolen or copied codebooks.
Once codes have been created, codebook distribution is logistically clumsy, and increases chances the code will be compromised. There is a saying that "Three people can keep a secret if two of them are dead," (Benjamin Franklin - Wikiquote) and though it may be something of an exaggeration, a secret becomes harder to keep if it is shared among several people. Codes can be thought reasonably secure if they are only used by a few careful people, but if whole armies use the same codebook, security becomes much more difficult.
In contrast, the security of ciphers is generally dependent on protecting the cipher keys. Cipher keys can be stolen and people can betray them, but they are much easier to change and distribute.
It was common to encipher a message after first encoding it, to increase the difficulty of cryptanalysis. With a numerical code, this was commonly done with an "additive" - simply a long key number which was digit-by-digit added to the code groups, modulo 10. Unlike the codebooks, additives would be changed frequently. The famous Japanese Navy code,JN-25, was of this design.
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Apersonal identification number(PIN; sometimesredundantlyaPIN codeorPIN number) is a numeric (sometimes alpha-numeric)passcodeused in the process of authenticating a user accessing a system.
The PIN has been the key to facilitating theprivate dataexchange between different data-processing centers in computer networks for financial institutions, governments, and enterprises.[1]PINs may be used to authenticate banking systems with cardholders, governments with citizens, enterprises with employees, and computers with users, among other uses.
In common usage, PINs are used in ATM or PO transactions,[2]secure access control (e.g. computer access, door access, car access),[3]internet transactions,[4]or to log into a restricted website.
The PIN originated with the introduction of theautomated teller machine(ATM) in 1967, as an efficient way for banks to dispense cash to their customers. The first ATM system was that ofBarclaysin London, in 1967; it acceptedchequeswith machine-readable encoding, rather than cards, and matched the PIN to the cheque.[5][6][7]In 1972,Lloyds Bankissued the first bank card to feature an information-encoding magnetic strip, using a PIN for security.[8]James Goodfellow, the inventor who patented the first personal identification number, was awarded anOBEin the 2006Queen's Birthday Honours.[9][10]
Mohamed M. Atallainvented the first PIN-basedhardware security module(HSM),[11]dubbed the "Atalla Box," a security system that encrypted PIN andATMmessages and protected offline devices with an un-guessable PIN-generating key.[12]In 1972, Atalla filedU.S. patent 3,938,091for his PIN verification system, which included an encodedcard readerand described a system that utilizedencryptiontechniques to assure telephone link security while entering personal ID information that was transmitted to a remote location for verification.[13]
He foundedAtalla Corporation(nowUtimaco Atalla) in 1972,[14]and commercially launched the "Atalla Box" in 1973.[12]The product was released as the Identikey. It was a card reader andcustomer identification system, providing a terminal withplastic cardand PIN capabilities. The system was designed to letbanksandthrift institutionsswitch to a plastic card environment from apassbookprogram. The Identikey system consisted of a card reader console, two customerPIN pads, intelligent controller and built-in electronic interface package.[15]The device consisted of twokeypads, one for the customer and one for the teller. It allowed the customer to type in a secret code, which is transformed by the device, using amicroprocessor, into another code for the teller.[16]During atransaction, the customer'saccount number was read by the card reader. This process replaced manual entry and avoided possible key stroke errors. It allowed users to replace traditional customer verification methods such as signature verification and test questions with a secure PIN system.[15]In recognition of his work on the PIN system ofinformation security management, Atalla has been referred to as the "Father of the PIN".[17][18][19]
The success of the "Atalla Box" led to the wide adoption of PIN-based hardware security modules.[20]Its PIN verification process was similar to the laterIBM 3624.[21]By 1998 an estimated 70% of all ATM transactions in the United States were routed through specialized Atalla hardware modules,[22]and by 2003 the Atalla Box secured 80% of all ATM machines in the world,[17]increasing to 85% as of 2006.[23]Atalla's HSM products protect 250millioncard transactionsevery day as of 2013,[14]and still secure the majority of the world's ATM transactions as of 2014.[11]
In the context of a financial transaction, usually both a private "PIN code" and public user identifier are required to authenticate a user to the system. In these situations, typically the user is required to provide a non-confidential user identifier or token (theuser ID) and a confidential PIN to gain access to the system. Upon receiving the user ID and PIN, the system looks up the PIN based upon the user ID and compares the looked-up PIN with the received PIN. The user is granted access only when the number entered matches the number stored in the system. Hence, despite the name, a PIN does notpersonallyidentify the user.[24]The PIN is not printed or embedded on the card but is manually entered by the cardholder duringautomated teller machine(ATM) andpoint of sale(PO) transactions (such as those that comply withEMV), and incard not presenttransactions, such as over the Internet or for phone banking.
The international standard for financial services PIN management,ISO 9564-1, allows for PINs from four up to twelve digits, but recommends that for usability reasons the card issuer not assign a PIN longer than six digits.[25]The inventor of the ATM,John Shepherd-Barron, had at first envisioned a six-digit numeric code, but his wife could only remember four digits, and that has become the most commonly used length in many places,[6]although banks in Switzerland and many other countries require a six-digit PIN.
There are several main methods of validating PINs. The operations discussed below are usually performed within ahardware security module(HSM).
One of the earliest ATM models was theIBM 3624, which used the IBM method to generate what is termed anatural PIN. The natural PIN is generated by encrypting the primary account number (PAN), using an encryption key generated specifically for the purpose.[26]This key is sometimes referred to as the PIN generation key (PGK). This PIN is directly related to the primary account number. To validate the PIN, the issuing bank regenerates the PIN using the above method, and compares this with the entered PIN.
Natural PINs cannot be user selectable because they are derived from the PAN. If the card is reissued with a new PAN, a new PIN must be generated.
Natural PINs allow banks to issue PIN reminder letters as the PIN can be generated.
To allow user-selectable PINs it is possible to store a PIN offset value. The offset is found by subtracting the natural PIN from the customer selected PIN usingmodulo10.[27]For example, if the natural PIN is 1234, and the user wishes to have a PIN of 2345, the offset is 1111.
The offset can be stored either on the card track data,[28]or in a database at the card issuer.
To validate the PIN, the issuing bank calculates the natural PIN as in the above method, then adds the offset and compares this value to the entered PIN.
The VISA method is used by many card schemes and is not VISA-specific. The VISA method generates a PIN verification value (PVV). Similar to the offset value, it can be stored on the card's track data, or in a database at the card issuer. This is called the reference PVV.
The VISA method takes the rightmost eleven digits of the PAN excluding the checksum value, a PIN validation key index (PVKI, chosen from one to six, a PVKI of 0 indicates that the PIN cannot be verified through PVS[29]) and the required PIN value to make a 64-bit number, the PVKI selects a validation key (PVK, of 128 bits) to encrypt this number. From this encrypted value, the PVV is found.[30]
To validate the PIN, the issuing bank calculates a PVV value from the entered PIN and PAN and compares this value to the reference PVV. If the reference PVV and the calculated PVV match, the correct PIN was entered.
Unlike the IBM method, the VISA method does not derive a PIN. The PVV value is used to confirm the PIN entered at the terminal, was also used to generate the reference PVV. The PIN used to generate a PVV can be randomly generated, user-selected or even derived using the IBM method.
Financial PINs are often four-digit numbers in the range 0000–9999, resulting in 10,000 possible combinations. Switzerland issues six-digit PINs by default.[31]
Some systems set up default PINs and most allow the customer to set up a PIN or to change the default one, and on some a change of PIN on first access is mandatory. Customers are usually advised not to set up a PIN-based on their or their spouse's birthdays, on driver license numbers, consecutive or repetitive numbers, or some other schemes. Some financial institutions do not give out or permit PINs where all digits are identical (such as 1111, 2222, ...), consecutive (1234, 2345, ...), numbers that start with one or more zeroes, or the last four digits of the cardholder'ssocial security numberor birth date.[citation needed]
Many PIN verification systems allow three attempts, thereby giving a card thief a putative 0.03%probabilityof guessing the correct PIN before the card is blocked. This holds only if all PINs are equally likely and the attacker has no further information available, which has not been the case with some of the many PIN generation and verification algorithms that financial institutions and ATM manufacturers have used in the past.[32]
Research has been done on commonly used PINs.[33]The result is that without forethought, a sizable portion of users may find their PIN vulnerable. "Armed with only four possibilities, hackers can crack 20% of all PINs. Allow them no more than fifteen numbers, and they can tap the accounts of more than a quarter of card-holders."[34]
Breakable PINs can worsen with length, to wit:
The problem with guessable PINs surprisingly worsens when customers are forced to use additional digits, moving from about a 25% probability with fifteen numbers to more than 30% (not counting 7-digits with all those phone numbers). In fact, about half of all 9-digit PINs can be reduced to two dozen possibilities, largely because more than 35% of all people use the all too tempting 123456789. As for the remaining 64%, there's a good chance they're using theirSocial Security Number, which makes them vulnerable. (Social Security Numbers contain their own well-known patterns.)[34]
In 2002, two PhD students atCambridge University, Piotr Zieliński and Mike Bond, discovered a security flaw in the PIN generation system of theIBM 3624, which was duplicated in most later hardware. Known as thedecimalization table attack, the flaw would allow someone who has access to a bank's computer system to determine the PIN for an ATM card in an average of 15 guesses.[35][36]
Rumours have been in e-mail and Internet circulation claiming that in the event of entering a PIN into an ATM backwards, law enforcement will be instantly alerted as well as money being ordinarily issued as if the PIN had been entered correctly.[37]The intention of this scheme would be to protect victims of muggings; however, despite thesystembeing proposed for use in some US states,[38][39]there are no ATMs currently in existence that employ this software.[40]
A mobile phone may be PIN protected. If enabled, the PIN (also called a passcode) forGSMmobile phones can be between four and eight digits[41]and is recorded in theSIM card. If such a PIN is entered incorrectly three times, the SIM card is blocked until apersonal unblocking code(PUC or PUK), provided by the service operator, is entered.[42]If the PUC is entered incorrectly ten times, the SIM card is permanently blocked, requiring a new SIM card from the mobile carrier service.
Note that this should not be confused with software-based passcodes that are often used on smartphones withlock screens: these are not related to the device's cellular SIM card, PIN and PUC.
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AQR code,quick-response code,[1]is a type oftwo-dimensionalmatrix barcodeinvented in 1994 byMasahiro HaraofJapanesecompanyDenso Wavefor labelling automobile parts.[2][3]It features black squares on a white background withfiducial markers, readable by imaging devices like cameras, and processed usingReed–Solomon error correctionuntil the image can be appropriately interpreted. The required data is then extracted from patterns that are present in both the horizontal and the vertical components of the QR image.[4]
Whereas a barcode is a machine-readable optical image that contains information specific to the labeled item, the QR code contains the data for a locator, an identifier, and web-tracking. To store data efficiently, QR codes use four standardized modes of encoding:numeric,alphanumeric,byteorbinary, andkanji.[5]Compared to standardUPC barcodes, the QR labeling system was applied beyond the automobile industry because of faster reading of the optical image and greater data-storage capacity in applications such as product tracking, item identification, time tracking, document management, and general marketing.[4]
The QR code system was invented in 1994, at theDenso Waveautomotive products company, in Japan.[6][7][8]The initial alternating-square design presented by the team of researchers, headed byMasahiro Hara, was influenced by the black counters and the white counters played on aGo board;[9]the pattern of the position detection markers was determined by finding the least-used sequence of alternating black-white areas on printed matter, which was found to be (1:1:3:1:1).[10][6]The functional purpose of the QR code system was to facilitate keeping track of the types and numbers of automobile parts, by replacing individually-scanned bar-code labels on each box of auto parts with a single label that contained the data of each label. The quadrangular configuration of the QR code system consolidated the data of the various bar-code labels with Kanji,Kana, and alphanumeric codes printed onto a single label.[11][10][6]
As of 2024,[update]QR codes are used in a much broader context, including both commercial tracking applications and convenience-oriented applications aimed at mobile phone users (termed mobile tagging). QR codes may be used to display text to the user, to open awebpageon the user's device, to add avCardcontact to the user's device, to open aUniform Resource Identifier(URI), to connect to a wireless network, or to compose an email or text message. There are a great many QR code generators available as software or as online tools that are either free or require a paid subscription.[12]The QR code has become one of the most-used types of two-dimensional code.[13]
During June 2011, 14 million American mobile users scanned a QR code or a barcode. Some 58% of those users scanned a QR or barcode from their homes, while 39% scanned from retail stores; 53% of the 14 million users were men between the ages of 18 and 34.[14]
In 2022, 89 million people in theUnited Statesscanned a QR code using their mobile devices, up by 26 percent compared to 2020. The majority of QR code users used them to makepaymentsor to access product andmenuinformation.[15]
In September 2020, a survey found that 18.8 percent of consumers in the United States and theUnited Kingdomstrongly agreed that they had noticed an increase in QR code use since the then-activeCOVID-19-related restrictions had begun several months prior.[16]
Several standards cover the encoding of data as QR codes:[17]
At theapplication layer, there is some variation between most of the implementations. Japan'sNTT DoCoMohas establishedde factostandards for the encoding of URLs, contact information, and several other data types.[22]The open-source "ZXing" project maintains a list of QR code data types.[23]
QR codes have become common in consumer advertising. Typically, asmartphoneis used as a QR code scanner, displaying the code and converting it to some useful form (such as a standardURLfor a website, thereby obviating the need for a user to type it into aWeb browser).
QR codes have become a focus of advertising strategy to provide a way to access a brand's website more quickly than by manually entering a URL.[24][25]Beyond mere convenience to the consumer, the importance of this capability is the belief that it increases theconversion rate: the chance that contact with the advertisement will convert to a sale. It coaxes interested prospects further down theconversion funnelwith little delay or effort, bringing the viewer to the advertiser's website immediately, whereas a longer and more targeted sales pitch may lose the viewer's interest.
Although initially used to track parts in vehicle manufacturing, QR codes are used over a much wider range of applications. These include commercial tracking, warehouse stock control, entertainment and transport ticketing, product and loyalty marketing, and in-store product labeling.[citation needed]Examples of marketing include where a company's discounted and percent discount can be captured using a QR code decoder that is a mobile app, or storing a company's information such as address and related information alongside its alpha-numeric text data as can be seen in telephone directoryyellow pages.[citation needed]
They can also be used to store personal information for organizations. An example of this is thePhilippinesNational Bureau of Investigation(NBI) where NBI clearances now come with a QR code. Many of these applications targetmobile-phoneusers (viamobile tagging). Users may receive text, add a vCard contact to their device, open a URL, or compose ane-mailor text message after scanning QR codes. They can generate and print their own QR codes for others to scan and use by visiting one of several pay or free QR code-generating sites or apps.Googlehad anAPI, now deprecated, to generate QR codes,[26]and apps for scanning QR codes can be found on nearly all smartphone devices.[27]
QR codes storing addresses and URLs may appear in magazines, on signs, on buses, on business cards, or on almost any object about which users might want information. Users with acamera phoneequipped with the correct reader application can scan the image of the QR code to display text and contact information, connect to awireless network, or open a web page in the phone's browser. This act of linking from physical world objects is termedhardlinkingorobject hyperlinking. QR codes also may be linked to a location to track where a code has been scanned. Either the application that scans the QR code retrieves the geo information by using GPS and cell tower triangulation (aGPS) or the URL encoded in the QR code itself is associated with a location. In 2008, a Japanese stonemason announced plans to engrave QR codes on gravestones, allowing visitors to view information about the deceased, and family members to keep track of visits.[29]PsychologistRichard Wisemanwas one of the first authors to include QR codes in a book, inParanormality: Why We See What Isn't There(2011).[30][failed verification]Microsoft OfficeandLibreOfficehave a functionality to insert QR code into documents.[31][32]
QR codes have been incorporated into currency. In June 2011, TheRoyal Dutch Mint(Koninklijke Nederlandse Munt) issued the world's first official coin with a QR code to celebrate the centenary of its current building and premises. The coin can be scanned by a smartphone and originally linked to a special website with content about the historical event and design of the coin.[33]In 2014, the Central Bank of Nigeria issued a 100-naira banknote to commemorate its centennial, the first banknote to incorporate a QR code in its design. When scanned with an internet-enabled mobile device, the code goes to a website that tells the centenary story of Nigeria.[34]
In 2015, theCentral Bank of the Russian Federationissued a 100-rublesnote to commemorate theannexation of Crimea by the Russian Federation.[35]It contains a QR code into its design, and when scanned with an internet-enabled mobile device, the code goes to a website that details the historical and technical background of the commemorative note. In 2017, theBank of Ghanaissued a 5-cedisbanknote to commemorate 60 years ofcentral bankinginGhana. It contains a QR code in its design which, when scanned with an internet-enabled mobile device, goes to the official Bank of Ghana website.
Credit card functionality is under development. In September 2016, theReserve Bank of India(RBI) launched the eponymously namedBharatQR, a common QR code jointly developed by all the four major card payment companies –National Payments Corporation of Indiathat runsRuPaycards along withMastercard,Visa, andAmerican Express. It will also have the capability of accepting payments on theUnified Payments Interface(UPI) platform.[36][37]
QR codes are used in someaugmented realitysystems to determine the positions of objects in 3-dimensional space.[11]
QR codes can be used on various mobile device operating systems. While initially requiring the installation and use of third-party apps, both Android and iOS (since iOS 11[38][39]) devices can now natively scan QR codes, without requiring an external app to be used.[40]The camera app can scan and display the kind of QR code along with the link. These devices supportURL redirection, which allows QR codes to sendmetadatato existing applications on the device.
QR codes have been used to establish "virtual stores", where a gallery of product information and QR codes is presented to the customer, e.g. on a train station wall. The customers scan the QR codes, and the products are delivered to their homes. This use started inSouth Korea,[41]and Argentina,[42]but is currently expanding globally.[43]Walmart, Procter & Gamble and Woolworths have already adopted the Virtual Store concept.[44]
QR codes can be used to store bank account information or credit card information, or they can be specifically designed to work with particular payment provider applications. There are several trial applications of QR code payments across the world.[45][46]In developing countries including China,[47][48]India[49]QR code payment is a very popular and convenient method of making payments. SinceAlipaydesigned a QR code payment method in 2011,[50]mobile payment has been quickly adopted in China. As of 2018, around 83% of all payments were made via mobile payment.[51]
In November 2012, QR code payments were deployed on a larger scale in theCzech Republicwhen an open format for payment information exchange – aShort Payment Descriptor– was introduced and endorsed by theCzech Banking Associationas the official local solution for QR payments.[52][53]In 2013, theEuropean Payment Councilprovided guidelines for theEPC QR codeenablingSCTinitiation within theEurozone.
In 2017,Singaporecreated a task force including government agencies such as theMonetary Authority of SingaporeandInfocomm Media Development Authorityto spearhead a system for e-payments using standardized QR code specifications. These specific dimensions are specialized for Singapore.[54]
The e-payment system, Singapore Quick Response Code (SGQR), essentially merges various QR codes into one label that can be used by both parties in the payment system. This allows for various banking apps to facilitate payments between multiple customers and a merchant that displays a single QR code. The SGQR scheme is co-owned by MAS and IMDA.[55]A single SDQR label contains e-payments and combines multiple payment options. People making purchases can scan the code and see which payment options the merchant accepts.[55]
QR codes can be used to log into websites: a QR code is shown on the login page on acomputer screen, and when a registered user scans it with a verified smartphone, they will automatically be logged in. Authentication is performed by the smartphone, which contacts the server. Google deployed such a login scheme in 2012.[56]
There is a system whereby a QR code can be displayed on a device such as a smartphone and used as anadmission ticket.[57][58]Its use is common forJ1 LeagueandNippon Professional Baseballtickets in Japan.[59][60]In some cases, rights can be transferred via the Internet. InLatvia, QR codes can be scanned inRigapublic transport to validateRīgas Satiksmee-tickets.[61]
Restaurants can present a QR code near the front door or at the table allowing guests to view an online menu, or even redirect them to an online ordering website or app, allowing them to order and/or possibly pay for their meal without having to use a cashier or waiter. QR codes can also link to daily or weekly specials that are not printed on the standardized menus,[62]and enable the establishment to update the entire menu without needing to print copies. At table-serve restaurants, QR codes enable guests to order and pay for their meals without a waiter involved – the QR code contains the table number so servers know where to bring the food.[63]This application has grown especially since the need for social distancing during the2020 COVID-19 pandemicprompted reduced contact between service staff and customers.[63]
By specifying the SSID, encryption type, password/passphrase, and if the SSID is hidden or not, mobile device users can quickly scan and join networks without having to manually enter the data.[64]AMeCard-like format is supported by Android and iOS 11+.[65]
A QR code can link to anobituaryand can be placed on aheadstone. In 2008, Ishinokoe in Yamanashi Prefecture, Japan began to sell tombstones with QR codes produced by IT DeSign, where the code leads to a virtual grave site of the deceased.[66][67][68]Other companies, such as Wisconsin-based Interactive Headstones, have also begun implementing QR codes into tombstones.[69]In 2014, theJewish Cemetery of La Pazin Uruguay began implementing QR codes for tombstones.[70]
QR codes can be used to generatetime-based one-time passwordsforelectronic authentication.
QR codes have been used by various retail outlets that haveloyalty programs. Sometimes these programs are accessed with anappthat is loaded onto a phone and includes a process triggered by a QR code scan. The QR codes for loyalty programs tend to be found printed on thereceiptfor a purchase or on the products themselves. Users in these schemes collect award points by scanning a code.
Serialised QR codes have been used by brands[71]and governments[72]to let consumers, retailers and distributors verify the authenticity of the products and help with detecting counterfeit products, as part of abrand protectionprogram.[73]However, the security level of a regular QR code is limited since QR codes printed on original products are easily reproduced on fake products, even though the analysis of data generated as a result of QR code scanning can be used to detect counterfeiting and illicit activity.[74]A higher security level can be attained by embedding adigital watermarkorcopy detection patterninto the image of the QR code. This makes the QR code more secure against counterfeiting attempts; products that display a code which is counterfeit, although valid as a QR code, can be detected by scanning the secure QR code with the appropriate app.[75]
The treaty regulatingapostilles(documents bearing a seal of authenticity), has been updated to allow the issuance of digital apostilles by countries; a digital apostille is a PDF document with acryptographic signaturecontaining a QR code for a canonical URL of the original document, allowing users to verify the apostille from a printed version of the document.
Different studies have been conducted to assess the effectiveness of QR codes as a means of conveying labelling information and their use as part of a food traceability system. In a field experiment, it was found that when provided free access to a smartphone with a QR code scanning app, 52.6% of participants would use it to access labelling information.[76]A study made in South Korea showed that consumers appreciate QR code used in food traceability system, as they provide detailed information about food, as well as information that helps them in their purchasing decision.[77]If QR codes are serialised, consumers can access a web page showing the supply chain for each ingredient, as well as information specific to each related batch, including meat processors and manufacturers, which helps address the concerns they have about the origin of their food.[78]
After theCOVID-19 pandemicbegan spreading, QR codes began to be used as a "touchless" system to display information, show menus, or provide updated consumer information, especially in the hospitality industry. Restaurants replaced paper or laminated plastic menus with QR code decals on the table, which opened an online version of the menu. This prevented the need to dispose of single-use paper menus, or institute cleaning and sanitizing procedures for permanent menus after each use.[79]Local television stations have also begun to utilize codes onlocal newscaststo allow viewers quicker access to stories or information involving the pandemic, including testing and immunization scheduling websites, or for links within stories mentioned in the newscasts overall.
InAustralia, patrons were required to scan QR codes at shops, clubs, supermarkets, and other service and retail establishments on entry to assistcontact tracing. Singapore,Taiwan, the United Kingdom, andNew Zealandused similar systems.[80]
QR codes are also present on COVID-19 vaccination certificates in places such asCanadaand theEU(EU Digital COVID certificate), where they can be scanned to verify the information on the certificate.[81]
Unlike the older, one-dimensional barcodes that were designed to be mechanically scanned by a narrow beam of light, a QR code is detected by a two-dimensional digitalimage sensorand then digitally analyzed by a programmed processor. The processor locates the three distinctive squares at the corners of the QR code image, using a smaller square (or multiple squares) near the fourth corner to normalize the image for size, orientation, and angle of viewing. The small dots throughout the QR code are then converted to binary numbers and validated with an error-correcting algorithm.
The amount of data that can be represented by a QR code symbol depends on the data type (mode, or input character set), version (1, ..., 40, indicating the overall dimensions of the symbol, i.e. 4 × version number + 17 dots on each side), anderror correctionlevel. The maximum storage capacities occur for version 40 and error correction level L (low), denoted by 40-L:[13][82]
Here are some samples of QR codes:
QR codes useReed–Solomon error correctionover thefinite fieldF256{\displaystyle \mathbb {F} _{256}}orGF(28), the elements of which are encoded asbytes of 8 bits; the byteb7b6b5b4b3b2b1b0{\displaystyle b_{7}b_{6}b_{5}b_{4}b_{3}b_{2}b_{1}b_{0}}with a standard numerical value∑i=07bi2i{\displaystyle \textstyle \sum _{i=0}^{7}b_{i}2^{i}}encodes the field element∑i=07biαi{\displaystyle \textstyle \sum _{i=0}^{7}b_{i}\alpha ^{i}}whereα∈F256{\displaystyle \alpha \in \mathbb {F} _{256}}is taken to be a primitive element satisfyingα8+α4+α3+α2+1=0{\displaystyle \alpha ^{8}+\alpha ^{4}+\alpha ^{3}+\alpha ^{2}+1=0}. The primitive polynomial isx8+x4+x3+x2+1{\displaystyle x^{8}+x^{4}+x^{3}+x^{2}+1}, corresponding to the polynomial number 285, with initial root = 0 to obtain generator polynomials.
The Reed–Solomon code uses one of 37 different polynomials overF256{\displaystyle \mathbb {F} _{256}}, with degrees ranging from 7 to 68, depending on how many error correction bytes the code adds. It is implied by the form of Reed–Solomon used (systematic BCH view) that these polynomials are all on the form∏i=0n−1(x−αi){\textstyle \prod _{i=0}^{n-1}(x-\alpha ^{i})}. However, the rules for selecting the degreen{\displaystyle n}are specific to the QR standard.
For example, the generator polynomial used for the Version 1 QR code (21×21), when 7 error correction bytes are used, is:g(x)=x7+α87x6+α229x5+α146x4+α149x3+α238x2+α102x+α21{\displaystyle g(x)=x^{7}+\alpha ^{87}x^{6}+\alpha ^{229}x^{5}+\alpha ^{146}x^{4}+\alpha ^{149}x^{3}+\alpha ^{238}x^{2}+\alpha ^{102}x+\alpha ^{21}}.
This is obtained by multiplying the first seven terms:g(x)=(x+1)(x+α)(x+α2)(x+α3)(x+α4)(x+α5)(x+α6){\displaystyle g(x)=(x+1)(x+\alpha )(x+\alpha ^{2})(x+\alpha ^{3})(x+\alpha ^{4})(x+\alpha ^{5})(x+\alpha ^{6})}.
The same may also be expressed using decimal coefficients (overF256{\displaystyle \mathbb {F} _{256}}), as:g(x)=x7+127x6+122x5+154x4+164x3+11x2+68x+117{\displaystyle g(x)=x^{7}+127x^{6}+122x^{5}+154x^{4}+164x^{3}+11x^{2}+68x+117}.
The highest power ofx{\displaystyle x}in the polynomial (the degreen{\displaystyle n}, of the polynomial) determines the number of error correction bytes. In this case, the degree is 7.
When discussing the Reed–Solomon code phase there is some risk for confusion, in that the QR ISO/IEC standard uses the termcodewordfor the elements ofF256{\displaystyle \mathbb {F} _{256}}, which with respect to the Reed–Solomon code aresymbols, whereas it uses the termblockfor what with respect to the Reed–Solomon code are the codewords. The number of data versus error correction bytes within each block depends on (i) the version (side length) of the QR symbol and (ii) the error correction level, of which there are four. The higher the error correction level, the less storage capacity. The following table lists the approximate error correction capability at each of the four levels:
In larger QR symbols, the message is broken up into several Reed–Solomon code blocks. The block size is chosen so that no attempt is made at correcting more than 15 errors per block; this limits the complexity of the decoding algorithm. The code blocks are then interleaved together, making it less likely that localized damage to a QR symbol will overwhelm the capacity of any single block.
The Version 1 QR symbol with level L error correction, for example, consists of a single error correction block with a total of26 codebytes (made of19message bytes and seven error correction bytes). It can correct up to2byte errors. Hence, this code is known as a(26,19,2)error correction code overGF(28). It is also sometimes represented in short, as (26,19) code.
Due to error correction, it is possible to create artistic QR codes with embellishments to make them more readable or attractive to the human eye, and to incorporate colors, logos, and other features into the QR code block; the embellishments are treated as errors, but the codes still scan correctly.[84][85]
It is also possible to design artistic QR codes without reducing the error correction capacity by manipulating the underlying mathematical constructs.[86][87]Image processing algorithms are also used to reduce errors in QR-code.[88]
The format information records two things: the error correction level and the mask pattern used for the symbol. Masking is used to break up patterns in the data area that might confuse a scanner, such as large blank areas or misleading features that look like the locator marks. The mask patterns are defined on a grid that is repeated as necessary to cover the whole symbol. Modules corresponding to the dark areas of the mask are inverted. The 5-bit format information is protected from errors with aBCH code, and two complete copies are included in each QR symbol.[4]A (15,5) triple error-correcting BCH code overGF(24)is used, having the generator polynomialg(x)=x10+x8+x5+x4+x2+x+1{\displaystyle g(x)=x^{10}+x^{8}+x^{5}+x^{4}+x^{2}+x+1}. It can correct at most 3 bit-errors out of the 5 data bits. There are a total of 15 bits in this BCH code (10 bits are added for error correction). This 15-bit code is itself X-ORed with a fixed 15-bit mask pattern (101010000010010) to prevent an all-zero string.
To obtain the error correction (EC) bytes for a message "www.wikipedia.org", the following procedure may be carried out:
The message is 17 bytes long, hence it can be encoded using a (26,19,2) Reed-Solomon code to fit in a Ver1 (21×21) symbol, which has a maximum capacity of 19 bytes (for L level error correction).
The generator polynomial specified for the (26,19,2) code, is:g(x)=x7+127x6+122x5+154x4+164x3+11x2+68x+117{\displaystyle g(x)=x^{7}+127x^{6}+122x^{5}+154x^{4}+164x^{3}+11x^{2}+68x+117},
which may also be written in the form of a matrix of decimal coefficients:
The 17-byte long message "www.wikipedia.org" as hexadecimal coefficients (ASCII values), denoted by M1 through M17 is:
The encoding mode is "Byte encoding". Hence the 'Enc' field is [0100] (4 bits). The length of the above message is 17 bytes hence 'Len' field is [00010001] (8 bits). The 'End' field is End of message marker [0000] (4 bits).
The message code word (without EC bytes) is of the form:
Substituting the hexadecimal values, it can be expressed as:
This is rearranged as 19-byte blocks of 8 bits each:
Using the procedure forReed-Solomon systematic encoding, the 7 EC bytes obtained (E1 through E7, as shown in the symbol) which are the coefficients (in decimal) of the remainder after polynomial division are:
or in hexadecimal values:
These 7 EC bytes are then appended to the 19-byte message. The resulting coded message has 26 bytes (in hexadecimal):
Note: The bit values shown in the Ver1 QR symbol below do not match with the above values, as the symbol has been masked using a mask pattern (001).
The message dataset is placed from right to left in a zigzag pattern, as shown below. In larger symbols, this is complicated by the presence of the alignment patterns and the use of multiple interleaved error-correction blocks.
The general structure of a QR encoding is as a sequence of 4 bit indicators with payload length dependent on the indicator mode (e.g. byte encoding payload length is dependent on the first byte).[89]
Four-bit indicators are used to select the encoding mode and convey other information.
Encoding modes can be mixed as needed within a QR symbol. (e.g., a url with a long string of alphanumeric characters )
After every indicator that selects an encoding mode is a length field that tells how many characters are encoded in that mode. The number of bits in the length field depends on the encoding and the symbol version.
Alphanumeric encoding mode stores a message more compactly than the byte mode can, but cannot store lower-case letters and has only a limited selection of punctuation marks, which are sufficient for rudimentaryweb addresses. Two characters are coded in an 11-bit value by this formula:
This has the exception that the last character in an alphanumeric string with an odd length is read as a 6-bit value instead.
The following images offer more information about the QR code.
Model 1 QR codeis an older version of the specification. It is visually similar to the widely seen model 2 codes, but lacks alignment patterns. Differences are in the bottom right corner, and in the midsections of the bottom and right edges are additional functional regions.
Micro QR code is a smaller version of the QR code standard for applications where symbol size is limited. There are four different versions (sizes) of Micro QR codes: the smallest is 11×11 modules; the largest can hold 35 numeric characters,[90]or 21ASCIIalphanumeric characters, or 15 bytes (128 bits).
Rectangular Micro QR Code (also known as rMQR Code) is a two-dimensional (2D) matrix barcode invented and standardized in 2022 by Denso Wave as ISO/IEC 23941. rMQR Code is designed as a rectangular variation of the QR code and has the same parameters and applications as original QR codes; however, rMQR Code is more suitable for rectangular areas, and has a difference between width and height up to 19 in the R7x139 version.
iQR code is an alternative to existing square QR codes developed by Denso Wave. iQR codes can be created in square or rectangular formations; this is intended for situations where a longer and narrower rectangular shape is more suitable, such as on cylindrical objects. iQR codes can fit the same amount of information in 30% less space. There are 61 versions of square iQR codes, and 15 versions of rectangular codes. For squares, the minimum size is 9 × 9 modules; rectangles have a minimum of 19 × 5 modules. iQR codes add error correction level S, which allows for 50% error correction.[91]iQR Codes had not been given an ISO/IEC specification as of 2015, and only proprietary Denso Wave products could create or read iQR codes.[92]
Secure Quick Response (SQR) code is a QR code that contains a "private data" segment after the terminator instead of the specified filler bytes "ec 11".[93]This private data segment must be deciphered with an encryption key. This can be used to store private information and to manage a company's internal information.[94]
Frame QR is a QR code with a "canvas area" that can be flexibly used. In the center of this code is the canvas area, where graphics, letters, and more can be flexibly arranged, making it possible to lay out the code without losing the design of illustrations, photos, etc.[95]
Researchers have proposed a new High Capacity Colored 2-Dimensional (HCC2D) Code, which builds upon a QR code basis for preserving the QR robustness to distortions and uses colors for increasing data density (as of 2014 it is still in the prototyping phase). The HCC2D code specification is described in details in Queriniet al.(2011),[96]while techniques for color classification of HCC2D code cells are described in detail in Querini andItaliano(2014),[97]which is an extended version of Querini and Italiano (2013).[98]
Introducing colors into QR codes requires addressing additional issues. In particular, during QR code reading only the brightness information is taken into account, while HCC2D codes have to cope with chromatic distortions during the decoding phase. In order to ensure adaptation to chromatic distortions that arise in each scanned code, HCC2D codes make use of an additional field: the Color Palette Pattern. This is because color cells of a Color Palette Pattern are supposed to be distorted in the same way as color cells of the Encoding Region. Replicated color palettes are used for training machine-learning classifiers.
Accessible QRis a type of QR code that combines a standard QR code with a dot-dash pattern positioned around one corner of the code to provide product information for people who are blind and partially sighted. The codes, announce product categories and product details such as instructions, ingredients, safety warnings, and recycling information. The data is structured for the needs of users who are blind or partially sighted and offers larger text or audio output. It can read QR codes from a metre away, activating the smartphone's accessibility features like VoiceOver to announce product details.
The use of QR code technology is freely licensed as long as users follow the standards for QR code documented withJISorISO/IEC. Non-standardized codes may require special licensing.[99]
Denso Wave owns a number of patents on QR code technology, but has chosen to exercise them in a limited fashion.[99]In order to promote widespread usage of the technology Denso Wave chose to waive its rights to a key patent in its possession forstandardizedcodes only.[17]In the US, the granted QR code patent, 5726435, expired on March 14, 2015. In Japan, the corresponding patent, 2938338, expired on March 14, 2014. TheEuropean Patent Officegranted patent 0672994 to Denso Wave, which was then validated intoFrench, UK, and German patents, all of which expired in March 2015.[100]
The textQR Codeitself is aregistered trademarkandwordmarkof Denso Wave Incorporated.[101]In UK, the trademark is registered as E921775, the termQR Code, with a filing date of 3 September 1998.[102]The UK version of the trademark is based on the Kabushiki Kaisha Denso (DENSO CORPORATION) trademark, filed as Trademark 000921775, the termQR Code, on 3 September 1998 and registered on 16 December 1999 with the European Union OHIM (Office for Harmonization in the Internal Market).[103]The U.S. Trademark for the termQR Codeis Trademark 2435991 and was filed on 29 September 1998 with an amended registration date of 13 March 2001, assigned to Denso Corporation.[104]In South Korea, trademark application filed on 18 November 2011 was refused at 20 March 2012, because theKorean Intellectual Property Officeviewed that the phrase wasgenericizedamong South Korean people to refer to matrix barcodes in general.[105]
The only context in which common QR codes can carry executable data is the URL data type. These URLs may hostJavaScriptcode, which can be used to exploit vulnerabilities in applications on the host system, such as the reader, the web browser, or the image viewer, since a reader will typically send the data to the application associated with the data type used by the QR code.
In the case of no software exploits, malicious QR codes combined with a permissive reader can still put a computer's contents and user's privacy at risk. This practice is known as "attagging", aportmanteauof "attack tagging".[106]They are easily created and can be affixed over legitimate QR codes.[107][failed verification][108]On a smartphone, the reader's permissions may allow use of the camera, full Internet access, read/write contact data,GPS, read browser history, read/write local storage, and global system changes.[109][110][111][improper synthesis?]
Risks include linking to dangerous web sites with browser exploits, enabling the microphone/camera/GPS, and then streaming those feeds to a remote server, analysis of sensitive data (passwords, files, contacts, transactions),[112]and sending email/SMS/IM messages or packets forDDoSas part of abotnet, corrupting privacy settings, stealing identity,[113]and even containing malicious logic themselves such as JavaScript[114]or a virus.[115][116]These actions could occur in the background while the user is only seeing the reader opening a seemingly harmless web page.[117]In Russia, a malicious QR code caused phones that scanned it to send premium texts at a fee of$6 each.[106]QR codes have also been linked to scams in which stickers are placed onparking metersand other devices, posing as quick payment options, as seen inAustin,San AntonioandBoston, among other cities across the United States and Australia.[118][119][120]
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https://en.wikipedia.org/wiki/QR_Code
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S/KEYis aone-time passwordsystem developed forauthenticationtoUnix-likeoperating systems, especially fromdumb terminalsor untrusted public computers on which one does not want to type a long-term password. A user's real password is combined in an offline device with a short set of characters and a decrementing counter to form a single-use password. Because each password is only used once, they are useless topassword sniffers.
Because the short set of characters does not change until the counter reaches zero, it is possible to prepare a list of single-use passwords, in order, that can be carried by the user. Alternatively, the user can present the password, characters, and desired counter value to a local calculator to generate the appropriate one-time password that can then be transmitted over the network in the clear. The latter form is more common and practically amounts tochallenge–response authentication.
S/KEY is supported inLinux(viapluggable authentication modules),OpenBSD,NetBSD, andFreeBSD, and a generic open-source implementation can be used to enable its use on other systems.OpenSSHalso implements S/KEY since version OpenSSH 1.2.2 was released on December 1, 1999.[1]One common implementation is calledOPIE. S/KEY is a trademark ofTelcordia Technologies, formerly known as Bell Communications Research (Bellcore).
S/KEY is also sometimes referred to asLamport's scheme, after its author,Leslie Lamport. It was developed by Neil Haller,Phil Karnand John Walden at Bellcore in the late 1980s. With the expiration of the basic patents onpublic-key cryptographyand the widespread use oflaptop computersrunningSSHand
other cryptographic protocols that can secure an entire session, not just the password, S/KEY is falling
into disuse.[citation needed]Schemes that implementtwo-factor authentication, by comparison, are growing in use.[2]
Theserveris the computer that will perform the authentication.
After password generation, the user has a sheet of paper withnpasswords on it. Ifnis very large, either storing allnpasswords or calculate the given password fromH(W) become inefficient. There are methods to efficiently calculate the passwords in the required order, using only⌈logn2⌉{\displaystyle \left\lceil {\frac {\log n}{2}}\right\rceil }hash calculations per step and storing⌈logn⌉{\displaystyle \lceil \log n\rceil }passwords.[3]
More ideally, though perhaps less commonly in practice, the user may carry a small, portable, secure, non-networked computing device capable of regenerating any needed password given the secret passphrase, thesalt, and the number of iterations of the hash required, the latter two of which are conveniently provided by the server requesting authentication for login.
In any case, the first password will be the same password that the server has stored. This first password will not be used for authentication (the user should scratch this password on the sheet of paper), the second one will be used instead:
For subsequent authentications, the user will providepasswordi. (The last password on the printed list,passwordn, is the first password generated by the server,H(W), whereWis the initial secret).
The server will computeH(passwordi) and will compare the result topasswordi−1, which is stored as reference on the server.
The security of S/KEY relies on the difficulty of reversingcryptographic hash functions. Assume an attacker manages to get hold of a password that was used for a successful authentication. Supposing this ispasswordi, this password is already useless for subsequent authentications, because each password can only be used once. It would be interesting for the attacker to find outpasswordi−1, because this password is the one that will be used for the next authentication.
However, this would require inverting the hash function that producedpasswordi−1usingpasswordi(H(passwordi−1) =passwordi), which is extremely difficult to do with currentcryptographic hash functions.
Nevertheless, S/KEY is vulnerable to aman in the middle attackif used by itself. It is also vulnerable to certainrace conditions, such as where an attacker's software sniffs the network to learn the firstN− 1 characters in the password (whereNequals the password length), establishes its own TCP session to the server, and in rapid succession tries all valid characters in theN-th position until one succeeds. These types of vulnerabilities can be avoided by usingssh,SSL, SPKM, or other encrypted transport layer.
Since each iteration of S/KEY doesn't include the salt or count, it is feasible to find collisions directly without breaking the initial password. This has a complexity of 264, which can be pre-calculated with the same amount of space. The space complexity can be optimized by storing chains of values, although collisions might reduce the coverage of this method, especially for long chains.[4]
Someone with access to an S/KEY database can break all of them in parallel with a complexity of 264. While they wouldn't get the original password, they would be able to find valid credentials for each user. In this regard, it is similar to storing unsalted 64-bit hashes of strong, unique passwords.
The S/KEY protocol can loop. If such a loop were created in the S/KEY chain, an attacker could use user's key without finding the original value, and possibly without tipping off the valid user. The pathological case of this would be an OTP that hashes to itself.
Internally, S/KEY uses64-bitnumbers. For humanusabilitypurposes, each number is mapped to six short words, of one to four characters each, from a publicly accessible 2048-word dictionary. For example, one 64-bit number maps to "ROY HURT SKI FAIL GRIM KNEE".[5]
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Asecurity tokenis aperipheral deviceused to gain access to an electronically restricted resource. The token is used in addition to, or in place of, apassword.[1]Examples of security tokens include wirelesskey cardsused to open locked doors, a banking token used as a digital authenticator for signing in toonline banking, or signing transactions such aswire transfers.
Security tokens can be used to store information such aspasswords,cryptographic keysused to generatedigital signatures, orbiometricdata (such asfingerprints). Some designs incorporatetamper resistantpackaging, while others may include smallkeypadsto allow entry of aPINor a simple button to start a generation routine with some display capability to show a generated key number. Connected tokens utilize a variety of interfaces includingUSB,near-field communication(NFC),radio-frequency identification(RFID), orBluetooth. Some tokens have audio capabilities designed for those who are vision-impaired.
All tokens contain some secret information used to prove identity. There are four different ways in which this information can be used:
Time-synchronized, one-time passwords change constantly at a set time interval; e.g., once per minute. To do this, some sort of synchronization must exist between theclient's token and the authenticationserver. For disconnected tokens, this time-synchronization is done before the token is distributed to theclient. Other token types do the synchronization when the token is inserted into aninput device. The main problem with time-synchronized tokens is that they can, over time, become unsynchronized.[2]However, some such systems, such asRSA's SecurID, allow the user to re-synchronize the server with the token, sometimes by entering several consecutive passcodes. Most also cannot have replaceable batteries and only last up to 5 years before having to be replaced – so there is an additional cost.[3]Another type of one-time password uses a complex mathematical algorithm, such as ahash chain, to generate a series of one-time passwords from a secret shared key. Each password is unique, even when previous passwords are known. The open-sourceOATHalgorithm is standardized;[citation needed]other algorithms are covered by USpatents. Each password is observably unpredictable and independent of previous ones, whereby an adversary would be unable to guess what the next password may be, even with knowledge of all previous passwords.
Tokens can containchipswith functions varying from very simple to very complex, including multiple authentication methods.
The simplest security tokens do not need any connection to acomputer. The tokens have a physical display; the authenticating user simply enters the displayed number to log in. Other tokens connect to the computer using wireless techniques, such asBluetooth. These tokens transfer a key sequence to the local client or to a nearby access point.[4]
Alternatively, another form of token that has been widely available for many years is a mobile device which communicates using an out-of-band channel (like voice,SMS, orUSSD).
Still other tokens plug into the computer and may require a PIN. Depending on the type of the token, thecomputerOSwill then either read the key from the token and perform a cryptographic operation on it, or ask the token's firmware to perform this operation.[citation needed]
A related application is the hardwaredonglerequired by some computer programs to prove ownership of thesoftware. The dongle is placed in aninput deviceand thesoftwareaccesses theI/O devicein question toauthorizethe use of thesoftwarein question.
Commercial solutions are provided by a variety of vendors, each with their own proprietary (and often patented) implementation of variously used security features. Token designs meeting certain security standards are certified in theUnited Statesas compliant withFIPS 140, a federal security standard.[5]Tokens without any kind of certification are sometimes viewed as suspect, as they often do not meet accepted government or industry security standards, have not been put through rigorous testing, and likely cannot provide the same level of cryptographic security as token solutions which have had their designs independently audited by third-party agencies.[citation needed]
Disconnected tokens have neither a physical nor logical connection to the client computer. They typically do not require a special input device, and instead use a built-in screen to display the generated authentication data, which the user enters manually themselves via akeyboardorkeypad. Disconnected tokens are the most common type of security token used (usually in combination with a password) in two-factor authentication for online identification.[6]
Connected tokens are tokens that must be physically connected to the computer with which the user is authenticating. Tokens in this category automatically transmit the authentication information to the client computer once a physical connection is made, eliminating the need for the user to manually enter the authentication information. However, in order to use a connected token, the appropriate input device must be installed. The most common types of physical tokens aresmart cardsand USB tokens (also calledsecurity keys), which require a smart card reader and a USB port respectively. Increasingly,FIDO2tokens, supported by the open specification groupFIDO Alliancehave become popular for consumers with mainstream browser support beginning in 2015 and supported by popular websites and social media sites.[citation needed]
OlderPC cardtokens are made to work primarily withlaptops. Type II PC Cards are preferred as a token as they are half as thick as Type III.
The audio jack port is a relatively practical method to establish connection between mobile devices, such asiPhone,iPadandAndroid, and other accessories.[citation needed]The most well known device is calledSquare, a credit card reader foriOSand Android devices.
Some use a special purpose interface (e.g. thecrypto ignition keydeployed by the United StatesNational Security Agency). Tokens can also be used as a photoID card.Cell phonesandPDAscan also serve as security tokens with proper programming.
Many connected tokens use smart card technology. Smart cards can be very cheap (around ten cents)[citation needed]and contain proven security mechanisms (as used by financial institutions, like cash cards). However, computational performance of smart cards is often rather limited because of extreme low power consumption and ultra-thin form-factor requirements.
Smart-card-basedUSBtokens which contain asmart cardchip inside provide the functionality of both USB tokens and smart cards. They enable a broad range of security solutions and provide the abilities and security of a traditional smart card without requiring a unique input device. From thecomputer operating system's point of view such a token is a USB-connected smart card reader with one non-removable smart card present.[7]
Unlike connected tokens, contactless tokens form a logical connection to the client computer but do not require a physical connection. The absence of the need for physical contact makes them more convenient than both connected and disconnected tokens. As a result, contactless tokens are a popular choice forkeyless entrysystems and electronic payment solutions such asMobilSpeedpass, which usesRFIDto transmit authentication info from a keychain token.[citation needed]However, there have been various security concerns raised about RFID tokens after researchers atJohns Hopkins UniversityandRSA Laboratoriesdiscovered that RFID tags could be easily cracked and cloned.[8]
Another downside is that contactless tokens have relatively short battery lives; usually only 5–6 years, which is low compared toUSBtokens which may last more than 10 years.[citation needed]Some tokens however do allow the batteries to be changed, thus reducing costs.
TheBluetooth Low Energyprotocols provide long lasting battery lifecycle of wireless transmission.
Although, the automatic transmission power control attempts for radial distance estimates. The escape is available apart from the standardised Bluetooth power control algorithm to provide a calibration on minimally required transmission power.[9]
Bluetooth tokens are often combined with a USB token, thus working in both a connected and a disconnected state. Bluetooth authentication works when closer than 32 feet (9.8 meters). When the Bluetooth link is not properly operable, the token may be inserted into aUSBinput deviceto function.
Another combination is with asmart cardto store locally larger amounts of identity data and process information as well.[10]Another is a contactless BLE token that combines secure storage and tokenized release of fingerprint credentials.[11]
In the USB mode of operation sign-off requires care for the token while mechanically coupled to the USB plug. The advantage with the Bluetooth mode of operation is the option of combining sign-off with distance metrics. Respective products are in preparation, following the concepts of electronic leash.
Near-field communication(NFC) tokens combined with a Bluetooth token may operate in several modes, thus working in both a connected and a disconnected state. NFC authentication works when closer than 1 foot (0.3 meters).[citation needed]The NFC protocol bridges short distances to the reader while the Bluetooth connection serves for data provision with the token to enable authentication. Also when the Bluetooth link is not connected, the token may serve the locally stored authentication information in coarse positioning to the NFC reader and relieves from exact positioning to a connector.[citation needed]
Some types ofsingle sign-on(SSO) solutions, likeenterprise single sign-on, use the token to store software that allows for seamless authentication and password filling. As the passwords are stored on the token, users need not remember their passwords and therefore can select more secure passwords, or have more secure passwords assigned. Usually most tokens store a cryptographic hash of the password so that if the token is compromised, the password is still protected.[12]
Programmable tokens are marketed as "drop-in" replacement of mobile applications such asGoogle Authenticator(miniOTP[13]). They can be used as mobile app replacement, as well as in parallel as a backup.
The simplest vulnerability with any password container is theft or loss of the device. The chances of this happening, or happening unaware, can be reduced with physical security measures such as locks, electronic leash, or body sensor and alarm. Stolen tokens can be made useless by usingtwo factor authentication. Commonly, in order to authenticate, apersonal identification number(PIN) must be entered along with the information provided by the token the same time as the output of the token.
Any system which allows users to authenticate via an untrusted network (such asthe Internet) is vulnerable toman-in-the-middle attacks. In this type of attack, an attacker acts as the "go-between" of the user and the legitimate system, soliciting the token output from the legitimate user and then supplying it to the authentication system themselves. Since the token value is mathematically correct, the authentication succeeds and the fraudster is granted access. In 2006,Citibankwas the victim of an attack when its hardware-token-equipped business users became the victims of a large Ukrainian-based man-in-the-middlephishingoperation.[14][15]
In 2012, the Prosecco research team at INRIA Paris-Rocquencourt developed an efficient method of extracting the secret key from severalPKCS #11cryptographic devices.[16][17]These findings were documented in INRIA Technical Report RR-7944, ID hal-00691958,[18]and published at CRYPTO 2012.[19]
Trusted as a regular hand-written signature, thedigital signaturemust be made with a private key known only to the person authorized to make the signature. Tokens that allow secure on-board generation and storage of private keys enable secure digital signatures, and can also be used for user authentication, as the private key also serves as a proof of the user's identity.
For tokens to identify the user, all tokens must have some kind of number that is unique. Not all approaches fully qualify asdigital signaturesaccording to some national laws.[citation needed]Tokens with no on-board keyboard or anotheruser interfacecannot be used in somesigningscenarios, such as confirming a bank transaction based on the bank account number that the funds are to be transferred to.
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Time-based one-time password(TOTP) is a computer algorithm that generates a one-time password (OTP) using the current time as a source of uniqueness. As an extension of the HMAC-based one-time password algorithm HOTP, it has been adopted as Internet Engineering Task Force (IETF) standardRFC6238.[1]
TOTP is the cornerstone of Initiative for Open Authentication (OATH) and is used in a number of two-factor authentication[1](2FA) systems.
Through the collaboration of several OATH members, a TOTP draft was developed in order to create an industry-backed standard. It complements the event-based one-time standard HOTP, and it offers end user organizations and enterprises more choice in selecting technologies that best fit their application requirements andsecurityguidelines. In 2008, OATH submitted a draft version of the specification to the IETF. This version incorporates all the feedback and commentary that the authors received from the technical community based on the prior versions submitted to the IETF.[2]In May 2011, TOTP officially becameRFC6238.[1]
To establish TOTP authentication, the authenticatee and authenticator must pre-establish both theHOTP parametersand the following TOTP parameters:
Both the authenticator and the authenticatee compute the TOTP value, then the authenticator checks whether the TOTP value supplied by the authenticatee matches the locally generated TOTP value. Some authenticators allow values that should have been generated before or after the current time in order to account for slightclock skews, network latency and user delays.
TOTP uses the HOTP algorithm, replacing the counter with anon-decreasingvalue based on the current time:
TOTP value(K) =HOTP value(K,CT),
calculating counter valueCT=⌊T−T0TX⌋,{\displaystyle C_{T}=\left\lfloor {\frac {T-T_{0}}{T_{X}}}\right\rfloor ,}where
Unlikepasswords, TOTP codes are only valid for a limited time. However, users must enter TOTP codes into an authentication page, which creates the potential forphishing attacks. Due to the short window in which TOTP codes are valid, attackers must proxy the credentials in real time.[3]
TOTP credentials are also based on a shared secret known to both the client and the server, creating multiple locations from which a secret can be stolen. An attacker with access to this shared secret could generate new, valid TOTP codes at will. This can be a particular problem if the attacker breaches a large authentication database.[4]
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Multi-factor authentication(MFA;two-factor authentication, or2FA) is anelectronic authenticationmethod in which a user is granted access to awebsiteorapplicationonly after successfully presenting two or more distinct types of evidence (orfactors) to anauthenticationmechanism. MFA protectspersonal data—which may include personal identification orfinancial assets—from being accessed by an unauthorized third party that may have been able to discover, for example, a singlepassword.
Usage of MFA has increased in recent years. Security issues which can cause the bypass of MFA arefatigue attacks,phishingandSIM swapping.[1]
Accounts with MFA enabled are significantly less likely to be compromised.[2]
Authentication takes place when someone tries tolog intoa computer resource (such as acomputer network, device, or application). The resource requires the user to supply theidentityby which the user is known to the resource, along with evidence of the authenticity of the user's claim to that identity. Simple authentication requires only one such piece of evidence (factor), typically a password, or occasionally multiple pieces of evidence all of the same type, as with a credit card number and a card verification code (CVC). For additional security, the resource may require more than one factor—multi-factor authentication, or two-factor authentication in cases where exactly two types of evidence are to be supplied.[3]
The use of multiple authentication factors to prove one's identity is based on the premise that an unauthorized actor is unlikely to be able to supply all of the factors required for access. If, in an authentication attempt, at least one of the components is missing or supplied incorrectly, the user's identity is not established with sufficient certainty and access to the asset (e.g., a building, or data) being protected by multi-factor authentication then remains blocked. The authentication factors of a multi-factor authentication scheme may include:[4]
An example of two-factor authentication is the withdrawing of money from anATM; only the correct combination of a physically presentbank card(something the user possesses) and a PIN (something the user knows) allows the transaction to be carried out. Two other examples are to supplement a user-controlled password with aone-time password(OTP) or code generated or received by anauthenticator(e.g. asecurity tokenorsmartphone) that only the user possesses.[5]
Anauthenticatorapp enables two-factor authentication in a different way, by showing a randomly generated and constantly refreshing code, rather than sending anSMSor using another method.[6]This code is aTime-based one-time password(aTOTP)), and the authenticator app contains the key material that allows the generation of these codes.
Knowledge factors ("something only the user knows") are a form of authentication. In this form, the user is required to prove knowledge of a secret in order to authenticate.
A password is a secret word or string of characters that is used for user authentication. This is the most commonly used mechanism of authentication.[4]Many multi-factor authentication techniques rely on passwords as one factor of authentication. Variations include both longer ones formed from multiple words (apassphrase) and the shorter, purely numeric, PIN commonly used forATMaccess. Traditionally, passwords are expected to bememorized, but can also be written down on a hidden paper or text file.
Possession factors ("something only the user has") have been used for authentication for centuries, in the form of a key to a lock. The basic principle is that the key embodies a secret that is shared between the lock and the key, and the same principle underlies possession factor authentication in computer systems. Asecurity tokenis an example of a possession factor.
Disconnected tokenshave no connections to the client computer. They typically use a built-in screen to display the generated authentication data, which is manually typed in by the user. This type of token mostly uses aOTPthat can only be used for that specific session.[7]
Connected tokensaredevicesthat arephysicallyconnected to the computer to be used. Those devices transmit data automatically.[8]There are a number of different types, including USB tokens,smart cardsandwireless tags.[8]Increasingly,FIDO2capable tokens, supported by theFIDO Allianceand theWorld Wide Web Consortium(W3C), have become popular with mainstream browser support beginning in 2015.
Asoftware token(a.k.a.soft token) is a type of two-factor authentication security device that may be used to authorize the use of computer services. Software tokens are stored on a general-purpose electronic device such as adesktop computer,laptop,PDA, ormobile phoneand can be duplicated. (Contrasthardware tokens, where the credentials are stored on a dedicated hardware device and therefore cannot be duplicated, absent physical invasion of the device). A soft token may not be a device the user interacts with. Typically an X.509v3 certificate is loaded onto the device and stored securely to serve this purpose.[citation needed]
Multi-factor authenticationcan also be applied in physical security systems. These physical security systems are known and commonly referred to as access control. Multi-factor authentication is typically deployed in access control systems through the use, firstly, of a physical possession (such as a fob,keycard, orQR-codedisplayed on a device) which acts as the identification credential, and secondly, a validation of one's identity such as facial biometrics or retinal scan. This form of multi-factor authentication is commonly referred to as facial verification or facial authentication.
Inherent factors ("something the user is"), are factors associated with the user, and are usuallybiometricmethods, includingfingerprint,face,[9]voice, oririsrecognition. Behavioral biometrics such askeystroke dynamicscan also be used.
Increasingly, a fourth factor is coming into play involving the physical location of the user. While hard wired to the corporate network, a user could be allowed to login using only a pin code. Whereas if the user was off the network or working remotely, a more secure MFA method such as entering a code from a soft token as well could be required. Adapting the type of MFA method and frequency to a users' location will enable you to avoid risks common to remote working.[10]
Systems for network admission control work in similar ways where the level of network access can be contingent on the specific network a device is connected to, such asWi-Fivs wired connectivity. This also allows a user to move between offices and dynamically receivethe same level of network access[clarification needed]in each.[citation needed]
Two-factor authentication over text message was developed as early as 1996, when AT&T described a system for authorizing transactions based on an exchange of codes over two-way pagers.[11][12]
Many multi-factor authentication vendors offer mobile phone-based authentication. Some methods include push-based authentication,QR code-based authentication, one-time password authentication (event-based and time-based), and SMS-based verification. SMS-based verification suffers from some security concerns. Phones can be cloned, apps can run on several phones and cell-phone maintenance personnel can read SMS texts. Not least, cell phones can be compromised in general, meaning the phone is no longer something only the user has.
The major drawback of authentication including something the user possesses is that the user must carry around the physical token (the USB stick, the bank card, the key or similar), practically at all times. Loss and theft are risks. Many organizations forbid carrying USB and electronic devices in or out of premises owing tomalwareand data theft risks, and most important machines do not have USB ports for the same reason. Physical tokens usually do not scale, typically requiring a new token for each new account and system. Procuring and subsequently replacing tokens of this kind involves costs. In addition, there are inherent conflicts and unavoidable trade-offs between usability and security.[13]
Two-step authentication involvingmobile phonesandsmartphonesprovides an alternative to dedicated physical devices. To authenticate, people can use their personal access codes to the device (i.e. something that only the individual user knows) plus a one-time-valid, dynamic passcode, typically consisting of 4 to 6 digits. The passcode can be sent to their mobile device[3]bySMSor can be generated by a one-time passcode-generator app. In both cases, the advantage of using a mobile phone is that there is no need for an additional dedicated token, as users tend to carry theirmobile devicesaround at all times.
Notwithstanding the popularity of SMS verification, security advocates have publicly criticized SMS verification,[14]and in July 2016, a United StatesNISTdraft guideline proposed deprecating it as a form of authentication.[15]A year later NIST reinstated SMS verification as a valid authentication channel in the finalized guideline.[16]
As early as 2011, Duo Security was offeringpush notificationsfor MFA via a mobile app.[17]In 2016 and 2017 respectively, both Google and Apple started offering user two-step authentication with push notifications[4]as an alternative method.[18][19]
Security of mobile-delivered security tokens fully depends on the mobile operator's operational security and can be easily breached by wiretapping orSIM cloningby national security agencies.[20]
Advantages:
Disadvantages:
ThePayment Card Industry (PCI)Data Security Standard, requirement 8.3, requires the use of MFA for all remote network access that originates from outside the network to a Card Data Environment (CDE).[24]Beginning with PCI-DSS version 3.2, the use of MFA is required for all administrative access to the CDE, even if the user is within a trusted network.
The secondPayment Services Directiverequires "strong customer authentication" on most electronic payments in theEuropean Economic Areasince September 14, 2019.[25]
In India, theReserve Bank of Indiamandated two-factor authentication for all online transactions made using a debit or credit card using either a password or a one-time password sent overSMS. This requirement was removed in 2016 for transactions up to ₹2,000 after opting-in with the issuing bank.[26]Vendors such asUberhave been mandated by the bank to amend their payment processing systems in compliance with this two-factor authentication rollout.[27][28][29]
Details for authentication for federal employees and contractors in the U.S. are defined in Homeland Security Presidential Directive 12 (HSPD-12).[30]
IT regulatory standards for access to federal government systems require the use of multi-factor authentication to access sensitive IT resources, for example when logging on to network devices to perform administrative tasks[31]and when accessing any computer using a privileged login.[32]
NISTSpecial Publication 800-63-3 discusses various forms of two-factor authentication and provides guidance on using them in business processes requiring different levels of assurance.[33]
In 2005, the United States'Federal Financial Institutions Examination Councilissued guidance for financial institutions recommending financial institutions conduct risk-based assessments, evaluate customer awareness programs, and develop security measures to reliably authenticate customers remotely accessingonline financial services, officially recommending the use of authentication methods that depend on more than one factor (specifically, what a user knows, has, and is) to determine the user's identity.[34]In response to the publication, numerous authentication vendors began improperly promoting challenge-questions, secret images, and other knowledge-based methods as "multi-factor" authentication. Due to the resulting confusion and widespread adoption of such methods, on August 15, 2006, the FFIEC published supplemental guidelines—which state that by definition, a "true" multi-factor authentication system must use distinct instances of the three factors of authentication it had defined, and not just use multiple instances of a single factor.[35]
According to proponents, multi-factor authentication could drastically reduce the incidence of onlineidentity theftand other onlinefraud, because the victim's password would no longer be enough to give a thief permanent access to their information. However, many multi-factor authentication approaches remain vulnerable tophishing,[36]man-in-the-browser, andman-in-the-middle attacks.[37]Two-factor authentication in web applications are especially susceptible to phishing attacks, particularly in SMS and e-mails, and, as a response, many experts advise users not to share their verification codes with anyone,[38]and many web application providers will place an advisory in an e-mail or SMS containing a code.[39]
Multi-factor authentication may be ineffective[40]against modern threats, like ATM skimming, phishing, and malware.[41][vague][needs update?]
In May 2017,O2 Telefónica, a German mobile service provider, confirmed that cybercriminals had exploitedSS7vulnerabilities to bypass SMS based two-step authentication to do unauthorized withdrawals from users' bank accounts. The criminals firstinfectedthe account holder's computers in an attempt to steal their bank account credentials and phone numbers. Then the attackers purchased access to a fake telecom provider and set up a redirect for the victim's phone number to a handset controlled by them. Finally, the attackers logged into victims' online bank accounts and requested for the money on the accounts to be withdrawn to accounts owned by the criminals. SMS passcodes were routed to phone numbers controlled by the attackers and the criminals transferred the money out.[42]
An increasingly common approach to defeating MFA is to bombard the user with many requests to accept a log-in, until the user eventually succumbs to the volume of requests and accepts one.[43]This is called a multi-factor authentication fatigue attack (also MFA fatigue attack or MFA bombing) makes use ofsocial engineering.[44][45][46]When MFA applications are configured to send push notifications to end users, an attacker can send a flood of login attempts in the hope that a user will click on accept at least once.[44]
In 2022,Microsofthas deployed a mitigation against MFA fatigue attacks with their authenticator app.[47]
In September 2022Ubersecurity was breached by a member ofLapsus$using a multi-factor fatigue attack.[48][49]On March 24, 2023, YouTuberLinus Sebastiandeclared on theLinus Tech Tipschannel on theYouTubeplatform that he had suffered an Multi-factor authentication fatigue attack.[50]In early 2024, a small percentage ofAppleconsumers experienced a MFA fatigue attack that was caused by a hacker that bypassed the rate limit andCaptchaon Apple’s “Forgot Password” page.
Manymulti-factor authenticationproducts require users to deployclientsoftwareto make multi-factor authentication systems work. Some vendors have created separate installation packages fornetworklogin,Webaccesscredentials, andVPNconnectioncredentials. For such products, there may be four or five differentsoftwarepackages to push down to theclientPC in order to make use of thetokenorsmart card. This translates to four or five packages on which version control has to be performed, and four or five packages to check for conflicts with business applications. If access can be operated usingweb pages, it is possible to limit the overheads outlined above to a single application. With other multi-factor authentication technology such as hardware token products, no software must be installed by end-users.[citation needed]Some studies have shown that poorly implemented MFA recovery procedures can introduce new vulnerabilities that attackers may exploit.[51]
There are drawbacks to multi-factor authentication that are keeping many approaches from becoming widespread. Some users have difficulty keeping track of a hardware token or USB plug. Many users do not have the technical skills needed to install a client-side software certificate by themselves. Generally, multi-factor solutions require additional investment for implementation and costs for maintenance. Most hardware token-based systems are proprietary, and some vendors charge an annual fee per user. Deployment ofhardware tokensis logistically challenging. Hardwaretokensmay get damaged or lost, and issuance oftokensin large industries such as banking or even within large enterprises needs to be managed. In addition to deployment costs, multi-factor authentication often carries significant additional support costs.[citation needed]A 2008 survey[52]of over 120U.S. credit unionsby theCredit Union Journalreported on the support costs associated with two-factor authentication. In their report,software certificates and software toolbar approaches[clarification needed]were reported to have the highest support costs.
Research into deployments of multi-factor authentication schemes[53]has shown that one of the elements that tend to impact the adoption of such systems is the line of business of the organization that deploys the multi-factor authentication system. Examples cited include the U.S. government, which employs an elaborate system of physical tokens (which themselves are backed by robustPublic Key Infrastructure), as well as private banks, which tend to prefer multi-factor authentication schemes for their customers that involve more accessible, less expensive means of identity verification, such as an app installed onto a customer-owned smartphone. Despite the variations that exist among available systems that organizations may have to choose from, once a multi-factor authentication system is deployed within an organization, it tends to remain in place, as users invariably acclimate to the presence and use of the system and embrace it over time as a normalized element of their daily process of interaction with their relevant information system.
While the perception is that multi-factor authentication is within the realm of perfect security, Roger Grimes writes[54]that if not properly implemented and configured, multi-factor authentication can in fact be easily defeated.
In 2013,Kim Dotcomclaimed to have invented two-factor authentication in a 2000 patent,[55]and briefly threatened to sue all the major web services. However, the European Patent Office revoked his patent[56]in light of an earlier 1998 U.S. patent held by AT&T.[57]
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Inmathematicsandabstract algebra,group theorystudies thealgebraic structuresknown asgroups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such asrings,fields, andvector spaces, can all be seen as groups endowed with additionaloperationsandaxioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra.Linear algebraic groupsandLie groupsare two branches of group theory that have experienced advances and have become subject areas in their own right.
Various physical systems, such ascrystalsand thehydrogen atom, may be modelled bysymmetry groups. Thus group theory and the closely relatedrepresentation theoryhave many important applications inphysics,chemistry, andmaterials science. Group theory is also central topublic key cryptography.
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Inmathematics, agroupis asetwith abinary operationthat satisfies the following constraints: the operation isassociative, it has anidentity element, and every element of the set has aninverse element.
Manymathematical structuresare groups endowed with other properties. For example, theintegerswith theaddition operationform aninfinitegroup that isgenerated by a single elementcalled1{\displaystyle 1}(these properties fully characterize the integers).
The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers,geometric shapesandpolynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.[1][2]
Ingeometry, groups arise naturally in the study ofsymmetriesandgeometric transformations: The symmetries of an object form a group, called thesymmetry groupof the object, and the transformations of a given type form a general group.Lie groupsappear in symmetry groups in geometry, and also in theStandard Modelofparticle physics. ThePoincaré groupis a Lie group consisting of the symmetries ofspacetimeinspecial relativity.Point groupsdescribesymmetry in molecular chemistry.
The concept of a group arose in the study ofpolynomial equations, starting withÉvariste Galoisin the 1830s, who introduced the termgroup(French:groupe) for the symmetry group of therootsof an equation, now called aGalois group. After contributions from other fields such asnumber theoryand geometry, the group notion was generalized and firmly established around 1870. Moderngroup theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such assubgroups,quotient groupsandsimple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view ofrepresentation theory(that is, through therepresentations of the group) and ofcomputational group theory. A theory has been developed forfinite groups, which culminated with theclassification of finite simple groups, completed in 2004. Since the mid-1980s,geometric group theory, which studiesfinitely generated groupsas geometric objects, has become an active area in group theory.
One of the more familiar groups is the set ofintegersZ={…,−4,−3,−2,−1,0,1,2,3,4,…}{\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}together withaddition.[3]For any two integersa{\displaystyle a}andb{\displaystyle b}, thesuma+b{\displaystyle a+b}is also an integer; thisclosureproperty says that+{\displaystyle +}is abinary operationonZ{\displaystyle \mathbb {Z} }. The following properties of integer addition serve as a model for the group axioms in the definition below.
The integers, together with the operation+{\displaystyle +}, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.
The axioms for a group are short and natural ... Yet somehow hidden behind these axioms is themonster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.
A group is a non-emptysetG{\displaystyle G}together with abinary operationonG{\displaystyle G}, here denoted "⋅{\displaystyle \cdot }", that combines any twoelementsa{\displaystyle a}andb{\displaystyle b}ofG{\displaystyle G}to form an element ofG{\displaystyle G}, denoteda⋅b{\displaystyle a\cdot b}, such that the following three requirements, known asgroup axioms, are satisfied:[5][6][7][a]
Formally, a group is anordered pairof a set and a binary operation on this set that satisfies thegroup axioms. The set is called theunderlying setof the group, and the operation is called thegroup operationor thegroup law.
A group and its underlying set are thus two differentmathematical objects. To avoid cumbersome notation, it is common toabuse notationby using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.
For example, consider the set ofreal numbersR{\displaystyle \mathbb {R} }, which has the operations of additiona+b{\displaystyle a+b}andmultiplicationab{\displaystyle ab}. Formally,R{\displaystyle \mathbb {R} }is a set,(R,+){\displaystyle (\mathbb {R} ,+)}is a group, and(R,+,⋅){\displaystyle (\mathbb {R} ,+,\cdot )}is afield. But it is common to writeR{\displaystyle \mathbb {R} }to denote any of these three objects.
Theadditive groupof the fieldR{\displaystyle \mathbb {R} }is the group whose underlying set isR{\displaystyle \mathbb {R} }and whose operation is addition. Themultiplicative groupof the fieldR{\displaystyle \mathbb {R} }is the groupR×{\displaystyle \mathbb {R} ^{\times }}whose underlying set is the set of nonzero real numbersR∖{0}{\displaystyle \mathbb {R} \smallsetminus \{0\}}and whose operation is multiplication.
More generally, one speaks of anadditive groupwhenever the group operation is notated as addition; in this case, the identity is typically denoted0{\displaystyle 0}, and the inverse of an elementx{\displaystyle x}is denoted−x{\displaystyle -x}. Similarly, one speaks of amultiplicative groupwhenever the group operation is notated as multiplication; in this case, the identity is typically denoted1{\displaystyle 1}, and the inverse of an elementx{\displaystyle x}is denotedx−1{\displaystyle x^{-1}}. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition,ab{\displaystyle ab}instead ofa⋅b{\displaystyle a\cdot b}.
The definition of a group does not require thata⋅b=b⋅a{\displaystyle a\cdot b=b\cdot a}for all elementsa{\displaystyle a}andb{\displaystyle b}inG{\displaystyle G}. If this additional condition holds, then the operation is said to becommutative, and the group is called anabelian group. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.
Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements arefunctions, the operation is oftenfunction compositionf∘g{\displaystyle f\circ g}; then the identity may be denoted id. In the more specific cases ofgeometric transformationgroups,symmetrygroups,permutation groups, andautomorphism groups, the symbol∘{\displaystyle \circ }is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.
Two figures in theplanearecongruentif one can be changed into the other using a combination ofrotations,reflections, andtranslations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are calledsymmetries. Asquarehas eight symmetries. These are:
fh{\displaystyle f_{\mathrm {h} }}(horizontal reflection)
fd{\displaystyle f_{\mathrm {d} }}(diagonal reflection)
fc{\displaystyle f_{\mathrm {c} }}(counter-diagonal reflection)
These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example,r1{\displaystyle r_{1}}sends a point to its rotation 90° clockwise around the square's center, andfh{\displaystyle f_{\mathrm {h} }}sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called thedihedral groupof degree four, denotedD4{\displaystyle \mathrm {D} _{4}}. The underlying set of the group is the above set of symmetries, and the group operation is function composition.[8]Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing firsta{\displaystyle a}and thenb{\displaystyle b}is written symbolicallyfrom right to leftasb∘a{\displaystyle b\circ a}("apply the symmetryb{\displaystyle b}after performing the symmetrya{\displaystyle a}"). This is the usual notation for composition of functions.
ACayley tablelists the results of all such compositions possible. For example, rotating by 270° clockwise (r3{\displaystyle r_{3}}) and then reflecting horizontally (fh{\displaystyle f_{\mathrm {h} }}) is the same as performing a reflection along the diagonal (fd{\displaystyle f_{\mathrm {d} }}). Using the above symbols, highlighted in blue in the Cayley table:fh∘r3=fd.{\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.}
Given this set of symmetries and the described operation, the group axioms can be understood as follows.
Binary operation: Composition is a binary operation. That is,a∘b{\displaystyle a\circ b}is a symmetry for any two symmetriesa{\displaystyle a}andb{\displaystyle b}. For example,r3∘fh=fc,{\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },}that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (fc{\displaystyle f_{\mathrm {c} }}). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the Cayley table.
Associativity: The associativity axiom deals with composing more than two symmetries: Starting with three elementsa{\displaystyle a},b{\displaystyle b}andc{\displaystyle c}ofD4{\displaystyle \mathrm {D} _{4}}, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first composea{\displaystyle a}andb{\displaystyle b}into a single symmetry, then to compose that symmetry withc{\displaystyle c}. The other way is to first composeb{\displaystyle b}andc{\displaystyle c}, then to compose the resulting symmetry witha{\displaystyle a}. These two ways must give always the same result, that is,(a∘b)∘c=a∘(b∘c),{\displaystyle (a\circ b)\circ c=a\circ (b\circ c),}For example,(fd∘fv)∘r2=fd∘(fv∘r2){\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})}can be checked using the Cayley table:(fd∘fv)∘r2=r3∘r2=r1fd∘(fv∘r2)=fd∘fh=r1.{\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}}
Identity element: The identity element isid{\displaystyle \mathrm {id} }, as it does not change any symmetrya{\displaystyle a}when composed with it either on the left or on the right.
Inverse element: Each symmetry has an inverse:id{\displaystyle \mathrm {id} }, the reflectionsfh{\displaystyle f_{\mathrm {h} }},fv{\displaystyle f_{\mathrm {v} }},fd{\displaystyle f_{\mathrm {d} }},fc{\displaystyle f_{\mathrm {c} }}and the 180° rotationr2{\displaystyle r_{2}}are their own inverse, because performing them twice brings the square back to its original orientation. The rotationsr3{\displaystyle r_{3}}andr1{\displaystyle r_{1}}are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table.
In contrast to the group of integers above, where the order of the operation is immaterial, it does matter inD4{\displaystyle \mathrm {D} _{4}}, as, for example,fh∘r1=fc{\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }}butr1∘fh=fd{\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }}. In other words,D4{\displaystyle \mathrm {D} _{4}}is not abelian.
The modern concept of anabstract groupdeveloped out of several fields of mathematics.[9][10][11]The original motivation for group theory was the quest for solutions ofpolynomial equationsof degree higher than 4. The 19th-century French mathematicianÉvariste Galois, extending prior work ofPaolo RuffiniandJoseph-Louis Lagrange, gave a criterion for thesolvabilityof a particular polynomial equation in terms of thesymmetry groupof itsroots(solutions). The elements of such aGalois groupcorrespond to certainpermutationsof the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously.[12][13]More general permutation groups were investigated in particular byAugustin Louis Cauchy.Arthur Cayley'sOn the theory of groups, as depending on the symbolic equationθn=1{\displaystyle \theta ^{n}=1}(1854) gives the first abstract definition of afinite group.[14]
Geometry was a second field in which groups were used systematically, especially symmetry groups as part ofFelix Klein's 1872Erlangen program.[15]After novel geometries such ashyperbolicandprojective geometryhad emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas,Sophus Liefounded the study ofLie groupsin 1884.[16]
The third field contributing to group theory wasnumber theory. Certain abelian group structures had been used implicitly inCarl Friedrich Gauss's number-theoretical workDisquisitiones Arithmeticae(1798), and more explicitly byLeopold Kronecker.[17]In 1847,Ernst Kummermade early attempts to proveFermat's Last Theoremby developinggroups describing factorizationintoprime numbers.[18]
The convergence of these various sources into a uniform theory of groups started withCamille Jordan'sTraité des substitutions et des équations algébriques(1870).[19]Walther von Dyck(1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time.[20]As of the 20th century, groups gained wide recognition by the pioneering work ofFerdinand Georg FrobeniusandWilliam Burnside(who worked onrepresentation theoryof finite groups),Richard Brauer'smodular representation theoryandIssai Schur's papers.[21]The theory of Lie groups, and more generallylocally compact groupswas studied byHermann Weyl,Élie Cartanand many others.[22]Itsalgebraiccounterpart, the theory ofalgebraic groups, was first shaped byClaude Chevalley(from the late 1930s) and later by the work ofArmand BorelandJacques Tits.[23]
TheUniversity of Chicago's 1960–61 Group Theory Year brought together group theorists such asDaniel Gorenstein,John G. ThompsonandWalter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to theclassification of finite simple groups, with the final step taken byAschbacherand Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length ofproofand number of researchers. Research concerning this classification proof is ongoing.[24]Group theory remains a highly active mathematical branch,[b]impacting many other fields, as theexamples belowillustrate.
Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed underelementary group theory.[25]For example,repeatedapplications of the associativity axiom show that the unambiguity ofa⋅b⋅c=(a⋅b)⋅c=a⋅(b⋅c){\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)}generalizes to more than three factors. Because this implies thatparenthesescan be inserted anywhere within such a series of terms, parentheses are usually omitted.[26]
The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elementse{\displaystyle e}andf{\displaystyle f}of a group are equal, because the group axioms implye=e⋅f=f{\displaystyle e=e\cdot f=f}. It is thus customary to speak oftheidentity element of the group.[27]
The group axioms also imply that the inverse of each element is unique. Let a group elementa{\displaystyle a}have bothb{\displaystyle b}andc{\displaystyle c}as inverses. Then
Therefore, it is customary to speak oftheinverse of an element.[27]
Given elementsa{\displaystyle a}andb{\displaystyle b}of a groupG{\displaystyle G}, there is a unique solutionx{\displaystyle x}inG{\displaystyle G}to the equationa⋅x=b{\displaystyle a\cdot x=b}, namelya−1⋅b{\displaystyle a^{-1}\cdot b}.[c][28]It follows that for eacha{\displaystyle a}inG{\displaystyle G}, the functionG→G{\displaystyle G\to G}that maps eachx{\displaystyle x}toa⋅x{\displaystyle a\cdot x}is abijection; it is calledleft multiplicationbya{\displaystyle a}orleft translationbya{\displaystyle a}.
Similarly, givena{\displaystyle a}andb{\displaystyle b}, the unique solution tox⋅a=b{\displaystyle x\cdot a=b}isb⋅a−1{\displaystyle b\cdot a^{-1}}. For eacha{\displaystyle a}, the functionG→G{\displaystyle G\to G}that maps eachx{\displaystyle x}tox⋅a{\displaystyle x\cdot a}is a bijection calledright multiplicationbya{\displaystyle a}orright translationbya{\displaystyle a}.
The group axioms for identity and inverses may be "weakened" to assert only the existence of aleft identityandleft inverses. From theseone-sided axioms, one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are not weaker.[29]
In particular, assuming associativity and the existence of a left identitye{\displaystyle e}(that is,e⋅f=f{\displaystyle e\cdot f=f}) and a left inversef−1{\displaystyle f^{-1}}for each elementf{\displaystyle f}(that is,f−1⋅f=e{\displaystyle f^{-1}\cdot f=e}), it follows that every left inverse is also a right inverse of the same element as follows.[29]Indeed, one has
Similarly, the left identity is also a right identity:[29]
These results do not hold if any of these axioms (associativity, existence of left identity and existence of left inverse) is removed. For a structure with a looser definition (like asemigroup) one may have, for example, that a left identity is not necessarily a right identity.
The same result can be obtained by only assuming the existence of a right identity and a right inverse.
However, only assuming the existence of aleftidentity and arightinverse (or vice versa) is not sufficient to define a group. For example, consider the setG={e,f}{\displaystyle G=\{e,f\}}with the operator⋅{\displaystyle \cdot }satisfyinge⋅e=f⋅e=e{\displaystyle e\cdot e=f\cdot e=e}ande⋅f=f⋅f=f{\displaystyle e\cdot f=f\cdot f=f}. This structure does have a left identity (namely,e{\displaystyle e}), and each element has a right inverse (which ise{\displaystyle e}for both elements). Furthermore, this operation is associative (since the product of any number of elements is always equal to the rightmost element in that product, regardless of the order in which these operations are applied). However,(G,⋅){\displaystyle (G,\cdot )}is not a group, since it lacks a right identity.
When studying sets, one uses concepts such assubset, function, andquotient by an equivalence relation. When studying groups, one uses insteadsubgroups,homomorphisms, andquotient groups. These are the analogues that take the group structure into account.[d]
Group homomorphisms[e]are functions that respect group structure; they may be used to relate two groups. Ahomomorphismfrom a group(G,⋅){\displaystyle (G,\cdot )}to a group(H,∗){\displaystyle (H,*)}is a functionφ:G→H{\displaystyle \varphi :G\to H}such that
It would be natural to require also thatφ{\displaystyle \varphi }respect identities,φ(1G)=1H{\displaystyle \varphi (1_{G})=1_{H}}, and inverses,φ(a−1)=φ(a)−1{\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}}for alla{\displaystyle a}inG{\displaystyle G}. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.[30]
Theidentity homomorphismof a groupG{\displaystyle G}is the homomorphismιG:G→G{\displaystyle \iota _{G}:G\to G}that maps each element ofG{\displaystyle G}to itself. Aninverse homomorphismof a homomorphismφ:G→H{\displaystyle \varphi :G\to H}is a homomorphismψ:H→G{\displaystyle \psi :H\to G}such thatψ∘φ=ιG{\displaystyle \psi \circ \varphi =\iota _{G}}andφ∘ψ=ιH{\displaystyle \varphi \circ \psi =\iota _{H}}, that is, such thatψ(φ(g))=g{\displaystyle \psi {\bigl (}\varphi (g){\bigr )}=g}for allg{\displaystyle g}inG{\displaystyle G}and such thatφ(ψ(h))=h{\displaystyle \varphi {\bigl (}\psi (h){\bigr )}=h}for allh{\displaystyle h}inH{\displaystyle H}. Anisomorphismis a homomorphism that has an inverse homomorphism; equivalently, it is abijectivehomomorphism. GroupsG{\displaystyle G}andH{\displaystyle H}are calledisomorphicif there exists an isomorphismφ:G→H{\displaystyle \varphi :G\to H}. In this case,H{\displaystyle H}can be obtained fromG{\displaystyle G}simply by renaming its elements according to the functionφ{\displaystyle \varphi }; then any statement true forG{\displaystyle G}is true forH{\displaystyle H}, provided that any specific elements mentioned in the statement are also renamed.
The collection of all groups, together with the homomorphisms between them, form acategory, thecategory of groups.[31]
Aninjectivehomomorphismϕ:G′→G{\displaystyle \phi :G'\to G}factors canonically as an isomorphism followed by an inclusion,G′→∼H↪G{\displaystyle G'\;{\stackrel {\sim }{\to }}\;H\hookrightarrow G}for some subgroupH{\displaystyle H}ofG{\displaystyle G}.
Injective homomorphisms are themonomorphismsin the category of groups.
Informally, asubgroupis a groupH{\displaystyle H}contained within a bigger one,G{\displaystyle G}: it has a subset of the elements ofG{\displaystyle G}, with the same operation.[32]Concretely, this means that the identity element ofG{\displaystyle G}must be contained inH{\displaystyle H}, and wheneverh1{\displaystyle h_{1}}andh2{\displaystyle h_{2}}are both inH{\displaystyle H}, then so areh1⋅h2{\displaystyle h_{1}\cdot h_{2}}andh1−1{\displaystyle h_{1}^{-1}}, so the elements ofH{\displaystyle H}, equipped with the group operation onG{\displaystyle G}restricted toH{\displaystyle H}, indeed form a group. In this case, the inclusion mapH→G{\displaystyle H\to G}is a homomorphism.
In the example of symmetries of a square, the identity and the rotations constitute a subgroupR={id,r1,r2,r3}{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}, highlighted in red in the Cayley table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. Thesubgroup testprovides anecessary and sufficient conditionfor a nonempty subsetH{\displaystyle H}of a groupG{\displaystyle G}to be a subgroup: it is sufficient to check thatg−1⋅h∈H{\displaystyle g^{-1}\cdot h\in H}for all elementsg{\displaystyle g}andh{\displaystyle h}inH{\displaystyle H}. Knowing a group'ssubgroupsis important in understanding the group as a whole.[f]
Given any subsetS{\displaystyle S}of a groupG{\displaystyle G}, the subgroupgeneratedbyS{\displaystyle S}consists of all products of elements ofS{\displaystyle S}and their inverses. It is the smallest subgroup ofG{\displaystyle G}containingS{\displaystyle S}.[33]In the example of symmetries of a square, the subgroup generated byr2{\displaystyle r_{2}}andfv{\displaystyle f_{\mathrm {v} }}consists of these two elements, the identity elementid{\displaystyle \mathrm {id} }, and the elementfh=fv⋅r2{\displaystyle f_{\mathrm {h} }=f_{\mathrm {v} }\cdot r_{2}}. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.
In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroupH{\displaystyle H}determines left and right cosets, which can be thought of as translations ofH{\displaystyle H}by an arbitrary group elementg{\displaystyle g}. In symbolic terms, theleftandrightcosets ofH{\displaystyle H}, containing an elementg{\displaystyle g}, are
The left cosets of any subgroupH{\displaystyle H}form apartitionofG{\displaystyle G}; that is, theunionof all left cosets is equal toG{\displaystyle G}and two left cosets are either equal or have anemptyintersection.[35]The first caseg1H=g2H{\displaystyle g_{1}H=g_{2}H}happensprecisely wheng1−1⋅g2∈H{\displaystyle g_{1}^{-1}\cdot g_{2}\in H}, i.e., when the two elements differ by an element ofH{\displaystyle H}. Similar considerations apply to the right cosets ofH{\displaystyle H}. The left cosets ofH{\displaystyle H}may or may not be the same as its right cosets. If they are (that is, if allg{\displaystyle g}inG{\displaystyle G}satisfygH=Hg{\displaystyle gH=Hg}), thenH{\displaystyle H}is said to be anormal subgroup.
InD4{\displaystyle \mathrm {D} _{4}}, the group of symmetries of a square, with its subgroupR{\displaystyle R}of rotations, the left cosetsgR{\displaystyle gR}are either equal toR{\displaystyle R}, ifg{\displaystyle g}is an element ofR{\displaystyle R}itself, or otherwise equal toU=fcR={fc,fd,fv,fh}{\displaystyle U=f_{\mathrm {c} }R=\{f_{\mathrm {c} },f_{\mathrm {d} },f_{\mathrm {v} },f_{\mathrm {h} }\}}(highlighted in green in the Cayley table ofD4{\displaystyle \mathrm {D} _{4}}). The subgroupR{\displaystyle R}is normal, becausefcR=U=Rfc{\displaystyle f_{\mathrm {c} }R=U=Rf_{\mathrm {c} }}and similarly for the other elements of the group. (In fact, in the case ofD4{\displaystyle \mathrm {D} _{4}}, the cosets generated by reflections are all equal:fhR=fvR=fdR=fcR{\displaystyle f_{\mathrm {h} }R=f_{\mathrm {v} }R=f_{\mathrm {d} }R=f_{\mathrm {c} }R}.)
Suppose thatN{\displaystyle N}is a normal subgroup of a groupG{\displaystyle G}, andG/N={gN∣g∈G}{\displaystyle G/N=\{gN\mid g\in G\}}denotes its set of cosets.
Then there is a unique group law onG/N{\displaystyle G/N}for which the mapG→G/N{\displaystyle G\to G/N}sending each elementg{\displaystyle g}togN{\displaystyle gN}is a homomorphism.
Explicitly, the product of two cosetsgN{\displaystyle gN}andhN{\displaystyle hN}is(gh)N{\displaystyle (gh)N}, the coseteN=N{\displaystyle eN=N}serves as the identity ofG/N{\displaystyle G/N}, and the inverse ofgN{\displaystyle gN}in the quotient group is(gN)−1=(g−1)N{\displaystyle (gN)^{-1}=\left(g^{-1}\right)N}.
The groupG/N{\displaystyle G/N}, read as "G{\displaystyle G}moduloN{\displaystyle N}",[36]is called aquotient grouporfactor group.
The quotient group can alternatively be characterized by auniversal property.
The elements of the quotient groupD4/R{\displaystyle \mathrm {D} _{4}/R}areR{\displaystyle R}andU=fvR{\displaystyle U=f_{\mathrm {v} }R}. The group operation on the quotient is shown in the table. For example,U⋅U=fvR⋅fvR=(fv⋅fv)R=R{\displaystyle U\cdot U=f_{\mathrm {v} }R\cdot f_{\mathrm {v} }R=(f_{\mathrm {v} }\cdot f_{\mathrm {v} })R=R}. Both the subgroupR={id,r1,r2,r3}{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}and the quotientD4/R{\displaystyle \mathrm {D} _{4}/R}are abelian, butD4{\displaystyle \mathrm {D} _{4}}is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by thesemidirect productconstruction;D4{\displaystyle \mathrm {D} _{4}}is an example.
Thefirst isomorphism theoremimplies that anysurjectivehomomorphismϕ:G→H{\displaystyle \phi :G\to H}factors canonically as a quotient homomorphism followed by an isomorphism:G→G/kerϕ→∼H{\displaystyle G\to G/\ker \phi \;{\stackrel {\sim }{\to }}\;H}.
Surjective homomorphisms are theepimorphismsin the category of groups.
Every group is isomorphic to a quotient of afree group, in many ways.
For example, the dihedral groupD4{\displaystyle \mathrm {D} _{4}}is generated by the right rotationr1{\displaystyle r_{1}}and the reflectionfv{\displaystyle f_{\mathrm {v} }}in a vertical line (every element ofD4{\displaystyle \mathrm {D} _{4}}is a finite product of copies of these and their inverses).
Hence there is a surjective homomorphismϕ{\displaystyle \phi }from the free group⟨r,f⟩{\displaystyle \langle r,f\rangle }on two generators toD4{\displaystyle \mathrm {D} _{4}}sendingr{\displaystyle r}tor1{\displaystyle r_{1}}andf{\displaystyle f}tof1{\displaystyle f_{1}}.
Elements inkerϕ{\displaystyle \ker \phi }are calledrelations; examples includer4,f2,(r⋅f)2{\displaystyle r^{4},f^{2},(r\cdot f)^{2}}.
In fact, it turns out thatkerϕ{\displaystyle \ker \phi }is the smallest normal subgroup of⟨r,f⟩{\displaystyle \langle r,f\rangle }containing these three elements; in other words, all relations are consequences of these three.
The quotient of the free group by this normal subgroup is denoted⟨r,f∣r4=f2=(r⋅f)2=1⟩{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle }.
This is called apresentationofD4{\displaystyle \mathrm {D} _{4}}by generators and relations, because the first isomorphism theorem forϕ{\displaystyle \phi }yields an isomorphism⟨r,f∣r4=f2=(r⋅f)2=1⟩→D4{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle \to \mathrm {D} _{4}}.[37]
A presentation of a group can be used to construct theCayley graph, a graphical depiction of adiscrete group.[38]
Examples and applications of groups abound. A starting point is the groupZ{\displaystyle \mathbb {Z} }of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtainsmultiplicative groups. These groups are predecessors of important constructions inabstract algebra.
Groups are also applied in many other mathematical areas. Mathematical objects are often examined byassociatinggroups to them and studying the properties of the corresponding groups. For example,Henri Poincaréfounded what is now calledalgebraic topologyby introducing thefundamental group.[39]By means of this connection,topological propertiessuch asproximityandcontinuitytranslate into properties of groups.[g]
Elements of the fundamental group of atopological spaceareequivalence classesof loops, where loops are considered equivalent if one can besmoothly deformedinto another, and the group operation is "concatenation" (tracing one loop then the other). For example, as shown in the figure, if the topological space is the plane with one point removed, then loops which do not wrap around the missing point (blue)can be smoothly contracted to a single pointand are the identity element of the fundamental group. A loop which wraps around the missing pointk{\displaystyle k}times cannot be deformed into a loop which wrapsm{\displaystyle m}times (withm≠k{\displaystyle m\neq k}), because the loop cannot be smoothly deformed across the hole, so each class of loops is characterized by itswinding numberaround the missing point. The resulting group is isomorphic to the integers under addition.
In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background.[h]In a similar vein,geometric group theoryemploys geometric concepts, for example in the study ofhyperbolic groups.[40]Further branches crucially applying groups includealgebraic geometryand number theory.[41]
In addition to the above theoretical applications, many practical applications of groups exist.Cryptographyrelies on the combination of the abstract group theory approach together withalgorithmicalknowledge obtained incomputational group theory, in particular when implemented for finite groups.[42]Applications of group theory are not restricted to mathematics; sciences such asphysics,chemistryandcomputer sciencebenefit from the concept.
Many number systems, such as the integers and therationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known asringsand fields. Further abstract algebraic concepts such asmodules,vector spacesandalgebrasalso form groups.
The group of integersZ{\displaystyle \mathbb {Z} }under addition, denoted(Z,+){\displaystyle \left(\mathbb {Z} ,+\right)}, has been described above. The integers, with the operation of multiplication instead of addition,(Z,⋅){\displaystyle \left(\mathbb {Z} ,\cdot \right)}donotform a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example,a=2{\displaystyle a=2}is an integer, but the only solution to the equationa⋅b=1{\displaystyle a\cdot b=1}in this case isb=12{\displaystyle b={\tfrac {1}{2}}}, which is a rational number, but not an integer. Hence not every element ofZ{\displaystyle \mathbb {Z} }has a (multiplicative) inverse.[i]
The desire for the existence of multiplicative inverses suggests consideringfractionsab.{\displaystyle {\frac {a}{b}}.}
Fractions of integers (withb{\displaystyle b}nonzero) are known asrational numbers.[j]The set of all such irreducible fractions is commonly denotedQ{\displaystyle \mathbb {Q} }. There is still a minor obstacle for(Q,⋅){\displaystyle \left(\mathbb {Q} ,\cdot \right)}, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is nox{\displaystyle x}such thatx⋅0=1{\displaystyle x\cdot 0=1}),(Q,⋅){\displaystyle \left(\mathbb {Q} ,\cdot \right)}is still not a group.
However, the set of allnonzerorational numbersQ∖{0}={q∈Q∣q≠0}{\displaystyle \mathbb {Q} \smallsetminus \left\{0\right\}=\left\{q\in \mathbb {Q} \mid q\neq 0\right\}}does form an abelian group under multiplication, also denotedQ×{\displaystyle \mathbb {Q} ^{\times }}.[k]Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse ofa/b{\displaystyle a/b}isb/a{\displaystyle b/a}, therefore the axiom of the inverse element is satisfied.
The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – ifdivisionby other than zero is possible, such as inQ{\displaystyle \mathbb {Q} }– fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.[l]
Modular arithmetic for amodulusn{\displaystyle n}defines any two elementsa{\displaystyle a}andb{\displaystyle b}that differ by a multiple ofn{\displaystyle n}to be equivalent, denoted bya≡b(modn){\displaystyle a\equiv b{\pmod {n}}}. Every integer is equivalent to one of the integers from0{\displaystyle 0}ton−1{\displaystyle n-1}, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalentrepresentative. Modular addition, defined in this way for the integers from0{\displaystyle 0}ton−1{\displaystyle n-1}, forms a group, denoted asZn{\displaystyle \mathrm {Z} _{n}}or(Z/nZ,+){\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}, with0{\displaystyle 0}as the identity element andn−a{\displaystyle n-a}as the inverse element ofa{\displaystyle a}.
A familiar example is addition of hours on the face of aclock, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on9{\displaystyle 9}and is advanced4{\displaystyle 4}hours, it ends up on1{\displaystyle 1}, as shown in the illustration. This is expressed by saying that9+4{\displaystyle 9+4}is congruent to1{\displaystyle 1}"modulo12{\displaystyle 12}" or, in symbols,9+4≡1(mod12).{\displaystyle 9+4\equiv 1{\pmod {12}}.}
For any prime numberp{\displaystyle p}, there is also themultiplicative group of integers modulop{\displaystyle p}.[43]Its elements can be represented by1{\displaystyle 1}top−1{\displaystyle p-1}. The group operation, multiplication modulop{\displaystyle p}, replaces the usual product by its representative, theremainderof division byp{\displaystyle p}. For example, forp=5{\displaystyle p=5}, the four group elements can be represented by1,2,3,4{\displaystyle 1,2,3,4}. In this group,4⋅4≡1mod5{\displaystyle 4\cdot 4\equiv 1{\bmod {5}}}, because the usual product16{\displaystyle 16}is equivalent to1{\displaystyle 1}: when divided by5{\displaystyle 5}it yields a remainder of1{\displaystyle 1}. The primality ofp{\displaystyle p}ensures that the usual product of two representatives is not divisible byp{\displaystyle p}, and therefore that the modular product is nonzero.[m]The identity element is represented by1{\displaystyle 1}, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integera{\displaystyle a}not divisible byp{\displaystyle p}, there exists an integerb{\displaystyle b}such thata⋅b≡1(modp),{\displaystyle a\cdot b\equiv 1{\pmod {p}},}that is, such thatp{\displaystyle p}evenly dividesa⋅b−1{\displaystyle a\cdot b-1}. The inverseb{\displaystyle b}can be found by usingBézout's identityand the fact that thegreatest common divisorgcd(a,p){\displaystyle \gcd(a,p)}equals1{\displaystyle 1}.[44]In the casep=5{\displaystyle p=5}above, the inverse of the element represented by4{\displaystyle 4}is that represented by4{\displaystyle 4}, and the inverse of the element represented by3{\displaystyle 3}is represented by2{\displaystyle 2}, as3⋅2=6≡1mod5{\displaystyle 3\cdot 2=6\equiv 1{\bmod {5}}}. Hence all group axioms are fulfilled. This example is similar to(Q∖{0},⋅){\displaystyle \left(\mathbb {Q} \smallsetminus \left\{0\right\},\cdot \right)}above: it consists of exactly those elements in the ringZ/pZ{\displaystyle \mathbb {Z} /p\mathbb {Z} }that have a multiplicative inverse.[45]These groups, denotedFp×{\displaystyle \mathbb {F} _{p}^{\times }}, are crucial topublic-key cryptography.[n]
Acyclic groupis a group all of whose elements arepowersof a particular elementa{\displaystyle a}.[46]In multiplicative notation, the elements of the group are…,a−3,a−2,a−1,a0,a,a2,a3,…,{\displaystyle \dots ,a^{-3},a^{-2},a^{-1},a^{0},a,a^{2},a^{3},\dots ,}wherea2{\displaystyle a^{2}}meansa⋅a{\displaystyle a\cdot a},a−3{\displaystyle a^{-3}}stands fora−1⋅a−1⋅a−1=(a⋅a⋅a)−1{\displaystyle a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1}}, etc.[o]Such an elementa{\displaystyle a}is called a generator or aprimitive elementof the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as…,(−a)+(−a),−a,0,a,a+a,….{\displaystyle \dots ,(-a)+(-a),-a,0,a,a+a,\dots .}
In the groups(Z/nZ,+){\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}introduced above, the element1{\displaystyle 1}is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are1{\displaystyle 1}. Any cyclic group withn{\displaystyle n}elements is isomorphic to this group. A second example for cyclic groups is the group ofn{\displaystyle n}thcomplex roots of unity, given bycomplex numbersz{\displaystyle z}satisfyingzn=1{\displaystyle z^{n}=1}. These numbers can be visualized as theverticeson a regularn{\displaystyle n}-gon, as shown in blue in the image forn=6{\displaystyle n=6}. The group operation is multiplication of complex numbers. In the picture, multiplying withz{\displaystyle z}corresponds to acounter-clockwiserotation by 60°.[47]Fromfield theory, the groupFp×{\displaystyle \mathbb {F} _{p}^{\times }}is cyclic for primep{\displaystyle p}: for example, ifp=5{\displaystyle p=5},3{\displaystyle 3}is a generator since31=3{\displaystyle 3^{1}=3},32=9≡4{\displaystyle 3^{2}=9\equiv 4},33≡2{\displaystyle 3^{3}\equiv 2}, and34≡1{\displaystyle 3^{4}\equiv 1}.
Some cyclic groups have an infinite number of elements. In these groups, for every non-zero elementa{\displaystyle a}, all the powers ofa{\displaystyle a}are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to(Z,+){\displaystyle (\mathbb {Z} ,+)}, the group of integers under addition introduced above.[48]As these two prototypes are both abelian, so are all cyclic groups.
The study of finitely generated abelian groups is quite mature, including thefundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such ascenterandcommutator, describe the extent to which a given group is not abelian.[49]
Symmetry groupsare groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below).[51]Conceptually, group theory can be thought of as the study of symmetry.[p]Symmetries in mathematicsgreatly simplify the study ofgeometricaloranalyticalobjects. A group is said toacton another mathematical objectX{\displaystyle X}if every group element can be associated to some operation onX{\displaystyle X}and the composition of these operations follows the group law. For example, an element of the(2,3,7) triangle groupacts on a triangulartilingof thehyperbolic planeby permuting the triangles.[50]By a group action, the group pattern is connected to the structure of the object being acted on.
In chemistry,point groupsdescribemolecular symmetries, whilespace groupsdescribe crystal symmetries incrystallography. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification ofquantum mechanicalanalysis of these properties.[52]For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.[53]
Group theory helps predict the changes in physical properties that occur when a material undergoes aphase transition, for example, from a cubic to a tetrahedral crystalline form. An example isferroelectricmaterials, where the change from a paraelectric to a ferroelectric state occurs at theCurie temperatureand is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called softphononmode, a vibrational lattice mode that goes to zero frequency at the transition.[54]
Suchspontaneous symmetry breakinghas found further application in elementary particle physics, where its occurrence is related to the appearance ofGoldstone bosons.[55]
Finite symmetry groups such as theMathieu groupsare used incoding theory, which is in turn applied inerror correctionof transmitted data, and inCD players.[59]Another application isdifferential Galois theory, which characterizes functions havingantiderivativesof a prescribed form, giving group-theoretic criteria for when solutions of certaindifferential equationsare well-behaved.[q]Geometric properties that remain stable under group actions are investigated in(geometric)invariant theory.[60]
Matrix groupsconsist ofmatricestogether withmatrix multiplication. Thegeneral linear groupGL(n,R){\displaystyle \mathrm {GL} (n,\mathbb {R} )}consists of allinvertiblen{\displaystyle n}-by-n{\displaystyle n}matrices with real entries.[61]Its subgroups are referred to asmatrix groupsorlinear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is thespecial orthogonal groupSO(n){\displaystyle \mathrm {SO} (n)}. It describes all possible rotations inn{\displaystyle n}dimensions.Rotation matricesin this group are used incomputer graphics.[62]
Representation theoryis both an application of the group concept and important for a deeper understanding of groups.[63][64]It studies the group by its group actions on other spaces. A broad class ofgroup representationsare linear representations in which the group acts on a vector space, such as the three-dimensionalEuclidean spaceR3{\displaystyle \mathbb {R} ^{3}}. A representation of a groupG{\displaystyle G}on ann{\displaystyle n}-dimensionalreal vector space is simply a group homomorphismρ:G→GL(n,R){\displaystyle \rho :G\to \mathrm {GL} (n,\mathbb {R} )}from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.[r]
A group action gives further means to study the object being acted on.[s]On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups andtopological groups, especially (locally)compact groups.[63][65]
Galois groupswere developed to help solve polynomial equations by capturing their symmetry features.[66][67]For example, the solutions of thequadratic equationax2+bx+c=0{\displaystyle ax^{2}+bx+c=0}are given byx=−b±b2−4ac2a.{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}Each solution can be obtained by replacing the±{\displaystyle \pm }sign by+{\displaystyle +}or−{\displaystyle -}; analogous formulae are known forcubicandquartic equations, but donotexist in general fordegree 5and higher.[68]In thequadratic formula, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomial equations and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular theirsolvability) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, androotssimilar to the formula above.[69]
ModernGalois theorygeneralizes the above type of Galois groups by shifting to field theory and consideringfield extensionsformed as thesplitting fieldof a polynomial. This theory establishes—via thefundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.[70]
A group is calledfiniteif it has afinite number of elements. The number of elements is called theorderof the group.[71]An important class is thesymmetric groupsSN{\displaystyle \mathrm {S} _{N}}, the groups of permutations ofN{\displaystyle N}objects. For example, thesymmetric group on 3 lettersS3{\displaystyle \mathrm {S} _{3}}is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (factorialof 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric groupSN{\displaystyle \mathrm {S} _{N}}for a suitable integerN{\displaystyle N}, according toCayley's theorem. Parallel to the group of symmetries of the square above,S3{\displaystyle \mathrm {S} _{3}}can also be interpreted as the group of symmetries of anequilateral triangle.
The order of an elementa{\displaystyle a}in a groupG{\displaystyle G}is the least positive integern{\displaystyle n}such thatan=e{\displaystyle a^{n}=e}, wherean{\displaystyle a^{n}}representsa⋯a⏟nfactors,{\displaystyle \underbrace {a\cdots a} _{n{\text{ factors}}},}that is, application of the operation "⋅{\displaystyle \cdot }" ton{\displaystyle n}copies ofa{\displaystyle a}. (If "⋅{\displaystyle \cdot }" represents multiplication, thenan{\displaystyle a^{n}}corresponds to then{\displaystyle n}th power ofa{\displaystyle a}.) In infinite groups, such ann{\displaystyle n}may not exist, in which case the order ofa{\displaystyle a}is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.
More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups:Lagrange's Theoremstates that for a finite groupG{\displaystyle G}the order of any finite subgroupH{\displaystyle H}dividesthe order ofG{\displaystyle G}. TheSylow theoremsgive a partial converse.
The dihedral groupD4{\displaystyle \mathrm {D} _{4}}of symmetries of a square is a finite group of order 8. In this group, the order ofr1{\displaystyle r_{1}}is 4, as is the order of the subgroupR{\displaystyle R}that this element generates. The order of the reflection elementsfv{\displaystyle f_{\mathrm {v} }}etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. The groupsFp×{\displaystyle \mathbb {F} _{p}^{\times }}of multiplication modulo a primep{\displaystyle p}have orderp−1{\displaystyle p-1}.
Any finite abelian group is isomorphic to aproductof finite cyclic groups; this statement is part of thefundamental theorem of finitely generated abelian groups.
Any group of prime orderp{\displaystyle p}is isomorphic to the cyclic groupZp{\displaystyle \mathrm {Z} _{p}}(a consequence ofLagrange's theorem).
Any group of orderp2{\displaystyle p^{2}}is abelian, isomorphic toZp2{\displaystyle \mathrm {Z} _{p^{2}}}orZp×Zp{\displaystyle \mathrm {Z} _{p}\times \mathrm {Z} _{p}}.
But there exist nonabelian groups of orderp3{\displaystyle p^{3}}; the dihedral groupD4{\displaystyle \mathrm {D} _{4}}of order23{\displaystyle 2^{3}}above is an example.[72]
When a groupG{\displaystyle G}has a normal subgroupN{\displaystyle N}other than{1}{\displaystyle \{1\}}andG{\displaystyle G}itself, questions aboutG{\displaystyle G}can sometimes be reduced to questions aboutN{\displaystyle N}andG/N{\displaystyle G/N}. A nontrivial group is calledsimpleif it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by theJordan–Hölder theorem.
Computer algebra systemshave been used tolist all groups of order up to 2000.[t]Butclassifyingall finite groups is a problem considered too hard to be solved.
The classification of all finitesimplegroups was a major achievement in contemporary group theory. There areseveral infinite familiesof such groups, as well as 26 "sporadic groups" that do not belong to any of the families. The largestsporadic groupis called themonster group. Themonstrous moonshineconjectures, proved byRichard Borcherds, relate the monster group to certainmodular functions.[73]
The gap between the classification of simple groups and the classification of all groups lies in theextension problem.[74]
An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a setG{\displaystyle G}equipped with a binary operationG×G→G{\displaystyle G\times G\rightarrow G}(the group operation), aunary operationG→G{\displaystyle G\rightarrow G}(which provides the inverse) and anullary operation, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoidsexistential quantifiersand is used in computing with groups and forcomputer-aided proofs.
This way of defining groups lends itself to generalizations such as the notion ofgroup objectin a category. Briefly, this is an object withmorphismsthat mimic the group axioms.[75]
Sometopological spacesmay be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally,g⋅h{\displaystyle g\cdot h}andg−1{\displaystyle g^{-1}}must not vary wildly ifg{\displaystyle g}andh{\displaystyle h}vary only a little. Such groups are calledtopological groups,and they are the group objects in thecategory of topological spaces.[76]The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any othertopological field, such as the field of complex numbers or the field ofp-adic numbers. These examples arelocally compact, so they haveHaar measuresand can be studied viaharmonic analysis. Other locally compact topological groups include the group of points of an algebraic group over alocal fieldoradele ring; these are basic to number theory[77]Galois groups of infinite algebraic field extensions are equipped with theKrull topology, which plays a role ininfinite Galois theory.[78]A generalization used in algebraic geometry is theétale fundamental group.[79]
ALie groupis a group that also has the structure of adifferentiable manifold; informally, this means that itlooks locally likea Euclidean space of some fixed dimension.[80]Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to besmooth.
A standard example is the general linear group introduced above: it is anopen subsetof the space of alln{\displaystyle n}-by-n{\displaystyle n}matrices, because it is given by the inequalitydet(A)≠0,{\displaystyle \det(A)\neq 0,}whereA{\displaystyle A}denotes ann{\displaystyle n}-by-n{\displaystyle n}matrix.[81]
Lie groups are of fundamental importance in modern physics:Noether's theoremlinks continuous symmetries toconserved quantities.[82]Rotation, as well as translations inspaceandtime, are basic symmetries of the laws ofmechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.[u]Another example is the group ofLorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry ofMinkowski space. The latter serves—in the absence of significantgravitation—as a model ofspacetimeinspecial relativity.[83]The full symmetry group of Minkowski space, i.e., including translations, is known as thePoincaré group. By the above, it plays a pivotal role in special relativity and, by implication, forquantum field theories.[84]Symmetries that vary with locationare central to the modern description of physical interactions with the help ofgauge theory. An important example of a gauge theory is theStandard Model, which describes three of the four knownfundamental forcesand classifies all knownelementary particles.[85]
More general structures may be defined by relaxing some of the axioms defining a group.[31][86][87]The table gives a list of several structures generalizing groups.
For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called amonoid. Thenatural numbersN{\displaystyle \mathbb {N} }(including zero) under addition form a monoid, as do the nonzero integers under multiplication(Z∖{0},⋅){\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}. Adjoining inverses of all elements of the monoid(Z∖{0},⋅){\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}produces a group(Q∖{0},⋅){\displaystyle (\mathbb {Q} \smallsetminus \{0\},\cdot )}, and likewise adjoining inverses to any (abelian) monoidM{\displaystyle M}produces a group known as theGrothendieck groupofM{\displaystyle M}.
A group can be thought of as asmall categorywith one objectx{\displaystyle x}in which every morphism is an isomorphism: given such a category, the setHom(x,x){\displaystyle \operatorname {Hom} (x,x)}is a group; conversely, given a groupG{\displaystyle G}, one can build a small category with one objectx{\displaystyle x}in whichHom(x,x)≃G{\displaystyle \operatorname {Hom} (x,x)\simeq G}.
More generally, agroupoidis any small category in which every morphism is an isomorphism.
In a groupoid, the set of all morphisms in the category is usually not a group, because the composition is only partially defined:fg{\displaystyle fg}is defined only when the source off{\displaystyle f}matches the target ofg{\displaystyle g}.
Groupoids arise in topology (for instance, thefundamental groupoid) and in the theory ofstacks.
Finally, it is possible to generalize any of these concepts by replacing the binary operation with ann-aryoperation (i.e., an operation takingnarguments, for some nonnegative integern). With the proper generalization of the group axioms, this gives a notion ofn-ary group.[88]
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Bass–Serre theoryis a part of themathematicalsubject ofgroup theorythat deals with analyzing the algebraic structure ofgroupsactingby automorphisms on simplicialtrees. The theory relates group actions on trees with decomposing groups as iterated applications of the operations offree product with amalgamationandHNN extension, via the notion of thefundamental group of agraph of groups.Bass–Serre theory can be regarded as one-dimensional version of theorbifold theory.
Bass–Serre theory was developed byJean-Pierre Serrein the 1970s and formalized inTrees, Serre's 1977 monograph (developed in collaboration withHyman Bass) on the subject.[1][2]Serre's original motivation was to understand the structure of certainalgebraic groupswhoseBruhat–Tits buildingsare trees. However, the theory quickly became a standard tool ofgeometric group theoryandgeometric topology, particularly the study of3-manifolds. Subsequent work of Bass[3]contributed substantially to the formalization and development of basic tools of the theory and currently the term "Bass–Serre theory" is widely used to describe the subject.
Mathematically, Bass–Serre theory builds on exploiting and generalizing the properties of two older group-theoretic constructions:free product with amalgamationandHNN extension. However, unlike the traditional algebraic study of these two constructions, Bass–Serre theory uses the geometric language ofcovering theoryandfundamental groups.Graphs of groups, which are the basic objects of Bass–Serre theory, can be viewed as one-dimensional versions oforbifolds.
Apart from Serre's book,[2]the basic treatment of Bass–Serre theory is available in the article of Bass,[3]the article ofG. Peter ScottandC. T. C. Wall[4]and the books ofAllen Hatcher,[5]Gilbert Baumslag,[6]Warren Dicks andMartin Dunwoody[7]and Daniel E. Cohen.[8]
Serre's formalism ofgraphsis slightly different from the standard formalism fromgraph theory. Here agraphAconsists of avertex setV, anedge setE, anedge reversalmapE→E,e↦e¯{\displaystyle E\to E,\ e\mapsto {\overline {e}}}such thate≠eande¯¯=e{\displaystyle {\overline {\overline {e}}}=e}for everyeinE, and aninitial vertex mapo:E→V{\displaystyle o\colon E\to V}. Thus inAevery edgeecomes equipped with itsformal inversee. The vertexo(e) is called theoriginor theinitial vertexofeand the vertexo(e) is called theterminusofeand is denotedt(e). Both loop-edges (that is, edgesesuch thato(e) =t(e)) andmultiple edgesare allowed. AnorientationonAis a partition ofEinto the union of two disjoint subsetsE+andE−so that for every edgeeexactly one of the edges from the paire,ebelongs toE+and the other belongs toE−.
Agraph of groupsAconsists of the following data:
For everye∈E{\displaystyle e\in E}the mapαe¯:Ae→At(e){\displaystyle \alpha _{\overline {e}}\colon A_{e}\to A_{t(e)}}is also denoted byωe{\displaystyle \omega _{e}}.
There are two equivalent definitions of the notion of the fundamental group of a graph of groups: the first is a direct algebraic definition via an explicitgroup presentation(as a certain iterated application ofamalgamated free productsandHNN extensions), and the second using the language ofgroupoids.
The algebraic definition is easier to state:
First, choose aspanning treeTinA. The fundamental group ofAwith respect toT, denoted π1(A,T), is defined as the quotient of thefree product
whereF(E) is afree groupwith free basisE, subject to the following relations:
There is also a notion of the fundamental group ofAwith respect to a base-vertexvinV, denoted π1(A,v), which is defined using the formalism ofgroupoids. It turns out that for every choice of a base-vertexvand every spanning treeTinAthe groups π1(A,T) and π1(A,v) are naturallyisomorphic.
The fundamental group of a graph of groups has a natural topological interpretation as well: it is the fundamental group of agraph of spaceswhose vertex spaces and edge spaces have the fundamental groups of the vertex groups and edge groups, respectively, and whose gluing maps induce the homomorphisms of the edge groups into the vertex groups. One can therefore take this as a third definition of the fundamental group of a graph of groups.
The groupG= π1(A,T) defined above admits an algebraic description in terms of iteratedamalgamated free productsandHNN extensions. First, form a groupBas a quotient of the free product
subject to the relations
This presentation can be rewritten as
which shows thatBis an iteratedamalgamated free productof the vertex groupsAv.
Then the groupG= π1(A,T) has the presentation
which shows thatG= π1(A,T) is a multipleHNN extensionofBwith stable letters{e|e∈E+(A−T)}{\displaystyle \{e|e\in E^{+}(A-T)\}}.
An isomorphism between a groupGand the fundamental group of a graph of groups is called asplittingofG. If the edge groups in the splitting come from a particular class of groups (e.g. finite, cyclic, abelian, etc.), the splitting is said to be asplitting overthat class. Thus a splitting where all edge groups are finite is called a splitting over finite groups.
Algebraically, a splitting ofGwith trivial edge groups corresponds to a free product decomposition
whereF(X) is afree groupwith free basisX=E+(A−T) consisting of all positively oriented edges (with respect to some orientation onA) in the complement of some spanning treeTofA.
Letgbe an element ofG= π1(A,T) represented as a product of the form
wheree1, ...,enis a closed edge-path inAwith the vertex sequencev0,v1, ...,vn=v0(that isv0=o(e1),vn=t(en) andvi=t(ei) =o(ei+1) for 0 <i<n) and whereai∈Avi{\displaystyle a_{i}\in A_{v_{i}}}fori= 0, ...,n.
Suppose thatg= 1 inG. Then
The normal forms theorem immediately implies that the canonical homomorphismsAv→ π1(A,T) are injective, so that we can think of the vertex groupsAvas subgroups ofG.
Higgins has given a nice version of the normal form using the fundamentalgroupoidof a graph of groups.[9]This avoids choosing a base point or tree, and has been exploited by Moore.[10]
To every graph of groupsA, with a specified choice of a base-vertex, one can associate aBass–Serre covering treeA~{\displaystyle {\tilde {\mathbf {A} }}}, which is a tree that comes equipped with a naturalgroup actionof the fundamental group π1(A,v) without edge-inversions.
Moreover, thequotient graphA~/π1(A,v){\displaystyle {\tilde {\mathbf {A} }}/\pi _{1}(\mathbf {A} ,v)}is isomorphic toA.
Similarly, ifGis a group acting on a treeXwithout edge-inversions (that is, so that for every edgeeofXand everyginGwe havege≠e), one can define the natural notion of aquotient graph of groupsA. The underlying graphAofAis the quotient graphX/G. The vertex groups ofAare isomorphic to vertex stabilizers inGof vertices ofXand the edge groups ofAare isomorphic to edge stabilizers inGof edges ofX.
Moreover, ifXwas the Bass–Serre covering tree of a graph of groupsAand ifG= π1(A,v) then the quotient graph of groups for the action ofGonXcan be chosen to be naturally isomorphic toA.
LetGbe a group acting on a treeXwithout inversions. LetAbe the quotientgraph of groupsand letvbe a base-vertex inA. ThenGis isomorphic to the group π1(A,v) and there is an equivariant isomorphism between the treeXand the Bass–Serre covering treeA~{\displaystyle {\tilde {\mathbf {A} }}}. More precisely, there is agroup isomorphismσ:G→ π1(A,v) and a graph isomorphismj:X→A~{\displaystyle j:X\to {\tilde {\mathbf {A} }}}such that for everyginG, for every vertexxofXand for every edgeeofXwe havej(gx) =gj(x) andj(ge) =gj(e).
This result is also known as thestructure theorem.[2]
One of the immediate consequences is the classicKurosh subgroup theoremdescribing the algebraic structure of subgroups offree products.
Consider a graph of groupsAconsisting of a single non-loop edgee(together with its formal inversee) with two distinct end-verticesu=o(e) andv=t(e), vertex groupsH=Au,K=Av, an edge groupC=Aeand the boundary monomorphismsα=αe:C→H,ω=ωe:C→K{\displaystyle \alpha =\alpha _{e}:C\to H,\omega =\omega _{e}:C\to K}. ThenT=Ais a spanning tree inAand the fundamental group π1(A,T) is isomorphic to theamalgamated free product
In this case the Bass–Serre treeX=A~{\displaystyle X={\tilde {\mathbf {A} }}}can be described as follows. The vertex set ofXis the set ofcosets
Two verticesgKandfHare adjacent inXwhenever there existsk∈Ksuch thatfH=gkH(or, equivalently, whenever there ish∈Hsuch thatgK=fhK).
TheG-stabilizer of every vertex ofXof typegKis equal togKg−1and theG-stabilizer of every vertex ofXof typegHis equal togHg−1. For an edge [gH,ghK] ofXitsG-stabilizer is equal toghα(C)h−1g−1.
For everyc∈Candh∈ 'k∈K'the edges [gH,ghK] and [gH, ghα(c)K] are equal and the degree of the vertexgHinXis equal to theindex[H:α(C)]. Similarly, every vertex of typegKhas degree [K:ω(C)] inX.
LetAbe a graph of groups consisting of a single loop-edgee(together with its formal inversee), a single vertexv=o(e) =t(e), a vertex groupB=Av, an edge groupC=Aeand the boundary monomorphismsα=αe:C→B,ω=ωe:C→B{\displaystyle \alpha =\alpha _{e}:C\to B,\omega =\omega _{e}:C\to B}. ThenT=vis a spanning tree inAand the fundamental group π1(A,T) is isomorphic to theHNN extension
with the base groupB, stable lettereand the associated subgroupsH= α(C),K= ω(C) inB. The compositionϕ=ω∘α−1:H→K{\displaystyle \phi =\omega \circ \alpha ^{-1}:H\to K}is an isomorphism and the above HNN-extension presentation ofGcan be rewritten as
In this case the Bass–Serre treeX=A~{\displaystyle X={\tilde {\mathbf {A} }}}can be described as follows. The vertex set ofXis the set ofcosetsVX= {gB:g∈G}.
Two verticesgBandfBare adjacent inXwhenever there existsbinBsuch that eitherfB=gbeBorfB=gbe−1B. TheG-stabilizer of every vertex ofXis conjugate toBinGand the stabilizer of every edge ofXis conjugate toHinG. Every vertex ofXhas degree equal to [B:H] + [B:K].
LetAbe a graph of groups with underlying graphAsuch that all the vertex and edge groups inAare trivial. Letvbe a base-vertex inA. Thenπ1(A,v) is equal to thefundamental groupπ1(A,v) of the underlying graphAin the standard sense ofalgebraic topologyand the Bass–Serre covering treeA~{\displaystyle {\tilde {\mathbf {A} }}}is equal to the standarduniversal covering spaceA~{\displaystyle {\tilde {A}}}ofA. Moreover, the action ofπ1(A,v) onA~{\displaystyle {\tilde {\mathbf {A} }}}is exactly the standard action ofπ1(A,v) onA~{\displaystyle {\tilde {A}}}bydeck transformations.
A graph of groupsAis calledtrivialifA=Tis already a tree and there is some vertexvofAsuch thatAv= π1(A,A). This is equivalent to the condition thatAis a tree and that for every edgee= [u,z] ofA(witho(e) =u,t(e) =z) such thatuis closer tovthanzwe have [Az: ωe(Ae)] = 1, that isAz= ωe(Ae).
An action of a groupGon a treeXwithout edge-inversions is calledtrivialif there exists a vertexxofXthat is fixed byG, that is such thatGx=x. It is known that an action ofGonXis trivial if and only if thequotient graphof groups for that action is trivial.
Typically, only nontrivial actions on trees are studied in Bass–Serre theory since trivial graphs of groups do not carry any interesting algebraic information, although trivial actions in the above sense (e. g. actions of groups by automorphisms on rooted trees) may also be interesting for other mathematical reasons.
One of the classic and still important results of the theory is a theorem of Stallings aboutendsof groups. The theorem states that afinitely generated grouphas more than one end if and only if this group admits a nontrivial splitting over finite subgroups that is, if and only if the group admits a nontrivial action without inversions on a tree with finite edge stabilizers.[11]
An important general result of the theory states that ifGis a group withKazhdan's property (T)thenGdoes not admit any nontrivial splitting, that is, that any action ofGon a treeXwithout edge-inversions has a global fixed vertex.[12]
LetGbe a group acting on a treeXwithout edge-inversions.
For everyg∈Gput
ThenℓX(g) is called thetranslation lengthofgonX.
The function
is called thehyperbolic length functionor thetranslation length functionfor the action ofGonX.
The length-functionℓX:G→Zis said to beabelianif it is agroup homomorphismfromGtoZandnon-abelianotherwise. Similarly, the action ofGonXis said to beabelianif the associated hyperbolic length function is abelian and is said to benon-abelianotherwise.
In general, an action ofGon a treeXwithout edge-inversions is said to beminimalif there are no properG-invariant subtrees inX.
An important fact in the theory says that minimal non-abelian tree actions are uniquely determined by their hyperbolic length functions:[13]
LetGbe a group with two nonabelian minimal actions without edge-inversions on treesXandY. Suppose that the hyperbolic length functionsℓXandℓYonGare equal, that isℓX(g) =ℓY(g) for everyg∈G. Then the actions ofGonXandYare equal in the sense that there exists agraph isomorphismf:X→Ywhich isG-equivariant, that isf(gx) =gf(x) for everyg∈Gand everyx∈VX.
Important developments in Bass–Serre theory in the last 30 years include:
There have been several generalizations of Bass–Serre theory:
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Inmathematics, specificallyabstract algebra, theisomorphism theorems(also known asNoether's isomorphism theorems) aretheoremsthat describe the relationship amongquotients,homomorphisms, andsubobjects. Versions of the theorems exist forgroups,rings,vector spaces,modules,Lie algebras, and otheralgebraic structures. Inuniversal algebra, the isomorphism theorems can be generalized to the context of algebras andcongruences.
The isomorphism theorems were formulated in some generality for homomorphisms of modules byEmmy Noetherin her paperAbstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, which was published in 1927 inMathematische Annalen. Less general versions of these theorems can be found in work ofRichard Dedekindand previous papers by Noether.
Three years later,B.L. van der Waerdenpublished his influentialModerne Algebra, the firstabstract algebratextbook that took thegroups-rings-fieldsapproach to the subject. Van der Waerden credited lectures by Noether ongroup theoryandEmil Artinon algebra, as well as a seminar conducted by Artin,Wilhelm Blaschke,Otto Schreier, and van der Waerden himself onidealsas the main references. The three isomorphism theorems, calledhomomorphism theorem, andtwo laws of isomorphismwhen applied to groups, appear explicitly.
We first present the isomorphism theorems of thegroups.
LetG{\displaystyle G}andH{\displaystyle H}be groups, and letf:G→H{\displaystyle f:G\rightarrow H}be ahomomorphism. Then:
In particular, iff{\displaystyle f}issurjectivethenH{\displaystyle H}is isomorphic toG/kerf{\displaystyle G/\ker f}.
This theorem is usually called thefirst isomorphism theorem.
LetG{\displaystyle G}be a group. LetS{\displaystyle S}be a subgroup ofG{\displaystyle G}, and letN{\displaystyle N}be a normal subgroup ofG{\displaystyle G}. Then the following hold:
Technically, it is not necessary forN{\displaystyle N}to be a normal subgroup, as long asS{\displaystyle S}is a subgroup of thenormalizerofN{\displaystyle N}inG{\displaystyle G}. In this case,N{\displaystyle N}is not a normal subgroup ofG{\displaystyle G}, butN{\displaystyle N}is still a normal subgroup of the productSN{\displaystyle SN}.
This theorem is sometimes called thesecond isomorphism theorem,[1]diamond theorem[2]or theparallelogram theorem.[3]
An application of the second isomorphism theorem identifiesprojective linear groups: for example, the group on thecomplex projective linestarts with settingG=GL2(C){\displaystyle G=\operatorname {GL} _{2}(\mathbb {C} )}, the group ofinvertible2 × 2complexmatrices,S=SL2(C){\displaystyle S=\operatorname {SL} _{2}(\mathbb {C} )}, the subgroup ofdeterminant1 matrices, andN{\displaystyle N}the normal subgroup of scalar matricesC×I={(a00a):a∈C×}{\displaystyle \mathbb {C} ^{\times }\!I=\left\{\left({\begin{smallmatrix}a&0\\0&a\end{smallmatrix}}\right):a\in \mathbb {C} ^{\times }\right\}}, we haveS∩N={±I}{\displaystyle S\cap N=\{\pm I\}}, whereI{\displaystyle I}is theidentity matrix, andSN=GL2(C){\displaystyle SN=\operatorname {GL} _{2}(\mathbb {C} )}. Then the second isomorphism theorem states that:
LetG{\displaystyle G}be a group, andN{\displaystyle N}a normal subgroup ofG{\displaystyle G}.
Then
The last statement is sometimes referred to as thethird isomorphism theorem. The first four statements are often subsumed under Theorem D below, and referred to as thelattice theorem,correspondence theorem, orfourth isomorphism theorem.
LetG{\displaystyle G}be a group, andN{\displaystyle N}a normal subgroup ofG{\displaystyle G}.
The canonical projection homomorphismG→G/N{\displaystyle G\rightarrow G/N}defines a bijective correspondence
between the set of subgroups ofG{\displaystyle G}containingN{\displaystyle N}and the set of (all) subgroups ofG/N{\displaystyle G/N}. Under this correspondence normal subgroups correspond to normal subgroups.
This theorem is sometimes called thecorrespondence theorem, thelattice theorem, and thefourth isomorphism theorem.
TheZassenhaus lemma(also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem.[4]
The first isomorphism theorem can be expressed incategory theoreticallanguage by saying that thecategory of groupsis (normal epi, mono)-factorizable; in other words, thenormal epimorphismsand themonomorphismsform afactorization systemfor thecategory. This is captured in thecommutative diagramin the margin, which shows theobjectsandmorphismswhose existence can be deduced from the morphismf:G→H{\displaystyle f:G\rightarrow H}. The diagram shows that every morphism in the category of groups has akernelin the category theoretical sense; the arbitrary morphismffactors intoι∘π{\displaystyle \iota \circ \pi }, whereιis a monomorphism andπis an epimorphism (in aconormal category, all epimorphisms are normal). This is represented in the diagram by an objectkerf{\displaystyle \ker f}and a monomorphismκ:kerf→G{\displaystyle \kappa :\ker f\rightarrow G}(kernels are always monomorphisms), which complete theshort exact sequencerunning from the lower left to the upper right of the diagram. The use of theexact sequenceconvention saves us from having to draw thezero morphismsfromkerf{\displaystyle \ker f}toH{\displaystyle H}andG/kerf{\displaystyle G/\ker f}.
If the sequence is right split (i.e., there is a morphismσthat mapsG/kerf{\displaystyle G/\operatorname {ker} f}to aπ-preimage of itself), thenGis thesemidirect productof the normal subgroupimκ{\displaystyle \operatorname {im} \kappa }and the subgroupimσ{\displaystyle \operatorname {im} \sigma }. If it is left split (i.e., there exists someρ:G→kerf{\displaystyle \rho :G\rightarrow \operatorname {ker} f}such thatρ∘κ=idkerf{\displaystyle \rho \circ \kappa =\operatorname {id} _{{\text{ker}}f}}), then it must also be right split, andimκ×imσ{\displaystyle \operatorname {im} \kappa \times \operatorname {im} \sigma }is adirect productdecomposition ofG. In general, the existence of a right split does not imply the existence of a left split; but in anabelian category(such asthat of abelian groups), left splits and right splits are equivalent by thesplitting lemma, and a right split is sufficient to produce adirect sumdecompositionimκ⊕imσ{\displaystyle \operatorname {im} \kappa \oplus \operatorname {im} \sigma }. In an abelian category, all monomorphisms are also normal, and the diagram may be extended by a second short exact sequence0→G/kerf→H→cokerf→0{\displaystyle 0\rightarrow G/\operatorname {ker} f\rightarrow H\rightarrow \operatorname {coker} f\rightarrow 0}.
In the second isomorphism theorem, the productSNis thejoinofSandNin thelattice of subgroupsofG, while the intersectionS∩Nis themeet.
The third isomorphism theorem is generalized by thenine lemmatoabelian categoriesand more general maps between objects.
Below we present four theorems, labelled A, B, C and D. They are often numbered as "First isomorphism theorem", "Second..." and so on; however, there is no universal agreement on the numbering. Here we give some examples of the group isomorphism theorems in the literature. Notice that these theorems have analogs for rings and modules.
It is less common to include the Theorem D, usually known as thelattice theoremor thecorrespondence theorem, as one of isomorphism theorems, but when included, it is the last one.
The statements of the theorems forringsare similar, with the notion of a normal subgroup replaced by the notion of anideal.
LetR{\displaystyle R}andS{\displaystyle S}be rings, and letφ:R→S{\displaystyle \varphi :R\rightarrow S}be aring homomorphism. Then:
In particular, ifφ{\displaystyle \varphi }is surjective thenS{\displaystyle S}is isomorphic toR/kerφ{\displaystyle R/\ker \varphi }.[15]
LetR{\displaystyle R}be a ring. LetS{\displaystyle S}be a subring ofR{\displaystyle R}, and letI{\displaystyle I}be an ideal ofR{\displaystyle R}. Then:
LetRbe a ring, andIan ideal ofR. Then
LetI{\displaystyle I}be an ideal ofR{\displaystyle R}. The correspondenceA↔A/I{\displaystyle A\leftrightarrow A/I}is aninclusion-preservingbijectionbetween the set of subringsA{\displaystyle A}ofR{\displaystyle R}that containI{\displaystyle I}and the set of subrings ofR/I{\displaystyle R/I}. Furthermore,A{\displaystyle A}(a subring containingI{\displaystyle I}) is an ideal ofR{\displaystyle R}if and only ifA/I{\displaystyle A/I}is an ideal ofR/I{\displaystyle R/I}.[16]
The statements of the isomorphism theorems formodulesare particularly simple, since it is possible to form aquotient modulefrom anysubmodule. The isomorphism theorems forvector spaces(modules over afield) andabelian groups(modules overZ{\displaystyle \mathbb {Z} }) are special cases of these. Forfinite-dimensionalvector spaces, all of these theorems follow from therank–nullity theorem.
In the following, "module" will mean "R-module" for some fixed ringR.
LetM{\displaystyle M}andN{\displaystyle N}be modules, and letφ:M→N{\displaystyle \varphi :M\rightarrow N}be amodule homomorphism. Then:
In particular, ifφ{\displaystyle \varphi }is surjective thenN{\displaystyle N}is isomorphic toM/kerφ{\displaystyle M/\ker \varphi }.
LetM{\displaystyle M}be a module, and letS{\displaystyle S}andT{\displaystyle T}be submodules ofM{\displaystyle M}. Then:
LetMbe a module,Ta submodule ofM.
LetM{\displaystyle M}be a module,N{\displaystyle N}a submodule ofM{\displaystyle M}. There is a bijection between the submodules ofM{\displaystyle M}that containN{\displaystyle N}and the submodules ofM/N{\displaystyle M/N}. The correspondence is given byA↔A/N{\displaystyle A\leftrightarrow A/N}for allA⊇N{\displaystyle A\supseteq N}. This correspondence commutes with the processes of taking sums and intersections (i.e., is alattice isomorphismbetween the lattice of submodules ofM/N{\displaystyle M/N}and the lattice of submodules ofM{\displaystyle M}that containN{\displaystyle N}).[17]
To generalise this touniversal algebra, normal subgroups need to be replaced bycongruence relations.
Acongruenceon analgebraA{\displaystyle A}is anequivalence relationΦ⊆A×A{\displaystyle \Phi \subseteq A\times A}that forms a subalgebra ofA×A{\displaystyle A\times A}considered as an algebra with componentwise operations. One can make the set ofequivalence classesA/Φ{\displaystyle A/\Phi }into an algebra of the same type by defining the operations via representatives; this will bewell-definedsinceΦ{\displaystyle \Phi }is a subalgebra ofA×A{\displaystyle A\times A}. The resulting structure is thequotient algebra.
Letf:A→B{\displaystyle f:A\rightarrow B}be an algebrahomomorphism. Then the image off{\displaystyle f}is a subalgebra ofB{\displaystyle B}, the relation given byΦ:f(x)=f(y){\displaystyle \Phi :f(x)=f(y)}(i.e. thekerneloff{\displaystyle f}) is a congruence onA{\displaystyle A}, and the algebrasA/Φ{\displaystyle A/\Phi }andimf{\displaystyle \operatorname {im} f}areisomorphic. (Note that in the case of a group,f(x)=f(y){\displaystyle f(x)=f(y)}ifff(xy−1)=1{\displaystyle f(xy^{-1})=1}, so one recovers the notion of kernel used in group theory in this case.)
Given an algebraA{\displaystyle A}, a subalgebraB{\displaystyle B}ofA{\displaystyle A}, and a congruenceΦ{\displaystyle \Phi }onA{\displaystyle A}, letΦB=Φ∩(B×B){\displaystyle \Phi _{B}=\Phi \cap (B\times B)}be the trace ofΦ{\displaystyle \Phi }inB{\displaystyle B}and[B]Φ={K∈A/Φ:K∩B≠∅}{\displaystyle [B]^{\Phi }=\{K\in A/\Phi :K\cap B\neq \emptyset \}}the collection of equivalence classes that intersectB{\displaystyle B}. Then
LetA{\displaystyle A}be an algebra andΦ,Ψ{\displaystyle \Phi ,\Psi }two congruence relations onA{\displaystyle A}such thatΨ⊆Φ{\displaystyle \Psi \subseteq \Phi }. ThenΦ/Ψ={([a′]Ψ,[a″]Ψ):(a′,a″)∈Φ}=[]Ψ∘Φ∘[]Ψ−1{\displaystyle \Phi /\Psi =\{([a']_{\Psi },[a'']_{\Psi }):(a',a'')\in \Phi \}=[\ ]_{\Psi }\circ \Phi \circ [\ ]_{\Psi }^{-1}}is a congruence onA/Ψ{\displaystyle A/\Psi }, andA/Φ{\displaystyle A/\Phi }is isomorphic to(A/Ψ)/(Φ/Ψ).{\displaystyle (A/\Psi )/(\Phi /\Psi ).}
LetA{\displaystyle A}be an algebra and denoteConA{\displaystyle \operatorname {Con} A}the set of all congruences onA{\displaystyle A}. The setConA{\displaystyle \operatorname {Con} A}is acomplete latticeordered by inclusion.[18]IfΦ∈ConA{\displaystyle \Phi \in \operatorname {Con} A}is a congruence and we denote by[Φ,A×A]⊆ConA{\displaystyle \left[\Phi ,A\times A\right]\subseteq \operatorname {Con} A}the set of all congruences that containΦ{\displaystyle \Phi }(i.e.[Φ,A×A]{\displaystyle \left[\Phi ,A\times A\right]}is a principalfilterinConA{\displaystyle \operatorname {Con} A}, moreover it is a sublattice), then
the mapα:[Φ,A×A]→Con(A/Φ),Ψ↦Ψ/Φ{\displaystyle \alpha :\left[\Phi ,A\times A\right]\to \operatorname {Con} (A/\Phi ),\Psi \mapsto \Psi /\Phi }is a lattice isomorphism.[19][20]
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Inmathematics, theBoolean prime ideal theoremstates thatidealsin aBoolean algebracan be extended toprime ideals. A variation of this statement forfilters on setsis known as theultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example,ringsand prime ideals (of ring theory), ordistributive latticesandmaximalideals (oforder theory). This article focuses on prime ideal theorems from order theory.
Although the various prime ideal theorems may appear simple and intuitive, they cannot be deduced in general from the axioms ofZermelo–Fraenkel set theorywithout the axiom of choice (abbreviated ZF). Instead, some of the statements turn out to be equivalent to theaxiom of choice(AC), while others—the Boolean prime ideal theorem, for instance—represent a property that is strictly weaker than AC. It is due to this intermediate status between ZF and ZF + AC (ZFC) that the Boolean prime ideal theorem is often taken as an axiom of set theory. The abbreviations BPI or PIT (for Boolean algebras) are sometimes used to refer to this additional axiom.
Anorder idealis a (non-empty)directedlower set. If the consideredpartially ordered set(poset) has binarysuprema(a.k.a.joins), as do the posets within this article, then this is equivalently characterized as a non-empty lower setIthat is closed for binary suprema (that is,x,y∈I{\displaystyle x,y\in I}impliesx∨y∈I{\displaystyle x\vee y\in I}). An idealIis prime if its set-theoretic complement in the poset is afilter(that is,x∧y∈I{\displaystyle x\wedge y\in I}impliesx∈I{\displaystyle x\in I}ory∈I{\displaystyle y\in I}). Ideals are proper if they are not equal to the whole poset.
Historically, the first statement relating to later prime ideal theorems was in fact referring to filters—subsets that are ideals with respect to thedualorder. The ultrafilter lemma states that every filter on a set is contained within some maximal (proper) filter—anultrafilter. Recall that filters on sets are proper filters of the Boolean algebra of itspowerset. In this special case, maximal filters (i.e. filters that are not strict subsets of any proper filter) and prime filters (i.e. filters that with each union of subsetsXandYcontain alsoXorY) coincide. The dual of this statement thus assures that every ideal of a powerset is contained in a prime ideal.
The above statement led to various generalized prime ideal theorems, each of which exists in a weak and in a strong form.Weak prime ideal theoremsstate that everynon-trivialalgebra of a certain class has at least one prime ideal. In contrast,strong prime ideal theoremsrequire that every ideal that is disjoint from a given filter can be extended to a prime ideal that is still disjoint from that filter. In the case of algebras that are not posets, one uses different substructures instead of filters. Many forms of these theorems are actually known to be equivalent, so that the assertion that "PIT" holds is usually taken as the assertion that the corresponding statement for Boolean algebras (BPI) is valid.
Another variation of similar theorems is obtained by replacing each occurrence ofprime idealbymaximal ideal. The corresponding maximal ideal theorems (MIT) are often—though not always—stronger than their PIT equivalents.
The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras. Thus the formal statement is:
The weak prime ideal theorem for Boolean algebras simply states:
We refer to these statements as the weak and strongBPI. The two are equivalent, as the strong BPI clearly implies the weak BPI, and the reverse implication can be achieved by using the weak BPI to find prime ideals in the appropriate quotient algebra.
The BPI can be expressed in various ways. For this purpose, recall the following theorem:
For any idealIof a Boolean algebraB, the following are equivalent:
This theorem is a well-known fact for Boolean algebras. Its dual establishes the equivalence of prime filters and ultrafilters. Note that the last property is in fact self-dual—only the prior assumption thatIis an ideal gives the full characterization. All of the implications within this theorem can be proven in ZF.
Thus the following (strong) maximal ideal theorem (MIT) for Boolean algebras is equivalent to BPI:
Note that one requires "global" maximality, not just maximality with respect to being disjoint fromF. Yet, this variation yields another equivalent characterization of BPI:
The fact that this statement is equivalent to BPI is easily established by noting the following theorem: For anydistributive latticeL, if an idealIis maximal among all ideals ofLthat are disjoint to a given filterF, thenIis a prime ideal. The proof for this statement (which can again be carried out in ZF set theory) is included in the article on ideals. Since any Boolean algebra is a distributive lattice, this shows the desired implication.
All of the above statements are now easily seen to be equivalent. Going even further, one can exploit the fact the dual orders of Boolean algebras are exactly the Boolean algebras themselves. Hence, when taking the equivalent duals of all former statements, one ends up with a number of theorems that equally apply to Boolean algebras, but where every occurrence ofidealis replaced byfilter[citation needed]. It is worth noting that for the special case where the Boolean algebra under consideration is apowersetwith thesubsetordering, the "maximal filter theorem" is called the ultrafilter lemma.
Summing up, for Boolean algebras, the weak and strong MIT, the weak and strong PIT, and these statements with filters in place of ideals are all equivalent. It is known that all of these statements are consequences of theAxiom of Choice,AC, (the easy proof makes use ofZorn's lemma), but cannot be proven inZF(Zermelo-Fraenkel set theory withoutAC), if ZF isconsistent. Yet, the BPI is strictly weaker than the axiom of choice, though the proof of this statement, due to J. D. Halpern andAzriel Lévyis rather non-trivial.
The prototypical properties that were discussed for Boolean algebras in the above section can easily be modified to include more generallattices, such asdistributive latticesorHeyting algebras. However, in these cases maximal ideals are different from prime ideals, and the relation between PITs and MITs is not obvious.
Indeed, it turns out that the MITs for distributive lattices and even for Heyting algebras are equivalent to the axiom of choice. On the other hand, it is known that the strong PIT for distributive lattices is equivalent to BPI (i.e. to the MIT and PIT for Boolean algebras). Hence this statement is strictly weaker than the axiom of choice. Furthermore, observe that Heyting algebras are not self dual, and thus using filters in place of ideals yields different theorems in this setting. Maybe surprisingly, the MIT for the duals of Heyting algebras is not stronger than BPI, which is in sharp contrast to the abovementioned MIT for Heyting algebras.
Finally, prime ideal theorems do also exist for other (not order-theoretical) abstract algebras. For example, the MIT for rings implies the axiom of choice. This situation requires to replace the order-theoretic term "filter" by other concepts—for rings a "multiplicatively closed subset" is appropriate.
A filter on a setXis a nonempty collection of nonempty subsets ofXthat is closed under finite intersection and under superset. An ultrafilter is a maximal filter.
The ultrafilter lemma states that every filter on a setXis a subset of someultrafilteronX.[1]An ultrafilter that does not contain finite sets is called "non-principal". The ultrafilter lemma, and in particular the existence of non-principal ultrafilters (consider the filter of all sets with finite complements), can be proven fromZorn's lemma.
The ultrafilter lemma is equivalent to the Boolean prime ideal theorem, with the equivalence provable in ZF set theory without the axiom of choice. The idea behind the proof is that the subsets of any set form a Boolean algebra partially ordered by inclusion, and any Boolean algebra is representable as an algebra of sets byStone's representation theorem.
If the setXis finite then the ultrafilter lemma can be proven from the axioms ZF. This is no longer true for infinite sets; an additional axiommustbe assumed.Zorn's lemma, theaxiom of choice, andTychonoff's theoremcan all be used to prove the ultrafilter lemma. The ultrafilter lemma is strictly weaker than the axiom of choice.
The ultrafilter lemma has manyapplications in topology. The ultrafilter lemma can be used to prove theHahn-Banach theoremand theAlexander subbase theorem.
Intuitively, the Boolean prime ideal theorem states that there are "enough" prime ideals in a Boolean algebra in the sense that we can extendeveryideal to a maximal one. This is of practical importance for provingStone's representation theorem for Boolean algebras, a special case ofStone duality, in which one equips the set of all prime ideals with a certain topology and can indeed regain the original Boolean algebra (up toisomorphism) from this data. Furthermore, it turns out that in applications one can freely choose either to work with prime ideals or with prime filters, because every ideal uniquely determines a filter: the set of all Boolean complements of its elements. Both approaches are found in the literature.
Many other theorems of general topology that are often said to rely on the axiom of choice are in fact equivalent to BPI. For example, the theorem that a product of compactHausdorff spacesis compact is equivalent to it. If we leave out "Hausdorff" we get atheoremequivalent to the full axiom of choice.
Ingraph theory, thede Bruijn–Erdős theoremis another equivalent to BPI. It states that, if a given infinite graph requires at least some finite numberkin anygraph coloring, then it has a finite subgraph that also requiresk.[2]
A not too well known application of the Boolean prime ideal theorem is the existence of anon-measurable set[3](the example usually given is theVitali set, which requires the axiom of choice). From this and the fact that the BPI is strictly weaker than the axiom of choice, it follows that the existence of non-measurable sets is strictly weaker than the axiom of choice.
In linear algebra, the Boolean prime ideal theorem can be used to prove that any twobasesof a givenvector spacehave the samecardinality.
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Inmathematics,ideal theoryis the theory ofidealsincommutative rings. While the notion of an ideal exists also fornon-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.)
Throughout the articles, rings refer to commutative rings. See also the articleideal (ring theory)for basic operations such as sum or products of ideals.
Ideals in a finitely generated algebra over a field (that is, a quotient of apolynomial ringover a field) behave somehow nicer than those in a general commutative ring. First, in contrast to the general case, ifA{\displaystyle A}is a finitely generated algebra over a field, then theradical of an idealinA{\displaystyle A}is the intersection of all maximal ideals containing the ideal (becauseA{\displaystyle A}is aJacobson ring). This may be thought of as an extension ofHilbert's Nullstellensatz, which concerns the case whenA{\displaystyle A}is a polynomial ring.
IfIis an ideal in a ringA, then it determines the topology onAwhere a subsetUofAis open if, for eachxinU,
for some integern>0{\displaystyle n>0}. This topology is called theI-adic topology. It is also called ana-adic topology ifI=aA{\displaystyle I=aA}is generated by an elementa{\displaystyle a}.
For example, takeA=Z{\displaystyle A=\mathbb {Z} }, the ring of integers andI=pA{\displaystyle I=pA}an ideal generated by a prime numberp. For each integerx{\displaystyle x}, define|x|p=p−n{\displaystyle |x|_{p}=p^{-n}}whenx=pny{\displaystyle x=p^{n}y},y{\displaystyle y}prime top{\displaystyle p}. Then, clearly,
whereB(x,r)={z∈Z∣|z−x|p<r}{\displaystyle B(x,r)=\{z\in \mathbb {Z} \mid |z-x|_{p}<r\}}denotes an open ball of radiusr{\displaystyle r}with centerx{\displaystyle x}. Hence, thep{\displaystyle p}-adic topology onZ{\displaystyle \mathbb {Z} }is the same as themetric spacetopology given byd(x,y)=|x−y|p{\displaystyle d(x,y)=|x-y|_{p}}. As a metric space,Z{\displaystyle \mathbb {Z} }can becompleted. The resulting complete metric space has a structure of a ring that extended the ring structure ofZ{\displaystyle \mathbb {Z} }; this ring is denoted asZp{\displaystyle \mathbb {Z} _{p}}and is called thering ofp-adic integers.
In aDedekind domainA(e.g., a ring of integers in a number field or the coordinate ring of a smooth affine curve) with the field of fractionsK{\displaystyle K}, an idealI{\displaystyle I}is invertible in the sense: there exists afractional idealI−1{\displaystyle I^{-1}}(that is, anA-submodule ofK{\displaystyle K}) such thatII−1=A{\displaystyle I\,I^{-1}=A}, where the product on the left is a product of submodules ofK. In other words, fractional ideals form a group under a product. The quotient of the group of fractional ideals by the subgroup of principal ideals is then theideal class groupofA.
In a general ring, an ideal may not be invertible (in fact, already the definition of a fractional ideal is not clear). However, over a Noetherianintegral domain, it is still possible to develop some theory generalizing the situation in Dedekind domains. For example, Ch. VII of Bourbaki'sAlgèbre commutativegives such a theory.
The ideal class group ofA, when it can be defined, is closely related to thePicard groupof thespectrumofA(often the two are the same; e.g., for Dedekind domains).
Inalgebraic numbertheory, especially inclass field theory, it is more convenient to use a generalization of an ideal class group called anidele class group.
There are several operations on ideals that play roles of closures. The most basic one is theradical of an ideal. Another is theintegral closure of an ideal. Given an irredundant primary decompositionI=∩Qi{\displaystyle I=\cap Q_{i}}, the intersection ofQi{\displaystyle Q_{i}}'s whose radicals are minimal (don’t contain any of the radicals of otherQj{\displaystyle Q_{j}}'s) is uniquely determined byI{\displaystyle I}; this intersection is then called the unmixed part ofI{\displaystyle I}. It is also a closure operation.
Given idealsI,J{\displaystyle I,J}in a ringA{\displaystyle A}, the ideal
is called the saturation ofI{\displaystyle I}with respect toJ{\displaystyle J}and is a closure operation (this notion is closely related to the study of local cohomology).
See alsotight closure.
Local cohomology can sometimes be used to obtain information on an ideal. This section assumes some familiarity with sheaf theory and scheme theory.
LetM{\displaystyle M}be a module over a ringR{\displaystyle R}andI{\displaystyle I}an ideal. ThenM{\displaystyle M}determines the sheafM~{\displaystyle {\widetilde {M}}}onY=Spec(R)−V(I){\displaystyle Y=\operatorname {Spec} (R)-V(I)}(the restriction toYof the sheaf associated toM). Unwinding the definition, one sees:
Here,ΓI(M){\displaystyle \Gamma _{I}(M)}is called theideal transformofM{\displaystyle M}with respect toI{\displaystyle I}.[citation needed]
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Inmathematicalorder theory, anidealis a special subset of apartially ordered set(poset). Although this term historically was derived from the notion of aring idealofabstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order andlattice theory.
A subsetIof a partially ordered set(P,≤){\displaystyle (P,\leq )}is anideal, if the following conditions hold:[1][2]
While this is the most general way to define an ideal for arbitrary posets, it was originally defined forlatticesonly. In this case, the following equivalent definition can be given:
a subsetIof a lattice(P,≤){\displaystyle (P,\leq )}is an idealif and only ifit is a lower set that is closed under finitejoins(suprema); that is, it is nonempty and for allx,yinI, the elementx∨y{\displaystyle x\vee y}ofPis also inI.[3]
A weaker notion oforder idealis defined to be a subset of a posetPthat satisfies the above conditions 1 and 2. In other words, an order ideal is simply alower set. Similarly, an ideal can also be defined as a "directed lower set".
Thedualnotion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging∨{\displaystyle \vee }with∧,{\displaystyle \wedge ,}is afilter.
Frink ideals,pseudoidealsandDoyle pseudoidealsare different generalizations of the notion of a lattice ideal.
An ideal or filter is said to beproperif it is not equal to the whole setP.[3]
The smallest ideal that contains a given elementpis aprincipal idealandpis said to be aprincipal elementof the ideal in this situation. The principal ideal↓p{\displaystyle \downarrow p}for a principalpis thus given by↓p= {x∈P|x≤p}.
The above definitions of "ideal" and "order ideal" are the standard ones,[3][4][5]but there is some confusion in terminology. Sometimes the words and definitions such as "ideal", "order ideal", "Frink ideal", or "partial order ideal" mean one another.[6][7]
An important special case of an ideal is constituted by those ideals whose set-theoretic complements are filters, i.e. ideals in the inverse order. Such ideals are calledprime ideals. Also note that, since we require ideals and filters to be non-empty, every prime ideal is necessarily proper. For lattices, prime ideals can be characterized as follows:
A subsetIof a lattice(P,≤){\displaystyle (P,\leq )}is a prime ideal, if and only if
It is easily checked that this is indeed equivalent to stating thatP∖I{\displaystyle P\setminus I}is a filter (which is then also prime, in the dual sense).
For acomplete latticethe further notion of acompletely prime idealis meaningful.
It is defined to be a proper idealIwith the additional property that, whenever the meet (infimum) of some arbitrary setAis inI, some element ofAis also inI.
So this is just a specific prime ideal that extends the above conditions to infinite meets.
The existence of prime ideals is in general not obvious, and often a satisfactory amount of prime ideals cannot be derived within ZF (Zermelo–Fraenkel set theorywithout theaxiom of choice).
This issue is discussed in variousprime ideal theorems, which are necessary for many applications that require prime ideals.
An idealIis amaximal idealif it is proper and there is noproperidealJthat is a strict superset ofI. Likewise, a filterFis maximal if it is proper and there is no proper filter that is a strict superset.
When a poset is adistributive lattice, maximal ideals and filters are necessarily prime, while the converse of this statement is false in general.
Maximal filters are sometimes calledultrafilters, but this terminology is often reserved for Boolean algebras, where a maximal filter (ideal) is a filter (ideal) that contains exactly one of the elements {a, ¬a}, for each elementaof the Boolean algebra. In Boolean algebras, the termsprime idealandmaximal idealcoincide, as do the termsprime filterandmaximal filter.
There is another interesting notion of maximality of ideals: Consider an idealIand a filterFsuch thatIisdisjointfromF. We are interested in an idealMthat is maximal among all ideals that containIand are disjoint fromF. In the case of distributive lattices such anMis always a prime ideal. A proof of this statement follows.
Assume the idealMis maximal with respect to disjointness from the filterF. Suppose for a contradiction thatMis not prime, i.e. there exists a pair of elementsaandbsuch thata∧binMbut neitheranorbare inM. Consider the case that for allminM,m∨ais not inF. One can construct an idealNby taking the downward closure of the set of all binary joins of this form, i.e.N= {x|x≤m∨afor somem∈M}. It is readily checked thatNis indeed an ideal disjoint fromFwhich is strictly greater thanM. But this contradicts the maximality ofMand thus the assumption thatMis not prime.
For the other case, assume that there is someminMwithm∨ainF. Now if any elementninMis such thatn∨bis inF, one finds that(m∨n) ∨band(m∨n) ∨aare both inF. But then their meet is inFand, by distributivity,(m∨n) ∨ (a∧b)is inFtoo. On the other hand, this finite join of elements ofMis clearly inM, such that the assumed existence ofncontradicts the disjointness of the two sets. Hence all elementsnofMhave a join withbthat is not inF. Consequently one can apply the above construction withbin place ofato obtain an ideal that is strictly greater thanMwhile being disjoint fromF. This finishes the proof.
However, in general it is not clear whether there exists any idealMthat is maximal in this sense. Yet, if we assume theaxiom of choicein our set theory, then the existence ofMfor every disjoint filter–ideal-pair can be shown. In the special case that the considered order is aBoolean algebra, this theorem is called theBoolean prime ideal theorem. It is strictly weaker than the axiom of choice and it turns out that nothing more is needed for many order-theoretic applications of ideals.
The construction of ideals and filters is an important tool in many applications of order theory.
Ideals were introduced byMarshall H. Stonefirst forBoolean algebras,[8]where the name was derived from the ring ideals of abstract algebra. He adopted this terminology because, using theisomorphism of the categoriesofBoolean algebrasand ofBoolean rings, the two notions do indeed coincide.
Generalization to any posets was done byFrink.[9]
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Incommutative algebra, thenorm of an idealis a generalization of anormof an element in thefield extension. It is particularly important innumber theorysince it measures the size of anidealof a complicatednumber ringin terms of anidealin a less complicatedring. When the less complicated number ring is taken to be thering of integers,Z, then the norm of a nonzero idealIof a number ringRis simply the size of the finitequotient ringR/I.
LetAbe aDedekind domainwithfield of fractionsKandintegral closureofBin a finiteseparable extensionLofK. (this implies thatBis also a Dedekind domain.) LetIA{\displaystyle {\mathcal {I}}_{A}}andIB{\displaystyle {\mathcal {I}}_{B}}be theideal groupsofAandB, respectively (i.e., the sets of nonzerofractional ideals.) Following the technique developed byJean-Pierre Serre, thenorm map
is the uniquegroup homomorphismthat satisfies
for all nonzeroprime idealsq{\displaystyle {\mathfrak {q}}}ofB, wherep=q∩A{\displaystyle {\mathfrak {p}}={\mathfrak {q}}\cap A}is theprime idealofAlying belowq{\displaystyle {\mathfrak {q}}}.
Alternatively, for anyb∈IB{\displaystyle {\mathfrak {b}}\in {\mathcal {I}}_{B}}one can equivalently defineNB/A(b){\displaystyle N_{B/A}({\mathfrak {b}})}to be thefractional idealofAgenerated by the set{NL/K(x)|x∈b}{\displaystyle \{N_{L/K}(x)|x\in {\mathfrak {b}}\}}offield normsof elements ofB.[1]
Fora∈IA{\displaystyle {\mathfrak {a}}\in {\mathcal {I}}_{A}}, one hasNB/A(aB)=an{\displaystyle N_{B/A}({\mathfrak {a}}B)={\mathfrak {a}}^{n}}, wheren=[L:K]{\displaystyle n=[L:K]}.
The ideal norm of aprincipal idealis thus compatible with the field norm of an element:
LetL/K{\displaystyle L/K}be aGalois extensionofnumber fieldswithrings of integersOK⊂OL{\displaystyle {\mathcal {O}}_{K}\subset {\mathcal {O}}_{L}}.
Then the preceding applies withA=OK,B=OL{\displaystyle A={\mathcal {O}}_{K},B={\mathcal {O}}_{L}}, and for anyb∈IOL{\displaystyle {\mathfrak {b}}\in {\mathcal {I}}_{{\mathcal {O}}_{L}}}we have
which is an element ofIOK{\displaystyle {\mathcal {I}}_{{\mathcal {O}}_{K}}}.
The notationNOL/OK{\displaystyle N_{{\mathcal {O}}_{L}/{\mathcal {O}}_{K}}}is sometimes shortened toNL/K{\displaystyle N_{L/K}}, anabuse of notationthat is compatible with also writingNL/K{\displaystyle N_{L/K}}for the field norm, as noted above.
In the caseK=Q{\displaystyle K=\mathbb {Q} }, it is reasonable to use positiverational numbersas the range forNOL/Z{\displaystyle N_{{\mathcal {O}}_{L}/\mathbb {Z} }\,}sinceZ{\displaystyle \mathbb {Z} }has trivialideal class groupandunit group{±1}{\displaystyle \{\pm 1\}}, thus each nonzerofractional idealofZ{\displaystyle \mathbb {Z} }is generated by a uniquely determined positiverational number.
Under this convention the relative norm fromL{\displaystyle L}down toK=Q{\displaystyle K=\mathbb {Q} }coincides with theabsolute normdefined below.
LetL{\displaystyle L}be anumber fieldwithring of integersOL{\displaystyle {\mathcal {O}}_{L}}, anda{\displaystyle {\mathfrak {a}}}a nonzero (integral)idealofOL{\displaystyle {\mathcal {O}}_{L}}.
The absolute norm ofa{\displaystyle {\mathfrak {a}}}is
By convention, the norm of the zero ideal is taken to be zero.
Ifa=(a){\displaystyle {\mathfrak {a}}=(a)}is aprincipal ideal, then
The norm iscompletely multiplicative: ifa{\displaystyle {\mathfrak {a}}}andb{\displaystyle {\mathfrak {b}}}are ideals ofOL{\displaystyle {\mathcal {O}}_{L}}, then
Thus the absolute norm extends uniquely to agroup homomorphism
defined for all nonzerofractional idealsofOL{\displaystyle {\mathcal {O}}_{L}}.
The norm of anideala{\displaystyle {\mathfrak {a}}}can be used to give an upper bound on the field norm of the smallest nonzero element it contains:
there always exists a nonzeroa∈a{\displaystyle a\in {\mathfrak {a}}}for which
where
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Inmathematics, the interplay between theGalois groupGof aGalois extensionLof anumber fieldK, and the way theprime idealsPof thering of integersOKfactorise as products of prime ideals ofOL, provides one of the richest parts ofalgebraic number theory. Thesplitting of prime ideals in Galois extensionsis sometimes attributed toDavid Hilbertby calling itHilbert theory. There is a geometric analogue, forramified coveringsofRiemann surfaces, which is simpler in that only one kind of subgroup ofGneed be considered, rather than two. This was certainly familiar before Hilbert.
LetL/Kbe a finite extension of number fields, and letOKandOLbe the correspondingring of integersofKandL, respectively, which are defined to be theintegral closureof the integersZin the field in question.
Finally, letpbe a non-zero prime ideal inOK, or equivalently, amaximal ideal, so that the residueOK/pis afield.
From the basic theory of one-dimensionalrings follows the existence of a unique decomposition
of the idealpOLgenerated inOLbypinto a product of distinct maximal idealsPj, with multiplicitiesej.
The fieldF=OK/pnaturally embeds intoFj=OL/Pjfor everyj, the degreefj= [OL/Pj:OK/p] of thisresidue field extensionis calledinertia degreeofPjoverp.
The multiplicityejis calledramification indexofPjoverp. If it is bigger than 1 for somej, the field extensionL/Kis calledramifiedatp(or we say thatpramifies inL, or that it is ramified inL). Otherwise,L/Kis calledunramifiedatp. If this is the case then by theChinese remainder theoremthe quotientOL/pOLis a product of fieldsFj. The extensionL/Kis ramified in exactly those primes that divide therelative discriminant, hence the extension is unramified in all but finitely many prime ideals.
Multiplicativity ofideal normimplies
Iffj=ej= 1 for everyj(and thusg= [L:K]), we say thatpsplits completelyinL. Ifg= 1 andf1= 1 (and soe1= [L:K]), we say thatpramifies completelyinL. Finally, ifg= 1 ande1= 1 (and sof1= [L:K]), we say thatpisinertinL.
In the following, the extensionL/Kis assumed to be aGalois extension. Then theprime avoidance lemmacan be used to show theGalois groupG=Gal(L/K){\displaystyle G=\operatorname {Gal} (L/K)}acts transitivelyon thePj. That is, the prime ideal factors ofpinLform a singleorbitunder theautomorphismsofLoverK. From this and theunique factorisation theorem, it follows thatf=fjande=ejare independent ofj; something that certainly need not be the case for extensions that are not Galois. The basic relations then read
and
The relation above shows that [L:K]/efequals the numbergof prime factors ofpinOL. By theorbit-stabilizer formulathis number is also equal to|G|/|DPj|for everyj, whereDPj, thedecomposition groupofPj, is the subgroup of elements ofGsending a givenPjto itself. Since the degree ofL/Kand the order ofGare equal by basic Galois theory, it follows that the order of the decomposition groupDPjiseffor everyj.
This decomposition group contains a subgroupIPj, calledinertia groupofPj, consisting of automorphisms ofL/Kthat induce the identity automorphism onFj. In other words,IPjis the kernel of reduction mapDPj→Gal(Fj/F){\displaystyle D_{P_{j}}\to \operatorname {Gal} (F_{j}/F)}. It can be shown that this map is surjective, and it follows thatGal(Fj/F){\displaystyle \operatorname {Gal} (F_{j}/F)}is isomorphic toDPj/IPjand the order of the inertia groupIPjise.
The theory of theFrobenius elementgoes further, to identify an element ofDPj/IPjfor givenjwhich corresponds to the Frobenius automorphism in the Galois group of the finite field extensionFj/F. In the unramified case the order ofDPjisfandIPjis trivial, so the Frobenius element is in this case an element ofDPj, and thus also an element ofG. For varyingj, the groupsDPjare conjugate subgroups insideG: Recalling thatGacts transitively on thePj, one checks that ifσmapsPjtoPj′,σDPjσ−1=DPj′{\displaystyle \sigma D_{P_{j}}\sigma ^{-1}=D_{P_{j'}}}. Therefore, ifGis an abelian group, the Frobenius element of an unramified primePdoes not depend on whichPjwe take. Furthermore, in the abelian case, associating an unramified prime ofKto its Frobenius and extending multiplicatively defines a homomorphism from the group of unramified ideals ofKintoG. This map, known as theArtin map, is a crucial ingredient ofclass field theory, which studies the finite abelian extensions of a given number fieldK.[1]
In the geometric analogue, forcomplex manifoldsoralgebraic geometryover analgebraically closed field, the concepts ofdecomposition groupandinertia groupcoincide. There, given a Galois ramified cover, all but finitely many points have the same number ofpreimages.
The splitting of primes in extensions that are not Galois may be studied by using asplitting fieldinitially, i.e. a Galois extension that is somewhat larger. For example,cubic fieldsusually are 'regulated' by a degree 6 field containing them.
This section describes the splitting of prime ideals in the field extensionQ(i)/Q. That is, we takeK=QandL=Q(i), soOKis simplyZ, andOL=Z[i] is the ring ofGaussian integers. Although this case is far from representative — after all,Z[i] hasunique factorisation, andthere aren't many imaginary quadratic fields with unique factorization— it exhibits many of the features of the theory.
WritingGfor the Galois group ofQ(i)/Q, and σ for the complex conjugation automorphism inG, there are three cases to consider.
The prime 2 ofZramifies inZ[i]:
The ramification index here is thereforee= 2. The residue field is
which is the finite field with two elements. The decomposition group must be equal to all ofG, since there is only one prime ofZ[i] above 2. The inertia group is also all ofG, since
for any integersaandb, asa+bi=2bi+a−bi=(1+i)⋅(1−i)bi+a−bi≡a−bimod1+i{\displaystyle a+bi=2bi+a-bi=(1+i)\cdot (1-i)bi+a-bi\equiv a-bi{\bmod {1}}+i}.
In fact, 2 is theonlyprime that ramifies inZ[i], since every prime that ramifies must divide thediscriminantofZ[i], which is −4.
Any primep≡ 1 mod 4splitsinto two distinct prime ideals inZ[i]; this is a manifestation ofFermat's theorem on sums of two squares. For example:
The decomposition groups in this case are both the trivial group {1}; indeed the automorphism σswitchesthe two primes (2 + 3i) and (2 − 3i), so it cannot be in the decomposition group of either prime. The inertia group, being a subgroup of the decomposition group, is also the trivial group. There are two residue fields, one for each prime,
which are both isomorphic to the finite field with 13 elements. The Frobenius element is the trivial automorphism; this means that
for any integersaandb.
Any primep≡ 3 mod 4 remainsinertinZ[i]; that is, it doesnotsplit. For example, (7) remains prime inZ[i]. In this situation, the decomposition group is all ofG, again because there is only one prime factor. However, this situation differs from thep= 2 case, because now σ doesnotact trivially on the residue field
which is the finite field with 72= 49 elements. For example, the difference between1+i{\displaystyle 1+i}andσ(1+i)=1−i{\displaystyle \sigma (1+i)=1-i}is2i{\displaystyle 2i}, which is certainly not divisible by 7. Therefore, the inertia group is the trivial group {1}. The Galois group of this residue field over the subfieldZ/7Zhas order 2, and is generated by the image of the Frobenius element. The Frobenius element is none other than σ; this means that
for any integersaandb.
Suppose that we wish to determine the factorisation of a prime idealPofOKinto primes ofOL. The following procedure (Neukirch, p. 47) solves this problem in many cases. The strategy is to select an integer θ inOLso thatLis generated overKby θ (such a θ is guaranteed to exist by theprimitive element theorem), and then to examine theminimal polynomialH(X) of θ overK; it is a monic polynomial with coefficients inOK. Reducing the coefficients ofH(X) moduloP, we obtain a monic polynomialh(X) with coefficients inF, the (finite) residue fieldOK/P. Suppose thath(X) factorises in the polynomial ringF[X] as
where thehjare distinct monic irreducible polynomials inF[X]. Then, as long asPis not one of finitely many exceptional primes (the precise condition is described below), the factorisation ofPhas the following form:
where theQjare distinct prime ideals ofOL. Furthermore, the inertia degree of eachQjis equal to the degree of the corresponding polynomialhj, and there is an explicit formula for theQj:
wherehjdenotes here a lifting of the polynomialhjtoK[X].
In the Galois case, the inertia degrees are all equal, and the ramification indicese1= ... =enare all equal.
The exceptional primes, for which the above result does not necessarily hold, are the ones not relatively prime to theconductorof the ringOK[θ]. The conductor is defined to be the ideal
it measures how far theorderOK[θ] is from being the whole ring of integers (maximal order)OL.
A significant caveat is that there exist examples ofL/KandPsuch that there isnoavailable θ that satisfies the above hypotheses (see for example[2]). Therefore, the algorithm given above cannot be used to factor suchP, and more sophisticated approaches must be used, such as that described in.[3]
Consider again the case of the Gaussian integers. We take θ to be the imaginary uniti{\displaystyle i}, with minimal polynomialH(X) =X2+ 1. SinceZ[i{\displaystyle i}] is the whole ring of integers ofQ(i{\displaystyle i}), theconductoris the unit ideal, so there are no exceptional primes.
ForP= (2), we need to work in the fieldZ/(2)Z, which amounts to factorising the polynomialX2+ 1 modulo 2:
Therefore, there is only one prime factor, with inertia degree 1 and ramification index 2, and it is given by
The next case is forP= (p) for a primep≡ 3 mod 4. For concreteness we will takeP= (7). The polynomialX2+ 1 is irreducible modulo 7. Therefore, there is only one prime factor, with inertia degree 2 and ramification index 1, and it is given by
The last case isP= (p) for a primep≡ 1 mod 4; we will again takeP= (13). This time we have the factorisation
Therefore, there aretwoprime factors, both with inertia degree and ramification index 1. They are given by
and
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Inalgebraic geometryand other areas ofmathematics, anideal sheaf(orsheaf of ideals) is the global analogue of anidealin aring. The ideal sheaves on a geometric object are closely connected to its subspaces.
LetXbe atopological spaceandAasheafof rings onX. (In other words,(X,A)is aringed space.) An ideal sheafJinAis asubobjectofAin thecategoryof sheaves ofA-modules, i.e., asubsheafofAviewed as a sheaf of abelian groups such that
for all open subsetsUofX. In other words,Jis asheaf ofA-submodules ofA.
In the context ofschemes, the importance of ideal sheaves lies mainly in the correspondence between closedsubschemesandquasi-coherentideal sheaves. Consider a schemeXand a quasi-coherent ideal sheafJin OX. Then, the supportZof OX/Jis a closed subspace ofX, and(Z, OX/J)is a scheme (both assertions can be checked locally). It is called the closed subscheme ofXdefined byJ. Conversely, leti:Z→Xbe aclosed immersion, i.e., a morphism which is ahomeomorphismonto a closed subspace such that the associated map
is surjective on the stalks. Then, the kernelJofi#is a quasi-coherent ideal sheaf, andiinduces an isomorphism fromZonto the closed subscheme defined byJ.[1]
A particular case of this correspondence is the uniquereducedsubschemeXredofXhaving the same underlying space, which is defined by thenilradicalof OX(defined stalk-wise, or on open affine charts).[2]
For a morphismf:X→Yand a closed subschemeY′⊆Ydefined by an ideal sheafJ, the preimageY′×YXis defined by the ideal sheaf[3]
The pull-back of an ideal sheafJto the subschemeZdefined byJcontains important information, it is called theconormal bundleofZ. For example, the sheaf ofKähler differentialsmay be defined as the pull-back of the ideal sheaf defining the diagonalX→X×XtoX. (Assume for simplicity thatXisseparatedso that the diagonal is a closed immersion.)[4]
In the theory ofcomplex-analytic spaces, theOka-Cartan theoremstates that a closed subsetAof a complex space is analytic if and only if the ideal sheaf of functions vanishing onAiscoherent. This ideal sheaf also givesAthe structure of a reduced closed complex subspace.
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Ingroup theory, a subfield ofabstract algebra, acycle graphof agroupis anundirected graphthat illustrates the variouscyclesof that group, given a set ofgeneratorsfor the group. Cycle graphs are particularly useful in visualizing the structure of smallfinite groups.
A cycle is thesetof powers of a given group elementa, wherean, then-th power of an elementa, is defined as the product ofamultiplied by itselfntimes. The elementais said togeneratethe cycle. In a finite group, some non-zero power ofamust be thegroup identity, which we denote either aseor 1; the lowest such power is theorderof the elementa, the number of distinct elements in the cycle that it generates. In a cycle graph, the cycle is represented as a polygon, with its vertices representing the group elements and its edges indicating how they are linked together to form the cycle.
Each group element is represented by anodein the cycle graph, and enough cycles are represented as polygons in the graph so that every node lies on at least one cycle. All of those polygons pass through the node representing the identity, and some other nodes may also lie on more than one cycle.
Suppose that a group elementagenerates a cycle of order 6 (has order 6), so that the nodesa,a2,a3,a4,a5, anda6=eare the vertices of a hexagon in the cycle graph. The elementa2then has order 3; but making the nodesa2,a4, andebe the vertices of a triangle in the graph would add no new information. So, only theprimitivecycles need be considered, those that are notsubsetsof another cycle. Also, the nodea5, which also has order 6, generates the same cycle as doesaitself; so we have at least two choices for which element to use in generating a cycle --- often more.
To build a cycle graph for a group, we start with a node for each group element. For each primitive cycle, we then choose some elementathat generates that cycle, and we connect the node foreto the one fora,atoa2, ...,ak−1toak, etc., until returning toe. The result is a cycle graph for the group.
When a group elementahas order 2 (so that multiplication byais aninvolution), the rule above would connectetoaby two edges, one going out and the other coming back. Except when the intent is to emphasize the two edges of such a cycle, it is typically drawn[1]as a single line between the two elements.
Note that this correspondence between groups and graphs is not one-to-one in either direction: Two different groups can have the same cycle graph, and two different graphs can be cycle graphs for a single group. We give examples of each in thenon-uniqueness section.
As an example of a group cycle graph, consider thedihedral groupDih4. The multiplication table for this group is shown on the left, and the cycle graph is shown on the right, withespecifying the identity element.
Notice the cycle {e,a,a2,a3} in the multiplication table, witha4=e. The inversea−1=a3is also a generator of this cycle: (a3)2=a2,(a3)3=a, and(a3)4=e. Similarly, any cycle in any group has at least two generators, and may be traversed in either direction. More generally, the number ofgeneratorsof a cycle withnelements is given by theEuler φ functionofn, and any of these generators may be written as the first node in the cycle (next to the identitye); or more commonly the nodes are left unmarked. Two distinct cycles cannot intersect in a generator.
Cycles that contain a non-prime number of elements have cyclic subgroups that are not shown in the graph. For the group Dih4above, we could draw a line betweena2andesince(a2)2=e, but sincea2is part of a larger cycle, this is not an edge of the cycle graph.
There can be ambiguity when two cycles share a non-identity element. For example, the 8-elementquaternion grouphas cycle graph shown at right. Each of the elements in the middle row when multiplied by itself gives −1 (where 1 is the identity element). In this case we may use different colors to keep track of the cycles, although symmetry considerations will work as well.
As noted earlier, the two edges of a 2-element cycle are typically represented as a single line.
The inverse of an element is the node symmetric to it in its cycle, with respect to the reflection which fixes the identity.
The cycle graph of a group is not uniquely determined up tograph isomorphism; nor does it uniquely determine the group up togroup isomorphism. That is, the graph obtained depends on the set of generators chosen, and two different groups (with chosen sets of generators) can generate the same cycle graph.[2]
For some groups, choosing different elements to generate the various primitive cycles of that group can lead to different cycle graphs. There is an example of this for the abelian groupC5×C2×C2{\displaystyle C_{5}\times C_{2}\times C_{2}}, which has order 20.[2]We denote an element of that group as a triple of numbers(i;j,k){\displaystyle (i;j,k)}, where0≤i<5{\displaystyle 0\leq i<5}and each ofj{\displaystyle j}andk{\displaystyle k}is either 0 or 1. The triple(0;0,0){\displaystyle (0;0,0)}is the identity element. In the drawings below,i{\displaystyle i}is shown abovej{\displaystyle j}andk{\displaystyle k}.
This group has three primitive cycles, each of order 10. In the first cycle graph, we choose, as the generators of those three cycles, the nodes(1;1,0){\displaystyle (1;1,0)},(1;0,1){\displaystyle (1;0,1)}, and(1;1,1){\displaystyle (1;1,1)}. In the second, we generate the third of those cycles --- the blue one --- by starting instead with(2;1,1){\displaystyle (2;1,1)}.
The two resulting graphs are not isomorphic because they havediameters5 and 4 respectively.
Two different (non-isomorphic) groups can have cycle graphs that areisomorphic, where the latter isomorphism ignores the labels on the nodes of the graphs. It follows that the structure of a group is not uniquely determined by its cycle graph.
There is an example of this already for groups of order 16, the two groups beingC8×C2{\displaystyle C_{8}\times C_{2}}andC8⋊5C2{\displaystyle C_{8}\rtimes _{5}C_{2}}. TheabeliangroupC8×C2{\displaystyle C_{8}\times C_{2}}is the direct product of the cyclic groups of orders 8 and 2. The non-abelian groupC8⋊5C2{\displaystyle C_{8}\rtimes _{5}C_{2}}is thatsemidirect productofC8{\displaystyle C_{8}}andC2{\displaystyle C_{2}}in which the non-identity element ofC2{\displaystyle C_{2}}maps to the multiply-by-5automorphismofC8{\displaystyle C_{8}}.
In drawing the cycle graphs of those two groups, we takeC8×C2{\displaystyle C_{8}\times C_{2}}to be generated by elementssandtwith
where that latter relation makesC8×C2{\displaystyle C_{8}\times C_{2}}abelian. And we takeC8⋊5C2{\displaystyle C_{8}\rtimes _{5}C_{2}}to be generated by elements 𝜎 and 𝜏 with
Here are cycle graphs for those two groups, where we choosest{\displaystyle st}to generate the green cycle on the left andστ{\displaystyle \sigma \tau }to generate that cycle on the right:
In the right-hand graph, the green cycle, after moving from 1 toστ{\displaystyle \sigma \tau }, moves next toσ6,{\displaystyle \sigma ^{6},}because
Cycle graphs were investigated by the number theoristDaniel Shanksin the early 1950s as a tool to studymultiplicative groups of residue classes.[3]Shanks first published the idea in the 1962 first edition of his bookSolved and Unsolved Problems in Number Theory.[4]In the book, Shanks investigates which groups have isomorphic cycle graphs and when a cycle graph isplanar.[5]In the 1978 second edition, Shanks reflects on his research onclass groupsand the development of thebaby-step giant-stepmethod:[6]
The cycle graphs have proved to be useful when working with finite Abelian groups; and I have used them frequently in finding my way around an intricate structure [77, p. 852], in obtaining a wanted multiplicative relation [78, p. 426], or in isolating some wanted subgroup [79].
Cycle graphs are used as a pedagogical tool in Nathan Carter's 2009 introductory textbookVisual Group Theory.[7]
Certain group types give typical graphs:
Cyclic groupsZn, ordern, is a single cycle graphed simply as ann-sided polygon with the elements at the vertices:
Whennis aprime number, groups of the form (Zn)mwill have(nm− 1)/(n− 1)n-element cycles sharing the identity element:
Dihedral groupsDihn, order 2nconsists of ann-element cycle andn2-element cycles:
Dicyclic groups, Dicn= Q4n, order 4n:
Otherdirect products:
Symmetric groups– The symmetric group Sncontains, for any group of ordern, a subgroup isomorphic to that group. Thus the cycle graph of every group of ordernwill be found in the cycle graph of Sn.See example:Subgroups of S4
Thefull octahedral groupis thedirect productS4×Z2{\displaystyle S_{4}\times Z_{2}}of the symmetric group S4and the cyclic group Z2.Its order is 48, and it has subgroups of every order that divides 48.
In the examples below nodes that are related to each other are placed next to each other,so these are not the simplest possible cycle graphs for these groups (like those on the right).
Like all graphs a cycle graph can be represented in different ways to emphasize different properties. The two representations of the cycle graph of S4are an example of that.
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https://en.wikipedia.org/wiki/Cycle_graph_(group)
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Inmathematics, more specifically inring theory, acyclic moduleormonogenous module[1]is amodule over a ringthat is generated by one element. The concept is a generalization of the notion of acyclic group, that is, anAbelian group(i.e.Z-module) that is generated by one element.
A leftR-moduleMis calledcyclicifMcan be generated by a single element i.e.M= (x) =Rx= {rx|r∈R}for somexinM. Similarly, a rightR-moduleNis cyclic ifN=yRfor somey∈N.
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https://en.wikipedia.org/wiki/Cyclic_module
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Incombinatorial mathematics,cyclic sievingis a phenomenon in which an integerpolynomialevaluated at certainroots of unitycounts the rotational symmetries of afinite set.[1]Given a family of cyclic sieving phenomena, the polynomials give aq-analoguefor the enumeration of the sets, and often arise from an underlying algebraic structure, such as arepresentation.
The first study of cyclic sieving was published by Reiner, Stanton and White in 2004.[2]The phenomenon generalizes the "q= −1 phenomenon" ofJohn Stembridge, which considers evaluations of the polynomial only at the first and second roots of unity (that is,q= 1 andq= −1).[3]
For every positive integern{\displaystyle n}, letωn{\displaystyle \omega _{n}}denote the primitiven{\displaystyle n}throot of unitye2πi/n{\displaystyle e^{2\pi i/n}}.
LetX{\displaystyle X}be a finite set with anactionof the cyclic groupCn{\displaystyle C_{n}}, and letf(q){\displaystyle f(q)}be anintegerpolynomial. The triple(X,Cn,f(q)){\displaystyle (X,C_{n},f(q))}exhibits thecyclic sieving phenomenon(orCSP) if for every positive integerd{\displaystyle d}dividingn{\displaystyle n}, the number of elements inX{\displaystyle X}fixed by the action of the subgroupCd{\displaystyle C_{d}}ofCn{\displaystyle C_{n}}is equal tof(ωd){\displaystyle f(\omega _{d})}. IfCn{\displaystyle C_{n}}acts as rotation by2π/n{\displaystyle 2\pi /n}, this counts elements inX{\displaystyle X}withd{\displaystyle d}-foldrotational symmetry.
Equivalently, supposeσ:X→X{\displaystyle \sigma :X\to X}is abijectiononX{\displaystyle X}such thatσn=id{\displaystyle \sigma ^{n}={\rm {id}}}, whereid{\displaystyle {\rm {id}}}is the identity map. Thenσ{\displaystyle \sigma }induces an action ofCn{\displaystyle C_{n}}onX{\displaystyle X}, where a givengeneratorc{\displaystyle c}ofCn{\displaystyle C_{n}}acts byσ{\displaystyle \sigma }. Then(X,Cn,f(q)){\displaystyle (X,C_{n},f(q))}exhibits the cyclic sieving phenomenon if the number of elements inX{\displaystyle X}fixed byσd{\displaystyle \sigma ^{d}}is equal tof(ωnd){\displaystyle f(\omega _{n}^{d})}for every integerd{\displaystyle d}.
LetX{\displaystyle X}be the 2-element subsets of{1,2,3,4}{\displaystyle \{1,2,3,4\}}. Define a bijectionσ:X→X{\displaystyle \sigma :X\to X}which increases each element in the pair by one (and sends4{\displaystyle 4}back to1{\displaystyle 1}). This induces an action ofC4{\displaystyle C_{4}}onX{\displaystyle X}, which has anorbit{1,3}↦{2,4}↦{1,3}{\displaystyle \{1,3\}\mapsto \{2,4\}\mapsto \{1,3\}}of size two and an orbit{1,2}↦{2,3}↦{3,4}↦{1,4}↦{1,2}{\displaystyle \{1,2\}\mapsto \{2,3\}\mapsto \{3,4\}\mapsto \{1,4\}\mapsto \{1,2\}}of size four. Iff(q)=1+q+2q2+q3+q4{\displaystyle f(q)=1+q+2q^{2}+q^{3}+q^{4}}, thenf(1)=6{\displaystyle f(1)=6}is the number of elements inX{\displaystyle X},f(i)=0{\displaystyle f(i)=0}counts fixed points ofσ{\displaystyle \sigma },f(−1)=2{\displaystyle f(-1)=2}is the number of fixed points ofσ2{\displaystyle \sigma ^{2}}, andf(i)=0{\displaystyle f(i)=0}is the number of fixed points ofσ{\displaystyle \sigma }. Hence, the triple(X,C4,f(q)){\displaystyle (X,C_{4},f(q))}exhibits the cyclic sieving phenomenon.
More generally, set[n]q:=1+q+⋯+qn−1{\displaystyle [n]_{q}:=1+q+\cdots +q^{n-1}}and define theq-binomial coefficientby[nk]q=[n]q⋯[2]q[1]q[k]q⋯[2]q[1]q[n−k]q⋯[2]q[1]q.{\displaystyle \left[{n \atop k}\right]_{q}={\frac {[n]_{q}\cdots [2]_{q}[1]_{q}}{[k]_{q}\cdots [2]_{q}[1]_{q}[n-k]_{q}\cdots [2]_{q}[1]_{q}}}.}which is an integer polynomial evaluating to the usualbinomial coefficientatq=1{\displaystyle q=1}. For any positive integerd{\displaystyle d}dividingn{\displaystyle n},
IfXn,k{\displaystyle X_{n,k}}is the set of size-k{\displaystyle k}subsets of{1,…,n}{\displaystyle \{1,\dots ,n\}}withCn{\displaystyle C_{n}}acting by increasing each element in the subset by one (and sendingn{\displaystyle n}back to1{\displaystyle 1}), and iffn,k(q){\displaystyle f_{n,k}(q)}is theq-binomial coefficient above, then(Xn,k,Cn,fn,k(q)){\displaystyle (X_{n,k},C_{n},f_{n,k}(q))}exhibits the cyclic sieving phenomenon for every0≤k≤n{\displaystyle 0\leq k\leq n}.[4]
The cyclic sieving phenomenon can be naturally stated in the language of representation theory. The group action ofCn{\displaystyle C_{n}}onX{\displaystyle X}is linearly extended to obtain a representation, and the decomposition of this representation into irreducibles determines the required coefficients of the polynomialf(q){\displaystyle f(q)}.[5]
LetV=C(X){\displaystyle V=\mathbb {C} (X)}be thevector spaceover thecomplex numberswith abasisindexed by a finite setX{\displaystyle X}. If the cyclic groupCn{\displaystyle C_{n}}acts onX{\displaystyle X}, then linearly extending each action turnsV{\displaystyle V}into a representation ofCn{\displaystyle C_{n}}.
For a generatorc{\displaystyle c}ofCn{\displaystyle C_{n}}, the linear extension of its action onX{\displaystyle X}gives apermutation matrix[c]{\displaystyle [c]}, and thetraceof[c]d{\displaystyle [c]^{d}}counts the elements ofX{\displaystyle X}fixed bycd{\displaystyle c^{d}}. In particular, the triple(X,Cn,f(q)){\displaystyle (X,C_{n},f(q))}exhibits the cyclic sieving phenomenon if and only iff(ωnd)=χ(cd){\displaystyle f(\omega _{n}^{d})=\chi (c^{d})}for every0≤d<n{\displaystyle 0\leq d<n}, whereχ{\displaystyle \chi }is thecharacterofV{\displaystyle V}.
This gives a method for determiningf(q){\displaystyle f(q)}. For every integerk{\displaystyle k}, letV(k){\displaystyle V^{(k)}}be the one-dimensional representation ofCn{\displaystyle C_{n}}in whichc{\displaystyle c}acts asscalar multiplicationbyωnk{\displaystyle \omega _{n}^{k}}. For an integer polynomialf(q)=∑k≥0mkqk{\textstyle f(q)=\sum _{k\geq 0}m_{k}q^{k}}, the triple(X,Cn,f(q)){\displaystyle (X,C_{n},f(q))}exhibits the cyclic sieving phenomenon if and only ifV≅⨁k≥0mkV(k).{\displaystyle V\cong \bigoplus _{k\geq 0}m_{k}V^{(k)}.}
LetW{\displaystyle W}be a finite set ofwordsof the formw=w1⋯wn{\displaystyle w=w_{1}\cdots w_{n}}where each letterwj{\displaystyle w_{j}}is an integer andW{\displaystyle W}is closed under permutation (that is, ifw{\displaystyle w}is inW{\displaystyle W}, then so is anyanagramofw{\displaystyle w}). Themajor indexof a wordw{\displaystyle w}is the sum of all indicesj{\displaystyle j}such thatwj>wj+1{\displaystyle w_{j}>w_{j+1}}, and is denotedmaj(w){\displaystyle {\rm {maj}}(w)}.
IfCn{\displaystyle C_{n}}acts onW{\displaystyle W}by rotating the letters of each word, andf(q)=∑w∈Wqmaj(w){\displaystyle f(q)=\sum _{w\in W}q^{{\rm {maj}}(w)}}then(W,Cn,f(q)){\displaystyle (W,C_{n},f(q))}exhibits the cyclic sieving phenomenon.[6]
Letλ{\displaystyle \lambda }be apartitionof sizen{\displaystyle n}with rectangular shape, and letXλ{\displaystyle X_{\lambda }}be the set ofstandard Young tableauxwith shapeλ{\displaystyle \lambda }.Jeu de taquinpromotion gives an action ofCn{\displaystyle C_{n}}onX{\displaystyle X}. Letf(q){\displaystyle f(q)}be the followingq-analog of thehook length formula:fλ(q)=[n]q⋯[1]q∏(i,j)∈λ[h(i,j)]q.{\displaystyle f_{\lambda }(q)={\frac {[n]_{q}\cdots [1]_{q}}{\prod _{(i,j)\in \lambda }[h(i,j)]_{q}}}.}Then(Xλ,Cn,fλ(q)){\displaystyle (X_{\lambda },C_{n},f_{\lambda }(q))}exhibits the cyclic sieving phenomenon. Ifχλ{\displaystyle \chi _{\lambda }}is the character for theirreducible representation of the symmetric groupassociated toλ{\displaystyle \lambda }, thenfλ(ωnd)=±χλ(cd){\displaystyle f_{\lambda }(\omega _{n}^{d})=\pm \chi _{\lambda }(c^{d})}for every0≤d<n{\displaystyle 0\leq d<n}, wherec{\displaystyle c}is thelong cycle(12⋯n){\displaystyle (12\cdots n)}.[7]
IfY{\displaystyle Y}is the set of semistandard Young tableaux of shapeλ{\displaystyle \lambda }with entries in{1,…,k}{\displaystyle \{1,\dots ,k\}}, then promotion gives an action of the cyclic groupCk{\displaystyle C_{k}}onYλ{\displaystyle Y_{\lambda }}. Defineκ(λ)=∑i(i−1)λi{\textstyle \kappa (\lambda )=\sum _{i}(i-1)\lambda _{i}}andg(q)=q−κ(λ)sλ(1,q,…,qk−1),{\displaystyle g(q)=q^{-\kappa (\lambda )}s_{\lambda }(1,q,\dots ,q^{k-1}),}wheresλ{\displaystyle s_{\lambda }}is theSchur polynomial. Then(Y,Ck,g(q)){\displaystyle (Y,C_{k},g(q))}exhibits the cyclic sieving phenomenon.[8]
IfX{\displaystyle X}is the set of non-crossing (1,2)-configurations of{1,…,n−1}{\displaystyle \{1,\dots ,n-1\}}, thenCn−1{\displaystyle C_{n-1}}acts on these by rotation. Letf(q){\displaystyle f(q)}be the followingq-analog of then{\displaystyle n}thCatalan number:f(q)=1[n+1]q[2nn]q.{\displaystyle f(q)={\frac {1}{[n+1]_{q}}}\left[{2n \atop n}\right]_{q}.}Then(X,Cn−1,f(q)){\displaystyle (X,C_{n-1},f(q))}exhibits the cyclic sieving phenomenon.[9]
LetX{\displaystyle X}be the set of semi-standard Young tableaux of shape(n,n){\displaystyle (n,n)}with maximal entry2n−k{\displaystyle 2n-k}, where entries along each row and column are strictly increasing. IfC2n−k{\displaystyle C_{2n-k}}acts onX{\displaystyle X}byK{\displaystyle K}-promotion andf(q)=qn+(k2)[n−1k]q[2n−kn−k−1]q[n−k]q,{\displaystyle f(q)=q^{n+{\binom {k}{2}}}{\frac {\left[{n-1 \atop k}\right]_{q}\left[{2n-k \atop n-k-1}\right]_{q}}{[n-k]_{q}}},}then(X,C2n−k,f(q)){\displaystyle (X,C_{2n-k},f(q))}exhibits the cyclic sieving phenomenon.[10]
LetSλ,j{\displaystyle S_{\lambda ,j}}be the set ofpermutationsof cycle typeλ{\displaystyle \lambda }with exactlyj{\displaystyle j}exceedances. Conjugation gives an action ofCn{\displaystyle C_{n}}onSλ,j{\displaystyle S_{\lambda ,j}}, and ifaλ,j(q)=∑σ∈Sλ,jqmaj(σ)−j{\displaystyle a_{\lambda ,j}(q)=\sum _{\sigma \in S_{\lambda ,j}}q^{\operatorname {maj} (\sigma )-j}}then(Sλ,j,Cn,aλ,j(q)){\displaystyle (S_{\lambda ,j},C_{n},a_{\lambda ,j}(q))}exhibits the cyclic sieving phenomenon.[11]
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https://en.wikipedia.org/wiki/Cyclic_sieving
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Ring homomorphisms
Algebraic structures
Related structures
Algebraic number theory
Noncommutative algebraic geometry
Free algebra
Clifford algebra
In mathematics, specifically ingroup theory, thePrüferp-groupor thep-quasicyclic grouporp∞-group,Z(p∞), for aprime numberpis the uniquep-groupin which every element haspdifferentp-th roots.
The Prüferp-groups arecountableabelian groupsthat are important in the classification of infinite abelian groups: they (along with the group ofrational numbers) form the smallest building blocks of alldivisible groups.
The groups are named afterHeinz Prüfer, a German mathematician of the early 20th century.
The Prüferp-group may be identified with the subgroup of thecircle group, U(1), consisting of allpn-throots of unityasnranges over all non-negative integers:
The group operation here is the multiplication ofcomplex numbers.
There is apresentation
Here, the group operation inZ(p∞) is written as multiplication.
Alternatively and equivalently, the Prüferp-group may be defined as theSylowp-subgroupof thequotient groupQ/Z, consisting of those elements whose order is a power ofp:
(whereZ[1/p] denotes the group of all rational numbers whose denominator is a power ofp, using addition of rational numbers as group operation).
For each natural numbern, consider thequotient groupZ/pnZand the embeddingZ/pnZ→Z/pn+1Zinduced by multiplication byp. Thedirect limitof this system isZ(p∞):
If we perform the direct limit in thecategoryoftopological groups, then we need to impose a topology on each of theZ/pnZ{\displaystyle \mathbf {Z} /p^{n}\mathbf {Z} }, and take thefinal topologyonZ(p∞){\displaystyle \mathbf {Z} (p^{\infty })}. If we wish forZ(p∞){\displaystyle \mathbf {Z} (p^{\infty })}to beHausdorff, we must impose thediscrete topologyon each of theZ/pnZ{\displaystyle \mathbf {Z} /p^{n}\mathbf {Z} }, resulting inZ(p∞){\displaystyle \mathbf {Z} (p^{\infty })}to have the discrete topology.
We can also write
whereQpdenotes the additive group ofp-adic numbersandZpis the subgroup ofp-adic integers.
The complete list of subgroups of the Prüferp-groupZ(p∞) =Z[1/p]/Zis:
Here, each(1pnZ)/Z{\displaystyle \left({1 \over p^{n}}\mathbf {Z} \right)/\mathbf {Z} }is a cyclic subgroup ofZ(p∞) withpnelements; it contains precisely those elements ofZ(p∞) whoseorderdividespnand corresponds to the set ofpn-th roots of unity.
The Prüferp-groups are the only infinite groups whose subgroups aretotally orderedby inclusion. This sequence of inclusions expresses the Prüferp-group as thedirect limitof its finite subgroups. As there is nomaximal subgroupof a Prüferp-group, it is its ownFrattini subgroup.
Given this list of subgroups, it is clear that the Prüferp-groups areindecomposable(cannot be written as adirect sumof proper subgroups). More is true: the Prüferp-groups aresubdirectly irreducible. An abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclicp-group or to a Prüfer group.
The Prüferp-group is the unique infinitep-groupthat islocally cyclic(every finite set of elements generates a cyclic group). As seen above, all proper subgroups ofZ(p∞) are finite. The Prüferp-groups are the only infinite abelian groups with this property.[1]
The Prüferp-groups aredivisible. They play an important role in the classification of divisible groups; along with the rational numbers they are the simplest divisible groups. More precisely: an abelian group is divisible if and only if it is thedirect sumof a (possibly infinite) number of copies ofQand (possibly infinite) numbers of copies ofZ(p∞) for every primep. The (cardinal) numbers of copies ofQandZ(p∞) that are used in this direct sum determine the divisible group up to isomorphism.[2]
As an abelian group (that is, as aZ-module),Z(p∞) isArtinianbut notNoetherian.[3]It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas everyArtinianringis Noetherian).
Theendomorphism ringofZ(p∞) is isomorphic to the ring ofp-adic integersZp.[4]
In the theory oflocally compact topological groupsthe Prüferp-group (endowed with thediscrete topology) is thePontryagin dualof the compact group ofp-adic integers, and the group ofp-adic integers is the Pontryagin dual of the Prüferp-group.[5]
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https://en.wikipedia.org/wiki/Pr%C3%BCfer_group
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Inmathematics, thecircle group, denoted byT{\displaystyle \mathbb {T} }orS1{\displaystyle \mathbb {S} ^{1}}, is themultiplicative groupof allcomplex numberswithabsolute value1, that is, theunit circlein thecomplex planeor simply theunit complex numbers[1]T={z∈C:|z|=1}.{\displaystyle \mathbb {T} =\{z\in \mathbb {C} :|z|=1\}.}
The circle group forms asubgroupofC×{\displaystyle \mathbb {C} ^{\times }}, the multiplicative group of all nonzero complex numbers. SinceC×{\displaystyle \mathbb {C} ^{\times }}isabelian, it follows thatT{\displaystyle \mathbb {T} }is as well.
A unit complex number in the circle group represents arotationof the complex plane about the origin and can be parametrized by theangle measureθ{\displaystyle \theta }:θ↦z=eiθ=cosθ+isinθ.{\displaystyle \theta \mapsto z=e^{i\theta }=\cos \theta +i\sin \theta .}
This is theexponential mapfor the circle group.
The circle group plays a central role inPontryagin dualityand in the theory ofLie groups.
The notationT{\displaystyle \mathbb {T} }for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally,Tn{\displaystyle \mathbb {T} ^{n}}(thedirect productofT{\displaystyle \mathbb {T} }with itselfn{\displaystyle n}times) is geometrically ann{\displaystyle n}-torus.
The circle group isisomorphicto thespecial orthogonal groupSO(2){\displaystyle \mathrm {SO} (2)}.
One way to think about the circle group is that it describes how to addangles, where only angles between 0° and 360° or∈[0,2π){\displaystyle \in [0,2\pi )}or∈(−π,+π]{\displaystyle \in (-\pi ,+\pi ]}are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer is150° + 270° = 420°, but when thinking in terms of the circle group, we may "forget" the fact that we have wrapped once around the circle. Therefore, we adjust our answer by 360°, which gives420° ≡ 60° (mod360°).
Another description is in terms of ordinary (real) addition, where only numbers between 0 and 1 are allowed (with 1 corresponding to a full rotation: 360° or2π{\displaystyle 2\pi }), i.e. the real numbers modulo the integers:T≅R/Z{\displaystyle \mathbb {T} \cong \mathbb {R} /\mathbb {Z} }. This can be achieved by throwing away the digits occurring before the decimal point. For example, when we work out0.4166... + 0.75, the answer is 1.1666..., but we may throw away the leading 1, so the answer (in the circle group) is just0.16¯≡1.16¯≡−0.83¯(modZ){\displaystyle 0.1{\bar {6}}\equiv 1.1{\bar {6}}\equiv -0.8{\bar {3}}\;({\text{mod}}\,\mathbb {Z} )}, with some preference to 0.166..., because0.16¯∈[0,1){\displaystyle 0.1{\bar {6}}\in [0,1)}.
The circle group is more than just an abstract algebraic object. It has anatural topologywhen regarded as asubspaceof the complex plane. Since multiplication and inversion arecontinuous functionsonC×{\displaystyle \mathbb {C} ^{\times }}, the circle group has the structure of atopological group. Moreover, since the unit circle is aclosed subsetof the complex plane, the circle group is a closed subgroup ofC×{\displaystyle \mathbb {C} ^{\times }}(itself regarded as a topological group).
One can say even more. The circle is a 1-dimensional realmanifold, and multiplication and inversion arereal-analytic mapson the circle. This gives the circle group the structure of aone-parameter group, an instance of aLie group. In fact,up toisomorphism, it is the unique 1-dimensionalcompact,connectedLie group. Moreover, everyn{\displaystyle n}-dimensional compact, connected, abelian Lie group is isomorphic toTn{\displaystyle \mathbb {T} ^{n}}.
The circle group shows up in a variety of forms in mathematics. We list some of the more common forms here. Specifically, we show thatT≅U(1)≅R/Z≅SO(2),{\displaystyle \mathbb {T} \cong {\mbox{U}}(1)\cong \mathbb {R} /\mathbb {Z} \cong \mathrm {SO} (2),}where the slash (/{\displaystyle ~\!/~\!}) denotesgroup quotientand≅{\displaystyle \cong }the existence of anisomorphismbetween the groups.
The set of all1×1{\displaystyle 1\times 1}unitary matricescoincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1. Therefore, the circle group is canonically isomorphic to the firstunitary groupU(1){\displaystyle \mathrm {U} (1)}, i.e.,T≅U(1).{\displaystyle \mathbb {T} \cong {\mbox{U}}(1).}Theexponential functiongives rise to a mapexp:R→T{\displaystyle \exp :\mathbb {R} \to \mathbb {T} }from the additive real numbersR{\displaystyle \mathbb {R} }to the circle groupT{\displaystyle \mathbb {T} }known asEuler's formulaθ↦eiθ=cosθ+isinθ,{\displaystyle \theta \mapsto e^{i\theta }=\cos \theta +i\sin \theta ,}whereθ∈R{\displaystyle \theta \in \mathbb {R} }corresponds to the angle (inradians) on the unit circle as measured counterclockwise from the positivex-axis. The propertyeiθ1eiθ2=ei(θ1+θ2),∀θ1,θ2∈R,{\displaystyle e^{i\theta _{1}}e^{i\theta _{2}}=e^{i(\theta _{1}+\theta _{2})},\quad \forall \theta _{1},\theta _{2}\in \mathbb {R} ,}makesexp:R→T{\displaystyle \exp :\mathbb {R} \to \mathbb {T} }agroup homomorphism. While the map issurjective, it is notinjectiveand therefore not an isomorphism. Thekernelof this map is the set of allintegermultiples of2π{\displaystyle 2\pi }. By thefirst isomorphism theoremwe then have thatT≅R/2πZ.{\displaystyle \mathbb {T} \cong \mathbb {R} ~\!/~\!2\pi \mathbb {Z} .}After rescaling we can also say thatT{\displaystyle \mathbb {T} }is isomorphic toR/Z{\displaystyle \mathbb {R} /\mathbb {Z} }.
The unitcomplex numberscan be realized as 2×2 realorthogonal matrices, i.e.,eiθ=cosθ+isinθ↔[cosθ−sinθsinθcosθ]=f(eiθ),{\displaystyle e^{i\theta }=\cos \theta +i\sin \theta \leftrightarrow {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}}=f{\bigl (}e^{i\theta }{\bigr )},}associating thesquared modulusandcomplex conjugatewith thedeterminantandtranspose, respectively, of the corresponding matrix. As theangle sum trigonometric identitiesimply thatf(eiθ1eiθ2)=[cos(θ1+θ2)−sin(θ1+θ2)sin(θ1+θ2)cos(θ1+θ2)]=f(eiθ1)×f(eiθ2),{\displaystyle f{\bigl (}e^{i\theta _{1}}e^{i\theta _{2}}{\bigr )}={\begin{bmatrix}\cos(\theta _{1}+\theta _{2})&-\sin(\theta _{1}+\theta _{2})\\\sin(\theta _{1}+\theta _{2})&\cos(\theta _{1}+\theta _{2})\end{bmatrix}}=f{\bigl (}e^{i\theta _{1}}{\bigr )}\times f{\bigl (}e^{i\theta _{2}}{\bigr )},}where×{\displaystyle \times }is matrix multiplication, the circle group isisomorphicto thespecial orthogonal groupSO(2){\displaystyle \mathrm {SO} (2)}, i.e.,T≅SO(2).{\displaystyle \mathbb {T} \cong \mathrm {SO} (2).}This isomorphism has the geometric interpretation that multiplication by a unit complex number is a proper rotation in the complex (and real) plane, and every such rotation is of this form.
Every compact Lie groupG{\displaystyle \mathrm {G} }of dimension > 0 has asubgroupisomorphic to the circle group. This means that, thinking in terms ofsymmetry, a compact symmetry group actingcontinuouslycan be expected to have one-parameter circle subgroups acting; the consequences in physical systems are seen, for example, atrotational invarianceandspontaneous symmetry breaking.
The circle group has manysubgroups, but its only properclosedsubgroups consist ofroots of unity: For each integern>0{\displaystyle n>0}, then{\displaystyle n}th roots of unity form acyclic groupof ordern{\displaystyle n}, which is unique up to isomorphism.
In the same way that thereal numbersare acompletionof theb-adic rationalsZ[1b]{\displaystyle \mathbb {Z} {\bigl [}{\tfrac {1}{b}}{\bigr ]}}for everynatural numberb>1{\displaystyle b>1}, the circle group is the completion of thePrüfer groupZ[1b]/Z{\displaystyle \mathbb {Z} {\bigl [}{\tfrac {1}{b}}{\bigr ]}~\!/~\!\mathbb {Z} }forb{\displaystyle b}, given by thedirect limitlim→Z/bnZ{\displaystyle \varinjlim \mathbb {Z} ~\!/~\!b^{n}\mathbb {Z} }.
Therepresentationsof the circle group are easy to describe. It follows fromSchur's lemmathat theirreduciblecomplexrepresentations of an abelian group are all 1-dimensional. Since the circle group is compact, any representationρ:T→GL(1,C)≅C×{\displaystyle \rho :\mathbb {T} \to \mathrm {GL} (1,\mathbb {C} )\cong \mathbb {C} ^{\times }}must take values inU(1)≅T{\displaystyle {\mbox{U}}(1)\cong \mathbb {T} }. Therefore, the irreducible representations of the circle group are just thehomomorphismsfrom the circle group to itself.
For each integern{\displaystyle n}we can define a representationϕn{\displaystyle \phi _{n}}of the circle group byϕn(z)=zn{\displaystyle \phi _{n}(z)=z^{n}}. These representations are all inequivalent. The representationϕ−n{\displaystyle \phi _{-n}}isconjugatetoϕn{\displaystyle \phi _{n}}:ϕ−n=ϕn¯.{\displaystyle \phi _{-n}={\overline {\phi _{n}}}.}
These representations are just thecharactersof the circle group. Thecharacter groupofT{\displaystyle \mathbb {T} }is clearly aninfinite cyclic groupgenerated byϕ1{\displaystyle \phi _{1}}:Hom(T,T)≅Z.{\displaystyle \operatorname {Hom} (\mathbb {T} ,\mathbb {T} )\cong \mathbb {Z} .}
The irreduciblerealrepresentations of the circle group are thetrivial representation(which is 1-dimensional) and the representationsρn(eiθ)=[cosnθ−sinnθsinnθcosnθ],n∈Z+,{\displaystyle \rho _{n}{\bigl (}e^{i\theta }{\bigr )}={\begin{bmatrix}\cos n\theta &-\sin n\theta \\\sin n\theta &\cos n\theta \end{bmatrix}},\quad n\in \mathbb {Z} ^{+},}taking values inSO(2){\displaystyle \mathrm {SO} (2)}. Here we only have positive integersn{\displaystyle n}, since the representationρ−n{\displaystyle \rho _{-n}}is equivalent toρn{\displaystyle \rho _{n}}.
The circle groupT{\displaystyle \mathbb {T} }is adivisible group. Itstorsion subgroupis given by the set of alln{\displaystyle n}-throots of unityfor alln{\displaystyle n}and is isomorphic toQ/Z{\displaystyle \mathbb {Q} /\mathbb {Z} }. Thestructure theoremfor divisible groups and theaxiom of choicetogether tell us thatT{\displaystyle \mathbb {T} }is isomorphic to thedirect sumofQ/Z{\displaystyle \mathbb {Q} /\mathbb {Z} }with a number of copies ofQ{\displaystyle \mathbb {Q} }.[2]
The number of copies ofQ{\displaystyle \mathbb {Q} }must bec{\displaystyle {\mathfrak {c}}}(thecardinality of the continuum) in order for the cardinality of the direct sum to be correct. But the direct sum ofc{\displaystyle {\mathfrak {c}}}copies ofQ{\displaystyle \mathbb {Q} }is isomorphic toR{\displaystyle \mathbb {R} }, asR{\displaystyle \mathbb {R} }is avector spaceof dimensionc{\displaystyle {\mathfrak {c}}}overQ{\displaystyle \mathbb {Q} }. Thus,T≅R⊕(Q/Z).{\displaystyle \mathbb {T} \cong \mathbb {R} \oplus (\mathbb {Q} /\mathbb {Z} ).}
The isomorphismC×≅R⊕(Q/Z){\displaystyle \mathbb {C} ^{\times }\cong \mathbb {R} \oplus (\mathbb {Q} /\mathbb {Z} )}can be proved in the same way, sinceC×{\displaystyle \mathbb {C} ^{\times }}is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup ofT{\displaystyle \mathbb {T} }.
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https://en.wikipedia.org/wiki/Circle_group
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Intopology, aclopen set(aportmanteauofclosed-open set) in atopological spaceis a set which is bothopenandclosed. That this is possible may seem counterintuitive, as the common meanings ofopenandclosedare antonyms, but their mathematical definitions are notmutually exclusive. A set is closed if itscomplementis open, which leaves the possibility of an open set whose complement is also open, making both sets both openandclosed, and therefore clopen. As described by topologistJames Munkres, unlike adoor, "a set can be open, or closed, or both, or neither!"[1]emphasizing that the meaning of "open"/"closed" fordoorsis unrelated to their meaning forsets(and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "door spaces" their name.
In any topological spaceX,{\displaystyle X,}theempty setand the whole spaceX{\displaystyle X}are both clopen.[2][3]
Now consider the spaceX{\displaystyle X}which consists of theunionof the two openintervals(0,1){\displaystyle (0,1)}and(2,3){\displaystyle (2,3)}ofR.{\displaystyle \mathbb {R} .}ThetopologyonX{\displaystyle X}is inherited as thesubspace topologyfrom the ordinary topology on thereal lineR.{\displaystyle \mathbb {R} .}InX,{\displaystyle X,}the set(0,1){\displaystyle (0,1)}is clopen, as is the set(2,3).{\displaystyle (2,3).}This is a quite typical example: whenever a space is made up of a finite number ofdisjointconnected componentsin this way, the components will be clopen.
Now letX{\displaystyle X}be an infinite set under thediscrete metric– that is, two pointsp,q∈X{\displaystyle p,q\in X}have distance 1 if they're not the same point, and 0 otherwise. Under the resultingmetric space, anysingleton setis open; hence any set, being the union of single points, is open. Since any set is open, the complement of any set is open too, and therefore any set is closed. So, all sets in this metric space are clopen.
As a less trivial example, consider the spaceQ{\displaystyle \mathbb {Q} }of allrational numberswith their ordinary topology, and the setA{\displaystyle A}of all positive rational numbers whosesquareis bigger than 2. Using the fact that2{\displaystyle {\sqrt {2}}}is not inQ,{\displaystyle \mathbb {Q} ,}one can show quite easily thatA{\displaystyle A}is a clopen subset ofQ.{\displaystyle \mathbb {Q} .}(A{\displaystyle A}isnota clopen subset of the real lineR{\displaystyle \mathbb {R} }; it is neither open nor closed inR.{\displaystyle \mathbb {R} .})
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https://en.wikipedia.org/wiki/Clopen_set
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Inmathematics, more specifically intopology, anopen mapis afunctionbetween twotopological spacesthat mapsopen setsto open sets.[1][2][3]That is, a functionf:X→Y{\displaystyle f:X\to Y}is open if for any open setU{\displaystyle U}inX,{\displaystyle X,}theimagef(U){\displaystyle f(U)}is open inY.{\displaystyle Y.}Likewise, aclosed mapis a function that mapsclosed setsto closed sets.[3][4]A map may be open, closed, both, or neither;[5]in particular, an open map need not be closed and vice versa.[6]
Open[7]and closed[8]maps are not necessarilycontinuous.[4]Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property;[3]this fact remains true even if one restricts oneself to metric spaces.[9]Although their definitions seem more natural, open and closed maps are much less important than continuous maps.
Recall that, by definition, a functionf:X→Y{\displaystyle f:X\to Y}is continuous if thepreimageof every open set ofY{\displaystyle Y}is open inX.{\displaystyle X.}[2](Equivalently, if the preimage of every closed set ofY{\displaystyle Y}is closed inX{\displaystyle X}).
Early study of open maps was pioneered bySimion StoilowandGordon Thomas Whyburn.[10]
IfS{\displaystyle S}is a subset of a topological space then letS¯{\displaystyle {\overline {S}}}andClS{\displaystyle \operatorname {Cl} S}(resp.IntS{\displaystyle \operatorname {Int} S}) denote theclosure(resp.interior) ofS{\displaystyle S}in that space.
Letf:X→Y{\displaystyle f:X\to Y}be a function betweentopological spaces. IfS{\displaystyle S}is any set thenf(S):={f(s):s∈S∩domainf}{\displaystyle f(S):=\left\{f(s)~:~s\in S\cap \operatorname {domain} f\right\}}is called the image ofS{\displaystyle S}underf.{\displaystyle f.}
There are two different competing, but closely related, definitions of "open map" that are widely used, where both of these definitions can be summarized as: "it is a map that sends open sets to open sets."
The following terminology is sometimes used to distinguish between the two definitions.
A mapf:X→Y{\displaystyle f:X\to Y}is called a
Every strongly open map is a relatively open map. However, these definitions are not equivalent in general.
Asurjectivemap is relatively open if and only if it is strongly open; so for this important special case the definitions are equivalent.
More generally, a mapf:X→Y{\displaystyle f:X\to Y}is relatively open if and only if thesurjectionf:X→f(X){\displaystyle f:X\to f(X)}is a strongly open map.
BecauseX{\displaystyle X}is always an open subset ofX,{\displaystyle X,}the imagef(X)=Imf{\displaystyle f(X)=\operatorname {Im} f}of a strongly open mapf:X→Y{\displaystyle f:X\to Y}must be an open subset of its codomainY.{\displaystyle Y.}In fact, a relatively open map is a strongly open map if and only if its image is an open subset of its codomain.
In summary,
By using this characterization, it is often straightforward to apply results involving one of these two definitions of "open map" to a situation involving the other definition.
The discussion above will also apply to closed maps if each instance of the word "open" is replaced with the word "closed".
A mapf:X→Y{\displaystyle f:X\to Y}is called anopen mapor astrongly open mapif it satisfies any of the following equivalent conditions:
IfB{\displaystyle {\mathcal {B}}}is abasisforX{\displaystyle X}then the following can be appended to this list:
A mapf:X→Y{\displaystyle f:X\to Y}is called arelatively closed mapif wheneverC{\displaystyle C}is aclosed subsetof the domainX{\displaystyle X}thenf(C){\displaystyle f(C)}is a closed subset off{\displaystyle f}'simageImf:=f(X),{\displaystyle \operatorname {Im} f:=f(X),}where as usual, this set is endowed with thesubspace topologyinduced on it byf{\displaystyle f}'scodomainY.{\displaystyle Y.}
A mapf:X→Y{\displaystyle f:X\to Y}is called aclosed mapor astrongly closed mapif it satisfies any of the following equivalent conditions:
Asurjectivemap is strongly closed if and only if it is relatively closed. So for this important special case, the two definitions are equivalent.
By definition, the mapf:X→Y{\displaystyle f:X\to Y}is a relatively closed map if and only if thesurjectionf:X→Imf{\displaystyle f:X\to \operatorname {Im} f}is a strongly closed map.
If in the open set definition of "continuous map" (which is the statement: "every preimage of an open set is open"), both instances of the word "open" are replaced with "closed" then the statement of results ("every preimage of a closed set is closed") isequivalentto continuity.
This does not happen with the definition of "open map" (which is: "every image of an open set is open") since the statement that results ("every image of a closed set is closed") is the definition of "closed map", which is in generalnotequivalent to openness. There exist open maps that are not closed and there also exist closed maps that are not open. This difference between open/closed maps and continuous maps is ultimately due to the fact that for any setS,{\displaystyle S,}onlyf(X∖S)⊇f(X)∖f(S){\displaystyle f(X\setminus S)\supseteq f(X)\setminus f(S)}is guaranteed in general, whereas for preimages, equalityf−1(Y∖S)=f−1(Y)∖f−1(S){\displaystyle f^{-1}(Y\setminus S)=f^{-1}(Y)\setminus f^{-1}(S)}always holds.
The functionf:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} }defined byf(x)=x2{\displaystyle f(x)=x^{2}}is continuous, closed, and relatively open, but not (strongly) open. This is because ifU=(a,b){\displaystyle U=(a,b)}is any open interval inf{\displaystyle f}'s domainR{\displaystyle \mathbb {R} }that doesnotcontain0{\displaystyle 0}thenf(U)=(min{a2,b2},max{a2,b2}),{\displaystyle f(U)=(\min\{a^{2},b^{2}\},\max\{a^{2},b^{2}\}),}where this open interval is an open subset of bothR{\displaystyle \mathbb {R} }andImf:=f(R)=[0,∞).{\displaystyle \operatorname {Im} f:=f(\mathbb {R} )=[0,\infty ).}However, ifU=(a,b){\displaystyle U=(a,b)}is any open interval inR{\displaystyle \mathbb {R} }that contains0{\displaystyle 0}thenf(U)=[0,max{a2,b2}),{\displaystyle f(U)=[0,\max\{a^{2},b^{2}\}),}which is not an open subset off{\displaystyle f}'s codomainR{\displaystyle \mathbb {R} }butisan open subset ofImf=[0,∞).{\displaystyle \operatorname {Im} f=[0,\infty ).}Because the set of all open intervals inR{\displaystyle \mathbb {R} }is abasisfor theEuclidean topologyonR,{\displaystyle \mathbb {R} ,}this shows thatf:R→R{\displaystyle f:\mathbb {R} \to \mathbb {R} }is relatively open but not (strongly) open.
IfY{\displaystyle Y}has thediscrete topology(that is, all subsets are open and closed) then every functionf:X→Y{\displaystyle f:X\to Y}is both open and closed (but not necessarily continuous).
For example, thefloor functionfromR{\displaystyle \mathbb {R} }toZ{\displaystyle \mathbb {Z} }is open and closed, but not continuous.
This example shows that the image of aconnected spaceunder an open or closed map need not be connected.
Whenever we have aproductof topological spacesX=∏Xi,{\textstyle X=\prod X_{i},}the natural projectionspi:X→Xi{\displaystyle p_{i}:X\to X_{i}}are open[12][13](as well as continuous).
Since the projections offiber bundlesandcovering mapsare locally natural projections of products, these are also open maps.
Projections need not be closed however. Consider for instance the projectionp1:R2→R{\displaystyle p_{1}:\mathbb {R} ^{2}\to \mathbb {R} }on the first component; then the setA={(x,1/x):x≠0}{\displaystyle A=\{(x,1/x):x\neq 0\}}is closed inR2,{\displaystyle \mathbb {R} ^{2},}butp1(A)=R∖{0}{\displaystyle p_{1}(A)=\mathbb {R} \setminus \{0\}}is not closed inR.{\displaystyle \mathbb {R} .}However, for a compact spaceY,{\displaystyle Y,}the projectionX×Y→X{\displaystyle X\times Y\to X}is closed. This is essentially thetube lemma.
To every point on theunit circlewe can associate theangleof the positivex{\displaystyle x}-axis with the ray connecting the point with the origin. This function from the unit circle to the half-openinterval[0,2π) is bijective, open, and closed, but not continuous.
It shows that the image of acompact spaceunder an open or closed map need not be compact.
Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed. Specifying thecodomainis essential.
Everyhomeomorphismis open, closed, and continuous. In fact, abijectivecontinuous map is a homeomorphismif and only ifit is open, or equivalently, if and only if it is closed.
Thecompositionof two (strongly) open maps is an open map and the composition of two (strongly) closed maps is a closed map.[14][15]However, the composition of two relatively open maps need not be relatively open and similarly, the composition of two relatively closed maps need not be relatively closed.
Iff:X→Y{\displaystyle f:X\to Y}is strongly open (respectively, strongly closed) andg:Y→Z{\displaystyle g:Y\to Z}is relatively open (respectively, relatively closed) theng∘f:X→Z{\displaystyle g\circ f:X\to Z}is relatively open (respectively, relatively closed).
Letf:X→Y{\displaystyle f:X\to Y}be a map.
Given any subsetT⊆Y,{\displaystyle T\subseteq Y,}iff:X→Y{\displaystyle f:X\to Y}is a relatively open (respectively, relatively closed, strongly open, strongly closed, continuous,surjective) map then the same is true of its restrictionf|f−1(T):f−1(T)→T{\displaystyle f{\big \vert }_{f^{-1}(T)}~:~f^{-1}(T)\to T}to thef{\displaystyle f}-saturatedsubsetf−1(T).{\displaystyle f^{-1}(T).}
The categorical sum of two open maps is open, or of two closed maps is closed.[15]The categoricalproductof two open maps is open, however, the categorical product of two closed maps need not be closed.[14][15]
A bijective map is open if and only if it is closed.
The inverse of a bijective continuous map is a bijective open/closed map (and vice versa).
A surjective open map is not necessarily a closed map, and likewise, a surjective closed map is not necessarily an open map. Alllocal homeomorphisms, including allcoordinate chartsonmanifoldsand allcovering maps, are open maps.
Closed map lemma—Every continuous functionf:X→Y{\displaystyle f:X\to Y}from acompact spaceX{\displaystyle X}to aHausdorff spaceY{\displaystyle Y}is closed andproper(meaning that preimages of compact sets are compact).
A variant of the closed map lemma states that if a continuous function betweenlocally compactHausdorff spaces is proper then it is also closed.
Incomplex analysis, the identically namedopen mapping theoremstates that every non-constantholomorphic functiondefined on aconnectedopen subset of thecomplex planeis an open map.
Theinvariance of domaintheorem states that a continuous and locally injective function between twon{\displaystyle n}-dimensionaltopological manifoldsmust be open.
Invariance of domain—IfU{\displaystyle U}is anopen subsetofRn{\displaystyle \mathbb {R} ^{n}}andf:U→Rn{\displaystyle f:U\to \mathbb {R} ^{n}}is aninjectivecontinuous map, thenV:=f(U){\displaystyle V:=f(U)}is open inRn{\displaystyle \mathbb {R} ^{n}}andf{\displaystyle f}is ahomeomorphismbetweenU{\displaystyle U}andV.{\displaystyle V.}
Infunctional analysis, theopen mapping theoremstates that every surjective continuouslinear operatorbetweenBanach spacesis an open map.
This theorem has been generalized totopological vector spacesbeyond just Banach spaces.
A surjective mapf:X→Y{\displaystyle f:X\to Y}is called analmost open mapif for everyy∈Y{\displaystyle y\in Y}there exists somex∈f−1(y){\displaystyle x\in f^{-1}(y)}such thatx{\displaystyle x}is apoint of opennessforf,{\displaystyle f,}which by definition means that for every open neighborhoodU{\displaystyle U}ofx,{\displaystyle x,}f(U){\displaystyle f(U)}is aneighborhoodoff(x){\displaystyle f(x)}inY{\displaystyle Y}(note that the neighborhoodf(U){\displaystyle f(U)}is not required to be anopenneighborhood).
Every surjective open map is an almost open map but in general, the converse is not necessarily true.
If a surjectionf:(X,τ)→(Y,σ){\displaystyle f:(X,\tau )\to (Y,\sigma )}is an almost open map then it will be an open map if it satisfies the following condition (a condition that doesnotdepend in any way onY{\displaystyle Y}'s topologyσ{\displaystyle \sigma }):
If the map is continuous then the above condition is also necessary for the map to be open. That is, iff:X→Y{\displaystyle f:X\to Y}is a continuous surjection then it is an open map if and only if it is almost open and it satisfies the above condition.
Iff:X→Y{\displaystyle f:X\to Y}is a continuous map that is also openorclosed then:
In the first two cases, being open or closed is merely asufficient conditionfor the conclusion that follows.
In the third case, it isnecessaryas well.
Iff:X→Y{\displaystyle f:X\to Y}is a continuous (strongly) open map,A⊆X,{\displaystyle A\subseteq X,}andS⊆Y,{\displaystyle S\subseteq Y,}then:
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https://en.wikipedia.org/wiki/Closed_map
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Inmathematical analysis, adomainorregionis anon-empty,connected, andopen setin atopological space. In particular, it is any non-empty connected opensubsetof thereal coordinate spaceRnor thecomplex coordinate spaceCn. A connected open subset ofcoordinate spaceis frequently used for thedomain of a function.[1]
The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the termdomain,[2]some use the termregion,[3]some use both terms interchangeably,[4]and some define the two terms slightly differently;[5]some avoid ambiguity by sticking with a phrase such asnon-empty connected open subset.[6]
One common convention is to define adomainas a connected open set but aregionas theunionof a domain with none, some, or all of itslimit points.[7]Aclosed regionorclosed domainis the union of a domain and all of its limit points.
Various degrees of smoothness of theboundaryof the domain are required for various properties of functions defined on the domain to hold, such as integral theorems (Green's theorem,Stokes theorem), properties ofSobolev spaces, and to definemeasureson the boundary and spaces oftraces(generalized functions defined on the boundary). Commonly considered types of domains are domains withcontinuousboundary,Lipschitz boundary,C1boundary, and so forth.
Abounded domainis a domain that isbounded, i.e., contained in some ball.Bounded regionis defined similarly. Anexterior domainorexternal domainis a domain whosecomplementis bounded; sometimes smoothness conditions are imposed on its boundary.
Incomplex analysis, acomplex domain(or simplydomain) is any connected open subset of thecomplex planeC. For example, the entire complex plane is a domain, as is the openunit disk, the openupper half-plane, and so forth. Often, a complex domain serves as thedomain of definitionfor aholomorphic function. In the study ofseveral complex variables, the definition of a domain is extended to include any connected open subset ofCn.
InEuclidean spaces,one-,two-, andthree-dimensionalregions arecurves,surfaces, andsolids, whose extent are called, respectively,length,area, andvolume.
Definition. An open set is connected if it cannot be expressed as the sum of two open sets. An open connected set is called a domain.
German:Eine offene Punktmenge heißt zusammenhängend, wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann. Eine offene zusammenhängende Punktmenge heißt ein Gebiet.
According toHans Hahn,[8]the concept of a domain as an open connected set was introduced byConstantin Carathéodoryin his famous book (Carathéodory 1918).
In this definition, Carathéodory considers obviouslynon-emptydisjointsets.
Hahn also remarks that the word "Gebiet" ("Domain") was occasionally previously used as asynonymofopen set.[9]The rough concept is older. In the 19th and early 20th century, the termsdomainandregionwere often used informally (sometimes interchangeably) without explicit definition.[10]
However, the term "domain" was occasionally used to identify closely related but slightly different concepts. For example, in his influentialmonographsonelliptic partial differential equations,Carlo Mirandauses the term "region" to identify an open connected set,[11][12]and reserves the term "domain" to identify an internally connected,[13]perfect set, each point of which is an accumulation point of interior points,[11]following his former masterMauro Picone:[14]according to this convention, if a setAis a region then itsclosureAis a domain.[11]
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https://en.wikipedia.org/wiki/Closed_region
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Inmathematics, anopen setis ageneralizationof anopen intervalin thereal line.
In ametric space(asetwith adistancedefined between every two points), an open set is a set that, with every pointPin it, contains all points of the metric space that are sufficiently near toP(that is, all points whose distance toPis less than some value depending onP).
More generally, an open set is a member of a givencollectionofsubsetsof a given set, a collection that has the property of containing everyunionof its members, every finiteintersectionof its members, theempty set, and the whole set itself. A set in which such a collection is given is called atopological space, and the collection is called atopology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example,everysubset can be open (thediscrete topology), ornosubset can be open except the space itself and the empty set (theindiscrete topology).[1]
In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such ascontinuity,connectedness, andcompactness, which were originally defined by means of a distance.
The most common case of a topology without any distance is given bymanifolds, which are topological spaces that,neareach point, resemble an open set of aEuclidean space, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, theZariski topology, which is fundamental inalgebraic geometryandscheme theory.
Intuitively, an open set provides a method to distinguish twopoints. For example, if about one of two points in atopological space, there exists an open set not containing the other (distinct) point, the two points are referred to astopologically distinguishable. In this manner, one may speak of whether two points, or more generally twosubsets, of a topological space are "near" without concretely defining adistance. Therefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are calledmetric spaces.
In the set of allreal numbers, one has the naturalEuclidean metric; that is, a function which measures the distance between two real numbers:d(x,y) = |x−y|. Therefore, given a real numberx, one can speak of the set of all points close to that real number; that is, withinεofx. In essence, points within ε ofxapproximatexto an accuracy of degreeε. Note thatε> 0 always but asεbecomes smaller and smaller, one obtains points that approximatexto a higher and higher degree of accuracy. For example, ifx= 0 andε= 1, the points withinεofxare precisely the points of theinterval(−1, 1); that is, the set of all real numbers between −1 and 1. However, withε= 0.5, the points withinεofxare precisely the points of (−0.5, 0.5). Clearly, these points approximatexto a greater degree of accuracy than whenε= 1.
The previous discussion shows, for the casex= 0, that one may approximatexto higher and higher degrees of accuracy by definingεto be smaller and smaller. In particular, sets of the form (−ε,ε) give us a lot of information about points close tox= 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close tox. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (−ε,ε)), one may find different results regarding the distance between 0 and other real numbers. For example, if we were to defineRas the only such set for "measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member ofR. Thus, we find that in some sense, every real number is distance 0 away from 0. It may help in this case to think of the measure as being a binary condition: all things inRare equally close to 0, while any item that is not inRis not close to 0.
In general, one refers to the family of sets containing 0, used to approximate 0, as aneighborhood basis; a member of this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitrary set (X); rather than just the real numbers. In this case, given a point (x) of that set, one may define a collection of sets "around" (that is, containing)x, used to approximatex. Of course, this collection would have to satisfy certain properties (known asaxioms) for otherwise we may not have a well-defined method to measure distance. For example, every point inXshould approximatextosomedegree of accuracy. ThusXshould be in this family. Once we begin to define "smaller" sets containingx, we tend to approximatexto a greater degree of accuracy. Bearing this in mind, one may define the remaining axioms that the family of sets aboutxis required to satisfy.
Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one.
A subsetU{\displaystyle U}of theEuclideann-spaceRnisopenif, for every pointxinU{\displaystyle U},there existsa positive real numberε(depending onx) such that any point inRnwhoseEuclidean distancefromxis smaller thanεbelongs toU{\displaystyle U}.[2]Equivalently, a subsetU{\displaystyle U}ofRnis open if every point inU{\displaystyle U}is the center of anopen ballcontained inU.{\displaystyle U.}
An example of a subset ofRthat is not open is theclosed interval[0,1], since neither0 -εnor1 +εbelongs to[0,1]for anyε> 0, no matter how small.
A subsetUof ametric space(M,d)is calledopenif, for any pointxinU, there exists a real numberε> 0 such that any pointy∈M{\displaystyle y\in M}satisfyingd(x,y) <εbelongs toU. Equivalently,Uis open if every point inUhas a neighborhood contained inU.
This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.
Atopologyτ{\displaystyle \tau }on a setXis a set of subsets ofXwith the properties below. Each member ofτ{\displaystyle \tau }is called anopen set.[3]
Xtogether withτ{\displaystyle \tau }is called atopological space.
Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form(−1/n,1/n),{\displaystyle \left(-1/n,1/n\right),}wheren{\displaystyle n}is a positive integer, is the set{0}{\displaystyle \{0\}}which is not open in the real line.
A metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls. There are, however, topological spaces that are not metric spaces.
Theunionof any number of open sets, or infinitely many open sets, is open.[4]Theintersectionof a finite number of open sets is open.[4]
Acomplementof an open set (relative to the space that the topology is defined on) is called aclosed set. A set may be both open and closed (aclopen set). Theempty setand the full space are examples of sets that are both open and closed.[5]
A set can never been considered as open by itself. This notion is relative to a containing set and a specific topology on it.
Whether a set is open depends on thetopologyunder consideration. Having opted forgreater brevity over greater clarity, we refer to a setXendowed with a topologyτ{\displaystyle \tau }as "the topological spaceX" rather than "the topological space(X,τ){\displaystyle (X,\tau )}", despite the fact that all the topological data is contained inτ.{\displaystyle \tau .}If there are two topologies on the same set, a setUthat is open in the first topology might fail to be open in the second topology. For example, ifXis any topological space andYis any subset ofX, the setYcan be given its own topology (called the 'subspace topology') defined by "a setUis open in the subspace topology onYif and only ifUis the intersection ofYwith an open set from the original topology onX."[6]This potentially introduces new open sets: ifVis open in the original topology onX, butV∩Y{\displaystyle V\cap Y}isn't open in the original topology onX, thenV∩Y{\displaystyle V\cap Y}is open in the subspace topology onY.
As a concrete example of this, ifUis defined as the set of rational numbers in the interval(0,1),{\displaystyle (0,1),}thenUis an open subset of therational numbers, but not of thereal numbers. This is because when the surrounding space is the rational numbers, for every pointxinU, there exists a positive numberasuch that allrationalpoints within distanceaofxare also inU. On the other hand, when the surrounding space is the reals, then for every pointxinUthere isnopositiveasuch that allrealpoints within distanceaofxare inU(becauseUcontains no non-rational numbers).
Open sets have a fundamental importance intopology. The concept is required to define and make sense oftopological spaceand other topological structures that deal with the notions of closeness and convergence for spaces such asmetric spacesanduniform spaces.
EverysubsetAof a topological spaceXcontains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called theinteriorofA.
It can be constructed by taking the union of all the open sets contained inA.[7]
Afunctionf:X→Y{\displaystyle f:X\to Y}between two topological spacesX{\displaystyle X}andY{\displaystyle Y}iscontinuousif thepreimageof every open set inY{\displaystyle Y}is open inX.{\displaystyle X.}[8]The functionf:X→Y{\displaystyle f:X\to Y}is calledopenif theimageof every open set inX{\displaystyle X}is open inY.{\displaystyle Y.}
An open set on thereal linehas the characteristic property that it is a countable union of disjoint open intervals.
A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subsetanda closed subset. Such subsets are known asclopen sets. Explicitly, a subsetS{\displaystyle S}of a topological space(X,τ){\displaystyle (X,\tau )}is calledclopenif bothS{\displaystyle S}and its complementX∖S{\displaystyle X\setminus S}are open subsets of(X,τ){\displaystyle (X,\tau )}; or equivalently, ifS∈τ{\displaystyle S\in \tau }andX∖S∈τ.{\displaystyle X\setminus S\in \tau .}
Inanytopological space(X,τ),{\displaystyle (X,\tau ),}the empty set∅{\displaystyle \varnothing }and the setX{\displaystyle X}itself are always clopen. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist ineverytopological space. To see, it suffices to remark that, by definition of a topology,X{\displaystyle X}and∅{\displaystyle \varnothing }are both open, and that they are also closed, since each is the complement of the other.
The open sets of the usualEuclidean topologyof thereal lineR{\displaystyle \mathbb {R} }are the empty set, theopen intervalsand every union of open intervals.
If a topological spaceX{\displaystyle X}is endowed with thediscrete topology(so that by definition, every subset ofX{\displaystyle X}is open) then every subset ofX{\displaystyle X}is a clopen subset.
For a more advanced example reminiscent of the discrete topology, suppose thatU{\displaystyle {\mathcal {U}}}is anultrafilteron a non-empty setX.{\displaystyle X.}Then the unionτ:=U∪{∅}{\displaystyle \tau :={\mathcal {U}}\cup \{\varnothing \}}is a topology onX{\displaystyle X}with the property thateverynon-empty proper subsetS{\displaystyle S}ofX{\displaystyle X}iseitheran open subset or else a closed subset, but never both; that is, if∅≠S⊊X{\displaystyle \varnothing \neq S\subsetneq X}(whereS≠X{\displaystyle S\neq X}) thenexactly oneof the following two statements is true: either (1)S∈τ{\displaystyle S\in \tau }or else, (2)X∖S∈τ.{\displaystyle X\setminus S\in \tau .}Said differently,everysubset is open or closed but theonlysubsets that are both (i.e. that are clopen) are∅{\displaystyle \varnothing }andX.{\displaystyle X.}
A subsetS{\displaystyle S}of a topological spaceX{\displaystyle X}is called aregular open setifInt(S¯)=S{\displaystyle \operatorname {Int} \left({\overline {S}}\right)=S}or equivalently, ifBd(S¯)=BdS{\displaystyle \operatorname {Bd} \left({\overline {S}}\right)=\operatorname {Bd} S}, whereBdS{\displaystyle \operatorname {Bd} S},IntS{\displaystyle \operatorname {Int} S}, andS¯{\displaystyle {\overline {S}}}denote, respectively, the topologicalboundary,interior, andclosureofS{\displaystyle S}inX{\displaystyle X}. A topological space for which there exists abaseconsisting of regular open sets is called asemiregular space.
A subset ofX{\displaystyle X}is a regular open set if and only if its complement inX{\displaystyle X}is a regular closed set, where by definition a subsetS{\displaystyle S}ofX{\displaystyle X}is called aregular closed setifIntS¯=S{\displaystyle {\overline {\operatorname {Int} S}}=S}or equivalently, ifBd(IntS)=BdS.{\displaystyle \operatorname {Bd} \left(\operatorname {Int} S\right)=\operatorname {Bd} S.}Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general,[note 1]the converses arenottrue.
Throughout,(X,τ){\displaystyle (X,\tau )}will be a topological space.
A subsetA⊆X{\displaystyle A\subseteq X}of a topological spaceX{\displaystyle X}is called:
The complement of a preopen set is calledpreclosed.
The complement of a β-open set is calledβ-closed.
The complement of a sequentially open set is calledsequentially closed. A subsetS⊆X{\displaystyle S\subseteq X}is sequentially closed inX{\displaystyle X}if and only ifS{\displaystyle S}is equal to itssequential closure, which by definition is the setSeqClXS{\displaystyle \operatorname {SeqCl} _{X}S}consisting of allx∈X{\displaystyle x\in X}for which there exists a sequence inS{\displaystyle S}that converges tox{\displaystyle x}(inX{\displaystyle X}).
Using the fact that
whenever two subsetsA,B⊆X{\displaystyle A,B\subseteq X}satisfyA⊆B,{\displaystyle A\subseteq B,}the following may be deduced:
Moreover, a subset is a regular open set if and only if it is preopen and semi-closed.[10]The intersection of an α-open set and a semi-preopen (resp. semi-open, preopen, b-open) set is a semi-preopen (resp. semi-open, preopen, b-open) set.[10]Preopen sets need not be semi-open and semi-open sets need not be preopen.[10]
Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen).[10]However, finite intersections of preopen sets need not be preopen.[13]The set of all α-open subsets of a space(X,τ){\displaystyle (X,\tau )}forms a topology onX{\displaystyle X}that isfinerthanτ.{\displaystyle \tau .}[9]
A topological spaceX{\displaystyle X}isHausdorffif and only if everycompact subspaceofX{\displaystyle X}is θ-closed.[13]A spaceX{\displaystyle X}istotally disconnectedif and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen. Moreover, the space is totally disconnected if and only if theclosureof every preopen subset is open.[9]
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Intopologyand related areas ofmathematics, aneighbourhood(orneighborhood) is one of the basic concepts in atopological space. It is closely related to the concepts ofopen setandinterior. Intuitively speaking, a neighbourhood of a point is asetof points containing that point where one can move some amount in any direction away from that point without leaving the set.
IfX{\displaystyle X}is atopological spaceandp{\displaystyle p}is a point inX,{\displaystyle X,}then aneighbourhood[1]ofp{\displaystyle p}is a subsetV{\displaystyle V}ofX{\displaystyle X}that includes anopen setU{\displaystyle U}containingp{\displaystyle p},p∈U⊆V⊆X.{\displaystyle p\in U\subseteq V\subseteq X.}
This is equivalent to the pointp∈X{\displaystyle p\in X}belonging to thetopological interiorofV{\displaystyle V}inX.{\displaystyle X.}
The neighbourhoodV{\displaystyle V}need not be an open subset ofX.{\displaystyle X.}WhenV{\displaystyle V}is open (resp. closed, compact, etc.) inX,{\displaystyle X,}it is called anopen neighbourhood[2](resp. closed neighbourhood, compact neighbourhood, etc.). Some authors[3]require neighbourhoods to be open, so it is important to note their conventions.
A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points. A closed rectangle, as illustrated in the figure, is not a neighbourhood of all its points; points on the edges or corners of the rectangle are not contained in any open set that is contained within the rectangle.
The collection of all neighbourhoods of a point is called theneighbourhood systemat the point.
IfS{\displaystyle S}is asubsetof a topological spaceX{\displaystyle X}, then aneighbourhoodofS{\displaystyle S}is a setV{\displaystyle V}that includes an open setU{\displaystyle U}containingS{\displaystyle S},S⊆U⊆V⊆X.{\displaystyle S\subseteq U\subseteq V\subseteq X.}It follows that a setV{\displaystyle V}is a neighbourhood ofS{\displaystyle S}if and only if it is a neighbourhood of all the points inS.{\displaystyle S.}Furthermore,V{\displaystyle V}is a neighbourhood ofS{\displaystyle S}if and only ifS{\displaystyle S}is a subset of theinteriorofV.{\displaystyle V.}A neighbourhood ofS{\displaystyle S}that is also an open subset ofX{\displaystyle X}is called anopen neighbourhoodofS.{\displaystyle S.}The neighbourhood of a point is just a special case of this definition.
In ametric spaceM=(X,d),{\displaystyle M=(X,d),}a setV{\displaystyle V}is aneighbourhoodof a pointp{\displaystyle p}if there exists anopen ballwith centerp{\displaystyle p}and radiusr>0,{\displaystyle r>0,}such thatBr(p)=B(p;r)={x∈X:d(x,p)<r}{\displaystyle B_{r}(p)=B(p;r)=\{x\in X:d(x,p)<r\}}is contained inV.{\displaystyle V.}
V{\displaystyle V}is called auniform neighbourhoodof a setS{\displaystyle S}if there exists a positive numberr{\displaystyle r}such that for all elementsp{\displaystyle p}ofS,{\displaystyle S,}Br(p)={x∈X:d(x,p)<r}{\displaystyle B_{r}(p)=\{x\in X:d(x,p)<r\}}is contained inV.{\displaystyle V.}
Under the same condition, forr>0,{\displaystyle r>0,}ther{\displaystyle r}-neighbourhoodSr{\displaystyle S_{r}}of a setS{\displaystyle S}is the set of all points inX{\displaystyle X}that are at distance less thanr{\displaystyle r}fromS{\displaystyle S}(or equivalently,Sr{\displaystyle S_{r}}is the union of all the open balls of radiusr{\displaystyle r}that are centered at a point inS{\displaystyle S}):Sr=⋃p∈SBr(p).{\displaystyle S_{r}=\bigcup \limits _{p\in {}S}B_{r}(p).}
It directly follows that anr{\displaystyle r}-neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains anr{\displaystyle r}-neighbourhood for some value ofr.{\displaystyle r.}
Given the set ofreal numbersR{\displaystyle \mathbb {R} }with the usualEuclidean metricand a subsetV{\displaystyle V}defined asV:=⋃n∈NB(n;1/n),{\displaystyle V:=\bigcup _{n\in \mathbb {N} }B\left(n\,;\,1/n\right),}thenV{\displaystyle V}is a neighbourhood for the setN{\displaystyle \mathbb {N} }ofnatural numbers, but isnota uniform neighbourhood of this set.
The above definition is useful if the notion ofopen setis already defined. There is an alternative way to define a topology, by first defining theneighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points.
A neighbourhood system onX{\displaystyle X}is the assignment of afilterN(x){\displaystyle N(x)}of subsets ofX{\displaystyle X}to eachx{\displaystyle x}inX,{\displaystyle X,}such that
One can show that both definitions are compatible, that is, the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system.
In auniform spaceS=(X,Φ),{\displaystyle S=(X,\Phi ),}V{\displaystyle V}is called auniform neighbourhoodofP{\displaystyle P}if there exists anentourageU∈Φ{\displaystyle U\in \Phi }such thatV{\displaystyle V}contains all points ofX{\displaystyle X}that areU{\displaystyle U}-close to some point ofP;{\displaystyle P;}that is,U[x]⊆V{\displaystyle U[x]\subseteq V}for allx∈P.{\displaystyle x\in P.}
Adeleted neighbourhoodof a pointp{\displaystyle p}(sometimes called apunctured neighbourhood) is a neighbourhood ofp,{\displaystyle p,}without{p}.{\displaystyle \{p\}.}For instance, theinterval(−1,1)={y:−1<y<1}{\displaystyle (-1,1)=\{y:-1<y<1\}}is a neighbourhood ofp=0{\displaystyle p=0}in thereal line, so the set(−1,0)∪(0,1)=(−1,1)∖{0}{\displaystyle (-1,0)\cup (0,1)=(-1,1)\setminus \{0\}}is a deleted neighbourhood of0.{\displaystyle 0.}A deleted neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in thedefinition of the limit of a functionand in the definition of limit points (among other things).[4]
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Inmathematical analysis, adomainorregionis anon-empty,connected, andopen setin atopological space. In particular, it is any non-empty connected opensubsetof thereal coordinate spaceRnor thecomplex coordinate spaceCn. A connected open subset ofcoordinate spaceis frequently used for thedomain of a function.[1]
The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the termdomain,[2]some use the termregion,[3]some use both terms interchangeably,[4]and some define the two terms slightly differently;[5]some avoid ambiguity by sticking with a phrase such asnon-empty connected open subset.[6]
One common convention is to define adomainas a connected open set but aregionas theunionof a domain with none, some, or all of itslimit points.[7]Aclosed regionorclosed domainis the union of a domain and all of its limit points.
Various degrees of smoothness of theboundaryof the domain are required for various properties of functions defined on the domain to hold, such as integral theorems (Green's theorem,Stokes theorem), properties ofSobolev spaces, and to definemeasureson the boundary and spaces oftraces(generalized functions defined on the boundary). Commonly considered types of domains are domains withcontinuousboundary,Lipschitz boundary,C1boundary, and so forth.
Abounded domainis a domain that isbounded, i.e., contained in some ball.Bounded regionis defined similarly. Anexterior domainorexternal domainis a domain whosecomplementis bounded; sometimes smoothness conditions are imposed on its boundary.
Incomplex analysis, acomplex domain(or simplydomain) is any connected open subset of thecomplex planeC. For example, the entire complex plane is a domain, as is the openunit disk, the openupper half-plane, and so forth. Often, a complex domain serves as thedomain of definitionfor aholomorphic function. In the study ofseveral complex variables, the definition of a domain is extended to include any connected open subset ofCn.
InEuclidean spaces,one-,two-, andthree-dimensionalregions arecurves,surfaces, andsolids, whose extent are called, respectively,length,area, andvolume.
Definition. An open set is connected if it cannot be expressed as the sum of two open sets. An open connected set is called a domain.
German:Eine offene Punktmenge heißt zusammenhängend, wenn man sie nicht als Summe von zwei offenen Punktmengen darstellen kann. Eine offene zusammenhängende Punktmenge heißt ein Gebiet.
According toHans Hahn,[8]the concept of a domain as an open connected set was introduced byConstantin Carathéodoryin his famous book (Carathéodory 1918).
In this definition, Carathéodory considers obviouslynon-emptydisjointsets.
Hahn also remarks that the word "Gebiet" ("Domain") was occasionally previously used as asynonymofopen set.[9]The rough concept is older. In the 19th and early 20th century, the termsdomainandregionwere often used informally (sometimes interchangeably) without explicit definition.[10]
However, the term "domain" was occasionally used to identify closely related but slightly different concepts. For example, in his influentialmonographsonelliptic partial differential equations,Carlo Mirandauses the term "region" to identify an open connected set,[11][12]and reserves the term "domain" to identify an internally connected,[13]perfect set, each point of which is an accumulation point of interior points,[11]following his former masterMauro Picone:[14]according to this convention, if a setAis a region then itsclosureAis a domain.[11]
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A subsetS{\displaystyle S}of atopological spaceX{\displaystyle X}is called aregular open setif it is equal to theinteriorof itsclosure; expressed symbolically, ifInt(S¯)=S{\displaystyle \operatorname {Int} ({\overline {S}})=S}or, equivalently, if∂(S¯)=∂S,{\displaystyle \partial ({\overline {S}})=\partial S,}whereIntS,{\displaystyle \operatorname {Int} S,}S¯{\displaystyle {\overline {S}}}and∂S{\displaystyle \partial S}denote, respectively, the interior, closure andboundaryofS.{\displaystyle S.}[1]
A subsetS{\displaystyle S}ofX{\displaystyle X}is called aregular closed setif it is equal to the closure of its interior; expressed symbolically, ifIntS¯=S{\displaystyle {\overline {\operatorname {Int} S}}=S}or, equivalently, if∂(IntS)=∂S.{\displaystyle \partial (\operatorname {Int} S)=\partial S.}[1]
IfR{\displaystyle \mathbb {R} }has its usualEuclidean topologythen the open setS=(0,1)∪(1,2){\displaystyle S=(0,1)\cup (1,2)}is not a regular open set, sinceInt(S¯)=(0,2)≠S.{\displaystyle \operatorname {Int} ({\overline {S}})=(0,2)\neq S.}Everyopen intervalinR{\displaystyle \mathbb {R} }is a regular open set and every non-degenerate closed interval (that is, a closed interval containing at least two distinct points) is a regular closed set. A singleton{x}{\displaystyle \{x\}}is a closed subset ofR{\displaystyle \mathbb {R} }but not a regular closed set because its interior is the empty set∅,{\displaystyle \varnothing ,}so thatInt{x}¯=∅¯=∅≠{x}.{\displaystyle {\overline {\operatorname {Int} \{x\}}}={\overline {\varnothing }}=\varnothing \neq \{x\}.}
A subset ofX{\displaystyle X}is a regular open set if and only if its complement inX{\displaystyle X}is a regular closed set.[2]Every regular open set is anopen setand every regular closed set is aclosed set.
Eachclopen subsetofX{\displaystyle X}(which includes∅{\displaystyle \varnothing }andX{\displaystyle X}itself) is simultaneously a regular open subset and regular closed subset.
The interior of a closed subset ofX{\displaystyle X}is a regular open subset ofX{\displaystyle X}and likewise, the closure of an open subset ofX{\displaystyle X}is a regular closed subset ofX.{\displaystyle X.}[2]The intersection (but not necessarily the union) of two regular open sets is a regular open set. Similarly, the union (but not necessarily the intersection) of two regular closed sets is a regular closed set.[2]
The collection of all regular open sets inX{\displaystyle X}forms acomplete Boolean algebra; thejoinoperation is given byU∨V=Int(U∪V¯),{\displaystyle U\vee V=\operatorname {Int} ({\overline {U\cup V}}),}themeetisU∧V=U∩V{\displaystyle U\land V=U\cap V}and the complement is¬U=Int(X∖U).{\displaystyle \neg U=\operatorname {Int} (X\setminus U).}
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Inmathematics, in the field ofalgebraic number theory, anS-unitgeneralises the idea ofunitof thering of integersof the field. Many of the results which hold for units are also valid forS-units.
LetKbe anumber fieldwith ring of integersR. LetSbe a finite set ofprime idealsofR. An elementxofKis anS-unit if theprincipal fractional ideal(x) is a product of primes inS(to positive or negative powers). For theringofrational integersZone may takeSto be a finite set ofprime numbersand define anS-unit to be arational numberwhose numerator and denominator aredivisibleonly by the primes inS.
TheS-units form a multiplicativegroupcontaining the units ofR.
Dirichlet's unit theoremholds forS-units: the group ofS-units isfinitely generated, withrank(maximal number of multiplicatively independent elements) equal tor+s, whereris the rank of the unit group ands= |S|.
TheS-unit equationis aDiophantine equation
withuandvrestricted to beingS-units ofK(or more generally, elements of a finitely generated subgroup of the multiplicative group of any field of characteristic zero). The number of solutions of this equation is finite[1]and the solutions are effectively determined using estimates forlinear forms in logarithmsas developed intranscendental number theory. A variety of Diophantine equations are reducible in principle to some form of theS-unit equation: a notable example isSiegel's theoremon integral points onelliptic curves, and more generallysuperelliptic curvesof the formyn=f(x).
A computational solver forS-unit equation is available in the softwareSageMath.[2]
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Incommutative algebraandalgebraic geometry,localizationis a formal way to introduce the "denominators" to a givenringormodule. That is, it introduces a new ring/module out of an existing ring/moduleR, so that it consists offractionsms,{\displaystyle {\frac {m}{s}},}such that thedenominatorsbelongs to a given subsetSofR. IfSis the set of the non-zero elements of anintegral domain, then the localization is thefield of fractions: this case generalizes the construction of the fieldQ{\displaystyle \mathbb {Q} }ofrational numbersfrom the ringZ{\displaystyle \mathbb {Z} }ofintegers.
The technique has become fundamental, particularly inalgebraic geometry, as it provides a natural link tosheaftheory. In fact, the termlocalizationoriginated inalgebraic geometry: ifRis a ring offunctionsdefined on some geometric object (algebraic variety)V, and one wants to study this variety "locally" near a pointp, then one considers the setSof all functions that are not zero atpand localizesRwith respect toS. The resulting ringS−1R{\displaystyle S^{-1}R}contains information about the behavior ofVnearp, and excludes information that is not "local", such as thezeros of functionsthat are outsideV(cf. the example given atlocal ring).
The localization of acommutative ringRby amultiplicatively closed setSis a new ringS−1R{\displaystyle S^{-1}R}whose elements are fractions with numerators inRand denominators inS.
If the ring is anintegral domainthe construction generalizes and follows closely that of thefield of fractions, and, in particular, that of therational numbersas the field of fractions of the integers. For rings that havezero divisors, the construction is similar but requires more care.
Localization is commonly done with respect to amultiplicatively closed setS(also called amultiplicative setor amultiplicative system) of elements of a ringR, that is a subset ofRthat isclosedunder multiplication, and contains1.
The requirement thatSmust be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong toS. The localization by a setUthat is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements ofU. However, the same localization is obtained by using the multiplicatively closed setSof all products of elements ofU. As this often makes reasoning and notation simpler, it is standard practice to consider only localizations by multiplicative sets.
For example, the localization by a single elementsintroduces fractions of the formas,{\displaystyle {\tfrac {a}{s}},}but also products of such fractions, such asabs2.{\displaystyle {\tfrac {ab}{s^{2}}}.}So, the denominators will belong to the multiplicative set{1,s,s2,s3,…}{\displaystyle \{1,s,s^{2},s^{3},\ldots \}}of the powers ofs. Therefore, one generally talks of "the localization by the powers of an element" rather than of "the localization by an element".
The localization of a ringRby a multiplicative setSis generally denotedS−1R,{\displaystyle S^{-1}R,}but other notations are commonly used in some special cases: ifS={1,t,t2,…}{\displaystyle S=\{1,t,t^{2},\ldots \}}consists of the powers of a single element,S−1R{\displaystyle S^{-1}R}is often denotedRt;{\displaystyle R_{t};}ifS=R∖p{\displaystyle S=R\setminus {\mathfrak {p}}}is thecomplementof aprime idealp{\displaystyle {\mathfrak {p}}}, thenS−1R{\displaystyle S^{-1}R}is denotedRp.{\displaystyle R_{\mathfrak {p}}.}
In the remainder of this article, only localizations by a multiplicative set are considered.
When the ringRis anintegral domainandSdoes not contain0, the ringS−1R{\displaystyle S^{-1}R}is a subring of thefield of fractionsofR. As such, the localization of a domain is a domain.
More precisely, it is thesubringof the field of fractions ofR, that consists of the fractionsas{\displaystyle {\tfrac {a}{s}}}such thats∈S.{\displaystyle s\in S.}This is a subring since the sumas+bt=at+bsst,{\displaystyle {\tfrac {a}{s}}+{\tfrac {b}{t}}={\tfrac {at+bs}{st}},}and the productasbt=abst{\displaystyle {\tfrac {a}{s}}\,{\tfrac {b}{t}}={\tfrac {ab}{st}}}of two elements ofS−1R{\displaystyle S^{-1}R}are inS−1R.{\displaystyle S^{-1}R.}This results from the defining property of a multiplicative set, which implies also that1=11∈S−1R.{\displaystyle 1={\tfrac {1}{1}}\in S^{-1}R.}In this case,Ris a subring ofS−1R.{\displaystyle S^{-1}R.}It is shown below that this is no longer true in general, typically whenScontainszero divisors.
For example, thedecimal fractionsare the localization of the ring of integers by the multiplicative set of the powers of ten. In this case,S−1R{\displaystyle S^{-1}R}consists of the rational numbers that can be written asn10k,{\displaystyle {\tfrac {n}{10^{k}}},}wherenis an integer, andkis a nonnegative integer.
In the general case, a problem arises withzero divisors. LetSbe a multiplicative set in a commutative ringR. Suppose thats∈S,{\displaystyle s\in S,}and0≠a∈R{\displaystyle 0\neq a\in R}is a zero divisor withas=0.{\displaystyle as=0.}Thena1{\displaystyle {\tfrac {a}{1}}}is the image inS−1R{\displaystyle S^{-1}R}ofa∈R,{\displaystyle a\in R,}and one hasa1=ass=0s=01.{\displaystyle {\tfrac {a}{1}}={\tfrac {as}{s}}={\tfrac {0}{s}}={\tfrac {0}{1}}.}Thus some nonzero elements ofRmust be zero inS−1R.{\displaystyle S^{-1}R.}The construction that follows is designed for taking this into account.
GivenRandSas above, one considers theequivalence relationonR×S{\displaystyle R\times S}that is defined by(r1,s1)∼(r2,s2){\displaystyle (r_{1},s_{1})\sim (r_{2},s_{2})}if there exists at∈S{\displaystyle t\in S}such thatt(s1r2−s2r1)=0.{\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}
The localizationS−1R{\displaystyle S^{-1}R}is defined as the set of theequivalence classesfor this relation. The class of(r,s)is denoted asrs,{\displaystyle {\frac {r}{s}},}r/s,{\displaystyle r/s,}ors−1r.{\displaystyle s^{-1}r.}So, one hasr1s1=r2s2{\displaystyle {\tfrac {r_{1}}{s_{1}}}={\tfrac {r_{2}}{s_{2}}}}if and only if there is at∈S{\displaystyle t\in S}such thatt(s1r2−s2r1)=0.{\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}The reason for thet{\displaystyle t}is to handle cases such as the abovea1=01,{\displaystyle {\tfrac {a}{1}}={\tfrac {0}{1}},}wheres1r2−s2r1{\displaystyle s_{1}r_{2}-s_{2}r_{1}}is nonzero even though the fractions should be regarded as equal.
The localizationS−1R{\displaystyle S^{-1}R}is a commutative ring with addition
multiplication
additive identity01,{\displaystyle {\tfrac {0}{1}},}andmultiplicative identity11.{\displaystyle {\tfrac {1}{1}}.}
Thefunction
defines aring homomorphismfromR{\displaystyle R}intoS−1R,{\displaystyle S^{-1}R,}which isinjectiveif and only ifSdoes not contain any zero divisors.
If0∈S,{\displaystyle 0\in S,}thenS−1R{\displaystyle S^{-1}R}is thezero ringthat has only one unique element0.
IfSis the set of allregular elementsofR(that is the elements that are not zero divisors),S−1R{\displaystyle S^{-1}R}is called thetotal ring of fractionsofR.
The (above defined) ring homomorphismj:R→S−1R{\displaystyle j\colon R\to S^{-1}R}satisfies auniversal propertythat is described below. This characterizesS−1R{\displaystyle S^{-1}R}up to anisomorphism. So all properties of localizations can be deduced from the universal property, independently from the way they have been constructed. Moreover, many important properties of localization are easily deduced from the general properties of universal properties, while their direct proof may be more technical.
The universal property satisfied byj:R→S−1R{\displaystyle j\colon R\to S^{-1}R}is the following:
Usingcategory theory, this can be expressed by saying that localization is afunctorthat isleft adjointto aforgetful functor. More precisely, letC{\displaystyle {\mathcal {C}}}andD{\displaystyle {\mathcal {D}}}be the categories whose objects arepairsof a commutative ring and asubmonoidof, respectively, the multiplicativemonoidor thegroup of unitsof the ring. Themorphismsof these categories are the ring homomorphisms that map the submonoid of the first object into the submonoid of the second one. Finally, letF:D→C{\displaystyle {\mathcal {F}}\colon {\mathcal {D}}\to {\mathcal {C}}}be the forgetful functor that forgets that the elements of the second element of the pair are invertible.
Then the factorizationf=g∘j{\displaystyle f=g\circ j}of the universal property defines a bijection
This may seem a rather tricky way of expressing the universal property, but it is useful for showing easily many properties, by using the fact that the composition of two left adjoint functors is a left adjoint functor.
Localization is a rich construction that has many useful properties. In this section, only the properties relative to rings and to a single localization are considered. Properties concerningideals,modules, or several multiplicative sets are considered in other sections.
LetS⊆R{\displaystyle S\subseteq R}be a multiplicative set. ThesaturationS^{\displaystyle {\hat {S}}}ofS{\displaystyle S}is the set
The multiplicative setSissaturatedif it equals its saturation, that is, ifS^=S{\displaystyle {\hat {S}}=S}, or equivalently, ifrs∈S{\displaystyle rs\in S}implies thatrandsare inS.
IfSis not saturated, andrs∈S,{\displaystyle rs\in S,}thensrs{\displaystyle {\frac {s}{rs}}}is amultiplicative inverseof the image ofrinS−1R.{\displaystyle S^{-1}R.}So, the images of the elements ofS^{\displaystyle {\hat {S}}}are all invertible inS−1R,{\displaystyle S^{-1}R,}and the universal property implies thatS−1R{\displaystyle S^{-1}R}andS^−1R{\displaystyle {\hat {S}}{}^{-1}R}arecanonically isomorphic, that is, there is a unique isomorphism between them that fixes the images of the elements ofR.
IfSandTare two multiplicative sets, thenS−1R{\displaystyle S^{-1}R}andT−1R{\displaystyle T^{-1}R}are isomorphic if and only if they have the same saturation, or, equivalently, ifsbelongs to one of the multiplicative sets, then there existst∈R{\displaystyle t\in R}such thatstbelongs to the other.
Saturated multiplicative sets are not widely used explicitly, since, for verifying that a set is saturated, one must knowallunitsof the ring.
The termlocalizationoriginates in the general trend of modern mathematics to studygeometricalandtopologicalobjectslocally, that is in terms of their behavior near each point. Examples of this trend are the fundamental concepts ofmanifolds,germsandsheafs. Inalgebraic geometry, anaffine algebraic setcan be identified with aquotient ringof apolynomial ringin such a way that the points of the algebraic set correspond to themaximal idealsof the ring (this isHilbert's Nullstellensatz). This correspondence has been generalized for making the set of theprime idealsof acommutative ringatopological spaceequipped with theZariski topology; this topological space is called thespectrum of the ring.
In this context, alocalizationby a multiplicative set may be viewed as the restriction of the spectrum of a ring to the subspace of the prime ideals (viewed aspoints) that do not intersect the multiplicative set.
Two classes of localizations are more commonly considered:
Innumber theoryandalgebraic topology, when working over the ringZ{\displaystyle \mathbb {Z} }ofintegers, one refers to a property relative to an integernas a property trueatnorawayfromn, depending on the localization that is considered. "Away fromn" means that the property is considered after localization by the powers ofn, and, ifpis aprime number, "atp" means that the property is considered after localization at the prime idealpZ{\displaystyle p\mathbb {Z} }. This terminology can be explained by the fact that, ifpis prime, the nonzero prime ideals of the localization ofZ{\displaystyle \mathbb {Z} }are either thesingleton set{p}or its complement in the set of prime numbers.
LetSbe a multiplicative set in a commutative ringR, andj:R→S−1R{\displaystyle j\colon R\to S^{-1}R}be the canonical ring homomorphism. Given anidealIinR, letS−1I{\displaystyle S^{-1}I}the set of the fractions inS−1R{\displaystyle S^{-1}R}whose numerator is inI. This is an ideal ofS−1R,{\displaystyle S^{-1}R,}which is generated byj(I), and called thelocalizationofIbyS.
ThesaturationofIbySisj−1(S−1I);{\displaystyle j^{-1}(S^{-1}I);}it is an ideal ofR, which can also defined as the set of the elementsr∈R{\displaystyle r\in R}such that there existss∈S{\displaystyle s\in S}withsr∈I.{\displaystyle sr\in I.}
Many properties of ideals are either preserved by saturation and localization, or can be characterized by simpler properties of localization and saturation.
In what follows,Sis a multiplicative set in a ringR, andIandJare ideals ofR; the saturation of an idealIby a multiplicative setSis denotedsatS(I),{\displaystyle \operatorname {sat} _{S}(I),}or, when the multiplicative setSis clear from the context,sat(I).{\displaystyle \operatorname {sat} (I).}
LetRbe acommutative ring,Sbe amultiplicative setinR, andMbe anR-module. Thelocalization of the moduleMbyS, denotedS−1M, is anS−1R-module that is constructed exactly as the localization ofR, except that the numerators of the fractions belong toM. That is, as a set, it consists ofequivalence classes, denotedms{\displaystyle {\frac {m}{s}}}, of pairs(m,s), wherem∈M{\displaystyle m\in M}ands∈S,{\displaystyle s\in S,}and two pairs(m,s)and(n,t)are equivalent if there is an elementuinSsuch that
Addition and scalar multiplication are defined as for usual fractions (in the following formula,r∈R,{\displaystyle r\in R,}s,t∈S,{\displaystyle s,t\in S,}andm,n∈M{\displaystyle m,n\in M}):
Moreover,S−1Mis also anR-module with scalar multiplication
It is straightforward to check that these operations are well-defined, that is, they give the same result for different choices of representatives of fractions.
The localization of a module can be equivalently defined by usingtensor products:
The proof of equivalence (up to acanonical isomorphism) can be done by showing that the two definitions satisfy the same universal property.
IfMis asubmoduleof anR-moduleN, andSis a multiplicative set inR, one hasS−1M⊆S−1N.{\displaystyle S^{-1}M\subseteq S^{-1}N.}This implies that, iff:M→N{\displaystyle f\colon M\to N}is aninjectivemodule homomorphism, then
is also an injective homomorphism.
Since the tensor product is aright exact functor, this implies that localization bySmapsexact sequencesofR-modules to exact sequences ofS−1R{\displaystyle S^{-1}R}-modules. In other words, localization is anexact functor, andS−1R{\displaystyle S^{-1}R}is aflatR-module.
This flatness and the fact that localization solves auniversal propertymake that localization preserves many properties of modules and rings, and is compatible with solutions of other universal properties. For example, thenatural map
is an isomorphism. IfM{\displaystyle M}is afinitely presented module, the natural map
is also an isomorphism.[4]
If a moduleMis afinitely generatedoverR, one has
whereAnn{\displaystyle \operatorname {Ann} }denotesannihilator, that is the ideal of the elements of the ring that map to zero all elements of the module.[5]In particular,
that is, iftM=0{\displaystyle tM=0}for somet∈S.{\displaystyle t\in S.}[6]
The definition of aprime idealimplies immediately that thecomplementS=R∖p{\displaystyle S=R\setminus {\mathfrak {p}}}of a prime idealp{\displaystyle {\mathfrak {p}}}in a commutative ringRis a multiplicative set. In this case, the localizationS−1R{\displaystyle S^{-1}R}is commonly denotedRp.{\displaystyle R_{\mathfrak {p}}.}The ringRp{\displaystyle R_{\mathfrak {p}}}is alocal ring, that is calledthe local ring ofRatp.{\displaystyle {\mathfrak {p}}.}This means thatpRp=p⊗RRp{\displaystyle {\mathfrak {p}}\,R_{\mathfrak {p}}={\mathfrak {p}}\otimes _{R}R_{\mathfrak {p}}}is the uniquemaximal idealof the ringRp.{\displaystyle R_{\mathfrak {p}}.}Analogously one can define the localization of a moduleMat a prime idealp{\displaystyle {\mathfrak {p}}}ofR. Again, the localizationS−1M{\displaystyle S^{-1}M}is commonly denotedMp{\displaystyle M_{\mathfrak {p}}}.
Such localizations are fundamental for commutative algebra and algebraic geometry for several reasons. One is that local rings are often easier to study than general commutative rings, in particular because ofNakayama lemma. However, the main reason is that many properties are true for a ring if and only if they are true for all its local rings. For example, a ring isregularif and only if all its local rings areregular local rings.
Properties of a ring that can be characterized on its local rings are calledlocal properties, and are often the algebraic counterpart of geometriclocal propertiesofalgebraic varieties, which are properties that can be studied by restriction to a small neighborhood of each point of the variety. (There is another concept of local property that refers to localization to Zariski open sets; see§ Localization to Zariski open sets, below.)
Many local properties are a consequence of the fact that the module
is afaithfully flat modulewhen the direct sum is taken over all prime ideals (or over allmaximal idealsofR). See alsoFaithfully flat descent.
A propertyPof anR-moduleMis alocal propertyif the following conditions are equivalent:
The following are local properties:
On the other hand, some properties are not local properties. For example, an infinitedirect productoffieldsis not anintegral domainnor aNoetherian ring, while all its local rings are fields, and therefore Noetherian integral domains.
Localizingnon-commutative ringsis more difficult. While the localization exists for every setSof prospective units, it might take a different form to the one described above. One condition which ensures that the localization is well behaved is theOre condition.
One case for non-commutative rings where localization has a clear interest is for rings ofdifferential operators. It has the interpretation, for example, of adjoining a formal inverseD−1for a differentiation operatorD. This is done in many contexts in methods fordifferential equations. There is now a large mathematical theory about it, namedmicrolocalization, connecting with numerous other branches. Themicro-tag is to do with connections withFourier theory, in particular.
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Incryptography, theTiny Encryption Algorithm(TEA) is ablock ciphernotable for its simplicity of description andimplementation, typically a few lines of code. It was designed byDavid WheelerandRoger Needhamof theCambridge Computer Laboratory; it was first presented at theFast Software Encryptionworkshop inLeuvenin 1994, and first published in the proceedings of that workshop.[4]
The cipher is not subject to anypatents.
TEA operates on two 32-bitunsigned integers(could be derived from a 64-bit datablock) and uses a 128-bitkey. It has aFeistel structurewith a suggested 64 rounds, typically implemented in pairs termedcycles. It has an extremely simplekey schedule, mixing all of the key material in exactly the same way for each cycle. Different multiples of amagic constantare used to prevent simple attacks based on thesymmetryof the rounds. The magic constant, 2654435769 or 0x9E3779B9 is chosen to be⌊232⁄𝜙⌋, where𝜙is thegolden ratio(as anothing-up-my-sleeve number).[4]
TEA has a few weaknesses. Most notably, it suffers from equivalent keys—each key is equivalent to three others, which means that the effective key size is only 126bits.[5]As a result, TEA is especially bad as acryptographic hash function. This weakness led to a method forhackingMicrosoft'sXboxgame console, where the cipher was used as a hash function.[6]TEA is also susceptible to arelated-key attackwhich requires 223chosen plaintextsunder a related-key pair, with 232time complexity.[2]Because of these weaknesses, theXTEAcipher was designed.
The first published version of TEA was supplemented by a second version that incorporated extensions to make it more secure.Block TEA(which was specified along withXTEA) operates on arbitrary-size blocks in place of the 64-bit blocks of the original.
A third version (XXTEA), published in 1998, described further improvements for enhancing the security of the Block TEA algorithm.
Following is an adaptation of the reference encryption and decryption routines inC, released into the public domain by David Wheeler and Roger Needham:[4]
Note that the reference implementation acts on multi-byte numeric values. The original paper does not specify how to derive the numbers it acts on from binary or other content.
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Incryptography,XTEA(eXtended TEA) is ablock cipherdesigned to correct weaknesses inTEA. Thecipher's designers wereDavid WheelerandRoger Needhamof theCambridge Computer Laboratory, and the algorithm was presented in an unpublished technical report in 1997 (Needham and Wheeler, 1997). It is not subject to anypatents.[1]
Like TEA, XTEA is a64-bit blockFeistel cipherwith a128-bit keyand a suggested 64 rounds. Several differences from TEA are apparent, including a somewhat more complexkey-scheduleand a rearrangement of the shifts,XORs, and additions.
This standardCsource code, adapted from the reference code released into thepublic domainby David Wheeler and Roger Needham, encrypts and decrypts using XTEA:
The changes from the reference source code are minor:
The recommended value for the "num_rounds" parameter is 32, not 64, as each iteration of the loop does two Feistel-cipher rounds. To additionally improve speed, the loop can be unrolled by pre-computing the values of sum+key[].
In 2004, Ko et al. presented arelated-keydifferential attackon 27 out of 64 rounds of XTEA, requiring 220.5chosen plaintextsand atime complexityof 2115.15.[2][3]
In 2009, Lu presented a related-key rectangle attack on 36 rounds of XTEA, breaking more rounds than any previously published cryptanalytic results for XTEA. The paper presents two attacks, one without and with a weak key assumption, which corresponds to 264.98bytes of data and 2126.44operations, and 263.83bytes of data and 2104.33operations respectively.[4]
Presented along with XTEA was a variable-width block cipher termedBlock TEA, which uses the XTEA round function, but Block TEA applies it cyclically across an entire message for several iterations. Because it operates on the entire message, Block TEA has the property that it does not need amode of operation. An attack on the full Block TEA was described by Saarinen,[5]which also details a weakness in Block TEA's successor,XXTEA.
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Incryptography,Corrected Block TEA(often referred to asXXTEA) is ablock cipherdesigned to correct weaknesses in the originalBlock TEA.[2][3]
XXTEA is vulnerable to achosen-plaintext attackrequiring 259queries and negligible work. See cryptanalysis below.
Thecipher's designers wereRoger NeedhamandDavid Wheelerof theCambridge Computer Laboratory, and the algorithm was presented in an unpublished[clarification needed]technical report in October 1998 (Wheeler and Needham, 1998). It is not subject to anypatents.
Formally speaking, XXTEA is a consistent incomplete source-heavy heterogeneous UFN (unbalancedFeistel network) block cipher. XXTEA operates on variable-length blocks that are some arbitrary multiple of 32 bits in size (minimum 64 bits). The number of full cycles depends on the block size, but there are at least six (rising to 32 for small block sizes). The original Block TEA applies theXTEAround function to each word in the block and combines it additively with its leftmost neighbour. Slow diffusion rate of the decryption process was immediately exploited to break the cipher. Corrected Block TEA uses a more involved round function which makes use of both immediate neighbours in processing each word in the block.
XXTEA is likely to be more efficient than XTEA for longer messages.[citation needed]
Needham & Wheeler make the following comments on the use of Block TEA:
For ease of use and general security the large block version is to be preferred when applicable for the following reasons.
However, due to the incomplete nature of the round function, two large ciphertexts of 53 or more 32-bit words identical in all but 12 words can be found by a simple brute-force collision search requiring 296−Nmemory, 2Ntime and 2N+296−Nchosen plaintexts, in other words with a total time*memory complexity of 296, which is actually 2wordsize*fullcycles/2for any such cipher. It is currently unknown if such partial collisions pose any threat to the security of the cipher. Eight full cycles would raise the bar for such collision search above complexity of parallel brute-force attacks.[citation needed]
The unusually small size of the XXTEA algorithm would make it a viable option in situations where there are extreme constraints e.g. legacy hardware systems (perhaps embedded) where the amount of availableRAMis minimal, or alternativelysingle-board computerssuch as theRaspberry Pi,Banana PiorArduino.
An attack published in 2010 by E. Yarrkov presents achosen-plaintext attackagainst full-round XXTEA with wide block, requiring 259queries for a block size of 212 bytes or more, and negligible work. It is based ondifferential cryptanalysis.[1]
To cipher "212 bytes or more" algorithm performs just 6 rounds, and carefully chosen bit patterns allows to detect and analyze avalanche effect.
The original formulation of the Corrected Block TEA algorithm, published by David Wheeler and Roger Needham, is as follows:[4]
According to Needham and Wheeler:
BTEA will encode or decode n words as a single block wheren> 1
Note that the initialization of z isUndefined behaviorforn< 1 which may cause asegmentation faultor other unwanted behavior – it would be better placed inside the 'Coding Part' block. Also, in the definition of MX some programmers would prefer to use bracketing to clarify operator precedence.
A clarified version including those improvements is as follows:
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CipherSaberis a simplesymmetric encryptionprotocolbased on theRC4stream cipher. Its goals are both technical andpolitical: it gives reasonably strong protection of message confidentiality, yet it's designed to be simple enough that even noviceprogrammerscan memorize the algorithm and implement it from scratch. According to the designer, a CipherSaber version in theQBASIC programming languagetakes just sixteen lines of code. Its political aspect is that because it's so simple, it can be reimplemented anywhere at any time, and so it provides a way for users to communicate privately even ifgovernmentor other controls make distribution of normal cryptographic software completely impossible.
CipherSaber was invented byArnold Reinholdto keep strong cryptography in the hands of the public. Many governments have implemented legal restrictions on who can use cryptography, and many more have proposed them. By publicizing details on a secure yet easy-to-program encryption algorithm, Reinhold hopes to keep encryption technology accessible to everyone.
Unlike programs likePGPwhich are distributed as convenient-to-use prewritten software, Reinhold publishes CipherSaber only as a specification. The specification is intended to be so simple that even a beginning programmer can implement it easily. As the CipherSaber web site[1]explains:
The web site has a graphics file that displays as a "CipherKnight" certificate; however, that file is encrypted using CipherSaber with a known key published alongside the file. Users can view the graphic (and optionally print it out for framing) by first writing their own CipherSaber implementation to decrypt the file. By writing their own implementation and performing a few other small tasks, the user becomes a CipherKnight and the decrypted certificate attests to their knighthood. So, rather than providing a ready-made tool, CipherSaber's designer hopes to help computer users understand that they're capable of making their own strong cryptography programs without having to rely on professional developers or the permission of the government.
In the original version of CipherSaber (now called CipherSaber-1 or CS1), each encrypted message begins with a random ten-byteinitialization vector(IV). This IV is appended to the CipherSaber key to form the input to the RC4 key setup algorithm. The message, XORed with the RC4keystream, immediately follows.[1]
TheFluhrer, Mantin and Shamir attackon RC4 has rendered CipherSaber-1 vulnerable if a large number (>1000) messages are sent with the same CipherSaber key. To address this, the CipherSaber designer has made a modified protocol (called CipherSaber-2) in which the RC4 key setup loop is repeated multiple times (20 is recommended). In addition to agreeing on a secret key, parties communicating with CipherSaber-2 must agree on how many times to repeat this loop.[2]
The ciphertext output is a binary byte stream that is designed to be "indistinguishable from random noise".[3]For use with communications systems that can accept onlyASCIIdata, the author recommends encoding the byte stream as hexadecimal digits. This is less efficient than, for example,base64MIMEencoding, but it is much simpler to program, keeping with CipherSaber's goal of maximal ease of implementation.
CipherSaber is strong enough and usable enough to make its political point effectively. However, it falls markedly short of the security and convenience one would normally ask of such a cryptosystem. While CipherKnights can use CipherSaber to exchange occasional messages with each other reasonably securely, either for fun or in times of great distress, CipherSaber strips cryptography to its bare essentials and it does not offer enough features to be suitable for wide deployment and routine daily use. CipherSaber's author in fact asks users to download and installPGPas one of the steps of becoming a CipherKnight. CipherSaber can be seen as a last-resort fallback system to use if programs like PGP arebanned. Some, but not all of CipherSaber's sacrifices and shortcomings are unique to RC4.
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This article summarizes publicly knownattacksagainstcryptographic hash functions. Note that not all entries may be up to date. For a summary of other hash function parameters, seecomparison of cryptographic hash functions.
Hashes described here are designed for fast computation and have roughly similar speeds.[31]Because most users typically choose shortpasswordsformed in predictable ways, passwords can often be recovered from their hashed value if a fast hash is used. Searches on the order of 100 billion tests per second are possible with high-endgraphics processors.[32][33]Special hashes calledkey derivation functionshave been created to slow brute force searches. These includepbkdf2,bcrypt,scrypt,argon2, andballoon.
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TheInternational Association for Cryptologic Research(IACR) is a non-profit scientific organization that furthers research incryptologyand related fields. The IACR was organized at the initiative ofDavid Chaumat the CRYPTO '82 conference.[1]
The IACR organizes and sponsors three annual flagshipconferences, four area conferences in specific sub-areas of cryptography, and one symposium:[2]
Several other conferences and workshops are held in cooperation with the IACR. Starting in 2015, selected summer schools will be officially sponsored by the IACR. CRYPTO '83 was the first conference officially sponsored by the IACR.
The IACR publishes theJournal of Cryptology, in addition to the proceedings of its conference and workshops. The IACR also maintains theCryptology ePrint Archive, an online repository of cryptologic research papers aimed at providing rapid dissemination of results.[3]
Asiacrypt (also ASIACRYPT) is an international conference for cryptography research. The full name of the conference is currently International Conference on the Theory and Application of Cryptology and Information Security, though this has varied over time. Asiacrypt is a conference sponsored by the IACR since 2000, and is one of its three flagship conferences. Asiacrypt is now held annually in November or December at various locations throughoutAsiaandAustralia.
Initially, the Asiacrypt conferences were called AUSCRYPT, as the first one was held inSydney, Australia in 1990, and only later did the community decide that the conference should be held in locations throughout Asia. The first conference to be called "Asiacrypt" was held in 1991 inFujiyoshida,Japan.
Cryptographic Hardware and Embedded Systems (CHES) is a conference for cryptography research,[4]focusing on the implementation of cryptographic algorithms. The two general areas treated are the efficient and the secure implementation of algorithms. Related topics such as random number generators,physical unclonable functionor special-purpose cryptanalytical machines are also commonly covered at the workshop. It was first held inWorcester, Massachusettsin 1999 atWorcester Polytechnic Institute(WPI). It was founded by Çetin Kaya Koç and Christof Paar. CHES 2000 was also held at WPI; after that, the conference has been held at various locations worldwide. After the two CHES' at WPI, the locations in the first ten years were, in chronological order,Paris,San Francisco,Cologne,Boston,Edinburgh,Yokohama,Vienna,Washington, D.C., andLausanne. Since 2009, CHES rotates between the three continents Europe, North America and Asia.[5]The attendance record was set by CHES 2018 in Amsterdam with about 600 participants.
Eurocrypt (or EUROCRYPT) is a conference for cryptography research. The full name of the conference is now the Annual International Conference on the Theory and Applications of Cryptographic Techniques. Eurocrypt is one of the IACR flagship conferences, along with CRYPTO and ASIACRYPT.
Eurocrypt is held annually in the spring in various locations throughout Europe. The first workshop in the series of conferences that became known as Eurocrypt was held in 1982. In 1984, the name "Eurocrypt" was first used. Generally, there have been published proceedings including all papers at the conference every year, with two exceptions; in 1983, no proceedings was produced, and in 1986, the proceedings contained only abstracts.Springerhas published all the official proceedings, first as part of Advances in Cryptology in theLecture Notes in Computer Scienceseries.
Fast Software Encryption, often abbreviated FSE, is a workshop for cryptography research, focused onsymmetric-key cryptographywith an emphasis on fast, practical techniques, as opposed to theory. Though "encryption" is part of the conference title, it is not limited to encryption research; research on other symmetric techniques such asmessage authentication codesandhash functionsis often presented there. FSE has been an IACR workshop since 2002, though the first FSE workshop was held in 1993. FSE is held annually in various locations worldwide, mostly in Europe. The dates of the workshop have varied over the years, but recently, it has been held in February.
PKC or Public-Key Cryptography is the short name of the International Workshop on Theory and Practice in Public Key Cryptography (modified as International Conference on Theory and Practice in Public Key Cryptography since 2006).
The Theory of Cryptography Conference, often abbreviated TCC, is an annual conference for theoretical cryptography research.[6]It was first held in 2004 atMIT, and was also held at MIT in 2005, both times in February. TCC became an IACR-sponsored workshop in 2006. The founding steering committee consists of Mihir Bellare, Ivan Damgard, Oded Goldreich, Shafi Goldwasser, Johan Hastad, Russell Impagliazzo, Ueli Maurer, Silvio Micali, Moni Naor, and Tatsuaki Okamoto.
The importance of the theoretical study of Cryptography is widely recognized by now. This area has contributed much to the practice of cryptography and secure systems as well as to the theory of computation at large.
The needs of the theoretical cryptography (TC) community are best understood in relation to the two communities between which it resides: the Theory of Computation (TOC) community and the Cryptography/Security community. All three communities have grown in volume in recent years. This increase in volume makes the hosting of TC by the existing TOC and Crypto conferences quite problematic. Furthermore, the perspectives of TOC and Crypto on TC do not necessarily fit the internal perspective of TC and the interests of TC. All these indicate a value in the establishment of an independent specialized conference. A dedicated conference not only provides opportunities for research dissemination and interaction, but helps shape the field, give it a recognizable identity, and communicate its message.
The Real World Crypto Symposium is a conference for applied cryptography research, which was started in 2012 byKenny PatersonandNigel Smart. The winner of theLevchin Prizeis announced at RWC.[7][8]Announcements made at the symposium include the first knownchosen prefix attackon SHA-1[9][10]and the inclusion ofend-to-end encryptioninFacebook Messenger.[11]Also, the introduction of the E4 chip took place at RWC.[12]Flaws in messaging apps such asWhatsAppwere also presented there.[13]
CRYPTO, the International Cryptology Conference, is an academic conference on all aspects of cryptography andcryptanalysis. It is held yearly in August inSanta Barbara,Californiaat theUniversity of California, Santa Barbara.[14]
The first CRYPTO was held in 1981.[15]It was the first major conference on cryptology and was all the more important because relations between government, industry and academia were rather tense. Encryption was considered a very sensitive subject and the coming together of delegates from different countries was unheard-of at the time. The initiative for the formation of the IACR came during CRYPTO '82, and CRYPTO '83 was the first IACR sponsored conference.
The IACR Fellows Program (FIACR) has been established as an honor to bestow upon its exceptional members. There are currently 104 IACR Fellows.[16]
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TheSecure Hash Algorithmsare a family ofcryptographic hash functionspublished by theNational Institute of Standards and Technology(NIST) as aU.S.Federal Information Processing Standard(FIPS), including:
The corresponding standards areFIPSPUB 180 (original SHA), FIPS PUB 180-1 (SHA-1), FIPS PUB 180-2 (SHA-1, SHA-256, SHA-384, and SHA-512). NIST has updated Draft FIPS Publication 202, SHA-3 Standard separate from the Secure Hash Standard (SHS).
In the table below,internal statemeans the "internal hash sum" after each compression of a data block.
All SHA-family algorithms, as FIPS-approved security functions, are subject to official validation by theCMVP(Cryptographic Module Validation Program), a joint program run by the AmericanNational Institute of Standards and Technology(NIST) and the CanadianCommunications Security Establishment(CSE).
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HashClashwas avolunteer computingproject running on theBerkeley Open Infrastructure for Network Computing (BOINC)software platform to findcollisionsin theMD5hash algorithm.[1]It was based at Department ofMathematicsandComputer Scienceat theEindhoven University of Technology, andMarc Stevensinitiated the project as part of hismaster's degreethesis.
The project ended after Stevens defended his M.Sc. thesis in June 2007.[2]However,SHA1was added later, and the code repository was ported to git in 2017.[3]
The project was used to create a roguecertificate authoritycertificate in 2009.[4]
Thiscomputer-engineering-related article is astub. You can help Wikipedia byexpanding it.
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cryptis aPOSIX C libraryfunction. It is typically used to compute thehashof user account passwords. The function outputs a text string which alsoencodesthesalt(usually the first two characters are the salt itself and the rest is the hashed result), and identifies the hash algorithm used (defaulting to the "traditional" one explained below). This output string forms a password record, which is usually stored in a text file.
More formally, crypt provides cryptographic key derivation functions for password validation and storage on Unix systems.
There is an unrelatedcryptutility in Unix, which is often confused with the C library function. To distinguish between the two, writers often refer to the utility program ascrypt(1), because it is documented in section 1 of the Unixmanual pages, and refer to the C library function ascrypt(3), because its documentation is in manual section 3.[1]
This samecryptfunction is used both to generate a new hash for storage and also to hash a proffered password with a recordedsaltfor comparison.
Modern Unix implementations of the crypt library routine support a variety of hash schemes. The particular hash algorithm used can be identified by a unique code prefix in the resulting hashtext, following ade factostandard called Modular Crypt Format.[2][3][4]
Thecrypt()library function is also included in thePerl,[5]PHP,[6]Pike,[7]Python[8](although it is now deprecated as of 3.11), andRuby[9]programming languages.
Over time various algorithms have been introduced. To enablebackward compatibility, each scheme started using some convention ofserializingthepassword hashesthat was later called the Modular Crypt Format (MCF).[3]Old crypt(3) hashes generated before the de facto MCF standard may vary from scheme to scheme. A well-defined subset of the Modular Crypt Format was created during thePassword Hashing Competition.[3]The format is defined as:[10]
$<id>[$<param>=<value>(,<param>=<value>)*][$<salt>[$<hash>]]
where
The radix-64 encoding in crypt is called B64 and uses the alphabet./0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyzwhich is different than the more commonRFC 4648 base64
$8$aes256-gcm$hmac-sha2-256$100$y/4YMC4YDLU$fzYDI4jjN6YCyQsYLsaf8A$Ilu4jLcZarD9YnyD /Hejww$okhBlc0cGakSqYxKww
The PHC subset covers a majority of MCF hashes. A number of extra application-defined methods exist.[3]
The original implementation of the crypt() library function[11]in Third Edition Unix[12]mimicked theM-209cipher machine. Rather than encrypting thepasswordwith a key, which would have allowed the password to be recovered from the encrypted value and the key, it used the password itself as a key, and the password database contained the result of encrypting the password with this key.
The original password encryption scheme was found to be too fast and thus subject to brute force enumeration of the most likely passwords.[11]InSeventh Edition Unix,[13]the scheme was changed to a modified form of theDESalgorithm. A goal of this change was to make encryption slower. In addition, the algorithm incorporated a 12-bitsaltin order to ensure that an attacker would be forced to crack each password independently as opposed to being able to target the entire password database simultaneously.
In detail, the user's password is truncated to eight characters, and those are coerced down to only 7-bits each; this forms the 56-bit DES key. That key is then used to encrypt an all-bits-zero block, and then the ciphertext is encrypted again with the same key, and so on for a total of 25 DES encryptions. A 12-bit salt is used to perturb the encryption algorithm, so standard DES implementations can't be used to implement crypt(). The salt and the final ciphertext are encoded into a printable string in a form ofbase64.
This is technically not encryption since the data (all bits zero) is not being kept secret; it's widely known to all in advance. However, one of the properties of DES is that it's very resistant to key recovery even in the face ofknown plaintextsituations. It is theoretically possible that two different passwords could result in exactly the same hash. Thus the password is never "decrypted": it is merely used to compute a result, and the matching results are presumed to be proof that the passwords were "the same."
The advantages of this method have been that the hashtext can be stored and copied among Unix systems without exposing the corresponding plaintext password to the system administrators or other users. This portability has worked for over 30 years across many generations of computing architecture, and across many versions of Unix from many vendors.
The traditional DES-basedcryptalgorithm was originally chosen because DES was resistant to key recovery even in the face of "known plaintext" attacks, and because it was computationally expensive. On the earliest Unix machines it took over a full second to compute a password hash. This also made it reasonably resistant todictionary attacksin that era. At that time password hashes were commonly stored in an account file (/etc/passwd) which was readable to anyone on the system. (This account file was also used to map user ID numbers into names, and user names into full names, etc.).
In the three decades since that time, computers have become vastly more powerful.Moore's Lawhas generally held true, so the computer speed and capacity available for a given financial investment has doubled over 20 times since Unix was first written. This has long since left the DES-based algorithm vulnerable to dictionary attacks, and Unix and Unix-like systems such asLinuxhave used"shadow" filesfor a long time, migrating just the password hash values out of the account file (/etc/passwd) and into a file (conventionally named/etc/shadow) which can only be read by privileged processes.
To increase the computational cost of password breaking, some Unix sites privately started increasing the number of encryption rounds on an ad hoc basis.[citation needed]This had the side effect of making theircrypt()incompatible with the standardcrypt(): the hashes had the same textual form, but were now calculated using a different algorithm. Some sites also took advantage of this incompatibility effect, by modifying the initial block from the standard all-bits-zero.[citation needed]This did not increase the cost of hashing, but meant that precomputed hash dictionaries based on the standardcrypt()could not be applied.
BSDiused a slight modification of the classic DES-based scheme. BSDi extended the salt to 24 bits and made the number of rounds variable (up to 224-1). The chosen number of rounds is encoded in the stored password hash, avoiding the incompatibility that occurred when sites modified the number of rounds used by the original scheme. These hashes are identified by starting with an underscore (_), which is followed by 4 characters representing the number of rounds then 4 characters for the salt.
The BSDi algorithm also supports longer passwords, using DES to fold the initial long password down to the eight 7-bit bytes supported by the original algorithm.
Poul-Henning Kampdesigned a baroque and (at the time) computationally expensive algorithm based on theMD5message digest algorithm. MD5 itself would provide good cryptographic strength for the password hash, but it is designed to be quite quick to calculate relative to the strength it provides. The crypt() scheme is designed to be expensive to calculate, to slow down dictionary attacks. The printable form of MD5 password hashes starts with$1$.
This scheme allows users to have any length password, and they can use any characters supported by their platform (not just 7-bit ASCII). (In practice many implementations limit the password length, but they generally support passwords far longer than any person would be willing to type.) The salt is also an arbitrary string, limited only by character set considerations.
First the passphrase and salt are hashed together, yielding an MD5 message digest. Then a new digest is constructed, hashing together the passphrase, the salt, and the first digest, all in a rather complex form. Then this digest is passed through a thousand iterations of a function which rehashes it together with the passphrase and salt in a manner that varies between rounds. The output of the last of these rounds is the resulting passphrase hash.
The fixed iteration count has caused this scheme to lose the computational expense that it once enjoyed and variable numbers of rounds are now favoured. In June 2012, Poul-Henning Kamp declared the algorithm insecure and encouraged users to migrate to stronger password scramblers.[14]
Niels ProvosandDavid Mazièresdesigned a crypt() scheme calledbcryptbased onBlowfish, and presented it atUSENIXin 1999.[15]The printable form of these hashes starts with$2$,$2a$,$2b$,$2x$or$2y$depending on which variant of the algorithm is used:
Blowfish is notable among block ciphers for its expensive key setup phase. It starts off with subkeys in a standard state, then uses this state to perform a block encryption using part of the key, and uses the result of that encryption (really, a hashing) to replace some of the subkeys. Then it uses this modified state to encrypt another part of the key, and uses the result to replace more of the subkeys. It proceeds in this fashion, using a progressively modified state to hash the key and replace bits of state, until all subkeys have been set.
The number of rounds of keying is a power of two, which is an input to the algorithm. The number is encoded in the textual hash, e.g.$2y$10...
FreeBSD implemented support for theNT LAN Managerhash algorithm to provide easier compatibility with NT accounts viaMS-CHAP.[19]The NT-Hash algorithm is known to be weak, as it uses the deprecatedmd4hash algorithm without any salting.[20]FreeBSD used the$3$prefix for this. Its use is not recommended, as it is easily broken.[1]
The commonly used MD5 based scheme has become easier to attack as computer power has increased. Although the Blowfish-based system has the option of adding rounds and thus remain a challenging password algorithm, it does not use aNIST-approved algorithm. In light of these facts,Ulrich Drepper[de]ofRed Hatled an effort to create a scheme based on theSHA-2(SHA-256 and SHA-512) hash functions.[21]The printable form of these hashes starts with$5$(for SHA-256) or$6$(for SHA-512) depending on which SHA variant is used. Its design is similar to the MD5-based crypt, with a few notable differences:[21]
The specification and sample code have been released into the public domain; it is often referred to as "SHAcrypt".[24]
Additional formats, if any, are described in theman pagesof implementations.[27]
BigCryptis the modified version of DES-Crypt used on HP-UX, Digital Unix, and OSF/1. The main difference between it and DES is that BigCrypt uses all the characters of a password, not just the first 8, and has a variable length hash.[28]
Crypt16is the minor modification of DES, which allows passwords of up to 16 characters. Used on Ultrix and Tru64.[29]
TheGNU C Library(glibc) used by almost all Linux distributions provides an implementation of thecryptfunction which supports the DES, MD5, and (since version 2.7) SHA-2 based hashing algorithms mentioned above.
Ulrich Drepper, the glibc maintainer, rejected bcrypt (scheme 2) support since it isn't approved byNIST.[32]A public domaincrypt_blowfishlibrary is available for systems without bcrypt. It has been integrated into glibc inSUSE Linux.[33]
In August 2017, glibc announced plans to remove its crypt implementation completely. In response, a number of Linux distributions (including, but not limited to, Fedora and Debian) have switched tolibxcrypt, an ABI-compatible implementation that additionally supports new algorithms, including bcrypt and yescrypt.[34]
ThemuslC library supports schemes 1, 2, 5, and 6, plus the tradition DES scheme. The traditional DES code is based on the BSDFreeSec, with modification to be compatible with the glibcUFC-Crypt.[35]
Darwin's nativecrypt()provides limited functionality, supporting only DES and BSDi. OS X uses a few systems for its own password hashes, ranging from the old NeXTStepnetinfoto the newer directory services (ds) system.[36][37]
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md5deepis asoftwarepackage used in thecomputer security,system administrationandcomputer forensicscommunities to run large numbers offilesthrough any of several differentcryptographicdigests. It was originally authored by Jesse Kornblum, at the time a special agent of theAir Force Office of Special Investigations. As of 2017[update], he still maintains it.
The namemd5deepis misleading. Since version 2.0, themd5deeppackage contains several differentprogramsable to performMD5,SHA-1,SHA-256,Tiger192andWhirlpooldigests, each of them named by the digest type followed by the word "deep". Thus, the name may confuse some people into thinking it only provides the MD5 algorithm when the package supports many more.
md5deepcan be invoked in several different ways. Typically users operate itrecursively, wheremd5deepwalks through onedirectoryat a time giving digests of each file found, and recursing into any subdirectories within. Its recursive behavior is approximately adepth-first search, which has the benefit of presenting files inlexicographical order. OnUnix-likesystems, similar functionality can be often obtained by combiningfindwith hashing utilities such asmd5sum,sha256sumortthsum.
md5deepexists forWindowsand mostUnix-based systems, includingOS X. It is present in OS X'sFink,HomebrewandMacPortsprojects. Binary packages exist for mostfreeUnix systems. Many vendors initially resist includingmd5deepas they mistakenly[citation needed]believe its functions can be reproduced with one line of shell scripting.[1]The matching function of the program, however, cannot be done easily in shell.[citation needed]
Becausemd5deepwas written by anemployee of the U.S. government, on government time, it is in thepublic domain. Other software surrounding it, such as graphical front-ends, may be copyrighted.
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md5sumis acomputer programthat calculates and verifies 128-bitMD5hashes, as described in RFC 1321. The MD5 hash functions as a compact digital fingerprint of a file. As with all such hashing algorithms, there is theoretically an unlimited number of files that will have any given MD5 hash. However, it is very unlikely that any two non-identical files in the real world will have the same MD5 hash, unless they have been specifically created to have the same hash.[2]
The underlying MD5 algorithm isno longer deemed secure. Thus, whilemd5sumis well-suited for identifying known files in situations that are not security related, it should not be relied on if there is a chance that files have been purposefully and maliciously tampered. In the latter case, the use of a newer hashing tool such assha256sumis recommended.
md5sumis used to verify the integrity of files, as virtually any change to a file will cause its MD5 hash to change. Most commonly,md5sumis used to verify that a file has not changed as a result of a faulty file transfer, a disk error or non-malicious meddling. Themd5sumprogram is included in mostUnix-likeoperating systemsorcompatibility layerssuch asCygwin.
The original C code was written by Ulrich Drepper and extracted from a 2001 release ofglibc.[3]
All of the following files are assumed to be in the current directory.
File contains hash and filename pairs:
Please note:
md5sumis specific to systems that useGNU coreutilsor a clone such asBusyBox. OnFreeBSDandOpenBSDthe utilities are calledmd5,sha1,sha256, andsha512. These versions offer slightly different options and features. Additionally, FreeBSD offers the "SKEIN" family of message digests.[4]
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TheMD6 Message-Digest Algorithmis acryptographic hash function. It uses aMerkle tree-like structure to allow for immense parallel computation of hashes for very long inputs. Authors claim a performance of 28cycles per bytefor MD6-256 on anIntel Core 2 Duoand provable resistance againstdifferential cryptanalysis.[3]Thesource codeof thereference implementationwas released underMIT license.[4]
Speeds in excess of 1 GB/s have been reported to be possible for long messages on 16-core CPU architecture.[1]
In December 2008, Douglas Held ofFortify Softwarediscovered abuffer overflowin the original MD6 hash algorithm's reference implementation. This error was later made public byRon Riveston 19 February 2009, with a release of a corrected reference implementation in advance of the Fortify Report.[5]
MD6 was submitted to theNIST SHA-3 competition. However, on July 1, 2009, Rivest posted a comment at NIST that MD6 is not yet ready to be a candidate for SHA-3 because of speed issues, a "gap in the proof that the submitted version of MD6 is resistant to differential attacks", and an inability to supply such a proof for a faster reduced-round version,[6]although Rivest also stated at the MD6 website that it is not withdrawn formally.[7]MD6 did not advance to the second round of the SHA-3 competition. In September 2011, a paper presenting an improved proof that MD6 and faster reduced-round versions are resistant to differential attacks[8]was posted to the MD6 website.[9]
A change in even a single bit of the message will, with overwhelming probability, result in a completely different message digest due to theavalanche effect:
The hash of the zero-length string is:
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Pseudo-collision attack against up to 46 rounds of SHA-256.[2]
SHA-2(Secure Hash Algorithm 2) is a set ofcryptographic hash functionsdesigned by the United StatesNational Security Agency(NSA) and first published in 2001.[3][4]They are built using theMerkle–Damgård construction, from a one-way compression function itself built using theDavies–Meyer structurefrom a specialized block cipher.
SHA-2 includes significant changes from its predecessor,SHA-1. The SHA-2 family consists of six hash functions with digests (hash values) that are 224, 256, 384 or 512 bits:[5]SHA-224, SHA-256, SHA-384, SHA-512, SHA-512/224, SHA-512/256. SHA-256 and SHA-512 are hash functions whose digests are eight 32-bit and 64-bit words, respectively. They use different shift amounts and additive constants, but their structures are otherwise virtually identical, differing only in the number of rounds. SHA-224 and SHA-384 are truncated versions of SHA-256 and SHA-512 respectively, computed with different initial values. SHA-512/224 and SHA-512/256 are also truncated versions of SHA-512, but the initial values are generated using the method described inFederal Information Processing Standards(FIPS) PUB 180-4.
SHA-2 was first published by theNational Institute of Standards and Technology(NIST) as a U.S. federal standard. The SHA-2 family of algorithms are patented in the U.S.[6]The United States has released the patent under aroyalty-freelicense.[5]
As of 2011,[update]the best public attacks breakpreimage resistancefor 52 out of 64 rounds of SHA-256 or 57 out of 80 rounds of SHA-512, andcollision resistancefor 46 out of 64 rounds of SHA-256.[1][2]
With the publication of FIPS PUB 180-2, NIST added three additional hash functions in the SHA family. The algorithms are collectively known as SHA-2, named after their digest lengths (in bits): SHA-256, SHA-384, and SHA-512.
The algorithms were first published in 2001 in the draft FIPS PUB 180-2, at which time public review and comments were accepted. In August 2002, FIPS PUB 180-2 became the newSecure Hash Standard, replacing FIPS PUB 180-1, which was released in April 1995. The updated standard included the original SHA-1 algorithm, with updated technical notation consistent with that describing the inner workings of the SHA-2 family.[4]
In February 2004, a change notice was published for FIPS PUB 180-2, specifying an additional variant, SHA-224, defined to match the key length of two-keyTriple DES.[7]In October 2008, the standard was updated in FIPS PUB 180-3, including SHA-224 from the change notice, but otherwise making no fundamental changes to the standard. The primary motivation for updating the standard was relocating security information about the hash algorithms and recommendations for their use to Special Publications 800-107 and 800-57.[8][9][10]Detailed test data and example message digests were also removed from the standard, and provided as separate documents.[11]
In January 2011, NIST published SP800-131A, which specified a move from the then-current minimum of 80-bit security (provided by SHA-1) allowable for federal government use until the end of 2013, to 112-bit security (provided by SHA-2) being both the minimum requirement (starting in 2014) and the recommendedsecurity level(starting from the publication date in 2011).[12]
In March 2012, the standard was updated in FIPS PUB 180-4, adding the hash functions SHA-512/224 and SHA-512/256, and describing a method for generating initial values for truncated versions of SHA-512. Additionally, a restriction onpaddingthe input data prior to hash calculation was removed, allowing hash data to be calculated simultaneously with content generation, such as a real-time video or audio feed. Padding the final data block must still occur prior to hash output.[13]
In July 2012, NIST revised SP800-57, which provides guidance for cryptographic key management. The publication disallowed creation of digital signatures with a hash security lower than 112 bits after 2013. The previous revision from 2007 specified the cutoff to be the end of 2010.[10]In August 2012, NIST revised SP800-107 in the same manner.[9]
TheNIST hash function competitionselected a new hash function,SHA-3, in 2012.[14]The SHA-3 algorithm is not derived from SHA-2.
The SHA-2 hash function is implemented in some widely used security applications and protocols, includingTLSandSSL,PGP,SSH,S/MIME, andIPsec. The inherent computational demand of SHA-2 algorithms has driven the proposal of more efficient solutions, such as those based on application-specific integrated circuits (ASICs) hardware accelerators.[15]
SHA-256 is used for authenticatingDebiansoftware packages[16]and in theDKIMmessage signing standard; SHA-512 is part of a system to authenticate archival video from theInternational Criminal Tribunal of the Rwandan genocide.[17]SHA-256 and SHA-512 are used inDNSSEC.[18]Linux distributions usually use 512-bit SHA-2 for secure password hashing.[19][20]
Severalcryptocurrencies, includingBitcoin, use SHA-256 for verifying transactions and calculatingproof of work[21]orproof of stake.[22]The rise ofASICSHA-2 accelerator chips has led to the use ofscrypt-based proof-of-work schemes.
SHA-1 and SHA-2 are theSecure Hash Algorithmsrequired by law for use in certainU.S. Governmentapplications, including use within other cryptographic algorithms and protocols, for the protection of sensitive unclassified information. FIPS PUB 180-1 also encouraged adoption and use of SHA-1 by private and commercial organizations. SHA-1 is being retired for most government uses; the U.S. National Institute of Standards and Technology says, "Federal agenciesshouldstop using SHA-1 for...applications that require collision resistance as soon as practical, and must use the SHA-2 family of hash functions for these applications after 2010" (emphasis in original).[23]NIST's directive that U.S. government agencies ought to, but not explicitly must, stop uses of SHA-1 after 2010[24]was hoped to accelerate migration away from SHA-1.
The SHA-2 functions were not quickly adopted initially, despite better security than SHA-1. Reasons might include lack of support for SHA-2 on systems running Windows XP SP2 or older[25]and a lack of perceived urgency since SHA-1 collisions had not yet been found. TheGoogle Chrometeam announced a plan to make their web browser gradually stop honoring SHA-1-dependent TLS certificates over a period from late 2014 and early 2015.[26][27][28]Similarly, Microsoft announced[29]thatInternet ExplorerandEdge [Legacy]would stop honoring public SHA-1-signed TLS certificates from February 2017.Mozilladisabled SHA-1 in early January 2016, but had to re-enable it temporarily via aFirefoxupdate, after problems with web-based user interfaces of some router models andsecurity appliances.[30]
For a hash function for whichLis the number ofbitsin themessage digest, finding a message that corresponds to a given message digest can always be done using abrute forcesearch in 2Levaluations. This is called apreimage attackand may or may not be practical depending onLand the particular computing environment. The second criterion, finding two different messages that produce the same message digest, known as acollision, requires on average only 2L/2evaluations using abirthday attack.
Some of the applications that use cryptographic hashes, such as password storage, are only minimally affected by acollision attack. Constructing a password that works for a given account requires a preimage attack, as well as access to the hash of the original password (typically in theshadowfile) which may or may not be trivial. Reversing password encryption (e.g., to obtain a password to try against a user's account elsewhere) is not made possible by the attacks. (However, even a secure password hash cannot prevent brute-force attacks onweak passwords.)
In the case of document signing, an attacker could not simply fake a signature from an existing document—the attacker would have to produce a pair of documents, one innocuous and one damaging, and get the private key holder to sign the innocuous document. There are practical circumstances in which this is possible; until the end of 2008, it was possible to create forgedSSLcertificates using anMD5collision which would be accepted by widely used web browsers.[31]
Increased interest in cryptographic hash analysis during the SHA-3 competition produced several new attacks on the SHA-2 family, the best of which are given in the table below. Only the collision attacks are of practical complexity; none of the attacks extend to the full round hash function.
AtFSE2012, researchers atSonygave a presentation suggesting pseudo-collision attacks could be extended to 52 rounds on SHA-256 and 57 rounds on SHA-512 by building upon thebicliquepseudo-preimage attack.[32]
Implementations of all FIPS-approved security functions can be officially validated through theCMVP program, jointly run by theNational Institute of Standards and Technology(NIST) and theCommunications Security Establishment(CSE). For informal verification, a package to generate a high number of test vectors is made available for download on the NIST site; the resulting verification, however, does not replace the formal CMVP validation, which is required by law[citation needed]for certain applications.
As of December 2013,[update]there are over 1300 validated implementations of SHA-256 and over 900 of SHA-512, with only 5 of them being capable of handling messages with a length in bits not a multiple of eight while supporting both variants.[41]
Hash values of an empty string (i.e., a zero-length input text).
Even a small change in the message will (with overwhelming probability) result in a different hash, due to theavalanche effect. For example, adding a period to the end of the following sentence changes approximately half (111 out of 224) of the bits in the hash, equivalent to picking a new hash at random:
Pseudocode for the SHA-256 algorithm follows. Note the great increase in mixing between bits of thew[16..63]words compared to SHA-1.
The computation of thechandmajvalues can be optimized the same wayas described for SHA-1.
SHA-224 is identical to SHA-256, except that:
SHA-512 is identical in structure to SHA-256, but:
SHA-384 is identical to SHA-512, except that:
SHA-512/t is identical to SHA-512 except that:
TheSHA-512/t IV generation functionevaluates amodified SHA-512on the ASCII string "SHA-512/t", substituted with the decimal representation oft. Themodified SHA-512is the same as SHA-512 except its initial valuesh0throughh7have each beenXORedwith the hexadecimal constant0xa5a5a5a5a5a5a5a5.
Sample C implementation for SHA-2 family of hash functions can be found inRFC6234.
In the table below,internal statemeans the "internal hash sum" after each compression of a data block.
In the bitwise operations column, "Rot" stands forrotate no carry, and "Shr" stands forright logical shift. All of these algorithms employmodular additionin some fashion except for SHA-3.
More detailed performance measurements on modern processor architectures are given in the table below.
The performance numbers labeled 'x86' were running using 32-bit code on 64-bit processors, whereas the 'x86-64' numbers are native 64-bit code. While SHA-256 is designed for 32-bit calculations, it does benefit from code optimized for 64-bit processors on the x86 architecture. 32-bit implementations of SHA-512 are significantly slower than their 64-bit counterparts. Variants of both algorithms with different output sizes will perform similarly, since the message expansion and compression functions are identical, and only the initial hash values and output sizes are different. The best implementations of MD5 and SHA-1 perform between 4.5 and 6 cycles per byte on modern processors.
Testing was performed by theUniversity of Illinois at Chicagoon their hydra8 system running an Intel Xeon E3-1275 V2 at a clock speed of 3.5 GHz, and on their hydra9 system running an AMD A10-5800K APU at a clock speed of 3.8 GHz.[47]The referenced cycles per byte speeds above are the median performance of an algorithm digesting a 4,096 byte message using the SUPERCOP cryptographic benchmarking software.[48]The MiB/s performance is extrapolated from the CPU clockspeed on a single core; real-world performance will vary due to a variety of factors.
Cryptography libraries that support SHA-2:
Hardware acceleration is provided by the following processor extensions:
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TheMD2 Message-Digest Algorithmis acryptographic hash functiondeveloped byRonald Rivestin 1989.[2]The algorithm is optimized for8-bitcomputers. MD2 is specified inIETF RFC1319.[3]The "MD" in MD2 stands for "Message Digest".
Even though MD2 is not yet fully compromised, the IETF retired MD2 to "historic" status in 2011, citing "signs of weakness". It is deprecated in favor ofSHA-256and other strong hashing algorithms.[4]
Nevertheless, as of 2014[update], it remained in use inpublic key infrastructuresas part ofcertificatesgenerated with MD2 andRSA.[citation needed]
The 128-bit hash value of any message is formed by padding it to a multiple of the block length (128 bits or 16bytes) and adding a 16-bytechecksumto it. For the actual calculation, a 48-byte auxiliary block and a 256-byteS-tableare used. The constants were generated by shuffling the integers 0 through 255 using a variant ofDurstenfeld's algorithmwith apseudorandom number generatorbased on decimal digits ofπ(pi)[3][5](seenothing up my sleeve number). The algorithm runs through a loop where it permutes each byte in the auxiliary block 18 times for every 16 input bytes processed. Once all of the blocks of the (lengthened) message have been processed, the first partial block of the auxiliary block becomes the hash value of the message.
The S-table values in hex are:
The 128-bit (16-byte) MD2 hashes (also termedmessage digests) are typically represented as 32-digithexadecimalnumbers. The following demonstrates a 43-byteASCIIinput and the corresponding MD2 hash:
As the result of theavalanche effectin MD2, even a small change in the input message will (with overwhelming probability) result in a completely different hash. For example, changing the letterdtocin the message results in:
The hash of the zero-length string is:
Rogier and Chauvaud presented in 1995[6]collisions of MD2'scompression function, although they were unable to extend the attack to the full MD2. The described collisions was published in 1997.[7]
In 2004, MD2 was shown to be vulnerable to apreimage attackwithtime complexityequivalent to 2104applications of the compression function.[8]The author concludes, "MD2 can no longer be considered a secure one-way hash function".
In 2008, MD2 has further improvements on apreimage attackwithtime complexityof 273compression function evaluations and memory requirements of 273message blocks.[9]
In 2009, MD2 was shown to be vulnerable to acollision attackwithtime complexityof 263.3compression function evaluations and memory requirements of 252hash values. This is slightly better than thebirthday attackwhich is expected to take 265.5compression function evaluations.[10]
In 2009, security updates were issued disabling MD2 inOpenSSL,GnuTLS, andNetwork Security Services.[11]
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TheIntercontinental Dictionary Series(commonly abbreviated asIDS) is a large database of topical vocabulary lists in various world languages. The general editor of the database isBernard Comrieof theMax Planck Institute for Evolutionary Anthropology,Leipzig. Mary Ritchie Key of theUniversity of California, Irvineis the founding editor. The database has an especially large selection ofindigenous South American languagesandNortheast Caucasian languages.
The Intercontinental Dictionary Series' advanced browsing function allows users to make custom tables which compare languages in side-by-side columns.
Below are the languages that are currently included in the Intercontinental Dictionary Series. The languages are grouped by language families, some of which are still hypothetical.
It is part of theCross-Linguistic Linked Dataproject hosted by theMax Planck Institute for the Science of Human History.[1]
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Incryptography,key stretchingtechniques are used to make a possibly weak key, typically a password orpassphrase, more secure against abrute-force attackby increasing the resources (time and possibly space) it takes to test each possible key. Passwords or passphrases created by humans are often short or predictable enough to allow password cracking, and key stretching is intended to make such attacks more difficult by complicating a basic step of trying a single password candidate. Key stretching also improves security in some real-world applications where the key length has been constrained, by mimicking a longer key length from the perspective of a brute-force attacker.[1]
There are several ways to perform key stretching. One way is to apply acryptographic hash functionor ablock cipherrepeatedly in a loop. For example, in applications where the key is used for acipher, thekey schedulein the cipher may be modified so that it takes a specific length of time to perform. Another way is to use cryptographic hash functions that have large memory requirements – these can be effective in frustrating attacks by memory-bound adversaries.[2]
Key stretching algorithms depend on an algorithm which receives an input key and then expends considerable effort to generate a stretched cipher (called anenhanced key[citation needed]) mimicking randomness and longer key length. The algorithm must have no known shortcut, so the most efficient way to relate the input and cipher is to repeat the key stretching algorithm itself. This compels brute-force attackers to expend the same effort for each attempt. If this added effort compares to a brute-force key search of all keys with a certain key length, then the input key may be described asstretchedby that same length.[1]
Key stretching leaves an attacker with two options:
If the attacker uses the same class of hardware as the user, each guess will take the similar amount of time to process as it took the user (for example, one second). Even if the attacker has much greater computing resources than the user, the key stretching will still slow the attacker down while not seriously affecting the usability of the system for any legitimate user. This is because the user's computer only has to compute the stretching function once upon the user entering their password, whereas the attacker must compute it for every guess in the attack.
This process does not alter the original key-space entropy. The key stretching algorithm isdeterministic, allowing a weak input to always generate the same enhanced key, but therefore limiting the enhanced key to no more possible combinations than the input key space. Consequently, this attack remains vulnerable if unprotected against certaintime-memory tradeoffssuch as developingrainbow tablesto target multiple instances of the enhanced key space in parallel (effectively ashortcutto repeating the algorithm). For this reason, key stretching is often combined withsalting.[1]
Many libraries provide functions which perform key stretching as part of their function; seecrypt(3)for an example.[4]PBKDF2is for generating an encryption key from a password, and not necessarily for password authentication. PBKDF2 can be used for both if the number of output bits is less than or equal to the internal hashing algorithm used in PBKDF2, which is usuallySHA-2(up to 512 bits), or used as an encryption key to encrypt static data.[5]
These examples assume that a consumer CPU can do about 65,000SHA-1hashes in one second. Thus, a program that uses key stretching can use 65,000 rounds of hashes and delay the user for at most one second.
Testing a trial password or passphrase typically requires one hash operation. But if key stretching was used, the attacker must compute a strengthened key for each key they test, meaning there are 65,000 hashes to compute per test. This increases the attacker's workload by a factor of 65,000, approximately 216, which means the enhanced key is worth about 16 additional bits in key strength.
Moore's lawasserts that computer speed doubles roughly every 2 years. Under this assumption, every 2 years one more bit of key strength is plausibly brute-forcible. This implies that 16 extra bits of strength is worth about 16×2 = 32 years later cracking, but it also means that the number of key stretching rounds a system uses should be doubled about every 2 years to maintain the same level of security (since most keys are more secure than necessary, systems that require consistent deterministic key generation will likely not update the number of iterations used in key stretching. In such a case, the designer should take into consideration how long they wish for the key derivation system to go unaltered and should choose an appropriate number of hashes for the lifespan of the system).
CPU-bound hash functions are still vulnerable tohardware implementations. Such implementations of SHA-1 exist using as few as 5,000 gates, and 400 clock cycles.[6]With multi-million gateFPGAscosting less than $100,[7]an attacker can build a fullyunrolledhardware cracker for about $5,000.[citation needed]Such a design, clocked at 100 MHz can test about 300,000 keys/second. The attacker is free to choose a good price/speed compromise, for example a 150,000 keys/second design for $2,500.[citation needed]The key stretching still slows down the attacker in such a situation; a $5,000 design attacking a straight SHA-1 hash would be able to try 300,000÷216≈ 4.578 keys/second.[citation needed]
Similarly, modern consumerGPUscan speed up hashing considerably. For example, in a benchmark, aNvidiaRTX 2080 SUPER FE computes over 10 billion SHA1 hashes per second.[8]
To defend against the hardware approach,memory-boundcryptographic functions have been developed. These functions access a lot of memory in a way that makescachingineffective. Since large amounts of low latency memory are expensive, potential attackers are discouraged from pursuing such attacks.
The first deliberately slow password-based key derivation function"CRYPT"was described in 1978 byRobert Morrisfor encryptingUnixpasswords.[9]It used an iteration count of 25, a 12-bit salt and a variant ofDESas the sub-function. (DES proper was avoided in an attempt to frustrate attacks using standard DES hardware.) Passwords were limited to a maximum of eightASCIIcharacters. While it was a great advancement for its time, CRYPT(3) is now considered inadequate. The iteration count, designed for thePDP-11era, is too low, 12 bits of salt is an inconvenience but does not stop precomputed dictionary attacks, and the eight-character limit prevents the use of strongerpassphrases.
Modern password-based key derivation functions, such asPBKDF2, use a cryptographic hash, such asSHA-2, a longer salt (e.g. 64 bits) and a high iteration count. The U.S. National Institute of Standards and Technology (NIST) recommends a minimum iteration count of 10,000.[10]: 5.1.1.2"For especially critical keys, or for very powerful systems or systems where user-perceived performance is not critical, an iteration count of 10,000,000 may be appropriate.”[11]: 5.2
In 2009, a memory-intensive key strengthening algorithm,scrypt, was introduced with the intention of limiting the use of custom, highly parallel hardware to speed up key testing.[12]
In 2013, aPassword Hashing Competitionwas held to select an improved key stretching standard that would resist attacks from graphics processors and special purpose hardware. The winner,Argon2, was selected on July 1, 2015.[13]
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Password strengthis a measure of the effectiveness of apasswordagainst guessing orbrute-force attacks. In its usual form, it estimates how many trials an attacker who does not have direct access to the password would need, on average, to guess it correctly. The strength of a password is a function of length, complexity, and unpredictability.[1]
Using strong passwords lowers the overallriskof a security breach, but strong passwords do not replace the need for other effectivesecurity controls.[2]The effectiveness of a password of a given strength is strongly determined by the design and implementation of theauthentication factors(knowledge, ownership, inherence). The first factor is the main focus of this article.
The rate at which an attacker can submit guessed passwords to the system is a key factor in determining system security. Some systems impose a time-out of several seconds after a small number (e.g. three) of failed password entry attempts. In the absence of othervulnerabilities, such systems can be effectively secured with relatively simple passwords. However, the system store information about the user's passwords in some form and if that information is stolen, say by breaching system security, the user's passwords can be at risk.
In 2019, the United Kingdom'sNCSCanalyzed public databases of breached accounts to see which words, phrases, and strings people used. The most popular password on the list was 123456, appearing in more than 23 million passwords. The second-most popular string, 123456789, was not much harder to crack, while the top five included "qwerty", "password", and 1111111.[3]
Passwords are created either automatically (using randomizing equipment) or by a human; the latter case is more common. While the strength of randomly chosen passwords against abrute-force attackcan be calculated with precision, determining the strength of human-generated passwords is difficult.
Typically, humans are asked to choose a password, sometimes guided by suggestions or restricted by a set of rules, when creating a new account for a computer system or internet website. Only rough estimates of strength are possible since humans tend to follow patterns in such tasks, and those patterns can usually assist an attacker.[4]In addition, lists of commonly chosen passwords are widely available for use by password-guessing programs. Such lists include the numerous online dictionaries for various human languages, breached databases ofplaintextandhashedpasswords from various online business and social accounts, along with other common passwords. All items in such lists are considered weak, as are passwords that are simple modifications of them.
Although random password generation programs are available and are intended to be easy to use, they usually generate random, hard-to-remember passwords, often resulting in people preferring to choose their own. However, this is inherently insecure because the person's lifestyle, entertainment preferences, and other key individualistic qualities usually come into play to influence the choice of password, while the prevalence of onlinesocial mediahas made obtaining information about people much easier.
Systems that use passwords forauthenticationmust have some way to check any password entered to gain access. If the valid passwords are simply stored in a system file or database, an attacker who gains sufficient access to the system will obtain all user passwords, giving the attacker access to all accounts on the attacked system and possibly other systems where users employ the same or similar passwords. One way to reduce this risk is to store only acryptographic hashof each password instead of the password itself. Standard cryptographic hashes, such as theSecure Hash Algorithm(SHA) series, are very hard to reverse, so an attacker who gets hold of the hash value cannot directly recover the password. However, knowledge of the hash value lets the attacker quickly test guesses offline.Password crackingprograms are widely available that will test a large number of trial passwords against a purloined cryptographic hash.
Improvements in computing technology keep increasing the rate at which guessed passwords can be tested. For example, in 2010, theGeorgia Tech Research Institutedeveloped a method of usingGPGPUto crack passwords much faster.[5]Elcomsoftinvented the usage of common graphic cards for quicker password recovery in August 2007 and soon filed a corresponding patent in the US.[6]By 2011, commercial products were available that claimed the ability to test up to 112,000 passwords per second on a standard desktop computer, using a high-end graphics processor for that time.[7]Such a device will crack a six-letter single-case password in one day. The work can be distributed over many computers for an additional speedup proportional to the number of available computers with comparable GPUs. Specialkey stretchinghashes are available that take a relatively long time to compute, reducing the rate at which guessing can take place. Although it is considered best practice to use key stretching, many common systems do not.
Another situation where quick guessing is possible is when the password is used to form acryptographic key. In such cases, an attacker can quickly check to see if a guessed password successfully decodes encrypted data. For example, one commercial product claims to test 103,000WPAPSK passwords per second.[8]
If a password system only stores the hash of the password, an attacker can pre-compute hash values for common password variants and all passwords shorter than a certain length, allowing very rapid recovery of the password once its hash is obtained. Very long lists of pre-computed password hashes can be efficiently stored usingrainbow tables. This method of attack can be foiled by storing a random value, called acryptographic salt, along with the hash. The salt is combined with the password when computing the hash, so an attacker precomputing a rainbow table would have to store for each password its hash with every possible salt value. This becomes infeasible if the salt has a big enough range, say a 32-bit number. Many authentication systems in common use do not employ salts and rainbow tables are available on the Internet for several such systems.
Password strength is specified by the amount ofinformation entropy, which is measured inshannon(Sh) and is a concept frominformation theory. It can be regarded as the minimum number ofbitsnecessary to hold the information in a password of a given type. A related measure is thebase-2 logarithmof the number of guesses needed to find the password with certainty, which is commonly referred to as the "bits of entropy".[9]A password with 42 bits of entropy would be as strong as a string of 42 bits chosen randomly, for example by afair cointoss. Put another way, a password with 42 bits of entropy would require 242(4,398,046,511,104) attempts to exhaust all possibilities during abrute force search. Thus, increasing the entropy of the password by one bit doubles the number of guesses required, making an attacker's task twice as difficult. On average, an attacker will have to try half the possible number of passwords before finding the correct one.[4]
Random passwords consist of a string of symbols of specified length taken from some set of symbols using a random selection process in which each symbol is equally likely to be selected. The symbols can be individual characters from a character set (e.g., theASCIIcharacter set), syllables designed to form pronounceable passwords or even words from a word list (thus forming apassphrase).
The strength of random passwords depends on the actual entropy of the underlying number generator; however, these are often not truly random, but pseudorandom. Many publicly available password generators use random number generators found in programming libraries that offer limited entropy. However, most modern operating systems offer cryptographically strong random number generators that are suitable for password generation. It is also possible to use ordinarydiceto generate random passwords(seeRandom password generator § Stronger methods). Random password programs often can ensure that the resulting password complies with a localpassword policy; for instance, by always producing a mix of letters, numbers, and special characters.
For passwords generated by a process that randomly selects a string of symbols of length,L, from a set ofNpossible symbols, the number of possible passwords can be found by raising the number of symbols to the powerL, i.e.NL. Increasing eitherLorNwill strengthen the generated password. The strength of a random password as measured by theinformation entropyis just thebase-2 logarithmor log2of the number of possible passwords, assuming each symbol in the password is produced independently. Thus a random password's information entropy,H, is given by the formula:
H=log2NL=Llog2N=LlogNlog2{\displaystyle H=\log _{2}N^{L}=L\log _{2}N=L{\log N \over \log 2}}
whereNis the number of possible symbols andLis the number of symbols in the password.His measured inbits.[4][10]In the last expression,logcan be to anybase.
Abinarybyteis usually expressed using two hexadecimal characters.
To find the length,L,needed to achieve a desired strengthH,with a password drawn randomly from a set ofNsymbols, one computes:
L=⌈Hlog2N⌉{\displaystyle L={\left\lceil {\frac {H}{\log _{2}N}}\right\rceil }}
where⌈⌉{\displaystyle \left\lceil \ \right\rceil }denotes the mathematicalceiling function,i.e.rounding up to the next largestwhole number.
The following table uses this formula to show the required lengths of truly randomly generated passwords to achieve desired password entropies for common symbol sets:
People are notoriously poor at achieving sufficient entropy to produce satisfactory passwords. According to one study involving half a million users, the average password entropy was estimated at 40.54 bits.[11]
Thus, in one analysis of over 3 million eight-character passwords, the letter "e" was used over 1.5 million times, while the letter "f" was used only 250,000 times. Auniform distributionwould have had each character being used about 900,000 times. The most common number used is "1", whereas the most common letters are a, e, o, and r.[12]
Users rarely make full use of larger character sets in forming passwords. For example, hacking results obtained from a MySpace phishing scheme in 2006 revealed 34,000 passwords, of which only 8.3% used mixed case, numbers, and symbols.[13]
The full strength associated with using the entire ASCII character set (numerals, mixed case letters, and special characters) is only achieved if each possible password is equally likely. This seems to suggest that all passwords must contain characters from each of several character classes, perhaps upper and lower-case letters, numbers, and non-alphanumeric characters. Such a requirement is a pattern in password choice and can be expected to reduce an attacker's "work factor" (in Claude Shannon's terms). This is a reduction in password "strength". A better requirement would be to require a passwordnotto contain any word in an online dictionary, or list of names, or any license plate pattern from any state (in the US) or country (as in the EU). If patterned choices are required, humans are likely to use them in predictable ways, such as capitalizing a letter, adding one or two numbers, and a special character. This predictability means that the increase in password strength is minor when compared to random passwords.
Password Safety Awareness Projects
Google developed Interland teach the kid internet audience safety on internet. On the chapter calledTower Of Tresureit is advised to use unusual names paired with characters like (₺&@#%) with a game.[14]
NISTSpecial Publication 800-63 of June 2004 (revision two) suggested a scheme to approximate the entropy of human-generated passwords:[4]
Using this scheme, an eight-character human-selected password without uppercase characters and non-alphabetic characters OR with either but of the two character sets is estimated to have eighteen bits of entropy. The NIST publication concedes that at the time of development, little information was available on the real-world selection of passwords. Later research into human-selected password entropy using newly available real-world data has demonstrated that the NIST scheme does not provide a valid metric for entropy estimation of human-selected passwords.[15]The June 2017 revision of SP 800-63 (Revision three) drops this approach.[16]
Because national keyboard implementations vary, not all 94 ASCII printable characters can be used everywhere. This can present a problem to an international traveler who wished to log into a remote system using a keyboard on a local computer(see article concerned withkeyboard layouts). Many handheld devices, such astablet computersandsmart phones, require complex shift sequences or keyboard app swapping to enter special characters.
Authentication programs can vary as to the list of allowable password characters. Some do not recognize case differences (e.g., the upper-case "E" is considered equivalent to the lower-case "e"), and others prohibit some of the other symbols. In the past few decades, systems have permitted more characters in passwords, but limitations still exist. Systems also vary as to the maximum length of passwords allowed.
As a practical matter, passwords must be both reasonable and functional for the end user as well as strong enough for the intended purpose. Passwords that are too difficult to remember may be forgotten and so are more likely to be written on paper, which some consider a security risk.[17]In contrast, others argue that forcing users to remember passwords without assistance can only accommodate weak passwords, and thus poses a greater security risk. According toBruce Schneier, most people are good at securing their wallets or purses, which is a "great place" to store a written password.[18]
The minimum number of bits of entropy needed for a password depends on thethreat modelfor the given application. Ifkey stretchingis not used, passwords with more entropy are needed. RFC 4086, "Randomness Requirements for Security", published June 2005, presents some example threat models and how to calculate the entropy desired for each one.[19]Their answers vary between 29 bits of entropy needed if only online attacks are expected, and up to 96 bits of entropy needed for important cryptographic keys used in applications like encryption where the password or key needs to be secure for a long period and stretching isn't applicable. A 2010Georgia Tech Research Institutestudy based on unstretched keys recommended a 12-character random password but as a minimum length requirement.[5][20]It pays to bear in mind that since computing power continually grows, to prevent offline attacks the required number of bits of entropy should also increase over time.
The upper end is related to the stringent requirements of choosing keys used in encryption. In 1999,an Electronic Frontier Foundation projectbroke 56-bitDESencryption in less than a day using specially designed hardware.[21]In 2002,distributed.netcracked a 64-bit key in 4 years, 9 months, and 23 days.[22]As of October 12, 2011,distributed.netestimates that cracking a 72-bit key using current hardware will take about 45,579 days or 124.8 years.[23]Due to currently understood limitations from fundamental physics, there is no expectation that anydigital computer(or combination) will be capable of breaking 256-bit encryption via a brute-force attack.[24]Whether or notquantum computerswill be able to do so in practice is still unknown, though theoretical analysis suggests such possibilities.[25]
Guidelines for choosing good passwords are typically designed to make passwords harder to discover by intelligent guessing. Common guidelines advocated by proponents of software system security have included:[26][27][28][29][30]
Forcing the inclusion of lowercase letters, uppercase letters, numbers, and symbols in passwords was a common policy but has been found to decrease security, by making it easier to crack. Research has shown how predictable the common use of such symbols are, and the US[34]and UK[35]government cyber security departments advise against forcing their inclusion in password policy. Complex symbols also make remembering passwords much harder, which increases writing down, password resets, and password reuse – all of which lower rather than improve password security. The original author of password complexity rules, Bill Burr, has apologized and admits they decrease security, as research has found; this was widely reported in the media in 2017.[36]Online security researchers[37]and consultants are also supportive of the change[38]in best practice advice on passwords.
Some guidelines advise against writing passwords down, while others, noting the large numbers of password-protected systems users must access, encourage writing down passwords as long as the written password lists are kept in a safe place, not attached to a monitor or in an unlocked desk drawer.[39]Use of apassword manageris recommended by the NCSC.[40]
The possible character set for a password can be constrained by different websites or by the range of keyboards on which the password must be entered.[41]
As with any security measure, passwords vary in strength; some are weaker than others. For example, the difference in strength between a dictionary word and a word with obfuscation (e.g. letters in the password are substituted by, say, numbers — a common approach) may cost a password-cracking device a few more seconds; this adds little strength. The examples below illustrate various ways weak passwords might be constructed, all of which are based on simple patterns which result in extremely low entropy, allowing them to be tested automatically at high speeds.:[12]
There are many other ways a password can be weak,[44]corresponding to the strengths of various attack schemes; the core principle is that a password should have high entropy (usually taken to be equivalent to randomness) andnotbe readily derivable by any "clever" pattern, nor should passwords be mixed with information identifying the user. Online services often provide a restore password function that a hacker can figure out and by doing so bypass a password.
In the landscape of 2012, as delineated byWilliam Cheswickin an article for ACM magazine, password security predominantly emphasized an alpha-numeric password of eight characters or more. Such a password, it was deduced, could resist ten million attempts per second for a duration of 252 days. However, with the assistance of contemporary GPUs at the time, this period was truncated to just about 9 hours, given a cracking rate of 7 billion attempts per second. A 13-character password was estimated to withstand GPU-computed attempts for over 900,000 years.[45][46]
In the context of 2023 hardware technology, the 2012 standard of an eight-character alpha-numeric password has become vulnerable, succumbing in a few hours. The time needed to crack a 13-character password is reduced to a few years. The current emphasis, thus, has shifted. Password strength is now gauged not just by its complexity but its length, with recommendations leaning towards passwords comprising at least 13-16 characters. This era has also seen the rise of Multi-Factor Authentication (MFA) as a crucial fortification measure. The advent and widespread adoption of password managers have further aided users in cultivating and maintaining an array of strong, unique passwords.[47]
A password policy is a guide to choosing satisfactory passwords. It is intended to:
Previous password policies used to prescribe the characters which passwords must contain, such as numbers, symbols, or upper/lower case. While this is still in use, it has been debunked as less secure by university research,[48]by the original instigator[49]of this policy, and by the cyber security departments (and other related government security bodies[50]) of USA[51]and UK.[52]Password complexity rules of enforced symbols were previously used by major platforms such as Google[53]and Facebook,[54]but these have removed the requirement following the discovery that they actually reduced security. This is because the human element is a far greater risk than cracking, and enforced complexity leads most users to highly predictable patterns (number at the end, swap 3 for E, etc.) which helps crack passwords. So password simplicity and length (passphrases) are the new best practice and complexity is discouraged. Forced complexity rules also increase support costs, and user friction and discourage user signups.
Password expiration was in some older password policies but has been debunked[36]as best practice and is not supported by USA or UK governments, or Microsoft which removed[55]the password expiry feature. Password expiration was previously trying to serve two purposes:[56]
However, password expiration has its drawbacks:[57][58]
The hardest passwords to crack, for a given length and character set, are random character strings; if long enough they resist brute force attacks (because there are many characters) and guessing attacks (due to high entropy). However, such passwords are typically the hardest to remember. The imposition of a requirement for such passwords in a password policy may encourage users to write them down, store them inmobile devices, or share them with others as a safeguard against memory failure. While some people consider each of these user resorts to increase security risks, others suggest the absurdity of expecting users to remember distinct complex passwords for each of the dozens of accounts they access. For example, in 2005, security expertBruce Schneierrecommended writing down one's password:
Simply, people can no longer remember passwords good enough to reliably defend against dictionary attacks, and are much more secure if they choose a password too complicated to remember and then write it down. We're all good at securing small pieces of paper. I recommend that people write their passwords down on a small piece of paper, and keep it with their other valuable small pieces of paper: in their wallet.[39]
The following measures may increase acceptance of strong password requirements if carefully used:
Password policies sometimes suggestmemory techniquesto assist remembering passwords:
A reasonable compromise for using large numbers of passwords is to record them in a password manager program, which include stand-alone applications, web browser extensions, or a manager built into the operating system. A password manager allows the user to use hundreds of different passwords, and only have to remember a single password, the one which opens the encrypted password database.[65]Needless to say, this single password should be strong and well-protected (not recorded anywhere). Most password managers can automatically create strong passwords using a cryptographically securerandom password generator, as well as calculating the entropy of the generated password. A good password manager will provide resistance against attacks such askey logging, clipboard logging and various other memory spying techniques.
6 Types of Password Attacks & How to Stop Them | OneLogin. (n.d.). Retrieved April 24, 2024, fromhttps://www.google.com/
Franchi, E., Poggi, A., & Tomaiuolo, M. (2015). Information and Password Attacks on Social Networks: An Argument for Cryptography. Journal of Information Technology Research, 8(1), 25–42.https://doi.org/10.4018/JITR.2015010103
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Keystroke logging, often referred to askeyloggingorkeyboard capturing, is the action of recording (logging) the keys struck on akeyboard,[1][2]typically covertly, so that a person using the keyboard is unaware that their actions are being monitored. Data can then be retrieved by the person operating the logging program. Akeystroke recorderorkeyloggercan be eithersoftwareorhardware.
While the programs themselves are legal,[3]with many designed to allow employers to oversee the use of their computers, keyloggers are most often used for stealing passwords and otherconfidential information.[4][5]Keystroke logging can also be utilized to monitor activities of children in schools or at home and by law enforcement officials to investigate malicious usage.[6]
Keylogging can also be used to studykeystroke dynamics[7]orhuman-computer interaction. Numerous keylogging methods exist, ranging from hardware andsoftware-based approaches toacoustic cryptanalysis.
In the mid-1970s, theSoviet Uniondeveloped and deployed a hardware keylogger targetingUS Embassytypewriters. Termed the "selectric bug", it transmitted the typed characters onIBM Selectrictypewriters via magnetic detection of the mechanisms causing rotation of the print head.[8]An early keylogger was written byPerry Kivolowitzand posted to theUsenet newsgroupnet.unix-wizards, net.sources on November 17, 1983.[9]The posting seems to be a motivating factor in restricting access to/dev/kmemonUnixsystems. Theuser-modeprogram operated by locating and dumping character lists (clients) as they were assembled in the Unix kernel.
In the 1970s, spies installed keystroke loggers in theUS Embassyand Consulate buildings inMoscow.[10][11]They installed the bugs inSelectricII and Selectric III electric typewriters.[12]
Soviet embassies used manual typewriters, rather than electric typewriters, forclassified information—apparently because they are immune to such bugs.[12]As of 2013, Russian special services still use typewriters.[11][13][14]
A software-based keylogger is a computer program designed to record any input from the keyboard.[15]Keyloggers are used inITorganizations to troubleshoot technical problems with computers and business networks. Families and businesspeople use keyloggers legally to monitor network usage without their users' direct knowledge.Microsoftpublicly stated thatWindows 10has a built-in keylogger in its final version "to improve typing and writing services".[16]However, malicious individuals can use keyloggers on public computers to steal passwords or credit card information. Most keyloggers are not stopped byHTTPSencryption because that only protectsdata in transitbetween computers; software-based keyloggers run on the affected user's computer, reading keyboard inputs directly as the user types.
From a technical perspective, there are several categories:
Since 2006, keystroke logging has been an established research method for the study of writing processes.[22][23]Different programs have been developed to collect online process data of writing activities,[24]includingInputlog, Scriptlog, Translog and GGXLog.
Keystroke logging is used legitimately as a suitable research instrument in several writing contexts. These include studies on cognitive writing processes, which include
Keystroke logging can be used to research writing, specifically. It can also be integrated into educational domains for second language learning, programming skills, and typing skills.
Software keyloggers may be augmented with features that capture user information without relying on keyboard key presses as the sole input. Some of these features include:
Hardware-based keyloggers do not depend upon any software being installed as they exist at a hardware level in a computer system.
Writing simple software applications for keylogging can be trivial, and like any nefarious computer program, can be distributed as atrojan horseor as part of avirus. What is not trivial for an attacker, however, is installing a covert keystroke logger without getting caught and downloading data that has been logged without being traced. An attacker that manually connects to a host machine to download logged keystrokes risks being traced. A trojan that sends keylogged data to a fixed e-mail address orIP addressrisks exposing the attacker.
Researchers Adam Young and Moti Yung discussed several methods of sending keystroke logging. They presented a deniable password snatching attack in which the keystroke logging trojan is installed using a virus orworm. An attacker who is caught with the virus or worm can claim to be a victim. Thecryptotrojanasymmetrically encrypts the pilfered login/password pairs using thepublic keyof the trojan author and covertly broadcasts the resultingciphertext. They mentioned that the ciphertext can besteganographicallyencoded and posted to a public bulletin board such asUsenet.[45][46]
In 2000, theFBIused FlashCrest iSpy to obtain thePGPpassphraseofNicodemo Scarfo, Jr., son of mob bossNicodemo Scarfo.[47]Also in 2000, the FBI lured two suspected Russiancybercriminalsto the US in an elaborate ruse, and captured their usernames and passwords with a keylogger that was covertly installed on a machine that they used to access their computers inRussia. The FBI then used these credentials to gain access to the suspects' computers in Russia to obtain evidence to prosecute them.[48]
The effectiveness of countermeasures varies because keyloggers use a variety of techniques to capture data and the countermeasure needs to be effective against the particular data capture technique. In the case of Windows 10 keylogging by Microsoft, changing certain privacy settings may disable it.[49]An on-screen keyboard will be effective against hardware keyloggers; transparency[clarification needed]will defeat some—but not all—screen loggers. Ananti-spywareapplication that can only disable hook-based keyloggers will be ineffective against kernel-based keyloggers.
Keylogger program authors may be able to update their program's code to adapt to countermeasures that have proven effective against it.
Ananti-keyloggeris a piece ofsoftwarespecifically designed to detect keyloggers on a computer, typically comparing all files in the computer against a database of keyloggers, looking for similarities which might indicate the presence of a hidden keylogger. As anti-keyloggers have been designed specifically to detect keyloggers, they have the potential to be more effective than conventional antivirus software; some antivirus software do not consider keyloggers to be malware, as under some circumstances a keylogger can be considered a legitimate piece of software.[50]
Rebooting the computer using aLive CDor write-protectedLive USBis a possible countermeasure against software keyloggers if the CD is clean of malware and the operating system contained on it is secured and fully patched so that it cannot be infected as soon as it is started. Booting a different operating system does not impact the use of a hardware or BIOS based keylogger.
Manyanti-spywareapplications can detect some software based keyloggers and quarantine, disable, or remove them. However, because many keylogging programs are legitimate pieces of software under some circumstances, anti-spyware often neglects to label keylogging programs as spyware or a virus. These applications can detect software-based keyloggers based on patterns inexecutable code,heuristicsand keylogger behaviors (such as the use ofhooksand certainAPIs).
No software-based anti-spyware application can be 100% effective against all keyloggers.[51]Software-based anti-spyware cannot defeat non-software keyloggers (for example, hardware keyloggers attached to keyboards will always receive keystrokes before any software-based anti-spyware application).
The particular technique that the anti-spyware application uses will influence its potential effectiveness against software keyloggers. As a general rule, anti-spyware applications withhigher privilegeswill defeat keyloggers with lower privileges. For example, a hook-based anti-spyware application cannot defeat a kernel-based keylogger (as the keylogger will receive the keystroke messages before the anti-spyware application), but it could potentially defeat hook- and API-based keyloggers.
Network monitors(also known as reverse-firewalls) can be used to alert the user whenever an application attempts to make a network connection. This gives the user the chance to prevent the keylogger from "phoning home" with their typed information.
Automatic form-filling programs may prevent keylogging by removing the requirement for a user to type personal details and passwords using the keyboard.Form fillersare primarily designed forWeb browsersto fill in checkout pages and log users into their accounts. Once the user's account andcredit cardinformation has been entered into the program, it will be automatically entered into forms without ever using the keyboard orclipboard, thereby reducing the possibility that private data is being recorded. However, someone with physical access to the machine may still be able to install software that can intercept this information elsewhere in the operating system or while in transit on the network. (Transport Layer Security(TLS) reduces the risk that data in transit may be intercepted bynetwork sniffersandproxy tools.)
Usingone-time passwordsmay prevent unauthorized access to an account which has had its login details exposed to an attacker via a keylogger, as each password is invalidated as soon as it is used. This solution may be useful for someone using a public computer. However, an attacker who has remote control over such a computer can simply wait for the victim to enter their credentials before performing unauthorized transactions on their behalf while their session is active.
Another common way to protect access codes from being stolen by keystroke loggers is by asking users to provide a few randomly selected characters from their authentication code. For example, they might be asked to enter the 2nd, 5th, and 8th characters. Even if someone is watching the user or using a keystroke logger, they would only get a few characters from the code without knowing their positions.[52]
Use ofsmart cardsor othersecurity tokensmay improve security againstreplay attacksin the face of a successful keylogging attack, as accessing protected information would require both the (hardware) security token as well as the appropriate password/passphrase. Knowing the keystrokes, mouse actions, display, clipboard, etc. used on one computer will not subsequently help an attacker gain access to the protected resource. Some security tokens work as a type of hardware-assisted one-time password system, and others implement a cryptographicchallenge–response authentication, which can improve security in a manner conceptually similar to one time passwords.Smartcard readersand their associated keypads forPINentry may be vulnerable to keystroke logging through a so-calledsupply chain attack[53]where an attacker substitutes the card reader/PIN entry hardware for one which records the user's PIN.
Most on-screen keyboards (such as the on-screen keyboard that comes withWindows XP) send normal keyboard event messages to the external target program to type text. Software key loggers can log these typed characters sent from one program to another.[54]
Keystroke interference software is also available.[55]These programs attempt to trick keyloggers by introducing random keystrokes, although this simply results in the keylogger recording more information than it needs to. An attacker has the task of extracting the keystrokes of interest—the security of this mechanism, specifically how well it stands up tocryptanalysis, is unclear.
Similar to on-screen keyboards,speech-to-text conversionsoftware can also be used against keyloggers, since there are no typing or mouse movements involved. The weakest point of using voice-recognition software may be how the software sends the recognized text to target software after the user's speech has been processed.
ManyPDAsand latelytablet PCscan already convert pen (also called stylus) movements on theirtouchscreensto computer understandable text successfully.Mouse gesturesuse this principle by using mouse movements instead of a stylus. Mouse gesture programs convert these strokes to user-definable actions, such as typing text. Similarly,graphics tabletsandlight penscan be used to input these gestures, however, these are becoming less common.[timeframe?]
The same potential weakness of speech recognition applies to this technique as well.
With the help of many programs, a seemingly meaningless text can be expanded to a meaningful text and most of the time context-sensitively, e.g. "en.wikipedia.org" can be expanded when a web browser window has the focus. The biggest weakness of this technique is that these programs send their keystrokes directly to the target program. However, this can be overcome by usingthe 'alternating' technique described below, i.e. sending mouse clicks to non-responsive areas of the target program, sending meaningless keys, sending another mouse click to the target area (e.g. password field) and switching back-and-forth.
Alternating between typing the login credentials and typing characters somewhere else in the focus window[56]can cause a keylogger to record more information than it needs to, but this could be easily filtered out by an attacker. Similarly, a user can move their cursor using the mouse while typing, causing the logged keystrokes to be in the wrong order e.g., by typing a password beginning with the last letter and then using the mouse to move the cursor for each subsequent letter. Lastly, someone can also usecontext menusto remove,cut, copy, and pasteparts of the typed text without using the keyboard. An attacker who can capture only parts of a password will have a largerkey spaceto attack if they choose to execute abrute-force attack.
Another very similar technique uses the fact that any selected text portion is replaced by the next key typed. e.g., if the password is "secret", one could type "s", then some dummy keys "asdf". These dummy characters could then be selected with the mouse, and the next character from the password "e" typed, which replaces the dummy characters "asdf".
These techniques assume incorrectly that keystroke logging software cannot directly monitor the clipboard, the selected text in a form, or take a screenshot every time a keystroke or mouse click occurs. They may, however, be effective against some hardware keyloggers.
Media related toKeystroke loggingat Wikimedia Commons
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Apassphraseis a sequence of words or other text used to control access to acomputersystem, program ordata. It is similar to apasswordin usage, but a passphrase is generally longer for added security. Passphrases are often used to control both access to, and the operation of,cryptographicprograms and systems, especially those that derive anencryptionkey from a passphrase. The origin of the term is by analogy withpassword. The modern concept of passphrases is believed to have been invented by Sigmund N. Porter in 1982.[1]
Source:[2]
Considering that theentropyof written English is less than 1.1 bits per character,[3]passphrases can be relatively weak.NISThas estimated that the 23-character passphrase "IamtheCapitanofthePina4" contains a 45-bit strength. The equation employed here is:[4]
(This calculation does not take into account that this is a well-known quote from the operettaH.M.S. Pinafore. AnMD5hash of this passphrase can be cracked in 4 seconds using crackstation.net, indicating that the phrase is found in password cracking databases.)
Using this guideline, to achieve the 80-bit strength recommended for high security (non-military) byNIST, a passphrase would need to be 58 characters long, assuming a composition that includes uppercase and alphanumeric.
There is room for debate regarding the applicability of this equation, depending on the number of bits of entropy assigned. For example, the characters in five-letter words each contain 2.3 bits of entropy, which would mean only a 35-character passphrase is necessary to achieve 80 bit strength.[5]
If the words or components of a passphrase may be found in a language dictionary—especially one available as electronic input to a software program—the passphrase is rendered more vulnerable todictionary attack. This is a particular issue if the entire phrase can be found in a book of quotations or phrase compilations. However, the required effort (in time and cost) can be made impracticably high if there are enough words in the passphrase and if they arerandomlychosen and ordered in the passphrase. The number of combinations which would have to be tested under sufficient conditions make a dictionary attack so difficult as to be infeasible. These are difficult conditions to meet, and selecting at least one word that cannot be found inanydictionary significantly increases passphrase strength.
If passphrases are chosen by humans, they are usually biased by the frequency of particular words in natural language. In the case of four word phrases, actual entropy rarely exceeds 30 bits. On the other hand, user-selected passwordstend to be much weaker than that, and encouraging users to use even 2-word passphrases may be able to raise entropy from below 10 bits to over 20 bits.[6]
For example, the widely used cryptography standardOpenPGPrequires that a user make up a passphrase that must be entered whenever decrypting or signing messages. Internet services likeHushmailprovide free encrypted e-mail or file sharing services, but the security present depends almost entirely on the quality of the chosen passphrase.
Passphrases differ from passwords. Apasswordis usually short—six to ten characters. Such passwords may be adequate for various applications if frequently changed, chosen using an appropriate policy, not found in dictionaries, sufficiently random, and/or if the system prevents online guessing, etc.[citation needed], such as:
But passwords are typically not safe to use as keys for standalone security systems such as encryption systems that expose data to enable offline password guessing by an attacker.[7]Passphrases are theoretically stronger, and so should make a better choice in these cases. First, they usually are and always should be much longer—20 to 30 characters or more is typical—making some kinds of brute force attacks entirely impractical. Second, if well chosen, they will not be found in any phrase or quote dictionary, so such dictionary attacks will be almost impossible. Third, they can be structured to be more easily memorable than passwords without being written down, reducing the risk of hardcopy theft. However, if a passphrase is not protected appropriately by the authenticator and the clear-text passphrase is revealed its use is no better than other passwords. For this reason it is recommended that passphrases not be reused across different or unique sites and services.
In 2012, two Cambridge University researchers analyzed passphrases from theAmazon PayPhrasesystem and found that a significant percentage are easy to guess due to common cultural references such as movie names and sports teams, losing much of the potential of using long passwords.[8]
When used in cryptography, commonly the passphrase protects a long machine generatedkey, and the key protects the data. The key is so long a brute force attack directly on the data is impossible. Akey derivation functionis used, involving many thousands of iterations (salted& hashed), to slow downpassword crackingattacks.
Typical advice about choosing a passphrase includes suggestions that it should be:[9]
One method to create a strong passphrase is to usediceto select words at random from a long list, a technique often referred to asdiceware. While such a collection of words might appear to violate the "not from any dictionary" rule, the security is based entirely on the large number of possible ways to choose from the list of words and not from any secrecy about the words themselves. For example, if there are 7776 words in the list and six words are chosen randomly, then there are7,7766= 221,073,919,720,733,357,899,776combinations, providing about 78 bits ofentropy. (The number7776was chosen to allow words to be selected by throwing five dice.7776 = 65) Random word sequences may then be memorized using techniques such as thememory palace.
Another is to choose two phrases, turn one into anacronym, and include it in the second, making the final passphrase. For instance, using two English language typing exercises, we have the following.The quick brown fox jumps over the lazy dog, becomestqbfjotld. Including it in,Now is the time for all good men to come to the aid of their country, might produce,Now is the time for all good tqbfjotld to come to the aid of their countryas the passphrase.
There are several points to note here, all relating to why this example passphrase is not a good one.
ThePGPPassphrase FAQ[10]suggests a procedure that attempts a better balance between theoretical security and practicality than this example. All procedures for picking a passphrase involve a tradeoff between security and ease of use; security should be at least "adequate" while not "too seriously" annoying users. Both criteria should be evaluated to match particular situations.
Another supplementary approach to frustrating brute-force attacks is to derive the key from the passphrase using adeliberately slow hash function, such asPBKDF2as described in RFC 2898.
If backward compatibility withMicrosoft LAN Manageris not needed, in versions ofWindows NT(includingWindows 2000,Windows XPand later), a passphrase can be used as a substitute for a Windows password. If the passphrase is longer than 14 characters, this will also avoid the generation of averyweakLM hash.
In recent versions ofUnix-likeoperating systems such asLinux,OpenBSD,NetBSD,SolarisandFreeBSD, up to 255-character passphrases can be used.[citation needed]
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Phishingis a form ofsocial engineeringand ascamwhere attackers deceive people into revealingsensitive information[1]or installingmalwaresuch asviruses,worms,adware, orransomware. Phishing attacks have become increasingly sophisticated and often transparently mirror the site being targeted, allowing the attacker to observe everything while the victim navigates the site, and transverses any additional security boundaries with the victim.[2]As of 2020, it is the most common type ofcybercrime, with theFederal Bureau of Investigation'sInternet Crime Complaint Centerreporting more incidents of phishing than any other type of cybercrime.[3]
The term "phishing" was first recorded in 1995 in thecrackingtoolkitAOHell, but may have been used earlier in the hacker magazine2600.[4][5][6]It is a variation offishingand refers to the use of lures to "fish" for sensitive information.[5][7][8]
Measures to prevent or reduce the impact of phishing attacks includelegislation, user education, public awareness, and technical security measures.[9]The importance of phishing awareness has increased in both personal and professional settings, with phishing attacks among businesses rising from 72% in 2017 to 86% in 2020,[10]already rising to 94% in 2023.[11]
Phishing attacks, often delivered viaemail spam, attempt to trick individuals into giving away sensitive information or login credentials. Most attacks are "bulk attacks" that are not targeted and are instead sent in bulk to a wide audience.[12]The goal of the attacker can vary, with common targets including financial institutions, email and cloud productivity providers, and streaming services.[13]The stolen information or access may be used to steal money, installmalware, or spear phish others within the target organization.[14]Compromised streaming service accounts may also be sold ondarknet markets.[15]
This type ofsocial engineeringattack can involve sending fraudulent emails or messages that appear to be from a trusted source, such as a bank or government agency. These messages typically redirect to a fake login page where users are prompted to enter their credentials.
Spear phishing is a targeted phishing attack that uses personalized messaging, especially e‑mails,[16]to trick a specific individual or organization into believing they are legitimate. It often utilizes personal information about the target to increase the chances of success.[17][18][19][20]These attacks often target executives or those in financial departments with access to sensitive financial data and services. Accountancy and audit firms are particularly vulnerable to spear phishing due to the value of the information their employees have access to.[21]
The Russian government-runThreat Group-4127 (Fancy Bear)(GRU Unit 26165) targetedHillary Clinton's2016 presidential campaignwith spear phishing attacks on over 1,800Googleaccounts, using theaccounts-google.comdomain to threaten targeted users.[22][23]
A study on spear phishing susceptibility among different age groups found that 43% of youth aged 18–25 years and 58% of older users clicked on simulated phishing links in daily e‑mails over 21 days. Older women had the highest susceptibility, while susceptibility in young users declined during the study, but remained stable among older users.[24]
Voice over IP(VoIP) is used in vishing or voice phishing attacks,[25]where attackers make automated phone calls to large numbers of people, often usingtext-to-speechsynthesizers, claiming fraudulent activity on their accounts. The attackers spoof the calling phone number to appear as if it is coming from a legitimate bank or institution. The victim is then prompted to enter sensitive information or connected to a live person who usessocial engineeringtactics to obtain information.[25]Vishing takes advantage of the public's lower awareness and trust in voice telephony compared to email phishing.[26]
SMS phishing[27]or smishing[28][29]is a type of phishing attack that usestext messagesfrom a cell phone orsmartphoneto deliver a bait message.[30]The victim is usually asked to click a link, call a phone number, or contact anemailaddress provided by the attacker. They may then be asked to provideprivate information, such as login credentials for other websites.
The difficulty in identifying illegitimate links can be compounded on mobile devices due to the limited display of URLs in mobile browsers.[31]
Smishing can be just as effective as email phishing, as many smartphones have fast internet connectivity. Smishing messages may also come from unusual phone numbers.[32]
Page hijacking involves redirecting users to malicious websites orexploit kitsthrough the compromise of legitimate web pages, often usingcross site scripting.Hackersmay insert exploit kits such asMPackinto compromised websites to exploit legitimate users visiting the server. Page hijacking can also involve the insertion of maliciousinline frames, allowing exploit kits to load. This tactic is often used in conjunction withwatering holeattacks on corporate targets.[33]
A relatively new trend in online scam activity is "quishing", which meansQR Codephishing. The term is derived from "QR" (Quick Response) codes and "phishing", as scammers exploit the convenience of QR Codes to trick users into giving up sensitive data, by scanning a code containing an embedded malicious web site link. Unlike traditional phishing, which relies on deceptive emails or websites, quishing uses QR Codes to bypass email filters[34][35]and increase the likelihood that victims will fall for the scam, as people tend to trust QR Codes and may not scrutinize them as carefully as a URL or email link. The bogus codes may be sent by email, social media, or in some cases hard copy stickers are placed over legitimate QR Codes on such things as advertising posters and car park notices.[36][37]When victims scan the QR Code with their phone or device, they are redirected to a fake website designed to steal personal information, login credentials, or financial details.[34]
As QR Codes become more widely used for things like payments, event check-ins, and product information, quishing is emerging as a significant concern for digital security. Users are advised to exercise caution when scanning unfamiliar QR Codes and ensure they are from trusted sources, although the UK'sNational Cyber Security Centrerates the risk as lower than other types of lure.[38]
Traditional phishing attacks are typically limited to capturing user credentials directly inputted into fraudulent websites. However, the advent ofMan-in-the-Middle(MitM) phishing techniques has significantly advanced the sophistication of these attacks, enabling cybercriminals to bypasstwo-factor authentication(2FA) mechanisms during a user's active session on a web service. MitM phishing attacks employ intermediary tools that intercept communication between the user and the legitimate service.
Evilginx, originally created as an open-source tool for penetration testing and ethical hacking, has been repurposed by cybercriminals for MitM attacks. Evilginx works like a middleman, passing information between the victim and the real website without saving passwords or login codes. This makes it harder for security systems to detect, since they usually look for phishing sites that store stolen data. By grabbing login tokens and session cookies instantly, attackers can break into accounts and use them just like the real user, for as long as the session stays active.
Attackers employ various methods, including phishing emails, social engineering tactics, or distributing malicious links via social media platforms. Once the victim interacts with the counterfeit site, the MitM tool intercepts the authentication process, effectively bypassing 2FA protections.[39]
Phishing attacks often involve creating fakelinksthat appear to be from a legitimate organization.[40]These links may usemisspelled URLsorsubdomainsto deceive the user. In the following example URL,http://www.yourbank.example.com/, it can appear to the untrained eye as though the URL will take the user to theexamplesection of theyourbankwebsite; this URL points to the "yourbank" (i.e. phishing subdomain) section of theexamplewebsite (fraudster's domain name). Another tactic is to make the displayed text for a link appear trustworthy, while the actual link goes to the phisher's site. To check the destination of a link, many email clients and web browsers will show the URL in the status bar when themouseis hovering over it. However, some phishers may be able to bypass this security measure.[41]
Internationalized domain names(IDNs) can be exploited viaIDN spoofing[42]orhomograph attacks[43]to allow attackers to create fake websites with visually identical addresses to legitimate ones. These attacks have been used by phishers to disguise malicious URLs using openURL redirectorson trusted websites.[44][45][46]An example of this is inhttp://www.exаmple.com/, where the third character is not theLatinletter 'a', but instead theCyrilliccharacter 'а'. When the victim clicks on the link, unaware that the third character is actually the Cyrillic letter 'а', they get redirected to the malicious sitehttp://www.xn--exmple-4nf.com/Even digital certificates, such asSSL, may not protect against these attacks as phishers can purchase valid certificates and alter content to mimic genuine websites or host phishing sites without SSL.[47]
Phishing often usessocial engineeringtechniques to trick users into performing actions such as clicking a link or opening an attachment, or revealing sensitive information. It often involves pretending to be a trusted entity and creating a sense of urgency,[48]like threatening to close or seize a victim's bank or insurance account.[49]
An alternative technique to impersonation-based phishing is the use offake newsarticles to trick victims into clicking on a malicious link. These links often lead to fake websites that appear legitimate,[50]but are actually run by attackers who may try to install malware or presentfake "virus" notificationsto the victim.[51]
Early phishing techniques can be traced back to the 1990s, whenblack hathackers and thewarezcommunity usedAOLto steal credit card information and commit other online crimes. The term "phishing" is said to have been coined by Khan C. Smith, a well-known spammer and hacker,[52]and its first recorded mention was found in the hacking toolAOHell, which was released in 1994. AOHell allowed hackers to impersonate AOL staff and sendinstant messagesto victims asking them to reveal their passwords.[53][54]In response, AOL implemented measures to prevent phishing and eventually shut down thewarez sceneon their platform.[55][56]
In the 2000s, phishing attacks became more organized and targeted. The first known direct attempt against a payment system,E-gold, occurred in June 2001, and shortly after theSeptember 11 attacks, a "post-9/11 id check" phishing attack followed.[57]The first known phishing attack against a retail bank was reported in September 2003.[58]Between May 2004 and May 2005, approximately 1.2 million computer users in the United States suffered losses caused by phishing, totaling approximatelyUS$929 million.[59]Phishing was recognized as a fully organized part of the black market, and specializations emerged on a global scale that provided phishing software for payment, which were assembled and implemented into phishing campaigns by organized gangs.[60][61]TheUnited Kingdombanking sector suffered from phishing attacks, with losses from web banking fraud almost doubling in 2005 compared to 2004.[62][63]In 2006, almost half of phishing thefts were committed by groups operating through the Russian Business Network based in St. Petersburg.[64]Email scams posing as theInternal Revenue Servicewere also used to steal sensitive data from U.S. taxpayers.[65]Social networking sitesare a prime target of phishing, since the personal details in such sites can be used inidentity theft;[66]In 2007, 3.6 million adults lostUS$3.2 billiondue to phishing attacks.[67]The Anti-Phishing Working Group reported receiving 115,370 phishing email reports from consumers with US and China hosting more than 25% of the phishing pages each in the third quarter of 2009.[68]
Phishing in the 2010s saw a significant increase in the number of attacks. In 2011, the master keys forRSASecurID security tokens were stolen through a phishing attack.[69][70]Chinese phishing campaigns also targeted high-ranking officials in the US and South Korean governments and military, as well as Chinese political activists.[71][72]According to Ghosh, phishing attacks increased from 187,203 in 2010 to 445,004 in 2012. In August 2013, Outbrain suffered a spear-phishing attack,[73]and in November 2013, 110 million customer and credit card records were stolen fromTargetcustomers through a phished subcontractor account.[74]CEO and IT security staff subsequently fired.[75]In August 2014, iCloud leaks of celebrity photos were based on phishing e-mails sent to victims that looked like they came from Apple or Google.[76]In November 2014, phishing attacks onICANNgained administrative access to the Centralized Zone Data System; also gained was data about users in the system - and access to ICANN's public Governmental Advisory Committee wiki, blog, and whois information portal.[77]Fancy Bear was linked to spear-phishing attacks against thePentagonemail system in August 2015,[78][79]and the group used a zero-day exploit of Java in a spear-phishing attack on the White House and NATO.[80][81]Fancy Bear carried out spear phishing attacks on email addresses associated with the Democratic National Committee in the first quarter of 2016.[82][83]In August 2016, members of theBundestagand political parties such asLinken-faction leaderSahra Wagenknecht,Junge Union, and theCDUofSaarlandwere targeted by spear-phishing attacks suspected to be carried out by Fancy Bear. In August 2016, theWorld Anti-Doping Agencyreported the receipt of phishing emails sent to users of its database claiming to be official WADA, but consistent with the Russian hacking group Fancy Bear.[84][85][86]In 2017, 76% of organizations experienced phishing attacks, with nearly half of theinformation securityprofessionals surveyed reporting an increase from 2016. In the first half of 2017, businesses and residents of Qatar were hit with over 93,570 phishing events in a three-month span.[87]In August 2017, customers ofAmazonfaced the Amazon Prime Day phishing attack, when hackers sent out seemingly legitimate deals to customers of Amazon. When Amazon's customers attempted to make purchases using the "deals", the transaction would not be completed, prompting the retailer's customers to input data that could be compromised and stolen.[88]In 2018, the company block.one, which developed the EOS.IO blockchain, was attacked by a phishing group who sent phishing emails to all customers aimed at intercepting the user's cryptocurrency wallet key, and a later attack targeted airdrop tokens.[89]
Phishing attacks have evolved in the 2020s to include elements of social engineering, as demonstrated by the July 15, 2020,Twitterbreach. In this case, a 17-year-old hacker and accomplices set up a fake website resembling Twitter's internalVPNprovider used by remote working employees. Posing as helpdesk staff, they called multiple Twitter employees, directing them to submit their credentials to the fake VPN website.[90]Using the details supplied by the unsuspecting employees, they were able to seize control of several high-profile user accounts, including those ofBarack Obama,Elon Musk,Joe Biden, andApple Inc.'s company account. The hackers then sent messages to Twitter followers solicitingBitcoin, promising to double the transaction value in return. The hackers collected 12.86 BTC (about $117,000 at the time).[91]In the 2020s, phishingas a service(PhaaS) platforms likeDarculaallow attackers to easily fake trusted websites.[92]
There are anti-phishing websites which publish exact messages that have been recently circulating the internet, such asFraudWatch Internationaland Millersmiles. Such sites often provide specific details about the particular messages.[93][94]
As recently as 2007, the adoption of anti-phishing strategies by businesses needing to protect personal and financial information was low.[95]There are several different techniques to combat phishing, including legislation and technology created specifically to protect against phishing. These techniques include steps that can be taken by individuals, as well as by organizations. Phone, web site, and email phishing can now be reported to authorities, as describedbelow.
Effective phishing education, including conceptual knowledge[96]and feedback,[97][98]is an important part of any organization's anti-phishing strategy. While there is limited data on the effectiveness of education in reducing susceptibility to phishing,[99]much information on the threat is available online.[49]
Simulated phishingcampaigns, in which organizations test their employees' training by sending fake phishing emails, are commonly used to assess their effectiveness. One example is a study by theNational Library of Medicine, in which an organization received 858,200 emails during a 1-month testing period, with 139,400 (16%) being marketing and 18,871 (2%) being identified as potential threats. These campaigns are often used in the healthcare industry, as healthcare data is a valuable target for hackers. These campaigns are just one of the ways that organizations are working to combat phishing.[100]
Nearly all legitimate e-mail messages from companies to their customers contain an item of information that is not readily available to phishers. Some companies, for examplePayPal, always address their customers by their username in emails, so if an email addresses the recipient in a generic fashion ("Dear PayPal customer") it is likely to be an attempt at phishing.[101]Furthermore, PayPal offers various methods to determine spoof emails and advises users to forward suspicious emails to their spoof@PayPal.com domain to investigate and warn other customers. However it is unsafe to assume that the presence of personal information alone guarantees that a message is legitimate,[102]and some studies have shown that the presence of personal information does not significantly affect the success rate of phishing attacks;[103]which suggests that most people do not pay attention to such details.
Emails from banks and credit card companies often include partial account numbers, but research has shown that people tend to not differentiate between the first and last digits.[104]
A study on phishing attacks in game environments found thateducational gamescan effectively educate players against information disclosures and can increase awareness on phishing risk thus mitigating risks.[105]
TheAnti-Phishing Working Group, one of the largest anti-phishing organizations in the world, produces regular report on trends in phishing attacks.[106]
A wide range of technical approaches are available to prevent phishing attacks reaching users or to prevent them from successfully capturing sensitive information.
Specializedspam filterscan reduce the number of phishing emails that reach their addressees' inboxes. These filters use a number of techniques includingmachine learning[107]andnatural language processingapproaches to classify phishing emails,[108][109]and reject email with forged addresses.[110]
Another popular approach to fighting phishing is to maintain a list of known phishing sites and to check websites against the list. One such service is theSafe Browsingservice.[111]Web browsers such asGoogle Chrome,Internet Explorer7,Mozilla Firefox2.0,Safari3.2, andOperaall contain this type of anti-phishing measure.[112][113][114][115][116]Firefox 2usedGoogleanti-phishing software. Opera 9.1 uses liveblacklistsfromPhishtank,cysconandGeoTrust, as well as livewhitelistsfrom GeoTrust. Some implementations of this approach send the visited URLs to a central service to be checked, which has raised concerns aboutprivacy.[117]According to a report by Mozilla in late 2006, Firefox 2 was found to be more effective thanInternet Explorer 7at detecting fraudulent sites in a study by an independent software testing company.[118]
An approach introduced in mid-2006 involves switching to a special DNS service that filters out known phishing domains.[119]
To mitigate the problem of phishing sites impersonating a victim site by embedding its images (such aslogos), several site owners have altered the images to send a message to the visitor that a site may be fraudulent. The image may be moved to a new filename and the original permanently replaced, or a server can detect that the image was not requested as part of normal browsing, and instead send a warning image.[120][121]
TheBank of Americawebsite[122][123]was one of several that asked users to select a personal image (marketed asSiteKey) and displayed this user-selected image with any forms that request a password. Users of the bank's online services were instructed to enter a password only when they saw the image they selected. The bank has since discontinued the use of SiteKey. Several studies suggest that few users refrain from entering their passwords when images are absent.[124][125]In addition, this feature (like other forms oftwo-factor authentication) is susceptible to other attacks, such as those suffered by Scandinavian bankNordeain late 2005,[126]andCitibankin 2006.[127]
A similar system, in which an automatically generated "Identity Cue" consisting of a colored word within a colored box is displayed to each website user, is in use at other financial institutions.[128]
Security skins[129][130]are a related technique that involves overlaying a user-selected image onto the login form as a visual cue that the form is legitimate. Unlike the website-based image schemes, however, the image itself is shared only between the user and the browser, and not between the user and the website. The scheme also relies on amutual authenticationprotocol, which makes it less vulnerable to attacks that affect user-only authentication schemes.
Still another technique relies on a dynamic grid of images that is different for each login attempt. The user must identify the pictures that fit their pre-chosen categories (such as dogs, cars and flowers). Only after they have correctly identified the pictures that fit their categories are they allowed to enter their alphanumeric password to complete the login. Unlike the static images used on the Bank of America website, a dynamic image-based authentication method creates a one-time passcode for the login, requires active participation from the user, and is very difficult for a phishing website to correctly replicate because it would need to display a different grid of randomly generated images that includes the user's secret categories.[131]
Several companies offer banks and other organizations likely to suffer from phishing scams round-the-clock services to monitor, analyze and assist in shutting down phishing websites.[132]Automated detection of phishing content is still below accepted levels for direct action, with content-based analysis reaching between 80% and 90% of success[133]so most of the tools include manual steps to certify the detection and authorize the response.[134]Individuals can contribute by reporting phishing to both volunteer and industry groups,[135]such ascysconorPhishTank.[136]Phishing web pages and emails can be reported to Google.[137][138]
Organizations can implement two factor ormulti-factor authentication(MFA), which requires a user to use at least 2 factors when logging in. (For example, a user must both present asmart cardand apassword). This mitigates some risk, in the event of a successful phishing attack, the stolen password on its own cannot be reused to further breach the protected system. However, there are several attack methods which can defeat many of the typical systems.[139]MFA schemes such asWebAuthnaddress this issue by design.
On January 26, 2004, the U.S.Federal Trade Commissionfiled the first lawsuit against aCalifornianteenager suspected of phishing by creating a webpage mimickingAmerica Onlineand stealing credit card information.[140]Other countries have followed this lead by tracing and arresting phishers. A phishing kingpin, Valdir Paulo de Almeida, was arrested inBrazilfor leading one of the largest phishingcrime rings, which in two years stole betweenUS$18 millionandUS$37 million.[141]UK authorities jailed two men in June 2005 for their role in a phishing scam,[142]in a case connected to theU.S. Secret ServiceOperation Firewall, which targeted notorious "carder" websites.[143]In 2006, Japanese police arrested eight people for creating fake Yahoo Japan websites, netting themselves¥100 million(US$870,000)[144]and theFBIdetained a gang of sixteen in the U.S. and Europe in Operation Cardkeeper.[145]
SenatorPatrick Leahyintroduced the Anti-Phishing Act of 2005 toCongressin theUnited Stateson March 1, 2005. Thisbillaimed to impose fines of up to $250,000 and prison sentences of up to five years on criminals who used fake websites and emails to defraud consumers.[146]In the UK, theFraud Act 2006[147]introduced a general offense of fraud punishable by up to ten years in prison and prohibited the development or possession of phishing kits with the intention of committing fraud.[148]
Companies have also joined the effort to crack down on phishing. On March 31, 2005,Microsoftfiled 117 federal lawsuits in theU.S. District Court for the Western District of Washington. The lawsuits accuse "John Doe" defendants of obtaining passwords and confidential information. March 2005 also saw a partnership between Microsoft and theAustralian governmentteaching law enforcement officials how to combat various cyber crimes, including phishing.[149]Microsoft announced a planned further 100 lawsuits outside the U.S. in March 2006,[150]followed by the commencement, as of November 2006, of 129 lawsuits mixing criminal and civil actions.[151]AOLreinforced its efforts against phishing[152]in early 2006 with three lawsuits[153]seeking a total ofUS$18 millionunder the 2005 amendments to the Virginia Computer Crimes Act,[154][155]andEarthlinkhas joined in by helping to identify six men subsequently charged with phishing fraud inConnecticut.[156]
In January 2007, Jeffrey Brett Goodin of California became the first defendant convicted by a jury under the provisions of theCAN-SPAM Act of 2003. He was found guilty of sending thousands of emails toAOLusers, while posing as the company's billing department, which prompted customers to submit personal and credit card information. Facing a possible 101 years in prison for the CAN-SPAM violation and ten other counts includingwire fraud, the unauthorized use of credit cards, and the misuse of AOL's trademark, he was sentenced to serve 70 months. Goodin had been in custody since failing to appear for an earlier court hearing and began serving his prison term immediately.[157][158][159][160]
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Vulnerabilitiesare flaws or weaknesses in a system's design, implementation, or management that can be exploited by a malicious actor to compromise its security.
Despite intentions to achieve complete correctness, virtually all hardware and software contain bugs where the system does not behave as expected. If the bug could enable an attacker to compromise the confidentiality, integrity, or availability of system resources, it is called a vulnerability. Insecuresoftware developmentpractices as well as design factors such as complexity can increase the burden of vulnerabilities. There are different types most common in different components such as hardware, operating systems, and applications.
Vulnerability managementis a process that includes identifying systems and prioritizing which are most important, scanning for vulnerabilities, and taking action to secure the system. Vulnerability management typically is a combination of remediation (fixing the vulnerability), mitigation (increasing the difficulty or reducing the danger of exploits), and accepting risks that are not economical or practical to eliminate. Vulnerabilities can be scored for risk according to theCommon Vulnerability Scoring Systemor other systems, and added to vulnerability databases. As of November 2024[update], there are more than 240,000 vulnerabilities[1]catalogued in theCommon Vulnerabilities and Exposures(CVE) database.
A vulnerability is initiated when it is introduced into hardware or software. It becomes active and exploitable when the software or hardware containing the vulnerability is running. The vulnerability may be discovered by the vendor or a third party. Disclosing the vulnerability (as apatchor otherwise) is associated with an increased risk of compromise because attackers often move faster than patches are rolled out. Regardless of whether a patch is ever released to remediate the vulnerability, its lifecycle will eventually end when the system, or older versions of it, fall out of use.
Despite developers' goal of delivering a product that works entirely as intended, virtually allsoftwareandhardwarecontain bugs.[2]If a bug creates a security risk, it is called a vulnerability.[3][4][5]Software patchesare often released to fix identified vulnerabilities, but those that remain unknown (zero days) as well as those that have not been patched are still liable for exploitation.[6]Vulnerabilities vary in their ability to beexploitedby malicious actors,[3]and the actual risk is dependent on the nature of the vulnerability as well as the value of the surrounding system.[7]Although some vulnerabilities can only be used fordenial of serviceattacks, more dangerous ones allow the attacker toinjectand run their own code (calledmalware), without the user being aware of it.[3]Only a minority of vulnerabilities allow forprivilege escalation, which is necessary for more severe attacks.[8]Without a vulnerability, the exploit cannot gain access.[9]It is also possible formalwareto be installed directly, without an exploit, if the attacker usessocial engineeringor implants the malware in legitimate software that is downloaded deliberately.[10]
Fundamental design factors that can increase the burden of vulnerabilities include:
Somesoftware developmentpractices can affect the risk of vulnerabilities being introduced to a code base. Lack of knowledge about secure software development or excessive pressure to deliver features quickly can lead to avoidable vulnerabilities to enter production code, especially if security is not prioritized by thecompany culture. This can lead to unintended vulnerabilities. The more complex the system is, the easier it is for vulnerabilities to go undetected. Some vulnerabilities are deliberately planted, which could be for any reason from a disgruntled employee selling access to cyber criminals, to sophisticated state-sponsored schemes to introduce vulnerabilities to software.[15]Inadequatecode reviewscan lead to missed bugs, but there are alsostatic code analysistools that can be used as part of code reviews and may find some vulnerabilities.[16]
DevOps, a development workflow that emphasizes automated testing and deployment to speed up the deployment of new features, often requires that many developers be granted access to change configurations, which can lead to deliberate or inadvertent inclusion of vulnerabilities.[17]Compartmentalizing dependencies, which is often part of DevOps workflows, can reduce theattack surfaceby paring down dependencies to only what is necessary.[18]Ifsoftware as a serviceis used, rather than the organization's own hardware and software, the organization is dependent on the cloud services provider to prevent vulnerabilities.[19]
TheNational Vulnerability Databaseclassifies vulnerabilities into eight root causes that may be overlapping, including:[20]
Deliberate security bugs can be introduced during or after manufacturing and cause theintegrated circuitnot to behave as expected under certain specific circumstances. Testing for security bugs in hardware is quite difficult due to limited time and the complexity of twenty-first century chips,[23]while the globalization of design and manufacturing has increased the opportunity for these bugs to be introduced by malicious actors.[24]
Althoughoperating system vulnerabilitiesvary depending on theoperating systemin use, a common problem isprivilege escalationbugs that enable the attacker to gain more access than they should be allowed.Open-sourceoperating systems such asLinuxandAndroidhave a freely accessiblesource codeand allow anyone to contribute, which could enable the introduction of vulnerabilities. However, the same vulnerabilities also occur in proprietary operating systems such asMicrosoft WindowsandApple operating systems.[25]All reputable vendors of operating systems provide patches regularly.[26]
Client–server applicationsare downloaded onto the end user's computers and are typically updated less frequently than web applications. Unlike web applications, they interact directly with a user'soperating system. Common vulnerabilities in these applications include:[27]
Web applicationsrun on many websites. Because they are inherently less secure than other applications, they are a leading source ofdata breachesand other security incidents.[28][29]They can include:
Attacks used against vulnerabilities in web applications include:
There is little evidence about the effectiveness and cost-effectiveness of different cyberattack prevention measures.[32]Although estimating the risk of an attack is not straightforward, the mean time to breach and expected cost can be considered to determine the priority for remediating or mitigating an identified vulnerability and whether it is cost effective to do so.[33]Although attention to security can reduce the risk of attack, achieving perfect security for a complex system is impossible, and many security measures have unacceptable cost or usability downsides.[34]For example, reducing the complexity and functionality of the system is effective at reducing theattack surface.[35]
Successful vulnerability management usually involves a combination of remediation (closing a vulnerability), mitigation (increasing the difficulty, and reducing the consequences, of exploits), and accepting some residual risk. Often adefense in depthstrategy is used for multiple barriers to attack.[36]Some organizations scan for only the highest-risk vulnerabilities as this enables prioritization in the context of lacking the resources to fix every vulnerability.[37]Increasing expenses is likely to havediminishing returns.[33]
Remediation fixes vulnerabilities, for example by downloading asoftware patch.[38]Software vulnerability scannersare typically unable to detect zero-day vulnerabilities, but are more effective at finding known vulnerabilities based on a database. These systems can find some known vulnerabilities and advise fixes, such as a patch.[39][40]However, they have limitations includingfalse positives.[38]
Vulnerabilities can only be exploited when they are active-the software in which they are embedded is actively running on the system.[41]Before the code containing the vulnerability is configured to run on the system, it is considered a carrier.[42]Dormant vulnerabilities can run, but are not currently running. Software containing dormant and carrier vulnerabilities can sometimes be uninstalled or disabled, removing the risk.[43]Active vulnerabilities, if distinguished from the other types, can be prioritized for patching.[41]
Vulnerability mitigation is measures that do not close the vulnerability, but make it more difficult to exploit or reduce the consequences of an attack.[44]Reducing theattack surface, particularly for parts of the system withroot(administrator) access, and closing off opportunities for exploits to engage inprivilege exploitationis a common strategy for reducing the harm that a cyberattack can cause.[38]If a patch for third-party software is unavailable, it may be possible to temporarily disable the software.[45]
Apenetration testattempts to enter the system via an exploit to see if the system is insecure.[46]If a penetration test fails, it does not necessarily mean that the system is secure.[47]Some penetration tests can be conducted with automated software that tests against existing exploits for known vulnerabilities.[48]Other penetration tests are conducted by trained hackers. Many companies prefer to contract out this work as it simulates an outsider attack.[47]
The vulnerability lifecycle begins when vulnerabilities are introduced into hardware or software.[49]Detection of vulnerabilities can be by the software vendor, or by a third party. In the latter case, it is considered most ethical to immediately disclose the vulnerability to the vendor so it can be fixed.[50]Government or intelligence agencies buy vulnerabilities that have not been publicly disclosed and may use them in an attack, stockpile them, or notify the vendor.[51]As of 2013, theFive Eyes(United States, United Kingdom, Canada, Australia, and New Zealand) captured the plurality of the market and other significant purchasers included Russia, India, Brazil, Malaysia, Singapore, North Korea, and Iran.[52]Organized criminal groups also buy vulnerabilities, although they typically preferexploit kits.[53]
Even vulnerabilities that are publicly known or patched are often exploitable for an extended period.[54][55]Security patches can take months to develop,[56]or may never be developed.[55]A patch can have negative effects on the functionality of software[55]and users may need totestthe patch to confirm functionality and compatibility.[57]Larger organizations may fail to identify and patch all dependencies, while smaller enterprises and personal users may not install patches.[55]Research suggests that risk of cyberattack increases if the vulnerability is made publicly known or a patch is released.[58]Cybercriminals canreverse engineerthe patch to find the underlying vulnerability and develop exploits,[59]often faster than users install the patch.[58]
Vulnerabilities become deprecated when the software or vulnerable versions fall out of use.[50]This can take an extended period of time; in particular, industrial software may not be feasible to replace even if the manufacturer stops supporting it.[60]
A commonly used scale for assessing the severity of vulnerabilities is the open-source specificationCommon Vulnerability Scoring System(CVSS). CVSS evaluates the possibility to exploit the vulnerability and compromise data confidentiality, availability, and integrity. It also considers how the vulnerability could be used and how complex an exploit would need to be. The amount of access needed for exploitation and whether it could take place without user interaction are also factored in to the overall score.[61][62]
Someone who discovers a vulnerability may disclose it immediately (full disclosure) or wait until a patch has been developed (responsible disclosure, or coordinated disclosure). The former approach is praised for its transparency, but the drawback is that the risk of attack is likely to be increased after disclosure with no patch available.[63]Some vendors paybug bountiesto those who report vulnerabilities to them.[64][65]Not all companies respond positively to disclosures, as they can cause legal liability and operational overhead.[66]There is no law requiring disclosure of vulnerabilities.[67]If a vulnerability is discovered by a third party that does not disclose to the vendor or the public, it is called azero-day vulnerability, often considered the most dangerous type because fewer defenses exist.[68]
The most commonly used vulnerability dataset isCommon Vulnerabilities and Exposures(CVE), maintained byMitre Corporation.[69]As of November 2024[update], it has over 240,000 entries[1]This information is shared into other databases, including the United States'National Vulnerability Database,[69]where each vulnerability is given a risk score usingCommon Vulnerability Scoring System(CVSS),Common Platform Enumeration(CPE) scheme, andCommon Weakness Enumeration.[citation needed]CVE and other databases typically do not track vulnerabilities insoftware as a serviceproducts.[39]Submitting a CVE is voluntary for companies that discovered a vulnerability.[67]
The software vendor is usually not legally liable for the cost if a vulnerability is used in an attack, which creates an incentive to make cheaper but less secure software.[70]Some companies are covered by laws, such asPCI,HIPAA, andSarbanes-Oxley, that place legal requirements on vulnerability management.[71]
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AS5678is a requirements specification created bySAE Internationalfor the production and test of passive onlyRadio Frequency Identification(RFID) tags for the Aerospace industry. This specification is also related to the Air Transportation Association SPEC2000 specification.
RFID tags are small devices that are used by a variety of industries in a variety of applications for the tracking of products, materials, assets and personnel. They vary in their ruggedness and durability from throwaway retail applications (retail compliance items such asWal-Mart, Metro, and other retailers specifying shipping carton labeling) to high durability tagging for items such as aircraft parts (SPEC2000 requirements forBoeing,Airbusand others tracking aircraft parts) that must withstand aerospace environments over their lifetimes.
Air Transport Association
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https://en.wikipedia.org/wiki/AS5678
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Abalise(/bəˈliːz/bə-LEEZ) is an electronicbeaconortransponderplaced between therailsof a railway as part of anautomatic train protection(ATP) system. TheFrenchwordbaliseis used to distinguish these beacons from other kinds of beacons.[1]
Balises are used in theKVBsignalling system installed on main lines of the French railway network, other than the high-speedLignes à Grande Vitesse.
Balises constitute an integral part of theEuropean Train Control System, where they serve as "beacons" giving the exact location of a train. The ETCS signalling system is gradually being introduced on railways throughout theEuropean Union.[2]
Balises are also used in theChinese Train Control Systemversions CTCS-2 and CTCS-3 installed on high-speed rail lines in China, which is based on theEuropean Train Control System.
A balise which complies with the European Train Control System specification is called aEurobalise.
A balise typically needs no power source. In response toradio frequencyenergy broadcast by aBalise Transmission Modulemounted under a passing train, the balise either transmits information to the train (uplink) or receives information from the train (downlink, although this function is rarely used). The transmission rate of Eurobalises is sufficient for a complete 'telegram' to be received by a train passing at any speed up to 500 km/h.
A balise may be either a 'Fixed Data Balise', or 'Fixed Balise' for short, transmitting the same data to every train, or a 'Transparent Data Balise' which transmits variable data, also called a 'Switchable' or 'Controllable Balise'. (Note that the word 'fixed' refers to the information transmitted by the balise, not to its physical location. All balises are immobile).
A fixed balise is programmed to transmit the same data to every train. Information transmitted by a fixed balise typically includes: the location of the balise; thegeometry of the line, such as curves and gradients; and any speed restrictions. The programming is performed using a wireless programming device. Thus a fixed balise can notify a train of its exact location, and the distance to the next signal, and can warn of any speed restrictions.
A controllable balise is connected to a Lineside Electronics Unit (LEU), which transmits dynamic data to the train, such as signal indications. Balises forming part of anETCSLevel 1 signalling system employ this capability.[3]The LEU integrates with the conventional (national) signal system either by connecting to the linesiderailway signalor to thesignalling controltower.
Balises must be deployed in pairs so that the train can distinguish the direction of travel 1→2 from direction 2→1, unless they are linked to a previous balise group in which case they can contain only one balise. Extra balises can be installed if the volume of data is too great.
Balises operate with equipment on the train to provide a system that enhances the safety of train operation: at the approaches to stations with multiple platforms fixed balises may be deployed, as a more accurate supplement toGPS, to enable safe operation of automaticselective door opening.[4]
The balise is typically mounted on or betweensleepersor ties in the centre line of the track.
A train travelling at maximum speed of 500 km/h (310 mph) will transmit and receive a minimum of three copies of the telegram while passing over each Eurobalise. The earlier KER balises (KVB, EBICAB, RSDD) were specified to work up to 350 km/h (220 mph).[5]
The train's on-board computer uses the data from the balises to determine the safe speed profile for the line ahead. Enough information is needed to allow the train to come to a safe standstill if required.
The data in the balise can include the distance to the next balise. This is used to check for missing balises which could otherwise lead to a potentialwrong-side failure.
At the start and end of ATP-equipped territory, a pair of fixed balises are often used to inform the onboard ATP equipment to start or stop supervision of the train movements.
Eurobalises are used in:
Balises other than Eurobalises are used in:
The earliest automatic train protection system were purely mechanical with atripcockwhich could be connected directly to the braking system by releasing the opening a switch in the hydraulic system. There were multiple incidents where trains had overrun a stop signal but due to excessive speed still crashed despite the automatic stop. Multiple systems were invented to show the speed in the driver's cab and to provide an electronic system on the train that would prevent speeding. With the advent of high-speed trains it was generally expected that a speed indicator on line-side signals is not sufficient beyond 160 km/h (99 mph) so that all these trains needcab signalling.
A combined solution to the requirements was the GermanLZBsystem that was presented in 1965. The original installations were all hard-wired logic. The first real cab electronics was presented in 1972 (named LZB L72) and a cab computer was introduced by 1980 (LZB 80). The LZB system uses a wire in the middle of the tracks that had loops at a distance of 100 m (330 ft) so that the position of a train was known more precisely than in any earlier system. As a result, the LZB system was not only used on high-speed tracks but also in commuter rail to increase throughput. Due to the deployment costs of the system however it was restricted to these application areas.
During the 1970s British Rail developed the C-APT, the system utilised passive transponders (balises) placed at no more than 1km track intervals, that would transmit the track speed (in an 80bit packet) to a passing train, for in cab display. If the trains control system failed to receive an update, within 1km of the last signal, the displayed speed limit would be blanked an audio tone the driver had to respond to generated, else the trains brakes were automatically applied, the system would be see revenue service from December 1981, with the introduction of theBritish Rail Class 370.[6]
The development of a system using the principle of passive balises with fixed or controlled information started in 1975 by LMEricson and SRT, following an incident in Norway in 1975 (Tretten). The LME/SRT system became the Ebicab system. The Ebicab system established the principles of using magnetic coupling, 27 MHz downlink from the antenna on the locomotive to energize the balises, and an uplink using 4,5 MHz to transmit information telegrams from the balises. The controlled information in the balises is encoded from statuses in the signalling system. The telegrams contains information about permitted speeds, and distances. The information is used in the on-board computer to calculate brake curves, monitor speed and eventually apply brakes. In Norway, the first line equipped with Ebicab as ATP was operational in 1983. The Ebicab principles are subsequently used in KVB and RSDD systems and also for the ERTMS ETCS balises. During the 1980s, other cab computers were introduced to read the older signalling and to overlay it with better control. The GermanPZ80was able to check the speed in steps of 10 km/h (6.2 mph). The FrenchKVBreplaced the external system with balises in the early 1990s to transmit a combined information for oncoming signal aspects and the allowed train speed. Siemens did also invent a successor to the PZB signalling that was deployed asZUB 121[de]in Switzerland since 1992 andZUB 123[de]in Denmark since 1992. ABB improved the external balises in the EBICAB 900 system which as then adopted in Spain and Italy.
Siemens had presented a study on balise systems in 1992[7]which influenced the choice of using a technology based on KVB and GSM instead of LZB when theEuropean Rail Traffic Management Systemwas researching a possible train signalling for Europe. The first Eurobalises were tested in 1996 and later train protection systems used them as a basis for their signalling needs.
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The term "bin bug" was coined in August 2006 by theBritish mediato refer to the use ofRadio Frequency Identification(RFID) chips by some local councils to monitor the amount ofdomestic wastecreated by each household. The system works by having a unique RFID chip for each household's non-recyclingwheelie bin(such households have two bins: one for general waste and onerecyclingbin). The chip is scanned by thedustbin lorryand, as it lifts the bin, records the weight of the contents. This is then stored in a centraldatabasethat monitors the non-recycled waste output of each household.[1][2]
In August 2006, it was reported that fiveUlstercouncils had installed chips in household wheelie bins,[3]and that three more local councils were about to trial the technology.[4]Paul Bettison, the chairman of theLocal Government Association's environment board, said that if pilot schemes received approval from the government and were successful, weighing schemes could be commonplace across the country within two years.[4]While some councils informed the householders of their intentions to monitor their waste output many others did not.[1]Worcester City Council, for example, detailed their plans through local newspaperWorcester Newsin August 2005.[5]Aberdeen City Councilkept the scheme quiet until a local newspaper ran the story; the council declared no intention to operate or bring the system online but did not rule out future use.[6]Some councillors said that the purpose of the "bin bugs" was to settle disputes about the ownership of the bins, but others mentioned that the system is a trial and means that they are more prepared should the government introduce a household waste tax. The tax would be in the form of a charge for households that exceed set limits of non-recycled waste.[1][7]With recycling in the UK amongst the lowest percentage in Europe at 18%, a new tax scheme would have the intention of encouraging domestic recycling and meeting European landfill reduction targets.[4]
Each RFID chip costs around £2, with each scanning system costing around £15,000. TheLocal Government Association(LGA) provided £5 million to councils to fund 40 pilot schemes.[4]They are supplied by two rival German companies: Sulo andDeister Electronic. Mr Bettison said that although removing a device from a bin "would not break any law", in the future a local authority might have grounds to refuse to empty the bin.[1]
The motivation behind the RFID chips are to monitor the production of landfill waste so that councils can comply with the European Landfill Directive 1999/31/EC.[8]The standard regulating RFID tags for the waste industry is EN 14803 Identification and/or determination of the quantity of waste.
TheRFIDtag is located in a recess under the front lip of the bin, either as a self-contained unit or behind a plastic cap.[9][10][11]
There is some debate as to the legality of removing the RFID chip.[12]
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Acampus credential, more commonly known as acampus cardor acampus ID cardis anidentification documentcertifying the status of aneducational institution's students, faculty, staff or other constituents as members of the institutional community and eligible for access to services and resources. Campus credentials are typically valid for the duration of a student's enrollment or an employee's service.
The functions of the campus credential, in addition to data storage for the student's identification, vary by University. Some examples of campus credential functions are:
Campus credentials with multiple functions can help simplify internal administrative processes.
Electronic card access has been available on campuses since as early as 1968. Early versions, such as the “VALI-DINE” system atRochester Institute of Technology, relied on cards with mechanically punched holes to allow access to their dining halls. In the years following, the use of campus credentials and technology has matured. In 1972,California State Polytechnic Universityinstalled the first known card-based system utilizingmagnetic stripetechnology. By 1985, the Harco multi-application, campus-wide system utilizingbar code, Prox contact-based chips and magnetic stripe technology was implemented by Duke University. Technological advances continued to pick up speed with both cashless payment systems introduced by Debitek Inc. and copy machine management introduced by DANYL Corporation in 1986. By the 1990s, universities began linking their campus cards to banks, with Florida State University being the first in 1990. DataCard introduced its first color digital imaging card production system in 1993. In 2001, contactless chip technology cards were introduced and CBORD released the first IP-addressablecard readerfor campus credential access systems. (Huber, 2007). Technology continued to ramp up, with cloud-based Campus Credential systems growing in popularity in 2005. By 2015, the use ofsmart devicesinstead of physical cards soared. And by 2020, wearable credentials, such as wristbands and fobs, gained popularity, along with mobile apps and digital wallets to manage credential functionality (Huber, n.d.).
Several universities throughout North America and Oceania use digital wallets such asApple WalletforiOS&watchOS,Google WalletforAndroid, andSamsung Walletfor Android &Wear OSto store campus cards as mobile credentials. Some universities have replaced physical cards entirely with digital IDs.[1]Samsung Wallet (on Android) and Apple Wallet (on iOS) allow the use of IDs when the phone's screen is off or the battery is depleted. Samsung allows up to 15 taps within 24 hours, and Apple rates their reserve feature as being available for 5 hours after initial depletion.
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https://en.wikipedia.org/wiki/Campus_card
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Chipless RFIDtags areRFIDtags that do not require amicrochipin the transponder.
RFIDs offer longer range and ability to be automated, unlike barcodes that require a human operator for interrogation. The main challenge to their adoption is the cost of RFIDs. The design and fabrication ofASICsneeded for RFID are the major component of their cost, so removing ICs altogether can significantly reduce its cost. The major challenges in designing chipless RFID is data encoding and transmission.[1]
To understand the development of chiplessRFIDtags, it is important to view it in comparison to classic RFID andbarcode. RFID benefits from a very wide spectrum of functionalities, related to the use ofradio-frequency(RF) waves for data exchange. The acquisition of the identifier (ID) is made much easier and volumetric readings are possible, all on tags containing modifiable information. These functions are impossible to implement with a barcode, but in reality, 70% of the items manufactured worldwide are equipped with it. The reasons for this enthusiasm are simple: barcode functions very well and is extremely cheap, the label as well as the reader. This is why barcodes remain the uncontested benchmark in terms of identification, with a cost-to-simplicity-of-use ratio that remains unequalled.
It is also true that RFID contributes other significant functionalities, and the question is therefore one of imagining a technology based on RF waves as a communication vector that would retain some of the advantages of barcodes. Pragmatically speaking, the question of system cost, and particularly of the tags that must be produced in large numbers, remains the central point. Due to the presence of electronic circuits, these tags have a non-negligible cost that is a very great deal higher than that of barcodes. It is logical therefore that a simple solution consists of producing chipless RF tags. The high cost of RFID tags is actually one of the principal reasons that chipped RFID is rare in the market for tags for widely distributed products, a market that numbers in the tens of thousands of billions of units sold per year. In this market, optic barcodes are very widely used.
However, technically speaking, chipped RFID offers significant advantages including increased reading distance and the ability to detect a target outside the field of vision, whatever its position. The concept of chipless RF label has been developed with the idea of competing with barcodes in certain areas of application. RFID has many arguments in its favor in terms of functionality, the only problem remaining being the price. The barcode offers no other feature than the ID recovery; however, the technology is time-tasted, widespread, and extremely low cost.
Chipless RFID has also good arguments in terms of functionality. Some functionalities are degraded versions of what RFID can do (read range / reading flexibility are reduced ...), others seem to be even more relevant in chipless (discretion, product integrity of the tag). The main advantage is the cost of chipless tags. As compared to the barcodes, the chipless technology must bring other features that are impossible to implement with the optical approach, while remaining a very low cost approach, that is to say, a potentially printable one. This is why the writing / rewriting and sensor capabilities are crucial features for the large-scale development of such a technology. For instance, the development of very low cost sensor - tags is now eagerly awaited for, for application reasons.[2]
Like various existingRFIDtechnologies, chipless RFID tags are associated with a specific RF reader, which questions the tag and recovers the information contained in it. The operating principle of the reader is based on the emission of a specific electromagnetic (EM) signal toward the tag, and the capture of the signal reflected by the tag. The processing of the signal received—notably via a decoding stage—makes it possible to recover the information contained in the tag.[3]
However, chipless RFID tags are fundamentally different from RFID tags. In the latter, a specific frame is sent by the reader[4]toward the tag according to a classic binary modulation schema. The tag demodulates this signal, processes the request, possibly writes data in its memory, and sends back a response, modulating its load.[5]Chipless RFID tags, on the other hand, function without a communication protocol. They employ a grid of dipole antennas that are tuned to different frequencies.[6]The interrogator generates a frequency sweep signal and scans for signal dips. Eachdipole antennacan encode one bit. The frequency swept will be determined by the antenna length. They can be viewed as radar targets possessing a specific, stationary temporal or frequential signature. With this technology, the remote reading of an identifier consists of analyzing the radar signature of the tag.
Currently, one of main challenges of the chipless technology is the robustness of tag detection in different environments.[6]It is useless to try to increase the quantity of information that a chipless tag can have if the tag ID cannot be read properly in real environments and without complex calibration techniques. The detection of a chipless tag in noisy environments is much more difficult in chipless than in UHF RFID due to the absence of modulation in time, that is, the absence of two different states in the backscattering signal.
In 2001,Roke Manor Researchcentre announced materials that emit characteristic radiation when moved. These may be exploited for storage of a few data bits encoded in the presence or absence of certain chemicals.[7]
Somark employed adielectricbarcode that may be read usingmicrowaves. The dielectric material reflects, transmits and scatters the incident radiation; the different position and orientation of these bars affects the incident radiation differently and thus encodes the spatial arrangement in the reflected wave. The dielectric material may be dispersed in a fluid to create a dielectric ink.[8]They were mainly used as tags for cattle, which were "painted" using a special needle. The ink may be visible or invisible according to the nature of the dielectric, Operating frequency of the tag may be changed by using different dielectrics.[9]
This system uses varying magnetism. Materials resonate at different frequencies when excited by radiation. The reader analyzes the spectrum of the reflected signal to identify the materials. 70 different materials were found. Each material's presence or absence may be used to encode a bit, enabling encoding up to 270unique binary strings. They work on frequencies between three and ten gigahertz.[10]
In 2004, Tapemark announced a chipless RFID that will have only a passive antenna with a diameter as small as 5μm. The antenna consists of small fibers called nano-resonant structures. Spatial difference in structure encode data. The interrogator sends out a coherent pulse and reads back an interference pattern that it decodes to identify a tag. They work from 24 GHz–60 GHz.[11]Tapemark later discontinued this technology.
Sagentia'sdevices are acousto-magnetic. They exploit the resonance features of magnetically softmagnetostrictivematerials and the data retention capability of hard magnetic materials. Data is written to the card using the contact method. The resonance of themagnetostrictivematerial is altered by the data stored in the hard material. Harmonics may be enabled or disabled corresponding to the state of the hard material, thus encoding the device state as a spectral signature. Tags built by Sagentia forAstraZenecafall into this category.[12][13][14]
Flying Null technology uses a series of passive magnetic structures, much like the lines used in conventional barcodes. These structures are made of soft magnetic material. The interrogator contains two permanent magnets with like poles. The resulting magnetic field has a null volume in the centre. Additionally, interrogating radiation is used. The magnetic field created by the interrogator is such that it drives the soft material to saturation except when it is at the null volume. When in the null volume the soft magnet interacts with the interrogating radiation thus giving away the position of the soft material. Spatial resolution of more than 50 μm may be attained.[15][16]
Surface acoustic wavedevices consists of apiezoelectriccrystal-likelithium niobateon which transducers are made by single-metal-layerphotolithographictechnology. The transducers usually areInter-Digital Transducers(IDT), which have a two-toothed comb-like structure. An antenna is attached to the IDT for reception and transmission. The transducers convert the incident radio wave to surface acoustic waves that travel on the crystal surface until it reaches the encoding reflectors that reflect some waves and transmit the rest. The IDT collects the reflected waves and transmits them to the reader. The first and last reflectors are used for calibration as the response may be affected by physical parameters such as temperature. A pair of reflectors may also be used for error correction. The reflections increase in size from nearest to farthest of the IDT to account for losses due to preceding reflectors and wave attenuation. Data is encoded usingPulse Position Modulation(PPM). The crystal is logically divided into groups, such that each group typically has a length equal to the inverse of the bandwidth. Each group is divided into slots of equal width. The reflector may be placed in any slot. The last slot in each group is usually unused, leaving n-1 positions for the reflector, thus encoding n-1 states. The repetition rate of the PPM is equal to the system bandwidth. The reflector's slot position may be used to encode phase. The devices' temperature dependence means they can also act as temperature sensors.[17]
They employ a grid of dipole antennas that are tuned to different frequencies. The interrogator generates a frequency sweep signal and scans for signal dips. Each dipole antenna can encode one bit. The frequency swept will be determined by the antenna length.[18]
Many improvements have been done in the past few years on communication systems, based on electronic devices where an integrated circuit is at the heart of the whole system. The democratization of these chipped based systems like the RFID one have however given rise to environmental issues.
Lately, new research projects, such asEuropean Research Council(ERC) funded project ScattererID,[19]have introduced the paradigm of RF communication system based on chipless labels, where new useful functionalities can be added. With comparable costs to a barcode, these labels should stand out by providing more functionalities than the optical approach. The objective ofScattererIDproject is to show that it is possible to associate the chipless label ID with other features like the ability to write and rewrite the information, to associate an ID with a sensor function and to associate an ID with gesture recognition.
The possibility of designing reconfigurable and low cost tags involves the development of original approaches at the forefront of progress, like the use of CBRAM from microelectronics, allowing to achieve reconfigurable elements based on Nano-switches.
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FASTagis anelectronic toll collectionsystem inIndia, operated by theNational Highways Authority of India(NHAI).[3][4]It employsRadio Frequency Identification(RFID) technology for making toll payments directly from the prepaid or savings account linked to it or directly toll owner. It is affixed on the windscreen of the vehicle and enables one to drive through toll plazas without stopping for transactions. The tag can be purchased from official Tag issuers or participating Banks[5]which are Made in India and if it is linked to a prepaid account, then recharging or top-up can be as per requirement.[6]The minimum recharge amount is₹100 and can be done online.[7]As perNHAI, FASTag has unlimited validity. 7.5% cashback offers were also provided to promote the use of FASTag. Dedicated Lanes at some Toll plazas have been built for FASTag.
In January 2019, state-run oil marketing companiesIOC,BPCLandHPCLhave signed MoUs enabling the use of FASTag to make purchases at petrol pumps.[8]
As of September 2019, FASTag lanes are available on over 500nationalandstate highwaysand over 54.6lakh(5.46 million) cars are enabled with FASTag.[9]Starting 1 January 2021, FASTag was to be mandatory for all vehicles but later the date was postponed to 15 February 2021.[10]FASTag transactions were valued at₹5,559.91crorein January 2024 with 33.138 crore in volume terms.[11]
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Internet of things(IoT) describes devices withsensors, processing ability,softwareand othertechnologiesthat connect and exchange data with other devices and systems over theInternetor other communication networks.[1][2][3][4][5]The IoT encompasseselectronics,communication, andcomputer scienceengineering. "Internet of things" has been considered amisnomerbecause devices do not need to be connected to the publicinternet; they only need to be connected to a network[6]and be individually addressable.[7][8]
The field has evolved due to the convergence of multipletechnologies, includingubiquitous computing,commoditysensors, and increasingly powerfulembedded systems, as well asmachine learning.[9]Older fields ofembedded systems,wireless sensor networks, control systems,automation(includinghomeandbuilding automation), independently and collectively enable the Internet of things.[10]In the consumer market, IoT technology is mostsynonymouswith "smart home" products, including devices andappliances(lighting fixtures,thermostats, homesecurity systems,cameras, and other home appliances) that support one or more common ecosystems and can be controlled via devices associated with that ecosystem, such assmartphonesandsmart speakers. IoT is also used inhealthcare systems.[11]
There are a number of concerns about the risks in the growth of IoT technologies and products, especially in the areas ofprivacyandsecurity, and consequently there have been industry and government moves to address these concerns, including the development of international and local standards, guidelines, and regulatory frameworks.[12]Because of their interconnected nature, IoT devices are vulnerable to security breaches and privacy concerns. At the same time, the way these devices communicate wirelessly creates regulatory ambiguities, complicating jurisdictional boundaries of the data transfer.[13]
Around 1972, for its remote site use,Stanford Artificial Intelligence Laboratorydeveloped a computer controlled vending machine, adapted from a machine rented fromCanteen Vending, which sold for cash or, though a computer terminal (Teletype Model 33 KSR),[14]on credit.[15]Products included, at least, beer, yogurt, and milk.[15][14]It was called thePrancing Pony, after the name of the room, named after an inn in Tolkien'sLord of the Rings,[15][16]as each room atStanford Artificial Intelligence Laboratorywas named after a place inMiddle Earth.[17]A successor version still operates in the Computer Science Department atStanford, with both hardware and software having been updated.[15]
In 1982,[18]an early concept of a network connectedsmart devicewas built as an Internet interface for sensors installed in theCarnegie Mellon UniversityComputer Science Department's departmentalCoca-Colavending machine, supplied by graduate student volunteers, provided a temperature model and an inventory status,[19][20]inspired by the computer controlled vending machine in thePrancing Ponyroom atStanford Artificial Intelligence Laboratory.[21]First accessible only on the CMU campus, it became the firstARPANET-connected appliance,[22][23]
Mark Weiser's 1991 paper onubiquitous computing, "The Computer of the 21st Century", as well as academic venues such as UbiComp and PerCom produced the contemporary vision of the IoT.[24][25]In 1994, Reza Raji described the concept inIEEE Spectrumas "[moving] small packets of data to a large set of nodes, so as to integrate and automate everything from home appliances to entire factories".[26]Between 1993 and 1997, several companies proposed solutions likeMicrosoft'sat WorkorNovell'sNEST. The field gained momentum whenBill Joyenvisioneddevice-to-devicecommunication as a part of his "Six Webs" framework, presented at the World Economic Forum at Davos in 1999.[27]
The concept of the "Internet of things" and the term itself, first appeared in a speech by Peter T. Lewis, to the Congressional Black Caucus Foundation15th AnnualLegislative Weekend inWashington, D.C., published in September 1985. According to Lewis, "The Internet of Things, or IoT, is the integration of people, processes andtechnologywith connectable devices and sensors to enable remote monitoring, status, manipulation and evaluation of trends of such devices."[28]
The term "Internet of things" was coined independently byKevin AshtonofProcter & Gamble, later ofMIT'sAuto-ID Center, in 1999,[29]though he prefers the phrase "Internetforthings".[30]At that point, he viewedradio-frequency identification(RFID) as essential to the Internet of things,[31]which would allowcomputersto manage all individual things.[32][33][34]The main theme of the Internet of things is to embed short-range mobile transceivers in various gadgets and daily necessities to enable new forms of communication between people and things, and between things themselves.[35]
In 2004 Cornelius "Pete" Peterson, CEO of NetSilicon, predicted that, "The next era of information technology will be dominated by [IoT] devices, and networked devices will ultimately gain in popularity and significance to the extent that they will far exceed the number of networked computers and workstations." Peterson believed that medical devices and industrial controls would become dominant applications of the technology.[36]
Defining the Internet of things as "simply the point in time when more 'things or objects' were connected to the Internet than people",Cisco Systemsestimated that the IoT was "born" between 2008 and 2009, with the things/people ratio growing from 0.08 in 2003 to 1.84 in 2010.[37]
The extensive set of applications for IoT devices[38]is often divided into consumer, commercial, industrial, and infrastructure spaces.[39][40]
A growing portion of IoT devices is created for consumer use, including connected vehicles,home automation,wearable technology, connected health, and appliances with remote monitoring capabilities.[41]
IoT devices are a part of the larger concept ofhome automation, which can include lighting, heating and air conditioning, media and security systems and camera systems.[42][43]Long-term benefits could include energy savings by automatically ensuring lights and electronics are turned off or by making the residents in the home aware of usage.[44]
A smart home or automated home could be based on a platform or hubs that control smart devices and appliances.[45]For instance, usingApple'sHomeKit, manufacturers can have their home products and accessories controlled by an application iniOSdevices such as theiPhoneand theApple Watch.[46][47]This could be a dedicated app or iOS native applications such asSiri.[48]This can be demonstrated in the case of Lenovo's Smart Home Essentials, which is a line of smart home devices that are controlled through Apple's Home app or Siri without the need for a Wi-Fi bridge.[48]There are also dedicated smart home hubs that are offered as standalone platforms to connect different smart home products. These include theAmazon Echo,Google Home, Apple'sHomePod, and Samsung'sSmartThings Hub.[49]In addition to the commercial systems, there are many non-proprietary, open source ecosystems, including Home Assistant, OpenHAB and Domoticz.[50]
One key application of a smart home is toassist the elderly and disabled. These home systems use assistive technology to accommodate an owner's specific disabilities.[51]Voice controlcan assist users with sight and mobility limitations while alert systems can be connected directly tocochlear implantsworn by hearing-impaired users.[52]They can also be equipped with additional safety features, including sensors that monitor for medical emergencies such as falls orseizures.[53]Smart home technology applied in this way can provide users with more freedom and a higher quality of life.[51]
The term "Enterprise IoT" refers to devices used in business and corporate settings.
TheInternet of Medical Things(IoMT) is an application of the IoT for medical and health-related purposes, data collection and analysis for research, and monitoring.[54][55][56][57][58]The IoMT has been referenced as "Smart Healthcare",[59]as the technology for creating a digitized healthcare system, connecting available medical resources and healthcare services.[60][61]
IoT devices can be used to enableremote health monitoringandemergency notification systems. These health monitoring devices can range from blood pressure and heart rate monitors to advanced devices capable of monitoring specialized implants, such as pacemakers, Fitbit electronic wristbands, or advanced hearing aids.[62]Some hospitals have begun implementing "smart beds" that can detect when they are occupied and when a patient is attempting to get up. It can also adjust itself to ensure appropriate pressure and support are applied to the patient without the manual interaction of nurses.[54]A 2015 Goldman Sachs report indicated that healthcare IoT devices "can save the United States more than $300 billion in annual healthcare expenditures by increasing revenue and decreasing cost."[63]Moreover, the use of mobile devices to support medical follow-up led to the creation of 'm-health', used analyzed health statistics.[64]
Specialized sensors can also be equipped within living spaces to monitor the health and general well-being of senior citizens, while also ensuring that proper treatment is being administered and assisting people to regain lost mobility via therapy as well.[65]These sensors create a network ofintelligent sensorsthat are able to collect, process, transfer, and analyze valuable information in different environments, such as connecting in-home monitoring devices to hospital-based systems.[59]Other consumer devices to encourage healthy living, such as connected scales orwearable heart monitors, are also a possibility with the IoT.[66]End-to-end health monitoring IoT platforms are also available for antenatal and chronic patients, helping one manage health vitals and recurring medication requirements.[67]
Advances in plastic and fabric electronics fabrication methods have enabled ultra-low cost, use-and-throw IoMT sensors. These sensors, along with the requiredRFIDelectronics, can be fabricated onpaperore-textilesfor wireless powered disposable sensing devices.[68]Applications have been established forpoint-of-care medical diagnostics, where portability and low system-complexity is essential.[69]
As of 2018[update]IoMT was being applied in theclinical laboratoryindustry.[56]
IoMT in the insurance industry provides access to better and new types of dynamic information. This includes sensor-based solutions such as biosensors, wearables, connected health devices, and mobile apps to track customer behavior. This can lead to more accurate underwriting and new pricing models.[70]
The application of the IoT in healthcare plays a fundamental role in managingchronic diseasesand in disease prevention and control. Remote monitoring is made possible through the connection of powerful wireless solutions. The connectivity enables health practitioners to capture patient's data and apply complex algorithms in health data analysis.[71]
The IoT can assist in the integration of communications, control, and information processing across varioustransportation systems. Application of the IoT extends to all aspects of transportation systems (i.e., the vehicle,[72]the infrastructure, and the driver or user). Dynamic interaction between these components of a transport system enables inter- and intra-vehicular communication,[73]smart traffic control, smart parking,electronic toll collection systems,logisticsandfleet management,vehicle control, safety, and road assistance.[62][74]
Invehicular communication systems,vehicle-to-everythingcommunication (V2X), consists of three main components: vehicle-to-vehicle communication (V2V), vehicle-to-infrastructure communication (V2I) and vehicle to pedestrian communications (V2P). V2X is the first step toautonomous drivingand connected road infrastructure.[75]
IoT devices can be used to monitor and control the mechanical, electrical and electronic systems used in various types of buildings (e.g., public and private, industrial, institutions, or residential)[62]inhome automationandbuilding automationsystems. In this context, three main areas are being covered in literature:[76]
Also known as IIoT, industrial IoT devices acquire and analyze data from connected equipment, operational technology (OT), locations, and people. Combined withoperational technology(OT) monitoring devices, IIoT helps regulate and monitor industrial systems.[77]Also, the same implementation can be carried out for automated record updates of asset placement in industrial storage units as the size of the assets can vary from a small screw to the whole motor spare part, and misplacement of such assets can cause a loss of manpower time and money.
The IoT can connect various manufacturing devices equipped with sensing, identification, processing, communication, actuation, and networking capabilities.[78]Network control and management ofmanufacturing equipment,assetand situation management, or manufacturingprocess controlallow IoT to be used for industrial applications and smart manufacturing.[79]IoT intelligent systems enable rapid manufacturing and optimization of new products and rapid response to product demands.[62]
Digital control systemsto automate process controls, operator tools and service information systems to optimize plant safety and security are within the purview of theIIoT.[80]IoT can also be applied to asset management viapredictive maintenance,statistical evaluation, and measurements to maximize reliability.[81]Industrial management systems can be integrated withsmart grids, enabling energy optimization. Measurements, automated controls, plant optimization, health and safety management, and other functions are provided by networked sensors.[62]
In addition to general manufacturing, IoT is also used for processes in the industrialization of construction.[82]
There are numerous IoT applications in farming[83]such as collecting data on temperature, rainfall, humidity, wind speed, pest infestation, and soil content. This data can be used to automate farming techniques, make informed decisions to improve quality and quantity, minimize risk and waste, and reduce the effort required to manage crops. For example, farmers can now monitor soil temperature and moisture from afar and even apply IoT-acquired data to precision fertilization programs.[84]The overall goal is that data from sensors, coupled with the farmer's knowledge and intuition about his or her farm, can help increase farm productivity, and also help reduce costs.
In August 2018,Toyota Tsushobegan a partnership withMicrosoftto createfish farmingtools using theMicrosoft Azureapplication suite for IoT technologies related to water management. Developed in part by researchers fromKindai University, the water pump mechanisms useartificial intelligenceto count the number of fish on aconveyor belt, analyze the number of fish, and deduce the effectiveness of water flow from the data the fish provide.[85]The FarmBeats project[86]from Microsoft Research that uses TV white space to connect farms is also a part of the Azure Marketplace now.[87]
IoT devices are in use to monitor the environments and systems of boats and yachts.[88]Many pleasure boats are left unattended for days in summer, and months in winter so such devices provide valuable early alerts of boat flooding, fire, and deep discharge of batteries. The use of global Internet data networks such asSigfox, combined with long-life batteries, and microelectronics allows the engine rooms, bilge, and batteries to be constantly monitored and reported to connected Android & Apple applications for example.
Monitoring and controlling operations of sustainable urban and rural infrastructures like bridges, railway tracks and on- and offshore wind farms is a key application of the IoT.[80]The IoT infrastructure can be used for monitoring any events or changes in structural conditions that can compromise safety and increase risk. The IoT can benefit the construction industry by cost-saving, time reduction, better quality workday, paperless workflow and increase in productivity. It can help in taking faster decisions and saving money in Real-TimeData Analytics. It can also be used for scheduling repair and maintenance activities efficiently, by coordinating tasks between different service providers and users of these facilities.[62]IoT devices can also be used to control critical infrastructure like bridges to provide access to ships. The usage of IoT devices for monitoring and operating infrastructure is likely to improve incident management and emergency response coordination, andquality of service,up-timesand reduce costs of operation in all infrastructure-related areas.[89]Even areas such as waste management can benefit.[90]
There are several planned or ongoing large-scale deployments of the IoT, to enable better management of cities and systems. For example,Songdo, South Korea, the first fully equipped and wiredsmart city, is gradually being built,[when?]with approximately 70 percent of the business district completed as of June 2018[update]. Much of the city, the first of its kind, is planned to be wired and automated to operate with little or no human intervention.[91]
In 2014 another application was undergoing a project inSantander, Spain. For this deployment, two approaches have been adopted. This city of 180,000 inhabitants has already seen 18,000 downloads of its city smartphone app. The app is connected to 10,000 sensors that enable services like parking search, and environmental monitoring. City context information is used in this deployment so as to benefit merchants through a spark deals mechanism based on city behavior that aims at maximizing the impact of each notification.[92]
Other examples of large-scale deployments underway include the Sino-Singapore Guangzhou Knowledge City;[93]work on improving air and water quality, reducing noise pollution, and increasing transportation efficiency in San Jose, California;[94]and smart traffic management in western Singapore.[95]Using its RPMA (Random Phase Multiple Access) technology, San Diego–basedIngenuhas built a nationwide public network[96]for low-bandwidthdata transmissions using the same unlicensed 2.4 gigahertz spectrum as Wi-Fi. Ingenu's "Machine Network" covers more than a third of the US population across 35 major cities including San Diego and Dallas.[97]French company,Sigfox, commenced building anUltra Narrowbandwireless data network in theSan Francisco Bay Areain 2014, the first business to achieve such a deployment in the U.S.[98][99]It subsequently announced it would set up a total of 4000base stationsto cover a total of 30 cities in the U.S. by the end of 2016, making it the largest IoT network coverage provider in the country thus far.[100][101]Cisco also participates in smart cities projects. Cisco has deployed technologies for Smart Wi-Fi, Smart Safety & Security,Smart Lighting, Smart Parking, Smart Transports, Smart Bus Stops, Smart Kiosks, Remote Expert for Government Services (REGS) and Smart Education in the five km area in the city of Vijaywada, India.[102][103]
Another example of a large deployment is the one completed by New York Waterways in New York City to connect all the city's vessels and be able to monitor them live 24/7. The network was designed and engineered byFluidmesh Networks, a Chicago-based company developing wireless networks for critical applications. The NYWW network is currently providing coverage on the Hudson River, East River, and Upper New York Bay. With the wireless network in place, NY Waterway is able to take control of its fleet and passengers in a way that was not previously possible. New applications can include security, energy and fleet management, digital signage, public Wi-Fi, paperless ticketing and others.[104]
Significant numbers of energy-consuming devices (e.g. lamps, household appliances, motors, pumps, etc.) already integrate Internet connectivity, which can allow them to communicate with utilities not only to balancepower generationbut also helps optimize the energy consumption as a whole.[62]These devices allow for remote control by users, or central management via acloud-based interface, and enable functions like scheduling (e.g., remotely powering on or off heating systems, controlling ovens, changing lighting conditions etc.).[62]Thesmart gridis a utility-side IoT application; systems gather and act on energy and power-related information to improve the efficiency of the production and distribution of electricity.[105]Usingadvanced metering infrastructure (AMI)Internet-connected devices, electric utilities not only collect data from end-users, but also manage distribution automation devices like transformers.[62]
Environmental monitoringapplications of the IoT typically use sensors to assist in environmental protection[106]by monitoringairorwater quality,[107]atmosphericorsoil conditions,[108]and can even include areas like monitoring themovements of wildlifeand theirhabitats.[109]Development of resource-constrained devices connected to the Internet also means that other applications likeearthquakeortsunami early-warning systemscan also be used by emergency services to provide more effective aid. IoT devices in this application typically span a large geographic area and can also be mobile.[62]It has been argued that the standardization that IoT brings to wireless sensing will revolutionize this area.[110]
Another example of integrating the IoT is Living Lab which integrates and combines research and innovation processes, establishing within a public-private-people-partnership.[111]Between 2006 and January 2024, there were over 440 Living Labs (though not all are currently active)[112]that use the IoT to collaborate and share knowledge between stakeholders to co-create innovative and technological products. For companies to implement and develop IoT services[113]for smart cities, they need to have incentives. The governments play key roles in smart city projects as changes in policies will help cities to implement the IoT which provides effectiveness, efficiency, and accuracy of the resources that are being used. For instance, the government provides tax incentives and cheap rent, improves public transports, and offers an environment where start-up companies, creative industries, and multinationals may co-create, share a common infrastructure and labor markets, and take advantage of locally embedded technologies, production process, and transaction costs.[111]
TheInternet of Military Things (IoMT)is the application of IoT technologies in the military domain for the purposes of reconnaissance, surveillance, and other combat-related objectives. It is heavily influenced by the future prospects of warfare in an urban environment and involves the use of sensors,munitions, vehicles, robots, human-wearable biometrics, and other smart technology that is relevant on the battlefield.[114]
One of the examples of IOT devices used in the military is Xaver 1000 system. The Xaver 1000 was developed by Israel's Camero Tech, which is the latest in the company's line of "through wall imaging systems". The Xaver line uses millimeter wave (MMW) radar, or radar in the range of 30-300 gigahertz. It is equipped with an AI-based life target tracking system as well as its own 3D 'sense-through-the-wall' technology.[115]
TheInternet of Battlefield Things(IoBT) is a project initiated and executed by theU.S. Army Research Laboratory (ARL)that focuses on the basic science related to the IoT that enhance the capabilities of Army soldiers.[116]In 2017, ARL launched theInternet of Battlefield Things Collaborative Research Alliance (IoBT-CRA), establishing a working collaboration between industry, university, and Army researchers to advance the theoretical foundations of IoT technologies and their applications to Army operations.[117][118]
TheOcean of Thingsproject is aDARPA-led program designed to establish an Internet of things across large ocean areas for the purposes of collecting, monitoring, and analyzing environmental and vessel activity data. The project entails the deployment of about 50,000 floats that house a passive sensor suite that autonomously detect and track military and commercial vessels as part of a cloud-based network.[119]
There are several applications of smart oractive packagingin which aQR codeorNFC tagis affixed on a product or its packaging. The tag itself is passive, however, it contains aunique identifier(typically aURL) which enables a user to access digital content about the product via a smartphone.[120]Strictly speaking, such passive items are not part of the Internet of things, but they can be seen as enablers of digital interactions.[121]The term "Internet of Packaging" has been coined to describe applications in which unique identifiers are used, to automate supply chains, and are scanned on large scale by consumers to access digital content.[122]Authentication of the unique identifiers, and thereby of the product itself, is possible via a copy-sensitivedigital watermarkorcopy detection patternfor scanning when scanning a QR code,[123]while NFC tags can encrypt communication.[124]
The IoT's major significant trend in recent years[when?]is the growth of devices connected and controlled via the Internet.[125]The wide range of applications for IoT technology mean that the specifics can be very different from one device to the next but there are basic characteristics shared by most.
The IoT creates opportunities for more direct integration of the physical world into computer-based systems, resulting in efficiency improvements, economic benefits, and reduced human exertions.[126][127][128][129]
IoT Analytics reported there were 16.6 billion IoT devices connected in 2023. In 2020, the same firm projected there would be 30 billion devices connected by 2025. As of October, 2024, there are around 17 billion.[130][131][132]
Ambient intelligenceand autonomous control are not part of the original concept of the Internet of things. Ambient intelligence and autonomous control do not necessarily require Internet structures, either. However, there is a shift in research (by companies such asIntel) to integrate the concepts of the IoT and autonomous control, with initial outcomes towards this direction considering objects as the driving force for autonomous IoT.[133]An approach in this context isdeep reinforcement learningwhere most of IoT systems provide a dynamic and interactive environment.[134]Training an agent (i.e., IoT device) to behave smartly in such an environment cannot be addressed by conventional machine learning algorithms such assupervised learning. By reinforcement learning approach, a learning agent can sense the environment's state (e.g., sensing home temperature), perform actions (e.g., turnHVACon or off) and learn through the maximizing accumulated rewards it receives in long term.
IoT intelligence can be offered at three levels: IoT devices,Edge/Fog nodes, andcloud computing.[135]The need for intelligent control and decision at each level depends on the time sensitiveness of the IoT application. For example, an autonomous vehicle's camera needs to make real-timeobstacle detectionto avoid an accident. This fast decision making would not be possible through transferring data from the vehicle to cloud instances and return the predictions back to the vehicle. Instead, all the operation should be performed locally in the vehicle. Integrating advanced machine learning algorithms includingdeep learninginto IoT devices is an active research area to make smart objects closer to reality. Moreover, it is possible to get the most value out of IoT deployments through analyzing IoT data, extracting hidden information, and predicting control decisions. A wide variety of machine learning techniques have been used in IoT domain ranging from traditional methods such asregression,support vector machine, andrandom forestto advanced ones such asconvolutional neural networks,LSTM, andvariational autoencoder.[136][135]
In the future, the Internet of things may be a non-deterministic and open network in which auto-organized or intelligent entities (web services,SOAcomponents) and virtual objects (avatars) will be interoperable and able to act independently (pursuing their own objectives or shared ones) depending on the context, circumstances or environments. Autonomous behavior through the collection and reasoning of context information as well as the object's ability to detect changes in the environment (faults affecting sensors) and introduce suitable mitigation measures constitutes a major research trend,[137]clearly needed to provide credibility to the IoT technology. Modern IoT products and solutions in the marketplace use a variety of different technologies to support suchcontext-awareautomation, but more sophisticated forms of intelligence are requested to permit sensor units and intelligentcyber-physical systemsto be deployed in real environments.[138]
IoT system architecture, in its simplistic view, consists of three tiers: Tier 1: Devices, Tier 2: theEdgeGateway, and Tier 3: the Cloud.[139]Devices include networked things, such as the sensors and actuators found in IoT equipment, particularly those that use protocols such asModbus,Bluetooth,Zigbee, or proprietary protocols, to connect to an Edge Gateway.[139]The Edge Gateway layer consists of sensor data aggregation systems called Edge Gateways that provide functionality, such as pre-processing of the data, securing connectivity to cloud, using systems such as WebSockets, the event hub, and, even in some cases, edge analytics orfog computing.[139]Edge Gateway layer is also required to give a common view of the devices to the upper layers to facilitate in easier management. The final tier includes the cloud application built for IoT using the microservices architecture, which are usually polyglot and inherently secure in nature using HTTPS/OAuth. It includes variousdatabasesystems that store sensor data, such as time series databases or asset stores using backend data storage systems (e.g. Cassandra, PostgreSQL).[139]The cloud tier in most cloud-based IoT system features event queuing and messaging system that handles communication that transpires in all tiers.[140]Some experts classified the three-tiers in the IoT system as edge, platform, and enterprise and these are connected by proximity network, access network, and service network, respectively.[141]
Building on the Internet of things, theweb of thingsis an architecture for the application layer of the Internet of things looking at the convergence of data from IoT devices into Web applications to create innovative use-cases. In order to program and control the flow of information in the Internet of things, a predicted architectural direction is being calledBPM Everywherewhich is a blending of traditional process management with process mining and special capabilities to automate the control of large numbers of coordinated devices.[citation needed]
The Internet of things requires huge scalability in the network space to handle the surge of devices.[142]IETF 6LoWPANcan be used to connect devices to IP networks. With billions of devices[143]being added to the Internet space,IPv6will play a major role in handling the network layer scalability.IETF's Constrained Application Protocol,ZeroMQ, andMQTTcan provide lightweight data transport. In practice many groups of IoT devices are hidden behind gateway nodes and may not have unique addresses. Also the vision of everything-interconnected is not needed for most applications as it is mainly the data which need interconnecting at a higher layer.[citation needed]
Fog computing is a viable alternative to prevent such a large burst of data flow through the Internet.[144]Theedge devices' computation power to analyze and process data is extremely limited. Limited processing power is a key attribute of IoT devices as their purpose is to supply data about physical objects while remaining autonomous. Heavy processing requirements use more battery power harming IoT's ability to operate. Scalability is easy because IoT devices simply supply data through the Internet to a server with sufficient processing power.[145]
Decentralized Internet of things, or decentralized IoT, is a modified IoT which utilizes fog computing to handle and balance requests of connected IoT devices in order to reduce loading on the cloud servers and improve responsiveness for latency-sensitive IoT applications like vital signs monitoring of patients, vehicle-to-vehicle communication of autonomous driving, and critical failure detection of industrial devices.[146]Performance is improved, especially for huge IoT systems with millions of nodes.[147]
Conventional IoT is connected via a mesh network and led by a major head node (centralized controller).[148]The head node decides how a data is created, stored, and transmitted.[149]In contrast, decentralized IoT attempts to divide IoT systems into smaller divisions.[150]The head node authorizes partial decision-making power to lower level sub-nodes under mutual agreed policy.[151]
Some approached to decentralized IoT attempts to address the limitedbandwidthand hashing capacity of battery powered or wireless IoT devices viablockchain.[152][153][154]
In semi-open or closed loops (i.e., value chains, whenever a global finality can be settled) the IoT will often be considered and studied as acomplex system[155]due to the huge number of different links, interactions between autonomous actors, and its capacity to integrate new actors. At the overall stage (full open loop) it will likely be seen as achaoticenvironment (sincesystemsalways have finality).
As a practical approach, not all elements on the Internet of things run in a global, public space. Subsystems are often implemented to mitigate the risks of privacy, control and reliability. For example, domestic robotics (domotics) running inside a smart home might only share data within and be available via alocal network.[156]Managing and controlling a high dynamic ad hoc IoT things/devices network is a tough task with the traditional networks architecture,software-defined networking(SDN) provides the agile dynamic solution that can cope with the special requirements of the diversity of innovative IoT applications.[157][158]
The exact scale of the Internet of things is unknown, with quotes of billions or trillions often quoted at the beginning of IoT articles. In 2015 there were 83 million smart devices in people's homes. This number is expected to grow to 193 million devices by 2020.[43][159]In 2023, the number of connected IoT devices will reach 16.6 billion.[160]
The figure of online capable devices grew 31% from 2016 to 2017 to reach 8.4 billion.[161]
In the Internet of things, the precise geographic location of a thing—and also the precise geographic dimensions of a thing—can be critical.[162]Therefore, facts about a thing, such as its location in time and space, have been less critical to track because the person processing the information can decide whether or not that information was important to the action being taken, and if so, add the missing information (or decide to not take the action). (Note that some things on the Internet of things will be sensors, and sensor location is usually important.[163]) TheGeoWebandDigital Earthare applications that become possible when things can become organized and connected by location. However, the challenges that remain include the constraints of variable spatial scales, the need to handle massive amounts of data, and an indexing for fast search and neighbour operations. On the Internet of things, if things are able to take actions on their own initiative, this human-centric mediation role is eliminated. Thus, the time-space context that we as humans take for granted must be given a central role in this informationecosystem. Just as standards play a key role on the Internet and the Web, geo-spatial standards will play a key role on the Internet of things.[164][165]
Many IoT devices have the potential to take a piece of this market.Jean-Louis Gassée(Apple initial alumni team, and BeOS co-founder) has addressed this topic in an article onMonday Note,[166]where he predicts that the most likely problem will be what he calls the "basket of remotes" problem, where we'll have hundreds of applications to interface with hundreds of devices that don't share protocols for speaking with one another.[166]For improved user interaction, some technology leaders are joining forces to create standards for communication between devices to solve this problem. Others are turning to the concept of predictive interaction of devices, "where collected data is used to predict and trigger actions on the specific devices" while making them work together.[167]
Social Internet of things (SIoT) is a new kind of IoT that focuses the importance of social interaction and relationship between IoT devices.[168]SIoT is a pattern of how cross-domain IoT devices enabling application to application communication and collaboration without human intervention in order to serve their owners with autonomous services,[169]and this only can be realized when gained low-level architecture support from both IoT software and hardware engineering.[170]
IoT defines a device with an identity like a citizen in a community and connect them to the Internet to provide services to its users.[171]SIoT defines a social network for IoT devices only to interact with each other for different goals that to serve human.[172]
SIoT is different from the original IoT in terms of the collaboration characteristics. IoT is passive, it was set to serve for dedicated purposes with existing IoT devices in predetermined system. SIoT is active, it was programmed and managed by AI to serve for unplanned purposes with mix and match of potential IoT devices from different systems that benefit its users.[173]
IoT devices built-in with sociability will broadcast their abilities or functionalities, and at the same time discovers, shares information, monitors, navigates and groups with other IoT devices in the same or nearby network realizing SIoT[174]and facilitating useful service compositions in order to help its users proactively in every day's life especially during emergency.[175]
There are many technologies that enable the IoT. Crucial to the field is the network used to communicate between devices of an IoT installation, a role that several wireless or wired technologies may fulfill:[182][183][184]
The original idea of theAuto-ID Centeris based on RFID-tags and distinct identification through theElectronic Product Code. This has evolved into objects having an IP address orURI.[185]An alternative view, from the world of theSemantic Web[186]focuses instead on making all things (not just those electronic, smart, or RFID-enabled) addressable by the existing naming protocols, such asURI. The objects themselves do not converse, but they may now be referred to by other agents, such as powerful centralised servers acting for their human owners.[187]Integration with the Internet implies that devices will use anIP addressas a distinct identifier. Due to thelimited address spaceofIPv4(which allows for 4.3 billion different addresses), objects in the IoT will have to usethe next generationof the Internet protocol (IPv6) to scale to the extremely large address space required.[188][189][190]Internet-of-things devices additionally will benefit from the stateless address auto-configuration present in IPv6,[191]as it reduces the configuration overhead on the hosts,[189]and theIETF 6LoWPANheader compression. To a large extent, the future of the Internet of things will not be possible without the support of IPv6; and consequently, the global adoption of IPv6 in the coming years will be critical for the successful development of the IoT in the future.[190]
Different technologies have different roles in aprotocol stack. Below is a simplified[notes 1]presentation of the roles of several popular communication technologies in IoT applications:
This is a list oftechnical standardsfor the IoT, most of which areopen standards, and thestandards organizationsthat aspire to successfully setting them.[206][207]
The GS1 digital link standard,[211]first released in August 2018, allows the use QR Codes, GS1 Datamatrix, RFID and NFC to enable various types of business-to-business, as well as business-to-consumers interactions.
Some scholars and activists argue that the IoT can be used to create new models ofcivic engagementif device networks can be open to user control and inter-operable platforms.Philip N. Howard, a professor and author, writes that political life in both democracies and authoritarian regimes will be shaped by the way the IoT will be used for civic engagement. For that to happen, he argues that any connected device should be able to divulge a list of the "ultimate beneficiaries" of its sensor data and that individual citizens should be able to add new organisations to the beneficiary list. In addition, he argues that civil society groups need to start developing their IoT strategy for making use of data and engaging with the public.[213]
One of the key drivers of the IoT is data. The success of the idea of connecting devices to make them more efficient is dependent upon access to and storage & processing of data. For this purpose, companies working on the IoT collect data from multiple sources and store it in their cloud network for further processing. This leaves the door wide open for privacy and security dangers and single point vulnerability of multiple systems.[214]The other issues pertain to consumer choice and ownership of data[215]and how it is used. Though still in their infancy, regulations and governance regarding these issues of privacy, security, and data ownership continue to develop.[216][217][218]IoT regulation depends on the country. Some examples of legislation that is relevant to privacy and data collection are: the US Privacy Act of 1974, OECD Guidelines on the Protection of Privacy and Transborder Flows of Personal Data of 1980, and the EU Directive 95/46/EC of 1995.[219]
Current regulatory environment:
A report published by theFederal Trade Commission(FTC) in January 2015 made the following three recommendations:[220]
However, the FTC stopped at just making recommendations for now. According to an FTC analysis, the existing framework, consisting of theFTC Act, theFair Credit Reporting Act, and theChildren's Online Privacy Protection Act, along with developing consumer education and business guidance, participation in multi-stakeholder efforts and advocacy to other agencies at the federal, state and local level, is sufficient to protect consumer rights.[222]
A resolution passed by the Senate in March 2015, is already being considered by the Congress.[223]This resolution recognized the need for formulating a National Policy on IoT and the matter of privacy, security and spectrum. Furthermore, to provide an impetus to the IoT ecosystem, in March 2016, a bipartisan group of four Senators proposed a bill, The Developing Innovation and Growing the Internet of Things (DIGIT) Act, to direct theFederal Communications Commissionto assess the need for more spectrum to connect IoT devices.
Approved on 28 September 2018, California Senate Bill No. 327[224]goes into effect on 1 January 2020. The bill requires "a manufacturer of a connected device, as those terms are defined, to equip the device with a reasonable security feature or features that are appropriate to the nature and function of the device, appropriate to the information it may collect, contain, or transmit, and designed to protect the device and any information contained therein from unauthorized access, destruction, use, modification, or disclosure,"
Several standards for the IoT industry are actually being established relating to automobiles because most concerns arising from use of connected cars apply to healthcare devices as well. In fact, theNational Highway Traffic Safety Administration(NHTSA) is preparing cybersecurity guidelines and a database of best practices to make automotive computer systems more secure.[225]
A recent report from the World Bank examines the challenges and opportunities in government adoption of IoT.[226]These include –
In early December 2021, the U.K. government introduced theProduct Security and Telecommunications Infrastructure bill(PST), an effort to legislate IoT distributors, manufacturers, and importers to meet certaincybersecurity standards. The bill also seeks to improve the security credentials of consumer IoT devices.[227]
The IoT suffers fromplatform fragmentation, lack of interoperability and commontechnical standards[228][229][230][231][232][233][234][excessive citations]a situation where the variety of IoT devices, in terms of both hardware variations and differences in the software running on them, makes the task of developing applications that work consistently between different inconsistent technologyecosystemshard.[1]For example, wireless connectivity for IoT devices can be done usingBluetooth,Wi-Fi,Wi-Fi HaLow,Zigbee,Z-Wave,LoRa,NB-IoT,Cat M1as well as completely custom proprietary radios – each with its own advantages and disadvantages; and unique support ecosystem.[235]
The IoT'samorphous computingnature is also a problem for security, since patches to bugs found in the core operating system often do not reach users of older and lower-price devices.[236][237][238]One set of researchers says that the failure of vendors to support older devices with patches and updates leaves more than 87% of active Android devices vulnerable.[239][240]
Philip N. Howard, a professor and author, writes that the Internet of things offers immense potential for empowering citizens, making government transparent, and broadeninginformation access. Howard cautions, however, that privacy threats are enormous, as is the potential for social control and political manipulation.[241]
Concerns about privacy have led many to consider the possibility thatbig datainfrastructures such as the Internet of things anddata miningare inherently incompatible with privacy.[242]Key challenges of increased digitalization in the water, transport or energy sector are related to privacy andcybersecuritywhich necessitate an adequate response from research and policymakers alike.[243]
WriterAdam Greenfieldclaims that IoT technologies are not only an invasion of public space but are also being used to perpetuate normative behavior, citing an instance of billboards with hidden cameras that tracked the demographics of passersby who stopped to read the advertisement.
The Internet of Things Council compared the increased prevalence ofdigital surveillancedue to the Internet of things to the concept of thepanopticondescribed byJeremy Benthamin the 18th century.[244]The assertion is supported by the works of French philosophersMichel FoucaultandGilles Deleuze. InDiscipline and Punish: The Birth of the Prison, Foucault asserts that the panopticon was a central element of the discipline society developed during theIndustrial Era.[245]Foucault also argued that the discipline systems established in factories and school reflected Bentham's vision ofpanopticism.[245]In his 1992 paper "Postscripts on the Societies of Control", Deleuze wrote that the discipline society had transitioned into a control society, with thecomputerreplacing thepanopticonas an instrument of discipline and control while still maintaining the qualities similar to that of panopticism.[246]
Peter-Paul Verbeek, a professor of philosophy of technology at theUniversity of Twente, Netherlands, writes that technology already influences our moral decision making, which in turn affects human agency, privacy and autonomy. He cautions against viewing technology merely as a human tool and advocates instead to consider it as an active agent.[247]
Justin Brookman, of theCenter for Democracy and Technology, expressed concern regarding the impact of the IoT onconsumer privacy, saying that "There are some people in the commercial space who say, 'Oh, big data – well, let's collect everything, keep it around forever, we'll pay for somebody to think about security later.' The question is whether we want to have some sort of policy framework in place to limit that."[248]
Tim O'Reillybelieves that the way companies sell the IoT devices on consumers are misplaced, disputing the notion that the IoT is about gaining efficiency from putting all kinds of devices online and postulating that the "IoT is really about human augmentation. The applications are profoundly different when you have sensors and data driving the decision-making."[249]
Editorials atWIREDhave also expressed concern, one stating "What you're about to lose is your privacy. Actually, it's worse than that. You aren't just going to lose your privacy, you're going to have to watch the very concept of privacy be rewritten under your nose."[250]
TheAmerican Civil Liberties Union(ACLU) expressed concern regarding the ability of IoT to erode people's control over their own lives. The ACLU wrote that "There's simply no way to forecast how these immense powers – disproportionately accumulating in the hands of corporations seeking financial advantage and governments craving ever more control – will be used. Chances are big data and the Internet of Things will make it harder for us to control our own lives, as we grow increasingly transparent to powerful corporations and government institutions that are becoming more opaque to us."[251]
In response to rising concerns about privacy andsmart technology, in 2007 theBritish Governmentstated it would follow formalPrivacy by Designprinciples when implementing their smart metering program. The program would lead to replacement of traditionalpower meterswith smart power meters, which could track and manage energy usage more accurately.[252]However theBritish Computer Societyis doubtful these principles were ever actually implemented.[253]In 2009 theDutch Parliamentrejected a similar smart metering program, basing their decision on privacy concerns. The Dutch program later revised and passed in 2011.[253]
A challenge for producers of IoT applications is toclean, process and interpret the vast amount of data which is gathered by the sensors. There is a solution proposed for the analytics of the information referred to as Wireless Sensor Networks.[254]These networks share data among sensor nodes that are sent to a distributed system for the analytics of the sensory data.[255]
Another challenge is the storage of this bulk data. Depending on the application, there could be high data acquisition requirements, which in turn lead to high storage requirements. In 2013, the Internet was estimated to be responsible for consuming 5% of the total energy produced,[254]and a "daunting challenge to power" IoT devices to collect and even store data still remains.[256]
Data silos, although a common challenge of legacy systems, still commonly occur with the implementation of IoT devices, particularly within manufacturing. As there are a lot of benefits to be gained from IoT and IIoT devices, the means in which the data is stored can present serious challenges without the principles of autonomy, transparency, and interoperability being considered.[257]The challenges do not occur by the device itself, but the means in which databases and data warehouses are set-up. These challenges were commonly identified in manufactures and enterprises which have begun upon digital transformation, and are part of the digital foundation, indicating that in order to receive the optimal benefits from IoT devices and for decision making, enterprises will have to first re-align their data storing methods. These challenges were identified by Keller (2021) when investigating the IT and application landscape of I4.0 implementation within German M&E manufactures.[257]
Security is the biggest concern in adopting Internet of things technology,[258]with concerns that rapid development is happening without appropriate consideration of the profound security challenges involved[259]and the regulatory changes that might be necessary.[260][261]The rapid development of the Internet of Things (IoT) has allowed billions of devices to connect to the network. Due to too many connected devices and the limitation of communication security technology, various security issues gradually appear in the IoT.[262]
Most of the technical security concerns are similar to those of conventional servers, workstations and smartphones.[263]These concerns include using weak authentication, forgetting to change default credentials, unencrypted messages sent between devices,SQL injections,man-in-the-middle attacks, and poor handling of security updates.[264][265]However, many IoT devices have severe operational limitations on the computational power available to them. These constraints often make them unable to directly use basic security measures such as implementing firewalls or using strong cryptosystems to encrypt their communications with other devices[266]- and the low price and consumer focus of many devices makes a robust security patching system uncommon.[267]
Rather than conventional security vulnerabilities, fault injection attacks are on the rise and targeting IoT devices. A fault injection attack is a physical attack on a device to purposefully introduce faults in the system to change the intended behavior. Faults might happen unintentionally by environmental noises and electromagnetic fields. There are ideas stemmed from control-flow integrity (CFI) to prevent fault injection attacks and system recovery to a healthy state before the fault.[268]
Internet of things devices also have access to new areas of data, and can often control physical devices,[269]so that even by 2014 it was possible to say that many Internet-connected appliances could already "spy on people in their own homes" including televisions, kitchen appliances,[270]cameras, and thermostats.[271]Computer-controlled devices in automobiles such as brakes, engine, locks, hood and trunk releases, horn, heat, and dashboard have been shown to be vulnerable to attackers who have access to the on-board network. In some cases, vehicle computer systems are Internet-connected, allowing them to be exploited remotely.[272]By 2008 security researchers had shown the ability to remotely control pacemakers without authority. Later hackers demonstrated remote control of insulin pumps[273]and implantable cardioverter defibrillators.[274]
Poorly secured Internet-accessible IoT devices can also be subverted to attack others. In 2016, adistributed denial of service attackpowered by Internet of things devices running theMiraimalwaretook down a DNS provider and major web sites.[275]TheMirai Botnethad infected roughly 65,000 IoT devices within the first 20 hours.[276]Eventually the infections increased to around 200,000 to 300,000 infections.[276]Brazil, Colombia and Vietnam made up of 41.5% of the infections.[276]The Mirai Botnet had singled out specific IoT devices that consisted of DVRs, IP cameras, routers and printers.[276]Top vendors that contained the most infected devices were identified as Dahua, Huawei, ZTE, Cisco, ZyXEL and MikroTik.[276]In May 2017, Junade Ali, a computer scientist atCloudflarenoted that native DDoS vulnerabilities exist in IoT devices due to a poor implementation of thePublish–subscribe pattern.[277][278]These sorts of attacks have caused security experts to view IoT as a real threat to Internet services.[279]
The U.S.National Intelligence Councilin an unclassified report maintains that it would be hard to deny "access to networks of sensors and remotely-controlled objects by enemies of the United States, criminals, and mischief makers... An open market for aggregated sensor data could serve the interests of commerce and security no less than it helps criminals and spies identify vulnerable targets. Thus, massively parallelsensor fusionmay undermine social cohesion, if it proves to be fundamentally incompatible with Fourth-Amendment guarantees against unreasonable search."[280]In general, the intelligence community views the Internet of things as a rich source of data.[281]
On 31 January 2019,The Washington Postwrote an article regarding the security and ethical challenges that can occur with IoT doorbells and cameras: "Last month, Ring got caught allowing its team in Ukraine to view and annotate certain user videos; the company says it only looks at publicly shared videos and those from Ring owners who provide consent. Just last week, a California family's Nest camera let a hacker take over and broadcast fake audio warnings about a missile attack, not to mention peer in on them, when they used a weak password."[282]
There have been a range of responses to concerns over security. The Internet of Things Security Foundation (IoTSF) was launched on 23 September 2015 with a mission to secure the Internet of things by promoting knowledge and best practice. Its founding board is made from technology providers and telecommunications companies. In addition, large IT companies are continually developing innovative solutions to ensure the security of IoT devices. In 2017, Mozilla launchedProject Things, which allows to route IoT devices through a safe Web of Things gateway.[283]As per the estimates from KBV Research,[284]the overall IoT security market[285]would grow at 27.9% rate during 2016–2022 as a result of growing infrastructural concerns and diversified usage of Internet of things.[286][287]
Governmental regulation is argued by some to be necessary to secure IoT devices and the wider Internet – as market incentives to secure IoT devices is insufficient.[288][260][261]It was found that due to the nature of most of the IoT development boards, they generate predictable and weak keys which make it easy to be utilized byman-in-the-middle attack. However, various hardening approaches were proposed by many researchers to resolve the issue of SSH weak implementation and weak keys.[289]
IoT security within the field of manufacturing presents different challenges, and varying perspectives. Within the EU and Germany, data protection is constantly referenced throughout manufacturing and digital policy particularly that of I4.0. However, the attitude towards data security differs from the enterprise perspective whereas there is an emphasis on less data protection in the form of GDPR as the data being collected from IoT devices in the manufacturing sector does not display personal details.[257]Yet, research has indicated that manufacturing experts are concerned about "data security for protecting machine technology from international competitors with the ever-greater push for interconnectivity".[257]
IoT systems are typically controlled by event-driven smart apps that take as input either sensed data, user inputs, or other external triggers (from the Internet) and command one or more actuators towards providing different forms of automation.[290]Examples of sensors include smoke detectors, motion sensors, and contact sensors. Examples of actuators include smart locks, smart power outlets, and door controls. Popular control platforms on which third-party developers can build smart apps that interact wirelessly with these sensors and actuators include Samsung's SmartThings,[291]Apple's HomeKit,[292]and Amazon's Alexa,[293]among others.
A problem specific to IoT systems is that buggy apps, unforeseen bad app interactions, or device/communication failures, can cause unsafe and dangerous physical states, e.g., "unlock the entrance door when no one is at home" or "turn off the heater when the temperature is below 0 degrees Celsius and people are sleeping at night".[290]Detecting flaws that lead to such states, requires a holistic view of installed apps, component devices, their configurations, and more importantly, how they interact. Recently, researchers from the University of California Riverside have proposed IotSan, a novel practical system that uses model checking as a building block to reveal "interaction-level" flaws by identifying events that can lead the system to unsafe states.[290]They have evaluated IotSan on the Samsung SmartThings platform. From 76 manually configured systems, IotSan detects 147 vulnerabilities (i.e., violations of safe physical states/properties).
Given widespread recognition of the evolving nature of the design and management of the Internet of things, sustainable and secure deployment of IoT solutions must design for "anarchic scalability".[294]Application of the concept of anarchic scalability can be extended to physical systems (i.e. controlled real-world objects), by virtue of those systems being designed to account for uncertain management futures. This hard anarchic scalability thus provides a pathway forward to fully realize the potential of Internet-of-things solutions by selectively constraining physical systems to allow for all management regimes without risking physical failure.[294]
Brown University computer scientistMichael Littmanhas argued that successful execution of the Internet of things requires consideration of the interface's usability as well as the technology itself. These interfaces need to be not only more user-friendly but also better integrated: "If users need to learn different interfaces for their vacuums, their locks, their sprinklers, their lights, and their coffeemakers, it's tough to say that their lives have been made any easier."[295]
A concern regarding Internet-of-things technologies pertains to the environmental impacts of the manufacture, use, and eventual disposal of all these semiconductor-rich devices.[296]Modern electronics are replete with a wide variety of heavy metals and rare-earth metals, as well as highly toxic synthetic chemicals. This makes them extremely difficult to properly recycle. Electronic components are often incinerated or placed in regular landfills. Furthermore, the human and environmental cost of mining the rare-earth metals that are integral to modern electronic components continues to grow. This leads to societal questions concerning the environmental impacts of IoT devices over their lifetime.[297]
TheElectronic Frontier Foundationhas raised concerns that companies can use the technologies necessary to support connected devices to intentionally disable or "brick" their customers' devices via a remote software update or by disabling a service necessary to the operation of the device. In one example,home automationdevices sold with the promise of a "Lifetime Subscription" were rendered useless afterNest Labsacquired Revolv and made the decision to shut down the central servers the Revolv devices had used to operate.[298]As Nest is a company owned byAlphabet(Google's parent company), the EFF argues this sets a "terrible precedent for a company with ambitions to sell self-driving cars, medical devices, and other high-end gadgets that may be essential to a person's livelihood or physical safety."[299]
Owners should be free to point their devices to a different server or collaborate on improved software. But such action violates the United StatesDMCAsection 1201, which only has an exemption for "local use". This forces tinkerers who want to keep using their own equipment into a legal grey area. EFF thinks buyers should refuse electronics and software that prioritize the manufacturer's wishes above their own.[299]
Examples of post-sale manipulations includeGoogle NestRevolv, disabled privacy settings onAndroid, Sony disablingLinuxonPlayStation 3, and enforcedEULAonWii U.[299]
Kevin Lonergan atInformation Age, a business technology magazine, has referred to the terms surrounding the IoT as a "terminology zoo".[300]The lack of clear terminology is not "useful from a practical point of view" and a "source of confusion for the end user".[300]A company operating in the IoT space could be working in anything related to sensor technology, networking, embedded systems, or analytics.[300]According to Lonergan, the term IoT was coined before smart phones, tablets, and devices as we know them today existed, and there is a long list of terms with varying degrees of overlap andtechnological convergence: Internet of things, Internet of everything (IoE), Internet of goods (supply chain), industrial Internet,pervasive computing, pervasive sensing,ubiquitous computing,cyber-physical systems(CPS),wireless sensor networks(WSN),smart objects,digital twin, cyberobjects or avatars,[155]cooperating objects,machine to machine(M2M), ambient intelligence (AmI),Operational technology(OT), andinformation technology(IT).[300]Regarding IIoT, an industrial sub-field of IoT, theIndustrial Internet Consortium's Vocabulary Task Group has created a "common and reusable vocabulary of terms"[301]to ensure "consistent terminology"[301][302]across publications issued by the Industrial Internet Consortium. IoT One has created an IoT Terms Database including a New Term Alert[303]to be notified when a new term is published. As of March 2020[update], this database aggregates 807 IoT-related terms, while keeping material "transparent and comprehensive".[304][305]
Despite a shared belief in the potential of the IoT, industry leaders and consumers are facing barriers to adopt IoT technology more widely. Mike Farley argued inForbesthat while IoT solutions appeal toearly adopters, they either lack interoperability or a clear use case for end-users.[306]A study by Ericsson regarding the adoption of IoT among Danish companies suggests that many struggle "to pinpoint exactly where the value of IoT lies for them".[307]
As for IoT, especially in regards to consumer IoT, information about a user's daily routine is collected so that the "things" around the user can cooperate to provide better services that fulfill personal preference.[308]When the collected information which describes a user in detail travels through multiplehops in a network, due to a diverse integration of services, devices and network, the information stored on a device is vulnerable toprivacy violationby compromising nodes existing in an IoT network.[309]
For example, on 21 October 2016, a multipledistributed denial of service(DDoS) attacks systems operated bydomain name systemprovider Dyn, which caused the inaccessibility of several websites, such asGitHub,Twitter, and others. This attack is executed through abotnetconsisting of a large number of IoT devices including IP cameras,gateways, and even baby monitors.[310]
Fundamentally there are 4 security objectives that the IoT system requires: (1) dataconfidentiality: unauthorised parties cannot have access to the transmitted and stored data; (2) dataintegrity: intentional and unintentionalcorruptionof transmitted and stored data must be detected; (3)non-repudiation: the sender cannot deny having sent a given message; (4) data availability: the transmitted and stored data should be available to authorised parties even with thedenial-of-service(DOS) attacks.[311]
Information privacy regulations also require organisations to practice "reasonable security".California's SB-327 Information privacy: connected devices"would require a manufacturer of a connected device, as those terms are defined, to equip the device with a reasonable security feature or features that are appropriate to the nature and function of the device, appropriate to the information it may collect, contain, or transmit, and designed to protect the device and any information contained therein from unauthorised access, destruction, use, modification, or disclosure, as specified".[312]As each organisation's environment is unique, it can prove challenging to demonstrate what "reasonable security" is and what potential risks could be involved for the business. Oregon's HB2395 also "requires [aperson that manufactures, sells or offers to sell connected device]manufacturerto equip connected device with reasonable security features that protectconnected deviceand information that connected device [collects, contains, stores or transmits]storesfrom access, destruction, modification, use or disclosure that consumer does not authorise."[313]
According to antivirus providerKaspersky, there were 639 million data breaches of IoT devices in 2020 and 1.5 billion breaches in the first six months of 2021.[227]
One method of overcoming the barrier of safety issues is the introduction of standards and certification of devices. In 2024, two voluntary and non-competing programs were proposed and launched in the United States: the US Cyber Trust Mark fromThe Federal Communications Commissionand CSA's IoT Device Security Specification from theConnectivity Standards Alliance. The programs incorporate international expertise, with the CSA mark recognized by the Singapore Cybersecurity Agency. Compliance means that IoT devices can resist hacking, control hijacking and theft of confidential data.
A study issued by Ericsson regarding the adoption of Internet of things among Danish companies identified a "clash between IoT and companies' traditionalgovernancestructures, as IoT still presents both uncertainties and a lack of historical precedence."[307]Among the respondents interviewed, 60 percent stated that they "do not believe they have the organizational capabilities, and three of four do not believe they have the processes needed, to capture the IoT opportunity."[307]This has led to a need to understandorganizational culturein order to facilitateorganizational designprocesses and to test newinnovation managementpractices. A lack of digital leadership in the age ofdigital transformationhas also stifled innovation and IoT adoption to a degree that many companies, in the face of uncertainty, "were waiting for the market dynamics to play out",[307]or further action in regards to IoT "was pending competitor moves, customer pull, or regulatory requirements".[307]Some of these companies risk being "kodaked" – "Kodak was a market leader until digital disruption eclipsed film photography with digital photos" – failing to "see the disruptive forces affecting their industry"[314]and "to truly embrace the new business models the disruptive change opens up".[314]Scott Anthony has written inHarvard Business Reviewthat Kodak "created a digital camera, invested in the technology, and even understood that photos would be shared online"[314]but ultimately failed to realize that "online photo sharingwasthe new business, not just a way to expand the printing business."[314]
According to 2018 study, 70–75% of IoT deployments were stuck in the pilot or prototype stage, unable to reach scale due in part to a lack of business planning.[315][page needed][316]
Even though scientists, engineers, and managers across the world are continuously working to create and exploit the benefits of IoT products, there are some flaws in the governance, management and implementation of such projects. Despite tremendous forward momentum in the field of information and other underlying technologies, IoT still remains a complex area and the problem of how IoT projects are managed still needs to be addressed. IoT projects must be run differently than simple and traditional IT, manufacturing or construction projects. Because IoT projects have longer project timelines, a lack of skilled resources and several security/legal issues, there is a need for new and specifically designed project processes. The following management techniques should improve the success rate of IoT projects:[317]
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https://en.wikipedia.org/wiki/Internet_of_Things
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Mass surveillanceis the intricatesurveillanceof an entire or a substantial fraction of a population in order to monitor that group of citizens.[1]The surveillance is often carried out bylocalandfederal governmentsorgovernmental organizations, but it may also be carried out by corporations (either on behalf of governments or at their own initiative). Depending on each nation's laws andjudicial systems, the legality of and the permission required to engage in mass surveillance varies. It is the single most indicative distinguishing trait oftotalitarian regimes. It is often distinguished fromtargeted surveillance.
Mass surveillance has often been cited by agencies like theNational Security Agency(NSA) as necessary to fightterrorism, preventcrimeandsocial unrest, protectnational security, and control the population.[2]At the same time, mass surveillance has equally often been criticized for violatingprivacyrights, limitingcivil and political rights and freedoms, and being illegal under some legal or constitutional systems.[3]Another criticism is that increasing mass surveillance could potentially lead to the development of asurveillance state, anelectronic police state, or atotalitarian statewhereincivil libertiesare infringed orpolitical dissentis undermined byCOINTELPRO-like programs.[4]
In 2013, the practice of mass surveillance by world governments[5]was called into question afterEdward Snowden's2013 global surveillance disclosureon the practices by the NSA of the United States. Reporting based on documents Snowden leaked to various media outlets triggered a debate about civil liberties and theright to privacyin theDigital Age.[6]Mass surveillance is considered a global issue.[7][8][9][10]The Aerospace Corporationof the United States describes a near-future event, theGEOINT Singularity, in which everything on Earth will be monitored at all times, analyzed byartificial intelligencesystems, and then redistributed and made available to the general public globally in real time.[11][12]
Privacy International's 2007 survey, covering 47 countries, indicated that there had been an increase in surveillance and a decline in the performance of privacy safeguards, compared to the previous year. Balancing these factors, eight countries were rated as being 'endemic surveillance societies'. Of these eight,China,MalaysiaandRussiascored lowest, followed jointly bySingaporeand theUnited Kingdom, then jointly byTaiwan,Thailandand theUnited States. The best ranking was given toGreece, which was judged to have 'adequate safeguards against abuse'.[13]
Many countries throughout the world have already been adding thousands ofsurveillance camerasto their urban, suburban and even rural areas.[14][15]For example, in September 2007 theAmerican Civil Liberties Union(ACLU) stated that we are "in danger of tipping into a genuine surveillance society completely alien to American values" with "the potential for a dark future where our every move, our every transaction, our every communication is recorded, compiled, and stored away, ready to be examined and used against us by the authorities whenever they want".[16]
On 12 March 2013,Reporters Without Borderspublished aSpecial report on Internet Surveillance. The report included a list of "State Enemies of the Internet", countries whose governments are involved in active, intrusive surveillance of news providers, resulting in grave violations of freedom of information and human rights. Five countries were placed on the initial list:Bahrain,China,Iran,Syria(until December 2024), andVietnam.[17]
Bahrainis one of the five countries on Reporters Without Borders' March 2013 list of "State Enemies of the Internet", countries whose governments are involved in active, intrusive surveillance of news providers, resulting in grave violations of freedom of information and human rights. The level of Internet filtering and surveillance in Bahrain is one of the highest in the world. The royal family is represented in all areas of Internet management and has sophisticated tools at its disposal for spying on its subjects. The online activities of dissidents and news providers are closely monitored and the surveillance is increasing.[17]
Media reports published in July 2021 exposed the use ofNSO Group's phone malware software,Pegasus, for spying on rights activists, lawyers, and journalists, globally, by authoritarian governments. Bahrain was among the many countries listed as the Israeli firm's clients accused of hacking and conducting unauthorized mass surveillance using phone malware despite a poor human rights record. The software is said to infect devices, allowing its operators to get access to the target's messages, photos, record calls, and activate the microphone and camera.[18]
Yusuf al-Jamri had no idea that even after fleeing Bahrain, he won't be able to escape the government's prying eyes, even after taking asylum in the UK. After moving to the UK and getting his asylum request accepted, Al-Jamri legally filed charges against Bahrain along with the notorious spyware firm, NSO Group for infecting his phone with a malware, built with military-grade technology in August 2019. The hacking was verified by the researchers at Toronto based CitizenLab. As a result of which Yusuf complained of being subjected to personal injury, loss of privacy, distress and anxiety. Al-Jamri's lawyers made claims about the same in a pre-claim letter sent to both the accused, NSO Group and the Bahraini government. No response was received from the two on being approached for comments.[19]
Before the Digital Revolution, one of the world's biggest mass surveillance operations was carried out by theStasi, thesecret policeof the formerEast Germany. By the time the state collapsed in 1989, the Stasi had built up an estimated civilian network of 189,000 informants, who monitored even minute hints of political dissent among other citizens.[27]ManyWest Germansvisiting friends and family in East Germany were also subject to Stasi spying, as well as many high-ranking West German politicians and persons in the public eye.
Most East German citizens were well aware that their government was spying on them, which led to a culture of mistrust: touchy political issues were only discussed in the comfort of their own four walls and only with the closest of friends and family members, while widely maintaining a façade of unquestioning followership in public.[citation needed]
The right toprivacyis a highly developed area of law in Europe. TheData Protection Directiveused to regulate the processing of personal data within theEuropean Union, before it was replaced by theGDPR. For comparison, the US has no data protection law that is comparable to this; instead, the US regulates data protection on a sectoral basis.[28]
Since early 2012, the European Union has been working on aGeneral Data Protection Regulationto replace the Data Protection Directive and harmonise data protection and privacy law. On 20 October 2013, a committee at theEuropean Parliamentbacked the measure, which, if it is enacted, could require American companies to seek clearance from European officials before complying with United States warrants seeking private data. The vote is part of efforts in Europe to shield citizens from online surveillance in the wake ofrevelations about a far-reaching spying programby theU.S. National Security Agency.[29]European Union justice and rights commissioner Viviane Reding said "The question has arisen whether the large-scale collection and processing of personal information under US surveillance programmes is necessary and proportionate to meet the interests of national security." The EU is also asking the US for changes to US legislation to match the legal redress offered in Europe; American citizens in Europe can go to the courts if they feel their rights are infringed but Europeans without right of residence in America cannot.[30]When the EU / US arrangement to implementInternational Safe Harbor Privacy Principleswere struck down by the European Court of Justice, a new framework for transatlantic data flows, called the "EU-US Privacy Shield", was adopted in July 2016.[31][32]
In April 2014, theEuropean Court of Justicedeclared invalid the EU Data Retention Directive. The Court said it violates two basic rights – respect for private life and protection of personal data.[33]Thelegislative bodyof theEuropean Unionpassed the Data Retention Directive on 15 December 2005. It required that telecommunication operators retain metadata for telephone, Internet, and other telecommunication services for periods of not less than six months and not more than two years from the date of the communication as determined by each EU member state and, upon request, to make the data available to various governmental bodies. Access to this information is not limited to investigation of serious crimes, nor was a warrant required for access.[34][35]Protection
Undertaken under theSeventh Framework Programmefor research and technological development(FP7 - Science in Society[36]) some multidisciplinary and mission oriented mass surveillance activities (for exampleINDECTandHIDE) were funded by theEuropean Commission[37]in association with industrial partners.[38][39][40][41]The INDECT Project ("Intelligent information system supporting observation, searching and detection for security of citizens in urban environment")[42]develops an intelligent urban environment observation system to register and exchange operational data for the automatic detection, recognition and intelligent processing of all information ofabnormal behaviouror violence.[43][44]
The main expected results of the INDECT project are:
HIDE ("Homeland Security, Biometric Identification & Personal Detection Ethics")[45]was a research project funded by the European Commission within the scope of theSeventh RTD Framework Programme(FP7). The consortium, coordinated by Emilio Mordini,[46]explored theethicalandprivacyimplications ofbiometricsand personal detection technologies, focusing on the continuum between personal detection, authentication, identification and mass surveillance.[47]
In 2002 German citizens were tipped off about wiretapping when a software error led to a phone number allocated to theGerman Secret Servicebeing listed on mobile telephone bills.[48]
The Indian parliament passed theInformation Technology Actof 2008 with no debate, giving the government fiat power to tap all communications without a court order or a warrant. Section 69 of the act states "Section 69 empowers the Central Government/State Government/ its authorized agency to intercept, monitor or decrypt any information generated, transmitted, received or stored in any computer resource if it is necessary or expedient so to do in the interest of the sovereignty or integrity of India, defence of India, security of the State, friendly relations with foreign States or public order or for preventing incitement to the commission of any cognizable offence or for investigation of any offence."
India is setting up a national intelligence grid calledNATGRID,[49]which would be fully set up by May 2011 where each individual's data ranging from land records, Internet logs, air and rail PNR, phone records, gun records, driving license, property records, insurance, and income tax records would be available in real time and with no oversight.[50]With a UID from theUnique Identification Authority of Indiabeing given to every Indian from February 2011, the government would be able track people in real time. A national population registry of all citizens will be established by the 2011 census, during which fingerprints and iris scans would be taken along with GPS records of each household.[51][52]
As per the initial plan, access to the combined data will be given to 11 agencies, including theResearch and Analysis Wing, theIntelligence Bureau, theEnforcement Directorate, theNational Investigation Agency, theCentral Bureau of Investigation, theDirectorate of Revenue Intelligenceand theNarcotics Control Bureau.[citation needed]
Several states within India have already installed CCTV surveillance systems with face matching capabilities using biometrics in Aadhaar.[53]Andhra Pradesh and Telangana are using information linked with Aadhaar across different agencies to create a 360-degree profile of a person, calling it the Integration Information Hub. Other states are now planning to follow this model.[54]
Iranis one of the five countries on Reporters Without Borders' March 2013 list of "State Enemies of the Internet", countries whose governments are involved in naturally active efforts to news providers . The government runs or controls almost all of the country's institutions for regulating, managing or legislating on telecommunications. The Supreme Council for Cyberspace, which was headed byPresident Ahmadinejad, was established in March 2012 and now determines digital policy. The construction of a parallel "Iranian Internet", with a high connection speed but fully monitored and censored, is almost complete.[17]
The tools used by the Iranian authorities to monitor and control the Internet include data interception tools capable ofdeep packet inspection. Interception products from leading Chinese companies such asZTEandHuaweiare in use. The products provided by Huawei toMobin Net, the leading national provider of mobile broadband, can be used to analyze email content, track browsing history and block access to sites. The products that ZTA sold to theTelecommunication Company of Iran(TCI) offer similar services plus the possibility of monitoring the mobile network. European companies are the source of other spying and data analysis tools. Products designed byEricssonandNokia Siemens Networks(later Trovicor) are in use. These companies soldSMSinterception and user location products to Mobile Communication Company of Iran andIrancell, Iran's two biggest mobile phone companies, in 2009 and they were used to identify Iranian citizens during thepost-election uprisingin 2009. The use of Israeli surveillance devices has also been detected in Iran. The network traffic management and surveillance device NetEnforcer was provided by Israel to Denmark and then resold to Iran. Similarly, US equipment has found its way to Iran via the Chinese company ZTE.[17]
In September 2023 The Iranian government approved a law enabling it to have instant undeniable access every single thing in digital online life of citizens including location /photos, data and other vital record tied to people's real identity, The persistent monitoring system is part of Iranian seventh quinquennial development program bill package.[55]
In July 2018, the Malaysian police announced the creation of the Malaysian Intercept Crimes Against Children Unit (icacu) that is equipped with real-time mass internet surveillance software developed in the United States and is tasked with the monitoring of all Malaysian internet users, with a focus on pornography and child pornography. The system creates a "data library" of users which includes details such as IP addresses, websites, locations, duration and frequency of use and files uploaded and downloaded.[56][57][58]
After struggling with drug trafficking and criminal groups for decades Mexico has been strengthening their military mass surveillance. Approximately half of the population in Mexico does not support democracy as a form of government, and believe an authoritarian system is better if social matters are solved through it.[59]The relevance of these political beliefs may make it easier for mass surveillance to spread within the country. "This does not necessarily mean the end of democratic institutions as a whole—such as free elections or the permanence of critical mass media—but it means strengthening the mechanisms for exercising power that exclude dialogue, transparency and social agreement."[60]
According to a 2003 report, the Netherlands has the second highest number of wiretaps per capita in the Western world.[61]The Dutch military intelligence service MIVD operates a satellite ground station to intercept foreign satellite links and also a facility to eavesdrop on foreign high-frequency radio traffic.
An example of mass surveillance carried out by corporations in the Netherlands is an initiative started by five Dutch banks (ABN AMRO, ING, Rabobank, Triodos Bank and de Volksbank). In July 2020 these five banks[62]have decided to establish Transaction Monitoring Netherlands (TMNL) in the collective fight against money laundering and the financing of terrorism.[63]The goal of TMNL-organization is to gather all transaction information provided by Dutch banks in a centralized database to enable full-scale collective transaction monitoring. Preparations have been started but the actual monitoring by TMNL can start after an amendment of the Dutch Anti-Money Laundering and Anti-Terrorist Financing Act.
Having attained the nickname 'surveillance state', North Korea's government has complete control over all forms of telecommunications and Internet. It is routine to be sent to a prison camp for communicating with the outside world. The government enforces restrictions around the types of appliances North Koreans may own in their home, in case radio or TV sets pick up signals from nearby South Korea, China and Russia.[64]There is no attempt to mask the way this government actively spies on their citizens. In North Korea, an increasing number of citizens do have smartphones. However, these devices are heavily controlled and are being used to censor and observe everything North Koreans do on their phones. Reuters reported in 2015 that Koryolink, North Korea's official mobile phone network, has around 3 million subscribers in a country of 24 million.[65]
TheSORM(and SORM-2) laws enable complete monitoring of anycommunication,electronicor traditional, by eight state agencies, without warrant. These laws seem to be in conflict with Article 23 of theConstitution of Russiawhich states:[66]
In 2015, theEuropean Court for Human Rightsruled that the legislation violatedArticle 8 of the European Convention on Human Rights(Zakharov v. Russia).
СAMERTONis a global vehicle tracking system, control and tracking, identification of probable routes and places of the most frequent appearance of a particular vehicle, integrated with a distributed network of radar complexes of photo-video fixation and road surveillance camera.[67]Developed and implemented by the"Advanced Scientific - Research Projects"enterprise St. Petersburg.[68]Within the framework of the practical use of the system of the Ministry of Internal Affairs of the Russian Federation, it has made it possible to identify and solve grave and especially grave crimes, the system is also operated by other state services and departments;
Singaporeis known as a city of sensors. Singapore's surveillance structure spreads widely fromclosed-circuit television(CCTV) in public areas even around the neighbourhood, internet monitoring/traffic monitoring and to the use of surveillancemetadatafor government initiatives. In Singapore, SIM card registration is mandatory even for prepaid card. Singapore's government have the rights to access communication data. Singapore's largest telecompany,Singtel, has close relations to the government and Singapore's laws are broadly phrased to allow the government to obtain sensitive data such as text-messages, email, call logs, and web surfing history from its people without the need for court permission.[69]
The installation of mass surveillance cameras in Singapore is an effort to act as a deterrence not only for terror attacks[70]but also for public security such as loan sharks, illegal parking, and more.[71]As part of Singapore'sSmart Nationinitiative to build a network of sensors to collect and connect data from city life (including the citizen's movement), the Singapore government rolled out 1000 sensors ranging from computer chips to surveillance cameras,[72]to track almost everything in Singapore from air quality to public safety in 2014.[73]
In 2016, in a bid to increase security, theSingapore Police Forceinstalled 62,000 police cameras in 10,000Housing and Development Board(HDB) blocks covering the lifts and multi-storey car parks.[74]With rising security concerns, the number of CCTV cameras in public areas such as monitoring of the public transport system and commercial/ government buildings in Singapore is set to increase.[70]
In 2018, the Singapore government rolled out new and more advanced surveillance systems. Starting with Singapore's maritime borders, new panoramic electro-optic sensors were put in place on the north and south coasts, monitoring a 360-degree view of the area.[75]A tetheredunmanned aerial vehicle(UAV) was unveiled by the Singapore Police Force to be used during search and rescue operations including hostage situations and public order incidents.[76]
According to a 2017 report byPrivacy International, Spain may be part of a group of 21 European countries that is withholding information, also known as data retention.[77]In 2014, many defense lawyers tried to overturn multiple cases that used mass storage as their evidence to convict, according to theEuropean Agency for Fundamental Rights.[78]
Prior to 2009, theNational Defence Radio Establishment(FRA) was limited to wirelesssignals intelligence(SIGINT), although it was left largely unregulated.[79]In December 2009, new legislation went into effect, allowing the FRA to monitor cable bound signals passing the Swedish border.[80]Communications service providers are legally required, under confidentiality, to transfer cable communications crossing Swedish borders to specific "interaction points", where data may be accessed after a court order.[81]
The FRA has been contested since the change in its legislation, mainly because of the public perception the change would enable mass surveillance.[82]The FRA categorically deny this allegation,[80][83]as they are not allowed to initialize any surveillance on their own,[84]and has no direct access to communication lines.[85]All SIGINT has to be authorized by aspecial courtand meet a set of narrow requirements, somethingMinister for DefenceSten Tolgforshave been quoted as saying, "should render the debate on mass surveillance invalid".[86][87][88]Due to the architecture of Internet backbones in the Nordic area, a large portion of Norwegian and Finnish traffic will also be affected by the Swedish wiretapping.[89]
Ba'athist government of Syria has been ruling the country as a totalitariansurveillance state, policing every aspect of Syrian society for decades.[90][91]Commanders of government's security forces – consisting ofSyrian Arab Army,secret police, Ba'athist paramilitaries – directly implement the executive functions of the Syrian state, with scant regard for legal processes andbureaucracy. Security services shut down civil society organizations, curtail freedom of movement within the country and bans non-Ba'athist political literature and symbols.[91][92]During theBa'athist rule,militarizationof the Syrian society intensified. The number of personnel in theSyrian militaryand various intelligence entities expanded drastically from 65,000 in 1965 to 530,000 in 1991; and surpassed 700,000 in 2004.[93]
Ba'athist secret police consists of four wings:general intelligenceand thepolitical securitydirectorates, which are supervised by theSyrian Ministry of Interior;military intelligenceand theair force intelligencedirectorates, which are supervised by theSyrian Ministry of Defence. The four directorates are directly controlled by theNational Security Bureau of the Arab Socialist Ba'ath Party, and heads of the four branches report directly to the Syrian president, who is also the secretary general of the Ba'ath party. The surveillance system of theMukhabaratis pervasive, and over 65,000 full-time officers were estimated to be working in its various branches during the 2000s. In addition, there are hundreds of thousands of part-time employees and informers in various Syrian intelligence departments.[94]According to estimates, there is one member of various branches of the Ba'athist secret police for every 158 citizens, which is one of the largest ratios in the world.[91]
Thegeneral intelligence,political security, andmilitary intelligencedivisions of the Ba'athist secret police have several branches in all governorates controlled by the Assad regime, with headquarters in Damascus. With state impunity granted by the Assad government,Mukhabaratofficers wield pervasive influence over local bodies, civil associations, and bureaucracy, playing a major role in shaping Ba'athist administrative decisions. Additionally, intense factional rivalries and power struggles exist among various branches of the secret police.[95]Several academics have described the military, bureaucratic, and secret police apparatus of the Ba'athist state as constituting a pyramidal socio-political structure with anOrwelliansurveillance system designed to neutralize independent civic activities and political dissent from its very onset.[96][97]
Syriais one of the five countries on Reporters Without Borders' March 2013 list of "State Enemies of the Internet", countries whose governments are involved in active, intrusive surveillance of news providers, resulting in grave violations of freedom of information and human rights. Syria has stepped up its web censorship and cyber-monitoring as thecountry's civil warhas intensified. At least 13Blue Coat proxy serversare in use,Skypecalls are intercepted, andsocial engineering techniques,phishing, andmalwareattacks are all in use.[17]
In October 2016,The Interceptreleased a report detailing the experience of an Italian security researcher Simone Margaritelli, of allegedly being hired for mass surveillance operations run byUnited Arab Emirates. According to Margaritelli, he was called for an interview with theAbu Dhabi-basedcybersecurityfirm calledDarkMatter. Margaritelli says he declined the offer and instead wrote a blog post titled "How the United Arab Emirates Intelligence Tried to Hire Me to Spy on Its People". In response to The Intercept inquiries, DarkMatter responded by stating: "No one from DarkMatter or its subsidiaries have ever interviewed Mr. Margaritelli." Kevin Healy, director of communications for DarkMatter, wrote in an email responding to The Intercept that the man Margaritelli says interviewed him was previously only an advisory consultant to DarkMatter and is currently no longer an advisor to the company. Dark Matter responded by saying "While we respect an author's right to express a personal opinion, we do not view the content in question as credible, and therefore have no further comment."[98]
In January 2019,Reutersreleased a detailed account of a 2014 state-surveillance operation – dubbed as Project Raven – led by theUnited Arab Emirateswith the help of formerNSAofficials like Lori Stroud, an ex-NSA cyberspy.Counter-terrorismstrategy was the primary motive of setting up the unit. However, soon the project began being used as asurveillanceprogram to spy on rival leaders, criticaldissidentsandjournalists.[99]
In December 2019,Google Play StoreandApple App Storeremoved anEmiratimessaging application calledToTokfollowing allegations that it was a state surveillance application, according toThe New York Timesreport.[100][101]The application's privacy policy clearly stated that it may share personal data of the users with "regulatory agencies, law enforcement, and other lawful access requests". The allegations were denied by the co-founders of ToTok, Giacomo Ziani and Long Ruan, respectively. The application was restored on Google Play Store later on.[102]
In July 2020, the United Arab Emirates came under renewed questions about mass surveillance amidst thecoronavirus outbreak. Experts highlighted that the country has one of the highest per capita concentrations of surveillance cameras in the world. In a statement, the Emirati government acknowledged that cameras are used to counter the threat of terrorism and have helped the country rank as one of the safest countries in the world.[103]
State surveillance in theUnited Kingdomhas formed part of the public consciousness since the 19th century. The postal espionage crisis of 1844 sparked the first panic over the privacy of citizens.[104]However, in the 20th century, electronic surveillance capabilities grew out of wartimesignal intelligenceand pioneeringcode breaking.[105]In 1946, theGovernment Communications Headquarters(GCHQ) was formed. The United Kingdom and the United States signed the bilateralUKUSA Agreementin 1948. It was later broadened to include Canada, Australia and New Zealand, as well as cooperation with several "third-party" nations. This became the cornerstone of Western intelligence gathering and the "Special Relationship" between the UK and the US.[106]
After the growth of the Internet and development of theWorld Wide Web, a series ofmedia reports in 2013revealed more recent programs and techniques involving GCHQ, such asTempora.[107]
The use of these capabilities is controlled by laws made in theUK Parliament. In particular, access to the content of private messages (that is, interception of a communication) must be authorized by a warrant signed by aSecretary of State.[108][109][110]In addition European Uniondata privacylaw applies in UK law. The UK exhibits governance and safeguards as well as use of electronic surveillance.[111][112][113]
TheInvestigatory Powers Tribunal, a judicial oversight body for the intelligence agencies, ruled in December 2014 that the legislative framework in the United Kingdom does not breach theEuropean Convention on Human Rights.[114][115][116]However, the Tribunal stated in February 2015 that one particular aspect, thedata sharingarrangement that allowed UK Intelligence services to request data from the US surveillance programsPrismandUpstream, had been in contravention of human rights law prior to this until two paragraphs of additional information, providing details about the procedures and safeguards, were disclosed to the public in December 2014.[117][118][119]
In its December 2014 ruling, the Investigatory Powers Tribunal found that the legislative framework in the United Kingdom does not permit mass surveillance and that while GCHQ collects and analyses data in bulk, it does not practice mass surveillance.[114][115][116]A report on Privacy and Security published by theIntelligence and Security Committee of Parliamentalso came to this view, although it found past shortcomings in oversight and said the legal framework should be simplified to improve transparency.[120][121][122]This view is supported by independent reports from theInterception of Communications Commissioner.[123]However, notable civil liberties groups continue to express strong views to the contrary and plan to appeal the ruling to theEuropean Court of Human Rights,[124]while others have criticised these viewpoints in turn.[125]
TheRegulation of Investigatory Powers Act 2000(RIP or RIPA) is a significant piece of legislation that granted and regulated the powers of public bodies to carry out surveillance and investigation.[126]In 2002 the UK government announced plans to extend the Regulation of Investigatory Powers Act so that at least 28 government departments would be given powers to accessmetadataabout citizens' web, e-mail, telephone and fax records, without a warrant and without a subject's knowledge.[127]
TheProtection of Freedoms Act 2012includes several provisions related to controlling and restricting the collection, storage, retention, and use of information in government databases.[128]
Supported by all three major political parties, the UK Parliament passed theData Retention and Investigatory Powers Actin July 2014 to ensure police and security services retain existing powers to access phone and Internet records.[129][130]
This was superseded by theInvestigatory Powers Act 2016, a comprehensive statute which made public a number of previously secret powers (equipment interference, bulk retention of metadata, intelligence agency use of bulk personal datasets), and enables the Government to require internet service providers and mobile phone companies to maintain records of (but not the content of) customers' Internet connections for 12 months. In addition, it created new safeguards, including a requirement for judges to approve the warrants authorised by a Secretary of State before they come into force.[131][132]The Act was informed by two reports byDavid Anderson QC, the UK'sIndependent Reviewer of Terrorism Legislation: A Question of Trust (2015)[133]and the report of his Bulk Powers Review (2016),[134]which contains a detailed appraisal (with 60 case studies) of the operational case for the powers often characterised as mass surveillance. It may yet require amendment as a consequence of legal cases brought before theCourt of Justice of the European Union[135]and theEuropean Court of Human Rights.[136]
Many advanced nation-states have implemented laws that partially protect citizens from unwarranted intrusion, such as theHuman Rights Act 1998, theData Protection Act 1998, (updated as theData Protection Act 2018, to include theGeneral Data Protection Regulation), and thePrivacy and Electronic Communications (EC Directive) Regulations 2003in the United Kingdom, and laws that require a formal warrant before private data may be gathered by a government.
The vast majority of video surveillance cameras in the UK are not operated by government bodies, but by private individuals or companies, especially to monitor the interiors of shops and businesses. According to 2011Freedom of Information Actrequests, the total number of local government operated CCTV cameras was around 52,000 over the entirety of the UK.[137]The prevalence of video surveillance in the UK is often overstated due to unreliable estimates being requoted;[138]for example one report in 2002 extrapolated from a very small sample to estimate the number of cameras in the UK at 4.2 million (of which 500,000 in London).[139]More reliable estimates put the number of private and local government operated cameras in the United Kingdom at around 1.85 million in 2011.[140]
Historically, mass surveillance was used as part of wartime censorship to control communications that could damage the war effort and aid the enemy. For example, during the world wars, every international telegram from or to the United States sent through companies such asWestern Unionwas reviewed by the US military. After the wars were over, surveillance continued in programs such as theBlack Chamberfollowing World War I andproject Shamrockfollowing World War II.[141]COINTELPROprojects conducted by the U.S. Federal Bureau of Investigation (FBI) between 1956 and 1971 targeted various "subversive" organizations, including peaceful anti-war and racial equality activists such asAlbert EinsteinandMartin Luther King Jr.
Billions of dollars per year are spent, by agencies such as theNational Security Agency(NSA) and theFederal Bureau of Investigation(FBI), to develop, purchase, implement, and operate systems such asCarnivore,ECHELON, andNarusInsightto intercept and analyze the immense amount of data that traverses the Internet and telephone system every day.[142]
Under theMail Isolation Control and Trackingprogram, the U.S. Postal Service photographs the exterior of every piece of paper mail that is processed in the United States – about 160 billion pieces in 2012. The U.S. Postmaster General stated that the system is primarily used for mail sorting, but the images are available for possible use by law enforcement agencies.[143]Created in 2001 following the anthrax attacks that killed five people, it is a sweeping expansion of a 100-year-old program called "mail cover" which targets people suspected of crimes.[144]
The FBI developed the computer programs "Magic Lantern" andCIPAV, which they can remotely install on a computer system, in order to monitor a person's computer activity.[145]
The NSA has been gathering information on financial records, Internet surfing habits, and monitoring e-mails. They have also performed extensive analysis ofsocial networkssuch asMyspace.[146]
ThePRISMspecial source operation system legally immunized private companies that cooperate voluntarily with U.S. intelligence collection. According toThe Register, theFISA Amendments Act of 2008"specifically authorizes intelligence agencies to monitor the phone, email, and other communications of U.S. citizens for up to a week without obtaining a warrant" when one of the parties is outside the U.S.[147]PRISM was first publicly revealed on 6 June 2013, after classified documents about the program were leaked toThe Washington PostandThe Guardianby AmericanEdward Snowden.
TheCommunications Assistance for Law Enforcement Act(CALEA) requires that all U.S. telecommunications and Internet service providers modify their networks to allow easywiretappingof telephone, VoIP, and broadband Internet traffic.[148][149][150]
In early 2006,USA Todayreported that several major telephone companies were providing the telephone call records of U.S. citizens to theNational Security Agency(NSA), which is storing them in a large database known as theNSA call database. This report came on the heels of allegations that theU.S. governmenthad been conducting electronic surveillance of domestic telephone calls without warrants.[151]In 2013, the existence of theHemisphere Project, through which AT&T provides telephone call data to federal agencies, became publicly known.
Traffic cameras, which were meant to help enforce traffic laws at intersections, may be used by law enforcement agencies for purposes unrelated to traffic violations.[152]Some cameras allow for the identification of individuals inside a vehicle and license plate data to be collected and time stamped for cross reference with other data used by police.[153]TheDepartment of Homeland Securityis funding networks of surveillance cameras in cities and towns as part of its efforts to combat terrorism.[154]
TheNew York City Police Departmentinfiltrated and compiled dossiers on protest groups before the2004 Republican National Convention, leading to over 1,800 arrests.[155]
Modern surveillance in the United States was thought of more of a wartime effort before Snowden disclosed in depth information about the National Security Agency in June 2013.[156]The constant development and improvements of the Internet and technology has made it easier for mass surveillance to take hold. Such revelations allow critical commentators to raise questions and scrutinize the implementation, use, and abuse of networking technologies, devices, and software systems that partake in a "global surveillant assemblage" (Bogard 2006; Collier and Ong 2004; Haggerty and Ericson 2000; Murakami Wood 2013).[156]The NSA collected millions of Verizon user's telephone records in between 2013 and 2014. The NSA also collected data through Google and Facebook with a program called 'Prism'. Journalists through Snowden published nearly 7,000 top-secret documents since then, yet the information disclosed is less than 1% of the entire information.
Vietnamis one of the five countries on Reporters Without Borders' March 2013 list of "State Enemies of the Internet", countries whose governments are involved in active, intrusive surveillance of news providers, resulting in grave violations of freedom of information and human rights. Most of the country's 16 service providers are directly or indirectly controlled by theVietnamese Communist Party. The industry leader,Vietnam Posts and Telecommunications Group, which controls 74 per cent of the market, is state-owned. So isViettel, an enterprise of theVietnamese armed forces. FPT Telecom is a private firm, but is accountable to the Party and depends on the market leaders for bandwidth.[17]
Service providers are the major instruments of control and surveillance. Bloggers monitored by the government frequently undergoman-in-the-middle attacks. These are designed to intercept data meant to be sent to secure (https) sites, allowing passwords and other communication to be intercepted.[17]According to a July 2012Freedom Housereport, 91 percent of survey respondents connected to the Internet on their mobile devices believe that the government monitors conversations and tracks the calls of "activists" or "reactionaries".[157]In 2018, theVietnam National Assemblyalso passed a cybersecurity law closely resemblingone passed in China, requiring localisation of user data and censorship of anti-state content.[158]
As a result of thedigital revolution, many aspects of life are now captured and stored in digital form. Concern has been expressed that governments may use this information to conduct mass surveillance on their populations. Commercial mass surveillance often makes use ofcopyrightlaws and "user agreements" to obtain (typically uninformed) 'consent' to surveillance from consumers who use their software or other related materials. This allows gathering of information which would be technically illegal if performed by government agencies. This data is then often shared with government agencies, thereby, in practice, defeating the purpose of such privacy protections.
One of the most common forms of mass surveillance is carried out by commercial organizations. Many people are willing to join supermarket and groceryloyalty cardprograms, trading their personal information and surveillance of their shopping habits in exchange for a discount on their groceries, although base prices might be increased to encourage participation in the program.
Through programs likeGoogle'sAdSense,OpenSocialand their increasing pool of so-called "web gadgets", "social gadgets", and other Google-hosted services many web sites on the Internet are effectively feeding user information about sites visited by the users, and now also their social connections, to Google.Facebookalso keep this information, although its acquisition is limited to page views within Facebook. This data is valuable for authorities, advertisers and others interested in profiling users, trends and web site marketing performance. Google, Facebook and others are increasingly becoming more guarded about this data as their reach increases and the data becomes more all inclusive, making it more valuable.[159]
New features likegeolocationgive an even increased admission of monitoring capabilities to large service providers like Google, where they also are enabled to track one's physical movements while users are using mobile devices, especially those which are syncing without any user interaction. Google'sGmailservice is increasingly employing features to work as a stand-alone application which also might activate while a web browser is not even active for synchronizing; a feature mentioned on theGoogle I/O2009 developer conference while showing the upcomingHTML5features which Google and others are actively defining and promoting.[160]
In 2008 at theWorld Economic ForuminDavos, Google CEO Eric Schmidt, said: "The arrival of a truly mobile Web, offering a new generation of location-based advertising, is set to unleash a 'huge revolution'".[161]At the Mobile World Congress in Barcelona on 16 February 2010, Google presented their vision of a new business model for mobile operators and trying to convince mobile operators to embracelocation-based servicesand advertising. With Google as the advertising provider, it would mean that every mobile operator using their location-based advertising service would be revealing the location of their mobile customers to Google.[162]
Google will also know more about the customer—because it benefits the customer to tell Google more about them. The more we know about the customer, the better the quality of searches, the better the quality of the apps. The operator one is "required", if you will, and the Google one will be optional. And today I would say, a minority choose to do that, but I think over time a majority will... because of the stored values in the servers and so forth and so on....
Organizations like theElectronic Frontier Foundationare constantly informing users on the importance of privacy, and considerations about technologies like geolocation.
Computer companyMicrosoftpatented in 2011 a product distribution system with a camera or capture device that monitors the viewers that consume the product, allowing the provider to take "remedial action" if the actual viewers do not match the distribution license.[164]
Reporters Without Borders' March 2013Special report on Internet Surveillancecontained a list of "Corporate Enemies of the Internet", companies that sell products that are liable to be used by governments to violate human rights and freedom of information. The five companies on the initial list were:Amesys(France),Blue Coat Systems(U.S.),Gamma Group(UK and Germany),Hacking Team(Italy), and Trovicor (Germany), but the list was not exhaustive and is likely to be expanded in the future.[17]
EFFfound that a company by the name ofFog Data Sciencepurchases location data from apps and sells it to the law enforcement agencies in the U.S. without requiring a warrant or a court order.[165]
A surveillance state is a country where the government engages in pervasive surveillance of large numbers of its citizens and visitors. Such widespread surveillance is usually justified as being necessary fornational security, such as to prevent crime or acts of terrorism, but may also be used tostifle criticism of and oppositionto the government.
Examples of early surveillance states include the formerSoviet Unionand the formerEast Germany, which had a large network of informers and an advanced technology base in computing and spy-camera technology.[166]However, these states did not have today's technologies for mass surveillance, such as the use ofdatabasesandpattern recognitionsoftware to cross-correlate information obtained bywire tapping, includingspeech recognitionand telecommunicationstraffic analysis, monitoring of financial transactions,automatic number plate recognition, thetracking of the position of mobile telephones, andfacial recognition systemsand the like which recognize people by their appearance,gait,DNA profiling, etc.
The development of smart cities has seen the increased adoption of surveillance technologies by governments, although the primary purpose of surveillance in such cities is touse information and communication technologiesto control the urban environment. The implementation of such technology bya number of citieshas resulted in increased efficiencies in urban infrastructure as well as improved community participation. Sensors and systems monitor a smart city's infrastructure, operations and activities and aim to help it run more efficiently. For example, the city could use less electricity; its traffic run more smoothly with fewer delays; its citizens use the city with more safety; hazards can be dealt with faster; citizen infractions of rules can be prevented, and the city's infrastructure; power distribution and roads with traffic lights for example, dynamically adjusted to respond to differing circumstances.[167]
The development of smart city technology has also led to an increase in potential unwarrantedintrusions into privacy and restrictions upon autonomy. The widespread incorporation of information and communication technologies within the daily life of urban residents results in increases in thesurveillance capacity of states– to the extent that individuals may be unaware of what information is being accessed, when the access occurs and for what purpose. It is possible that such conditions could give rise to thedevelopment of an electronic police state. Shanghai, Amsterdam, San Jose, Dubai, Barcelona, Madrid, Stockholm, and New York are all cities that use various techniques from smart city technology.
An electronicpolice stateis astatein which the government aggressively uses electronic technologies to record, collect, store, organize, analyze, search, and distribute information about its citizens. The first use of the term "electronic police state" was likely in a posting by Jim Davis.[168]Electronic police states also engage in mass government surveillance oflandlineandcellulartelephone traffic, mail, email, web surfing, Internet searches, radio, and other forms of electronic communication as well as widespread use of video surveillance. The information is usually collected in secret.
The crucial elements are not politically based, so long as the government can afford the technology and the populace will permit it to be used, an electronic police state can form. The continual use of electronic mass surveillance can result in constant low-level fear within the population, which can lead toself-censorshipand exerts a powerful coercive force upon the populace.[169]
The concept of being monitored by our governments collects a large audience of curious citizens. Mass surveillance has been prominently featured in a wide array of books, films, and other media. Advances in technology over the last century have led to possible social control through the Internet and the conditions of late capitalism. Many directors and writers have been enthralled with the potential stories that could come from mass surveillance. Perhaps the most iconic example of fictional mass surveillance isGeorge Orwell's 1949 novelNineteen Eighty-Four, which depicts adystopiansurveillance state.
Here are a few other works that focus on mass surveillance:
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https://en.wikipedia.org/wiki/Mass_surveillance
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Ahumanmicrochip implantis any electronic device implanted subcutaneously (subdermally) usually via an injection. Examples include an identifyingintegrated circuitRFID device encased insilicate glasswhich is implanted in the body of a human being. This type ofsubdermal implantusually contains a uniqueID numberthat can be linked to information contained in an external database, such asidentity document,criminal record,medical history, medications,address book, and otherpotential uses.
Several hobbyists, scientists and business personalities have placed RFID microchip implants into their hands or had them inserted by others.
For Microchip implants that are encapsulated in silicate glass, there exists multiple methods to embed the device subcutaneously ranging from placing the microchip implant in a syringe or trocar[39]and piercing under the flesh (subdermal) then releasing the syringe to using a cutting tool such as a surgical scalpel to cut open subdermal and positioning the implant in the open wound.
A list of popular uses for microchip implants are as follows;
Other uses either cosmetic or medical may also include;
RFID implants usingNFCtechnologies have been used as access cards ranging for car door entry to building access.[41]Secure identity has also been used to encapsulate or impersonate a users identity via secure element or related technologies.
Researchers have examined microchip implants in humans in the medical field and they indicate that there are potential benefits and risks to incorporating the device in the medical field. For example, it could be beneficial for noncompliant patients but still poses great risks for potential misuse of the device.[45]
Destron Fearing, a subsidiary ofDigital Angel, initially developed the technology for the VeriChip.[46]
In 2004, the VeriChip implanted device and reader were classified asClass II: General controls with special controlsby the FDA;[47]that year the FDA also published a draft guidance describing the special controls required to market such devices.[48]
About the size of a grain of rice, the device was typically implanted between the shoulder and elbow area of an individual's right arm. Once scanned at the proper frequency, the chip responded with a unique 16-digit number which could be then linked with information about the user held on a database for identity verification, medical records access and other uses. The insertion procedure was performed under local anesthetic in a physician's office.[49][50]
Privacy advocates raised concerns regarding potential abuse of the chip, with some warning that adoption by governments as a compulsory identification program could lead to erosion of civil liberties, as well asidentity theftif the device should be hacked.[50][51][52]Another ethical dilemma posed by the technology, is that people with dementia could possibly benefit the most from an implanted device that contained their medical records, but issues ofinformed consentare the most difficult in precisely such people.[53]
In June 2007, theAmerican Medical Associationdeclared that "implantable radio frequency identification (RFID) devices may help to identify patients, thereby improving the safety and efficiency of patient care, and may be used to enable secure access to patient clinical information",[54]but in the same year, news reports linking similar devices to cancer caused in laboratory animals.[55]
In 2010, the company, by then called PositiveID, withdrew the product from the market due to poor sales.[56]
In January 2012, PositiveID sold the chip assets to a company called VeriTeQ that was owned by Scott Silverman, the former CEO of Positive ID.[57]
In 2016, JAMM Technologies acquired the chip assets from VeriTeQ; JAMM's business plan was to partner with companies sellingimplanted medical devicesand use the RFID tags to monitor and identify the devices.[58]JAMM Technologies is co-located in the samePlymouth, Minnesotabuilding as Geissler Corporation with Randolph K. Geissler and Donald R. Brattain[59][60]listed as its principals.
The website also claims that Geissler was CEO of PositiveID Corporation, Destron Fearing Corporation, and Digital Angel Corporation.[61]
In 2018, a Danish firm called BiChip released a new generation of microchip implant[62]that is intended to be readable from a distance and connected to Internet. The company released an update for its microchip implant to associate it with the Ripplecryptocurrencyto allow payments to be made using the implanted microchip.[63]
Patients that undergo NFC implants do so for a variety of reasons ranging from, Biomedical diagnostics, health reasons to gaining new senses,[64]gain biological enhancement, to be part of existing growing movements, for workplace purposes, security, hobbyists and for scientific endeavour.[65]
In 2020, a London-based firm called Impli released a microchip implant that is intended to be used with an accompanying smartphone app. The primary functionality of the implant is as a storage of medical records. The implant can be scanned by any smartphone that has NFC capabilities.[66]
In February 2006, CityWatcher, Inc. of Cincinnati, OH became the first company in the world to implant microchips into their employees as part of their building access control and security system. The workers needed the implants to access the company's secure video tape room, as documented inUSA Today.[67]The project was initiated and implemented by Six Sigma Security, Inc. The VeriChip Corporation had originally marketed the implant as a way to restrict access to secure facilities such as power plants.
A major drawback for such systems is the relative ease with which the 16-digit ID number contained in a chip implant can be obtained and cloned using a hand-held device, a problem that has been demonstrated publicly by security researcher Jonathan Westhues[68]and documented in the May 2006 issue ofWiredmagazine,[69]among other places.
In 2017, Mike Miller, chief executive of theWorld Olympians Association, was widely reported as suggesting the use of such implants in athletes in an attempt to reduce problems in sports due to recreational drug use.[72]
Theoretically, a GPS-enabled chip could one day make it possible for individuals to be physically located by latitude, longitude, altitude, and velocity.[citation needed]Such implantable GPS devices are not technically feasible at this time. However, if widely deployed at some future point, implantable GPS devices could conceivably allow authorities to locatemissing people,fugitives, or those who fled a crime scene. Critics contend that the technology could lead topolitical repressionas governments could use implants to track and persecute human rights activists, labor activists, civil dissidents, and political opponents; criminals and domestic abusers could use them to stalk, harass, and/or abduct their victims.
Another suggested application for a tracking implant, discussed in 2008 by the legislature ofIndonesia'sIrian Jayawould be to monitor the activities of people infected withHIV, aimed at reducing their chances of infecting other people.[73][74]The microchipping section was not, however, included in the final version of the provincialHIV/AIDS Handling bylawpassed by the legislature in December 2008.[75]With current technology, this would not be workable anyway, since there is no implantable device on the market withGPS trackingcapability.
Some have theorized[who?]that governments could use implants for:
Infection has been cited as a source of failure within RFID and related microchip implanted individuals, either due to improper implantation techniques, implantrejectionsor corrosion of implant elements.[76]
Some chipped individuals have reported being turned away fromMRIsdue to the presence of magnets in their body.[77]No conclusive investigation has been done on the risks of each type of implant near MRIs, other than anecdotal reports ranging from no problems, requiring hand shielding before proximity, to being denied the MRI.[failed verification–see discussion]
Other medical imaging technologies likeX-rayandCT scannersdo not pose a similar risk. Rather, X-rays can be used to locate implants.
Electronics-based implants contain little material that can corrode.Magnetic implants, however, often contain a substantial amount of metallic elements by volume, and iron, a common implant element, is easily corroded by common elements such as oxygen and water. Implant corrosion occurs when these elements become trapped inside during the encapsulation process, which can cause slow corrosive effect, or the encapsulation fails and allows corrosive elements to come into contact with the magnet. Catastrophic encapsulation failures are usually obvious, resulting in tenderness, discoloration of the skin, and a slight inflammatory response. Small failures however can take much longer to become obvious, resulting in a slow degradation of field strength without many external signs that something is slowly going wrong with the magnet.[78]
In a self-published report,[79]anti-RFID advocateKatherine Albrecht, who refers to RFID devices as "spy chips", citesveterinaryandtoxicologicalstudies carried out from 1996 to 2006 which found lab rodents injected with microchips as an incidental part of unrelated experiments and dogs implanted with identification microchips sometimes developed cancerous tumors at the injection site (subcutaneoussarcomas) as evidence of a human implantation risk.[80]However, the link between foreign-body tumorigenesis in lab animals and implantation in humans has been publicly refuted as erroneous and misleading[81]and the report's author has been criticized[by whom?]over the use of "provocative" language "not based in scientific fact".[82]Notably, none of the studies cited specifically set out to investigate the cancer risk of implanted microchips and so none of the studies had a control group of animals that did not get implanted. While the issue is considered worthy of further investigation, one of the studies cited cautioned "Blind leaps from the detection of tumors to the prediction of human health risk should be avoided".[83][84][85]
The Council on Ethical and Judicial Affairs (CEJA) of theAmerican Medical Associationpublished a report in 2007 alleging that RFID implanted chips may compromiseprivacybecause even though no information can be stored in an RFID transponder, they allege that there is no assurance that the information contained in the chip can be properly protected.[86]
Stolen identity and privacy has been a major concern with microchip implants being cloned for various nefarious reasons in a process known asWireless identity theft. Incidents of forced removal of animal implants have been documented,[87]the concern lies in whether this same practice will be used to attack implanted microchipped patients also. Due to low adoption of microchip implants incidents of these physical attacks are rare. Nefarious RFID reprogramming of unprotected or unencrypted microchip tags are also a major security risk consideration.
There is concern technology can be abused.[88]Opponents have stated that such invasive technology has the potential to be used by governments to create an 'Orwellian'digital dystopiaand theorized that in such a world, self-determination, the ability to think freely, and all personal autonomy could be completely lost.[89][90][91]
In 2019,Elon Muskannounced that a company he had founded which deals with microchip implant research, called Neuralink, would be able to "solve"autismand other "brain diseases".[92]This led to a number of critics calling out Musk for his statements, with Dan Robitzski ofNeoscopesaying, "while schizophrenia can be a debilitating mental condition, autism is more tightly linked to a sense of identity — and listing it as a disease to be solved as Musk did risks further stigmatizing a community pushing for better treatment and representation."[93]Hilary Brueck ofInsideragreed, saying, "conditions like autism can't be neatly cataloged as things to "solve." Instead, they lead people to think differently". She went on to argue though that the technology shouldn't be discounted entirely, as it could potentially help people with a variety of disabilities such asblindnessandquadriplegia.[94]FellowInsiderwriter Isobel Asher Hamilton added, "it was not clear what Musk meant by saying Neuralink could "solve" autism, which is not a disease but a developmental disorder." She then cited The UK's National Autistic Society's website statement, which says, "Autism is not an illness or disease and cannot be 'cured.' Often people feel being autistic is a fundamental aspect of their identity."[95]Tristan Greene ofThe Next Webstated, in response to Musk, "there’s only one problem: autism isn’t a disease and it can’t be cured or solved. In fact, there’s some ethical debate in the medical community over whether autism, which is considered a disorder, should be treated as part of a person’s identity and not a ‘condition’ to be fixed... how freaking cool would it be to actually start yourTesla[electric vehicle] just by thinking? But, maybe... just maybe, the billionaire with access to the world's brightest medical minds who, even after founding a medical startup, still incorrectly thinks that autism is a disease that can be solved or cured shouldn't be someone we trust to shove wires or chips into our brains."[96]
Some autistic people also spoke out against Musk's statement about using microchips to "solve" autism, with Nera Birch ofThe Mighty, an autistic writer, stating, "autism is a huge part of who I am. It pervades every aspect of my life. Sure, there are days where being neurotypical would make everything so much easier. But I wouldn’t trade my autism for the world. I have the unique ability to view the world and experience senses in a way that makes all the negatives of autism worth it. The fact you think I would want to be “cured” is like saying I would rather be nothing than be myself. People with neurodiversity are proud of ourselves. For many of us, we wear our autism as a badge of pride. We have a culture within ourselves. It is not something that needs to be erased. The person with autism is not the problem. Neurotypical people need to stop molding us into somethingtheywant to interact with."[97]Florence Grant, an autistic writer forThe Independent, stated, "autistic people often have highly-focused interests, also known as special interests. I love my ability to hyperfocus and how passionate I get about things. I also notice small details and things that other people don’t see. I see the world differently, through a clear lens, and this means I can identify solutions where other people can’t. Does this sound familiar, Elon? My autism is a part of me, and it’s not something that can be separated from me. I should be able to exist freely autistic and proud. But for that to happen, the world needs to stop punishing difference and start embracing it." Grant noted that Musk himself had recently admitted that he had been diagnosed withAsperger's syndrome(itself an outdated diagnosis, the characteristics of which are currently recognized as part of the autism spectrum[98]) while hostingSaturday Night Live.[99]
Musk himself has not specified how Neuralink's microchip technology would "solve" autism, and has not commented publicly on the feedback from autistic people.
Despite a lack of evidence demonstrating invasive use or even technical capability of microchip implants, they have been the subject of many conspiracy theories.
TheSouthern Poverty Law Centerreported in 2010 that on the Christian right, there were concerns that implants could be the "mark of the beast" and amongst thePatriot movementthere were fears that implants could be used to track people.[100]The same yearNPRreported that a myth was circulating online that patients who signed up to receive treatment under theAffordable Care Act(Obamacare) would be implanted.[101]
In 2016,Snopesreported that being injected with microchips was a "perennial concern to the conspiracy-minded" and noted that a conspiracy theory was circulating in Australia at that time that the government was going to implant all of its citizens.[102]
A 2021 survey byYouGovfound that 20% of Americans believed microchips were inside theCOVID-19 vaccines.[103][104]A 2021Facebookpost byRT (Russia Today)claimedDARPAhad developed a COVID-19 detecting microchip implant.[105][106]
A few jurisdictions have researched or preemptively passed laws regarding human implantation of microchips.
In the United States, many states such asWisconsin(as of 2006),North Dakota(2007),California(2007),Oklahoma(2008), andGeorgia(2010) have laws making it illegal to force a person to have a microchip implanted, though politicians acknowledge they are unaware of cases of such forced implantation.[107][108][109][110]In 2010,Virginiapassed a bill forbidding companies from forcing employees to be implanted with tracking devices.[111]
In 2010,Washington'sHouse of Representativesintroduced a bill ordering the study of potential monitoring of sex offenders with implanted RFID or similar technology, but it did not pass.[112]
The general public are most familiar with microchips in the context ofidentifying pets.
Implanted individuals are considered to be grouped together as part of thetranshumanismmovement.
"Arkangel", an episode of the drama seriesBlack Mirror, considered the potential forhelicopter parentingof an imagined more advanced microchip.
Microchip implants have been explored incyberpunkmedia such asGhost in the Shell,Cyberpunk 2077, andDeus Ex.
Some Christians make a link between implants and the BiblicalMark of the Beast,[113][114]prophesied to be a future requirement for buying and selling, and a key element of theBook of Revelation.[115][116]Gary Wohlscheid, president of These Last Days Ministries, has argued that "Out of all the technologies with potential to be themark of the beast, VeriChip has got the best possibility right now."[117]
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Mobile RFID(M-RFID) are services that provide information on objects equipped with anRFID tagover a telecommunication network.[1]The reader or interrogator can be installed in a mobile device such as amobile phoneor PDA.[2]
Unlike ordinary fixed RFID, mobile RFID readers are mobile, and the tags fixed, instead of the other way around. The advantages of M-RFID over RFID include the absence of wires to fixed readers and the ability of a small number of mobile readers can cover a large area, instead of dozens of fixed readers.[3]
The main focus is on supporting supply chain management. But this application has also found its way inm-commerce.[citation needed]The customer in the supermarket can scan theElectronic Product Codefrom the tag and connects via the internet to get more information.[citation needed]
ISO/IEC 29143 "Information technology — Automatic Identification and Data Capture Technique — Air Interface specification for Mobile RFID interrogator"[4]is the first standard to be developed for Mobile RFID.[citation needed]
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Near-field communication(NFC) is a set ofcommunication protocolsthat enablescommunicationbetween two electronic devices over a distance of4 cm (1+1⁄2in) or less.[1]NFC offers a low-speed connection through a simple setup that can be used for thebootstrappingof capable wireless connections.[2]Like otherproximity cardtechnologies, NFC is based oninductive couplingbetween twoelectromagnetic coilspresent on a NFC-enabled device such as asmartphone. NFC communicating in one or both directions uses a frequency of 13.56 MHz in the globally available unlicensedradio frequencyISM band, compliant with theISO/IEC 18000-3air interface standard at data rates ranging from 106 to 848 kbit/s.
TheNFC Forumhas helped define and promote the technology, setting standards for certifying device compliance.[3][4]Secure communications are available by applying encryption algorithms as is done for credit cards[5]and if they fit the criteria for being considered apersonal area network.[6]
NFC standards cover communications protocols and data exchange formats and are based on existingradio-frequency identification(RFID) standards includingISO/IEC 14443andFeliCa.[7]The standards include ISO/IEC 18092[8]and those defined by the NFC Forum. In addition to the NFC Forum, theGSMAgroup defined a platform for the deployment of GSMA NFC Standards[9]within mobile handsets. GSMA's efforts include Trusted Services Manager,[10][11]Single Wire Protocol, testing/certification and secure element.[12]NFC-enabled portable devices can be provided withapplication software, for example to read electronic tags or make payments when connected to an NFC-compliant system. These are standardized to NFC protocols, replacing proprietary technologies used by earlier systems.
A patent licensing program for NFC is under deployment by France Brevets, a patent fund created in 2011. This program was under development by Via Licensing Corporation, an independent subsidiary ofDolby Laboratories, and was terminated in May 2012.[13]A platform-independentfree and open sourceNFC library,libnfc, is available under theGNU Lesser General Public License.[14][15]
Present and anticipated applications include contactless transactions, data exchange and simplified setup of more complex communications such asWi-Fi.[16]In addition, when one of the connected devices has Internet connectivity, the other can exchange data with online services.[citation needed]
Near-field communication (NFC) technology not only supports data transmission but also enables wireless charging, providing a dual-functionality that is particularly beneficial for small, portable devices. The NFC Forum has developed a specific wireless charging specification, known as NFC Wireless Charging (WLC), which allows devices to charge with up to 1W of power over distances of up to2 cm (3⁄4in).[17]This capability is especially suitable for smaller devices like earbuds, wearables, and other compact Internet of Things (IoT) appliances.[17]
Compared to the more widely knownQi wireless chargingstandard by theWireless Power Consortium, which offers up to 15W of power over distances up to4 cm (1+5⁄8in), NFC WLC provides a lower power output but benefits from a significantly smaller antenna size.[17]This makes NFC WLC an ideal solution for devices where space is at a premium and high power charging is less critical.[17]
The NFC Forum also facilitates a certification program, labeled as Test Release 13.1 (TR13.1), ensuring that products adhere to the WLC 2.0 specification. This certification aims to establish trust and consistency across NFC implementations, minimizing risks for manufacturers and providing assurance to consumers about the reliability and functionality of their NFC-enabled wireless charging devices.[17]
NFC is rooted inradio-frequency identificationtechnology (known as RFID) which allows compatible hardware to both supply power to and communicate with an otherwise unpowered and passive electronic tag using radio waves. This is used for identification, authentication andtracking. Similar ideas in advertising and industrial applications were not generally successful commercially, outpaced by technologies such asQR codes,barcodesandUHFRFIDtags.[citation needed]
Ultra-wideband(UWB) another radio technology has been hailed as a future possible alternatives to NFC technology due to further distances of data transmission, as well as Bluetooth and wireless technology.[55]
NFC is a set of short-range wireless technologies, typically requiring a separation of10 cm (3+7⁄8in) or less. NFC operates at 13.56MHzonISO/IEC 18000-3air interface and at rates ranging from 106 kbit/s to 424 kbit/s. NFC always involves an initiator and a target; the initiator actively generates anRFfield that can power a passive target. This enables NFC targets to take very simple form factors such as unpowered tags, stickers, key fobs, or cards. NFC peer-to-peer communication is possible, provided both devices are powered.[56]
NFC tags contain data and are typically read-only, but may be writable. They can be custom-encoded by their manufacturers or use NFC Forum specifications. The tags can securely store personal data such as debit and credit card information, loyalty program data, PINs and networking contacts, among other information. The NFC Forum defines five types of tags that provide different communication speeds and capabilities in terms of configurability, memory, security,data retentionand write endurance.[57]
As withproximity cardtechnology, NFC usesinductive couplingbetween two nearbyloop antennaseffectively forming an air-coretransformer. Because the distances involved are tiny compared to thewavelengthofelectromagnetic radiation(radio waves) of that frequency (about 22 metres), the interaction is described asnear field. An alternatingmagnetic fieldis the main coupling factor and almost no power is radiated in the form ofradio waves(which are electromagnetic waves, also involving an oscillatingelectric field); that minimises interference between such devices and any radio communications at the same frequency or with other NFC devices much beyond its intended range. NFC operates within the globally available and unlicensedradio frequencyISM bandof 13.56 MHz. Most of the RF energy is concentrated in the ±7 kHz bandwidth allocated for that band, but the emission'sspectral widthcan be as wide as 1.8 MHz[58]in order to support high data rates.
Working distance with compact standard antennas and realistic power levels could be up to about20 cm (7+7⁄8in) (but practically speaking, working distances never exceed10 cm or3+7⁄8in). Note that because the pickup antenna may be quenched in aneddy currentby nearby metallic surfaces, the tags may require a minimum separation from such surfaces.[59]
The ISO/IEC 18092 standard supports data rates of 106, 212 or 424kbit/s.
The communication takes place between an active "initiator" device and a target device which may either be:
NFC employs two differentcodingsto transfer data. If an active device transfers data at 106 kbit/s, a modifiedMiller codingwith 100 percentmodulationis used. In all other casesManchester codingis used with a modulation ratio of 10 percent.
Every active NFC device can work in one or more of three modes:
NFC tags are passive data stores which can be read, and under some circumstances written to, by an NFC device. They typically contain data (as of 2015[update]between 96 and 8,192 bytes) and are read-only in normal use, but may be rewritable. Applications include secure personal data storage (e.g.debitorcredit cardinformation,loyalty programdata,personal identification numbers(PINs), contacts). NFC tags can be custom-encoded by their manufacturers or use the industry specifications.
Although the range of NFC is limited to a few centimeters, standard plain NFC is not protected againsteavesdroppingand can be vulnerable to data modifications. Applications may use higher-layercryptographic protocolsto establish a secure channel.
The RF signal for the wireless data transfer can be picked up with antennas. The distance from which an attacker is able to eavesdrop the RF signal depends on multiple parameters, but is typically less than 10 meters.[60]Also, eavesdropping is highly affected by the communication mode. A passive device that doesn't generate its own RF field is much harder to eavesdrop on than an active device. An attacker can typically eavesdrop within 10 m of an active device and 1 m for passive devices.[61]
Because NFC devices usually includeISO/IEC 14443protocols,relay attacksare feasible.[62][63][64][page needed]For this attack the adversary forwards the request of the reader to the victim and relays its answer to the reader in real time, pretending to be the owner of the victim's smart card. This is similar to aman-in-the-middle attack.[62]Onelibnfccode example demonstrates a relay attack using two stock commercial NFC devices. This attack can be implemented using only two NFC-enabled mobile phones.[65]
NFC standards cover communications protocols and data exchange formats, and are based on existing RFID standards includingISO/IEC 14443andFeliCa.[7]The standards include ISO/IEC 18092[8]and those defined by the NFC Forum.
NFC is standardized in ECMA-340 and ISO/IEC 18092. These standards specify the modulation schemes, coding, transfer speeds and frame format of the RF interface of NFC devices, as well as initialization schemes and conditions required for data collision-control during initialization for both passive and active NFC modes. They also define thetransport protocol, including protocol activation and data-exchange methods. The air interface for NFC is standardized in:
NFC incorporates a variety of existing standards includingISO/IEC 14443Type A and Type B, andFeliCa(also simply named F or NFC-F). NFC-enabled phones work at a basic level with existing readers. In "card emulation mode" an NFC device should transmit, at a minimum, a unique ID number to a reader. In addition, NFC Forum defined a common data format calledNFC Data Exchange Format(NDEF) that can store and transport items ranging from anyMIME-typed object to ultra-short RTD-documents,[68]such asURLs. The NFC Forum added theSimple NDEF Exchange Protocol(SNEP) to the spec that allows sending and receiving messages between two NFC devices.[69]
TheGSM Association (GSMA)is a trade association representing nearly 800 mobile telephony operators and more than 200 product and service companies across 219 countries. Many of its members have led NFC trials and are preparing services for commercial launch.[70]
GSM is involved with several initiatives:
StoLPaN (Store Logistics and Payment with NFC) is a pan-European consortium supported by theEuropean Commission'sInformation Society Technologiesprogram. StoLPaN will examine the potential for NFC local wireless mobile communication.[74]
NFC Forum is a non-profit industry association formed on March 18, 2004, byNXP Semiconductors,SonyandNokiato advance the use of NFC wireless interaction in consumer electronics, mobile devices and PCs. Its specifications include the five distinct tag types that provide different communication speeds and capabilities covering flexibility, memory, security, data retention and write endurance. NFC Forum promotes implementation and standardization of NFC technology to ensure interoperability between devices and services. As of January 2020, the NFC Forum had over 120 member companies.[75]
NFC Forum promotes NFC and certifies device compliance[5]and whether it fits in apersonal area network.[5]
GSMA defined a platform for the deployment of GSMA NFC Standards[9]within mobile handsets. GSMA's efforts include,[76]Single Wire Protocol, testing and certification and secure element.[12]The GSMA standards surrounding the deployment of NFC protocols (governed byNFC Forum) on mobile handsets are neither exclusive nor universally accepted. For example, Google's deployment ofHost Card EmulationonAndroid KitKatprovides for software control of a universal radio. In this HCE Deployment[77]the NFC protocol is leveraged without the GSMA standards.
Other standardization bodies involved in NFC include:
NFC allows one- and two-way communication between endpoints, suitable for many applications.
NFC devices can act as electronicidentity documentsandkeycards.[2]They are used incontactless paymentsystems and allowmobile paymentreplacing or supplementing systems such as credit cards andelectronic ticketsmart cards. These are sometimes calledNFC/CTLSorCTLS NFC, withcontactlessabbreviated asCTLS. NFC can be used to share small files such as contacts and for bootstrapping fast connections to share larger media such as photos, videos, and other files.[78]
NFC devices can be used in contactless payment systems, similar to those used in credit cards andelectronic ticketsmart cards, and allow mobile payment to replace/supplement these systems.
InAndroid4.4, Google introduced platform support for secure NFC-based transactions throughHost Card Emulation(HCE), for payments, loyalty programs, card access, transit passes and other custom services. HCE allows any Android 4.4 app to emulate an NFC smart card, letting users initiate transactions with their device. Apps can use a new Reader Mode to act as readers for HCE cards and other NFC-based transactions.
On September 9, 2014,Appleannounced support for NFC-powered transactions as part ofApple Pay.[79]With the introduction of iOS 11, Apple devices allow third-party developers to read data from NFC tags.[80]
As of 2022, there are five major NFC apps available in the UK: Apple Pay, Google Pay, Samsung Pay, Barclays Contactless Mobile and Fitbit Pay. The UK Finance's UK Payment Markets Summary 2021 looked at Apple Pay, Google Pay and Samsung Pay and found 17.3 million UK adults had registered for mobile payment (up 75% from the year before) and of those, 84% had made a mobile payment.[81]
NFC offers a low-speed connection with simple setup that can be used tobootstrapmore capablewireless connections.[2]For example,Android Beamsoftware uses NFC to enable pairing and establish a Bluetooth connection when doing a file transfer and then disabling Bluetooth on both devices upon completion.[82]Nokia, Samsung, BlackBerry and Sony[83]have used NFC technology to pair Bluetooth headsets, media players and speakers with one tap.[84]The same principle can be applied to the configuration of Wi-Fi networks.Samsung Galaxydevices have a feature namedS-Beam—an extension of Android Beam that uses NFC (to shareMAC addressandIP addresses) and then usesWi-Fi Directto share files and documents. The advantage of using Wi-Fi Direct over Bluetooth is that it permits much faster data transfers, running up to 300 Mbit/s.[56]
NFC can be used forsocial networking, for sharing contacts, text messages and forums, links to photos, videos or files[78]and entering multiplayermobile games.[85]
NFC-enabled devices can act as electronicidentity documentsfound in passports and ID cards, andkeycardsfor the use infare cards,transit passes,login cards, car keys andaccess badges.[2]NFC's short range and encryption support make it more suitable than less private RFID systems.
NFC-equipped smartphones can be paired withNFC Tagsor stickers that can be programmed by NFC apps. These programs can allow a change of phone settings, texting, app launching, or command execution.
Such apps do not rely on a company or manufacturer, but can be utilized immediately with an NFC-equipped smartphone and an NFC tag.[86]
The NFC Forum published theSignature Record Type Definition(RTD) 2.0 in 2015 to add integrity and authenticity for NFC Tags. This specification allows an NFC device to verify tag data and identify the tag author.[87]
NFC has been used invideo gamesstarting withSkylanders: Spyro's Adventure.[88]These are customizable figurines which contain personal data with each figure, so no two figures are exactly alike. Nintendo'sWii U GamePadwas the first console system to include NFC technology out of the box. It was later included in theNintendo 3DSrange (being built into the New Nintendo 3DS/XL and in a separately sold reader which usesInfraredto communicate to older 3DS family consoles) and theNintendo Switchrange (being built within the rightJoy-Concontroller and directly in the Nintendo Switch Lite). Theamiiborange of accessories utilize NFC technology to unlock features.
Adidas Telstar 18is a soccer ball that contains an NFC chip within.[89]The chip enables users to interact with the ball using a smartphone.[90]
NFC and Bluetooth are both relatively short-range communication technologies available onmobile phones. NFC operates at slower speeds than Bluetooth and has a much shorter range, but consumes far less power and doesn't require pairing.[91]
NFC sets up more quickly than standard Bluetooth, but has a lower transfer rate thanBluetooth low energy. With NFC, instead of performing manual configurations to identify devices, the connection between two NFC devices is automatically established in less than .1 second. The maximum data transfer rate of NFC (424 kbit/s) is slower than that of Bluetooth V2.1 (2.1 Mbit/s).
NFC's maximum working distance of less than20 cm (7+7⁄8in) reduces the likelihood of unwanted interception, making it particularly suitable for crowded areas that complicate correlating a signal with its transmitting physical device (and by extension, its user).[92]
NFC is compatible with existing passive RFID (13.56 MHz ISO/IEC 18000-3) infrastructures. It requires comparatively low power, similar to the Bluetooth V4.0 low-energy protocol. However, when NFC works with an unpowered device (e.g. on a phone that may be turned off, a contactless smart credit card, a smart poster), the NFC power consumption is greater than that of Bluetooth V4.0 Low Energy, since illuminating the passive tag needs extra power.[91]
In 2011, handset vendors released more than 40 NFC-enabled handsets with theAndroidmobile operating system.BlackBerrydevices support NFC using BlackBerry Tag on devices running BlackBerry OS 7.0 and greater.[93]
MasterCardadded further NFC support for PayPass for the Android and BlackBerry platforms, enabling PayPass users to make payments using their Android or BlackBerry smartphones.[94]A partnership betweenSamsungandVisaadded a 'payWave' application on the Galaxy S4 smartphone.[95]
In 2012,Microsoftadded native NFC functionality in theirmobile OSwithWindows Phone 8, as well as theWindows 8operating system. Microsoft provides the "Wallet hub" in Windows Phone 8 for NFC payment, and can integrate multiple NFC payment services within a single application.[96]
In 2014,iPhone 6was released fromAppleto support NFC.[97]and since September 2019 iniOS 13Apple now allows NFC tags to be read out as well as labeled using an NFC app.[citation needed]
As of April 2011 hundreds of NFC trials had been conducted. Some firms moved to full-scale service deployments, spanning one or more countries. Multi-country deployments includeOrange's rollout of NFC technology to banks, retailers, transport, and service providers in multiple European countries,[42]andAirtel AfricaandOberthur Technologiesdeploying to 15 countries throughout Africa.[98]
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Ahumanmicrochip implantis any electronic device implanted subcutaneously (subdermally) usually via an injection. Examples include an identifyingintegrated circuitRFID device encased insilicate glasswhich is implanted in the body of a human being. This type ofsubdermal implantusually contains a uniqueID numberthat can be linked to information contained in an external database, such asidentity document,criminal record,medical history, medications,address book, and otherpotential uses.
Several hobbyists, scientists and business personalities have placed RFID microchip implants into their hands or had them inserted by others.
For Microchip implants that are encapsulated in silicate glass, there exists multiple methods to embed the device subcutaneously ranging from placing the microchip implant in a syringe or trocar[39]and piercing under the flesh (subdermal) then releasing the syringe to using a cutting tool such as a surgical scalpel to cut open subdermal and positioning the implant in the open wound.
A list of popular uses for microchip implants are as follows;
Other uses either cosmetic or medical may also include;
RFID implants usingNFCtechnologies have been used as access cards ranging for car door entry to building access.[41]Secure identity has also been used to encapsulate or impersonate a users identity via secure element or related technologies.
Researchers have examined microchip implants in humans in the medical field and they indicate that there are potential benefits and risks to incorporating the device in the medical field. For example, it could be beneficial for noncompliant patients but still poses great risks for potential misuse of the device.[45]
Destron Fearing, a subsidiary ofDigital Angel, initially developed the technology for the VeriChip.[46]
In 2004, the VeriChip implanted device and reader were classified asClass II: General controls with special controlsby the FDA;[47]that year the FDA also published a draft guidance describing the special controls required to market such devices.[48]
About the size of a grain of rice, the device was typically implanted between the shoulder and elbow area of an individual's right arm. Once scanned at the proper frequency, the chip responded with a unique 16-digit number which could be then linked with information about the user held on a database for identity verification, medical records access and other uses. The insertion procedure was performed under local anesthetic in a physician's office.[49][50]
Privacy advocates raised concerns regarding potential abuse of the chip, with some warning that adoption by governments as a compulsory identification program could lead to erosion of civil liberties, as well asidentity theftif the device should be hacked.[50][51][52]Another ethical dilemma posed by the technology, is that people with dementia could possibly benefit the most from an implanted device that contained their medical records, but issues ofinformed consentare the most difficult in precisely such people.[53]
In June 2007, theAmerican Medical Associationdeclared that "implantable radio frequency identification (RFID) devices may help to identify patients, thereby improving the safety and efficiency of patient care, and may be used to enable secure access to patient clinical information",[54]but in the same year, news reports linking similar devices to cancer caused in laboratory animals.[55]
In 2010, the company, by then called PositiveID, withdrew the product from the market due to poor sales.[56]
In January 2012, PositiveID sold the chip assets to a company called VeriTeQ that was owned by Scott Silverman, the former CEO of Positive ID.[57]
In 2016, JAMM Technologies acquired the chip assets from VeriTeQ; JAMM's business plan was to partner with companies sellingimplanted medical devicesand use the RFID tags to monitor and identify the devices.[58]JAMM Technologies is co-located in the samePlymouth, Minnesotabuilding as Geissler Corporation with Randolph K. Geissler and Donald R. Brattain[59][60]listed as its principals.
The website also claims that Geissler was CEO of PositiveID Corporation, Destron Fearing Corporation, and Digital Angel Corporation.[61]
In 2018, a Danish firm called BiChip released a new generation of microchip implant[62]that is intended to be readable from a distance and connected to Internet. The company released an update for its microchip implant to associate it with the Ripplecryptocurrencyto allow payments to be made using the implanted microchip.[63]
Patients that undergo NFC implants do so for a variety of reasons ranging from, Biomedical diagnostics, health reasons to gaining new senses,[64]gain biological enhancement, to be part of existing growing movements, for workplace purposes, security, hobbyists and for scientific endeavour.[65]
In 2020, a London-based firm called Impli released a microchip implant that is intended to be used with an accompanying smartphone app. The primary functionality of the implant is as a storage of medical records. The implant can be scanned by any smartphone that has NFC capabilities.[66]
In February 2006, CityWatcher, Inc. of Cincinnati, OH became the first company in the world to implant microchips into their employees as part of their building access control and security system. The workers needed the implants to access the company's secure video tape room, as documented inUSA Today.[67]The project was initiated and implemented by Six Sigma Security, Inc. The VeriChip Corporation had originally marketed the implant as a way to restrict access to secure facilities such as power plants.
A major drawback for such systems is the relative ease with which the 16-digit ID number contained in a chip implant can be obtained and cloned using a hand-held device, a problem that has been demonstrated publicly by security researcher Jonathan Westhues[68]and documented in the May 2006 issue ofWiredmagazine,[69]among other places.
In 2017, Mike Miller, chief executive of theWorld Olympians Association, was widely reported as suggesting the use of such implants in athletes in an attempt to reduce problems in sports due to recreational drug use.[72]
Theoretically, a GPS-enabled chip could one day make it possible for individuals to be physically located by latitude, longitude, altitude, and velocity.[citation needed]Such implantable GPS devices are not technically feasible at this time. However, if widely deployed at some future point, implantable GPS devices could conceivably allow authorities to locatemissing people,fugitives, or those who fled a crime scene. Critics contend that the technology could lead topolitical repressionas governments could use implants to track and persecute human rights activists, labor activists, civil dissidents, and political opponents; criminals and domestic abusers could use them to stalk, harass, and/or abduct their victims.
Another suggested application for a tracking implant, discussed in 2008 by the legislature ofIndonesia'sIrian Jayawould be to monitor the activities of people infected withHIV, aimed at reducing their chances of infecting other people.[73][74]The microchipping section was not, however, included in the final version of the provincialHIV/AIDS Handling bylawpassed by the legislature in December 2008.[75]With current technology, this would not be workable anyway, since there is no implantable device on the market withGPS trackingcapability.
Some have theorized[who?]that governments could use implants for:
Infection has been cited as a source of failure within RFID and related microchip implanted individuals, either due to improper implantation techniques, implantrejectionsor corrosion of implant elements.[76]
Some chipped individuals have reported being turned away fromMRIsdue to the presence of magnets in their body.[77]No conclusive investigation has been done on the risks of each type of implant near MRIs, other than anecdotal reports ranging from no problems, requiring hand shielding before proximity, to being denied the MRI.[failed verification–see discussion]
Other medical imaging technologies likeX-rayandCT scannersdo not pose a similar risk. Rather, X-rays can be used to locate implants.
Electronics-based implants contain little material that can corrode.Magnetic implants, however, often contain a substantial amount of metallic elements by volume, and iron, a common implant element, is easily corroded by common elements such as oxygen and water. Implant corrosion occurs when these elements become trapped inside during the encapsulation process, which can cause slow corrosive effect, or the encapsulation fails and allows corrosive elements to come into contact with the magnet. Catastrophic encapsulation failures are usually obvious, resulting in tenderness, discoloration of the skin, and a slight inflammatory response. Small failures however can take much longer to become obvious, resulting in a slow degradation of field strength without many external signs that something is slowly going wrong with the magnet.[78]
In a self-published report,[79]anti-RFID advocateKatherine Albrecht, who refers to RFID devices as "spy chips", citesveterinaryandtoxicologicalstudies carried out from 1996 to 2006 which found lab rodents injected with microchips as an incidental part of unrelated experiments and dogs implanted with identification microchips sometimes developed cancerous tumors at the injection site (subcutaneoussarcomas) as evidence of a human implantation risk.[80]However, the link between foreign-body tumorigenesis in lab animals and implantation in humans has been publicly refuted as erroneous and misleading[81]and the report's author has been criticized[by whom?]over the use of "provocative" language "not based in scientific fact".[82]Notably, none of the studies cited specifically set out to investigate the cancer risk of implanted microchips and so none of the studies had a control group of animals that did not get implanted. While the issue is considered worthy of further investigation, one of the studies cited cautioned "Blind leaps from the detection of tumors to the prediction of human health risk should be avoided".[83][84][85]
The Council on Ethical and Judicial Affairs (CEJA) of theAmerican Medical Associationpublished a report in 2007 alleging that RFID implanted chips may compromiseprivacybecause even though no information can be stored in an RFID transponder, they allege that there is no assurance that the information contained in the chip can be properly protected.[86]
Stolen identity and privacy has been a major concern with microchip implants being cloned for various nefarious reasons in a process known asWireless identity theft. Incidents of forced removal of animal implants have been documented,[87]the concern lies in whether this same practice will be used to attack implanted microchipped patients also. Due to low adoption of microchip implants incidents of these physical attacks are rare. Nefarious RFID reprogramming of unprotected or unencrypted microchip tags are also a major security risk consideration.
There is concern technology can be abused.[88]Opponents have stated that such invasive technology has the potential to be used by governments to create an 'Orwellian'digital dystopiaand theorized that in such a world, self-determination, the ability to think freely, and all personal autonomy could be completely lost.[89][90][91]
In 2019,Elon Muskannounced that a company he had founded which deals with microchip implant research, called Neuralink, would be able to "solve"autismand other "brain diseases".[92]This led to a number of critics calling out Musk for his statements, with Dan Robitzski ofNeoscopesaying, "while schizophrenia can be a debilitating mental condition, autism is more tightly linked to a sense of identity — and listing it as a disease to be solved as Musk did risks further stigmatizing a community pushing for better treatment and representation."[93]Hilary Brueck ofInsideragreed, saying, "conditions like autism can't be neatly cataloged as things to "solve." Instead, they lead people to think differently". She went on to argue though that the technology shouldn't be discounted entirely, as it could potentially help people with a variety of disabilities such asblindnessandquadriplegia.[94]FellowInsiderwriter Isobel Asher Hamilton added, "it was not clear what Musk meant by saying Neuralink could "solve" autism, which is not a disease but a developmental disorder." She then cited The UK's National Autistic Society's website statement, which says, "Autism is not an illness or disease and cannot be 'cured.' Often people feel being autistic is a fundamental aspect of their identity."[95]Tristan Greene ofThe Next Webstated, in response to Musk, "there’s only one problem: autism isn’t a disease and it can’t be cured or solved. In fact, there’s some ethical debate in the medical community over whether autism, which is considered a disorder, should be treated as part of a person’s identity and not a ‘condition’ to be fixed... how freaking cool would it be to actually start yourTesla[electric vehicle] just by thinking? But, maybe... just maybe, the billionaire with access to the world's brightest medical minds who, even after founding a medical startup, still incorrectly thinks that autism is a disease that can be solved or cured shouldn't be someone we trust to shove wires or chips into our brains."[96]
Some autistic people also spoke out against Musk's statement about using microchips to "solve" autism, with Nera Birch ofThe Mighty, an autistic writer, stating, "autism is a huge part of who I am. It pervades every aspect of my life. Sure, there are days where being neurotypical would make everything so much easier. But I wouldn’t trade my autism for the world. I have the unique ability to view the world and experience senses in a way that makes all the negatives of autism worth it. The fact you think I would want to be “cured” is like saying I would rather be nothing than be myself. People with neurodiversity are proud of ourselves. For many of us, we wear our autism as a badge of pride. We have a culture within ourselves. It is not something that needs to be erased. The person with autism is not the problem. Neurotypical people need to stop molding us into somethingtheywant to interact with."[97]Florence Grant, an autistic writer forThe Independent, stated, "autistic people often have highly-focused interests, also known as special interests. I love my ability to hyperfocus and how passionate I get about things. I also notice small details and things that other people don’t see. I see the world differently, through a clear lens, and this means I can identify solutions where other people can’t. Does this sound familiar, Elon? My autism is a part of me, and it’s not something that can be separated from me. I should be able to exist freely autistic and proud. But for that to happen, the world needs to stop punishing difference and start embracing it." Grant noted that Musk himself had recently admitted that he had been diagnosed withAsperger's syndrome(itself an outdated diagnosis, the characteristics of which are currently recognized as part of the autism spectrum[98]) while hostingSaturday Night Live.[99]
Musk himself has not specified how Neuralink's microchip technology would "solve" autism, and has not commented publicly on the feedback from autistic people.
Despite a lack of evidence demonstrating invasive use or even technical capability of microchip implants, they have been the subject of many conspiracy theories.
TheSouthern Poverty Law Centerreported in 2010 that on the Christian right, there were concerns that implants could be the "mark of the beast" and amongst thePatriot movementthere were fears that implants could be used to track people.[100]The same yearNPRreported that a myth was circulating online that patients who signed up to receive treatment under theAffordable Care Act(Obamacare) would be implanted.[101]
In 2016,Snopesreported that being injected with microchips was a "perennial concern to the conspiracy-minded" and noted that a conspiracy theory was circulating in Australia at that time that the government was going to implant all of its citizens.[102]
A 2021 survey byYouGovfound that 20% of Americans believed microchips were inside theCOVID-19 vaccines.[103][104]A 2021Facebookpost byRT (Russia Today)claimedDARPAhad developed a COVID-19 detecting microchip implant.[105][106]
A few jurisdictions have researched or preemptively passed laws regarding human implantation of microchips.
In the United States, many states such asWisconsin(as of 2006),North Dakota(2007),California(2007),Oklahoma(2008), andGeorgia(2010) have laws making it illegal to force a person to have a microchip implanted, though politicians acknowledge they are unaware of cases of such forced implantation.[107][108][109][110]In 2010,Virginiapassed a bill forbidding companies from forcing employees to be implanted with tracking devices.[111]
In 2010,Washington'sHouse of Representativesintroduced a bill ordering the study of potential monitoring of sex offenders with implanted RFID or similar technology, but it did not pass.[112]
The general public are most familiar with microchips in the context ofidentifying pets.
Implanted individuals are considered to be grouped together as part of thetranshumanismmovement.
"Arkangel", an episode of the drama seriesBlack Mirror, considered the potential forhelicopter parentingof an imagined more advanced microchip.
Microchip implants have been explored incyberpunkmedia such asGhost in the Shell,Cyberpunk 2077, andDeus Ex.
Some Christians make a link between implants and the BiblicalMark of the Beast,[113][114]prophesied to be a future requirement for buying and selling, and a key element of theBook of Revelation.[115][116]Gary Wohlscheid, president of These Last Days Ministries, has argued that "Out of all the technologies with potential to be themark of the beast, VeriChip has got the best possibility right now."[117]
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Privacy by designis an approach tosystems engineeringinitially developed byAnn Cavoukianand formalized in a joint report onprivacy-enhancing technologiesby a joint team of theInformation and Privacy Commissioner of Ontario(Canada), theDutch Data Protection Authority, and theNetherlands Organisation for Applied Scientific Researchin 1995.[1][2]The privacy by design framework was published in 2009[3]and adopted by the International Assembly of Privacy Commissioners and Data Protection Authorities in 2010.[4]Privacy by design calls forprivacyto be taken into account throughout the whole engineering process. The concept is an example ofvalue sensitive design, i.e., taking human values into account in a well-defined manner throughout the process.[5][6]
Cavoukian's approach to privacy has been criticized as being vague,[7]challenging to enforce its adoption,[8]difficult to apply to certain disciplines,[9][10]challenging to scale up to networked infrastructures,[10]as well as prioritizing corporate interests over consumers' interests[7]and placing insufficient emphasis on minimizing data collection.[9]Recent developments in computer science and data engineering, such as support for encoding privacy in data[11]and the availability and quality ofPrivacy-Enhancing Technologies(PET's) partly offset those critiques and help to make the principles feasible in real-world settings.
The EuropeanGDPRregulation incorporates privacy by design.[12]
The privacy by design framework was developed byAnn Cavoukian,Information and Privacy Commissioner of Ontario, following her joint work with theDutch Data Protection Authorityand theNetherlands Organisation for Applied Scientific Researchin 1995.[1][12]In 2009, theInformation and Privacy Commissioner of Ontarioco-hosted an event,Privacy by Design: The Definitive Workshop, with the Israeli Law, Information and Technology Authority at the 31st International Conference of Data Protection and Privacy Commissioner (2009).[13][14]
In 2010 the framework achieved international acceptance when the International Assembly of Privacy Commissioners and Data Protection Authorities unanimously passed a resolution on privacy by design[15]recognising it as an international standard at their annual conference.[14][16][17][4]Among other commitments, the commissioners resolved to promote privacy by design as widely as possible and foster the incorporation of the principle into policy and legislation.[4]
Privacy by design is based on seven "foundational principles":[3][18][19][20]
The principles have been cited in over five hundred articles[21]referring to thePrivacy by Design in Law, Policy and Practicewhite paper byAnn Cavoukian.[22]
The privacy by design approach is characterized by proactive rather than reactive measures. It anticipates and prevents privacy invasive events before they happen. Privacy by design does not wait for privacy risks to materialize, nor does it offer remedies for resolving privacy infractions once they have occurred — it aims to prevent them from occurring. In short, privacy by design comes before-the-fact, not after.[18][19][20]
Privacy by design seeks to deliver the maximum degree of privacy by ensuring that personal data are automatically protected in any given IT system or business practice. If an individual does nothing, their privacy still remains intact. No action is required on the part of the individual to protect their privacy — it is built into the system, by default.[18][19][20]
Privacy by design is embedded into the design and architecture of IT systems as well as business practices. It is not bolted on as an add-on, after the fact. The result is that privacy becomes an essential component of the core functionality being delivered. Privacy is integral to the system without diminishing functionality.[18][19][20]
Privacy by design seeks to accommodate all legitimate interests and objectives in a positive-sum “win-win” manner, not through a dated, zero-sum approach, where unnecessary trade-offs are made. Privacy by design avoids the pretense of false dichotomies, such as privacy versus security, demonstrating that it is possible to have both.[18][19][20]
Privacy by design, having been embedded into the system prior to the first element of information being collected, extends securely throughout the entire lifecycle of the data involved — strong security measures are essential to privacy, from start to finish. This ensures that all data are securely retained, and then securely destroyed at the end of the process, in a timely fashion. Thus, privacy by design ensures cradle-to-grave, secure lifecycle management of information, end-to-end.[18][19][20]
Privacy by design seeks to assure all stakeholders that whatever business practice or technology involved is in fact operating according to the stated promises and objectives, subject to independent verification. The component parts and operations remain visible and transparent, to users and providers alike. Remember to trust but verify.[18][19][20]
Above all, privacy by design requires architects and operators to keep the interests of the individual uppermost by offering such measures as strong privacy defaults, appropriate notice, and empowering user-friendly options. Keep it user-centric.[18][19][20]
The International Organization for Standardization(ISO) approved the Committee on Consumer Policy (COPOLCO) proposal for a new ISO standard:Consumer Protection: Privacy by Design for Consumer Goods and Services(ISO/PC317).[23]The standard will aim to specify the design process to provide consumer goods and services that meet consumers’ domestic processing privacy needs as well as the personal privacy requirements ofdata protection. The standard has the UK as secretariat with thirteen participating members[24]and twenty observing members.[24]
TheStandards Council of Canada(SCC) is one of the participating members and has established a mirror Canadian committee to ISO/PC317.[25]
The OASIS Privacy by Design Documentation for Software Engineers (PbD-SE)[26]Technical Committee provides a specification to operationalize privacy by design in the context of software engineering. Privacy by design, like security by design, is a normal part of the software development process and a risk reduction strategy for software engineers. The PbD-SE specification translates the PbD principles to conformance requirements within software engineering tasks and helps software development teams to produceartifactsas evidence of PbD principle adherence. Following the specification facilitates the documentation of privacy requirements from software conception to retirement, thereby providing a plan around adherence to privacy by design principles, and other guidance to privacy best practices, such as NIST's 800-53 Appendix J (NIST SP 800–53) and the Fair Information Practice Principles (FIPPs) (PMRM-1.0).[26]
Privacy by design originated fromprivacy-enhancing technologies(PETs) in a joint 1995 report by Ann Cavoukian and John Borking.[1]In 2007 theEuropean Commissionprovided a memo on PETs.[27]In 2008 the British Information Commissioner's Office commissioned a report titledPrivacy by Design – An Overview of Privacy Enhancing Technologies.[28]
There are many facets to privacy by design. There is the technical side like software and systems engineering,[29]administrative elements (e.g. legal, policy, procedural), other organizational controls, and operating contexts. Privacy by design evolved from early efforts to express fair information practice principles directly into the design and operation of information and communications technologies.[30]In his publicationPrivacy by Design: Delivering the Promises[2]Peter Hustinxacknowledges the key role played byAnn Cavoukianand John Borking, then Deputy Privacy Commissioners, in the joint 1995 publicationPrivacy-Enhancing Technologies: The Path to Anonymity.[1]This 1995 report focussed on exploring technologies that permit transactions to be conducted anonymously.
Privacy-enhancing technologies allow online users to protect the privacy of theirPersonally Identifiable Information(PII) provided to and handled by services or applications. Privacy by design evolved to consider the broader systems and processes in which PETs were embedded and operated.The U.S. Center for Democracy & Technology(CDT) inThe Role of Privacy by Design in Protecting Consumer Privacy[31]distinguishes PET from privacy by design noting that “PETs are most useful for users who already understand online privacy risks. They are essential user empowerment tools, but they form only a single piece of a broader framework that should be considered when discussing how technology can be used in the service of protecting privacy.”[31]
Germany released a statute (§ 3 Sec. 4Teledienstedatenschutzgesetz[Teleservices Data Protection Act]) back in July 1997.[32]The new EUGeneral Data Protection Regulation(GDPR) includes ‘data protection by design’ and ‘data protection by default’,[33][34][12]the second foundational principle of privacy by design. Canada's Privacy Commissioner included privacy by design in its report onPrivacy, Trust and Innovation – Building Canada’s Digital Advantage.[35][36]In 2012, U.S.Federal Trade Commission (FTC)recognized privacy by design as one of its three recommended practices for protecting online privacy in its report entitledProtecting Consumer Privacy in an Era of Rapid Change,[37]and the FTC included privacy by design as one of the key pillars in itsFinal Commissioner Report on Protecting Consumer Privacy.[38]In Australia, the Commissioner for Privacy and Data Protection for the State of Victoria (CPDP) has formally adopted privacy by design as a core policy to underpin information privacy management in the Victorian public sector.[39]The UK Information Commissioner's Office website highlights privacy by design[40]anddata protectionby design and default.[41]In October 2014, the Mauritius Declaration on the Internet of Things was made at the 36th International Conference of Data Protection and Privacy Commissioners and included privacy by design and default.[42]ThePrivacy Commissioner for Personal Data, Hong Kong held an educational conference on the importance of privacy by design.[43][44]
In the private sector,Sidewalk Torontocommits to privacy by design principles;[45]Brendon Lynch, Chief Privacy Officer atMicrosoft, wrote an article calledPrivacy by Design at Microsoft;[46]whilstDeloitterelates certifiably trustworthy to privacy by design.[47]
The privacy by design framework attracted academic debate, particularly following the 2010 International Data Commissioners resolution that provided criticism of privacy by design with suggestions by legal and engineering experts to better understand how to apply the framework into various contexts.[7][9][8]
Privacy by design has been critiqued as "vague"[7]and leaving "many open questions about their application when engineering systems." Suggestions have been made to instead start with and focus on minimizing data, which can be done through security engineering.[9]
In 2007, researchers at K.U. Leuven published Engineering Privacy by Design noting that “The design and implementation of privacy requirements in systems is a difficult problem and requires translation of complex social, legal and ethical concerns into systems requirements”. The principles of privacy by design "remain vague and leave many open questions about their application when engineering systems". The authors argue that "starting from data minimization is a necessary and foundational first step to engineer systems in line with the principles of privacy by design". The objective of their paper is to provide an "initial inquiry into the practice of privacy by design from an engineering perspective in order to contribute to the closing of the gap between policymakers’ and engineers’ understanding of privacy by design."[9]Extended peer consultations performed 10 years later in an EU project however confirmed persistent difficulties in translating legal principles into engineering requirements. This is partly a more structural problem due to the fact that legal principles are abstract, open-ended with different possible interpretations and exceptions, whereas engineering practices require unambiguous meanings and formal definitions of design concepts.[10]
In 2011, the Danish National It and Telecom Agency published a discussion paper in which they argued that privacy by design is a key goal for creating digital security models, by extending the concept to "Security by Design". The objective is to balance anonymity and surveillance by eliminating identification as much as possible.[48]
Another criticism is that current definitions of privacy by design do not address the methodological aspect of systems engineering, such as using decent system engineering methods, e.g. those which cover the complete system and data life cycle.[7]This problem is further exacerbated in the move to networked digital infrastructures initiatives such as thesmart cityor theInternet of Things. Whereas privacy by design has mainly been focused on the responsibilities of singular organisations for a certain technology, these initiatives often require the interoperability of many different technologies operated by different organisations. This requires a shift from organisational to infrastructural design.[10]
The concept of privacy by design also does not focus on the role of the actual data holder but on that of the system designer. This role is not known in privacy law, so the concept of privacy by design is not based on law. This, in turn, undermines the trust by data subjects, data holders and policy-makers.[7]Questions have been raised from science and technology studies of whether privacy by design will change the meaning and practice of rights through implementation in technologies, organizations, standards and infrastructures.[49]From a civil society perspective, some have even raised the possibility that a bad use of these design-based approaches can even lead to the danger ofbluewashing. This refers to the minimal instrumental use by organizations of privacy design without adequate checks, in order to portray themselves as more privacy-friendly than is factually justified.[10]
It has also been pointed out that privacy by design is similar tovoluntary complianceschemes in industries impacting the environment, and thus lacks the teeth necessary to be effective, and may differ per company. In addition, the evolutionary approach currently taken to the development of the concept will come at the cost of privacy infringements because evolution implies also letting unfit phenotypes (privacy-invading products) live until they are proven unfit.[7]Some critics have pointed out that certain business models are built around customer surveillance and data manipulation and therefore voluntary compliance is unlikely.[8]
In 2013, Rubinstein and Good usedGoogleandFacebookprivacy incidents to conduct a counterfactual analysis in order to identify lessons learned of value for regulators when recommending privacy by design. The first was that “more detailed principles and specific examples” would be more helpful to companies. The second is that “usability is just as important as engineering principles and practices”. The third is that there needs to be more work on “refining and elaborating on design principles–both in privacy engineering and usability design”. including efforts to define international privacy standards. The final lesson learned is that “regulators must do more than merely recommend the adoption and implementation of privacy by design.”[8]
The advent ofGDPRwith its maximum fine of 4% of global turnover now provides a balance between business benefit and turnover and addresses thevoluntary compliancecriticism and requirement from Rubinstein and Good that “regulators must do more than merely recommend the adoption and implementation of privacy by design”.[8]Rubinstein and Good also highlighted that privacy by design could result in applications that exemplified Privacy by Design and their work was well received.[50][8]
The May 2018European Data Protection SupervisorGiovanni Buttarelli's paperPreliminary Opinion on Privacy by Designstates, "While privacy by design has made significant progress in legal, technological and conceptual development, it is still far from unfolding its full potential for the protection of the fundamental rights of individuals. The following sections of this opinion provide an overview of relevant developments and recommend further efforts".[12]
The executive summary makes the following recommendations to EU institutions:
The EDPS will:
TheEuropean Data Protection SupervisorGiovanni Buttarelliset out the requirement to implement privacy by design in his article.[51]TheEuropean Union Agency for Network and Information Security(ENISA) provided a detailed reportPrivacy and Data Protection by Design – From Policy to Engineeringon implementation.[52]The Summer School on real-world crypto and privacy provided a tutorial on "Engineering Privacy by Design".[53]TheOWASP Top 10 Privacy Risks Projectfor web applications that gives hints on how to implement privacy by design in practice. The OASIS Privacy by Design Documentation for Software Engineers (PbD-SE)[26]offers a privacy extension/complement to OMG's Unified Modeling Language (UML) and serves as a complement to OASIS’ eXtensible Access Control Mark-up Language (XACML) and Privacy Management Reference Model (PMRM). Privacy by Design guidelines are developed to operationalise some of the high-level privacy-preserving ideas into more granular actionable advice.,[54][55]such as recommendations onhow to implement privacy by design into existing (data) systems. However, still the applications of privacy by design guidelines by software developers remains a challenge.[56]
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Aproximity cardorprox card[1]also known as a key card or keycard is a contactlesssmart cardwhich can be read without inserting it into a reader device, as required by earliermagnetic stripe cardssuch ascredit cardsand contact type smart cards.[2]The proximity cards are part of the contactless card technologies. Held near an electronic reader for a moment they enable the identification of an encoded number. The reader usually produces a beep or other sound to indicate the card has been read.
The term "proximity card" refers to the older 125 kHz devices as distinct from the newer 13.56 MHzcontactless smartcards.[citation needed]Second generation prox cards are used for mass and distance reading applications. Proximity cards typically have a read range of up to 50 cm (20 in)[1]which is the main difference from thecontactless smartcardwith a range of 2 to 10 cm (1 to 4 in). The card can often be left in a wallet or purse,[3]and read by simply holding the wallet or purse near the reader. These early proximity cards can't hold more data than amagnetic stripe card, and only cards with smart chips (ie,contactless smartcards) can hold other types of data like electronic funds balance forcontactless paymentsystems, history data for time and attendance or biometric templates. When used without encoding data, only with the card serial number, contactless smartcards have similar functionalities to proximity cards.
Passive 125 kHz cards, the more widely used type which were described above, are powered byradio frequencysignals from the reader device and so have a limited range and must be held close to the reader unit.[2]They are used askeycardsfor access control doors in office buildings. A version with more memory,contactless smartcards, are used for other applications:library cards,contactless paymentsystems, andpublic transitfare cards.
Active 125 kHz prox cards, sometimes calledvicinity cards[dubious–discuss], are powered by an internal lithium battery. They can have a greater range, up to 2 meters (6 ft). Other contactless technologies like UHF (Ultra High Frequency) smart cards can reach up to 150 meters (500 ft) and are often used for applications where the card is read inside a vehicle, such as security gates which open when a vehicle with the access card inside approaches, orautomated toll collection.[2]The battery eventually runs down, however, and the card must be replaced after 2 to 7 years.
The card and the reader unit communicate with each other through 125 kHz radio frequency fields (13.56 MHz for thecontactless smartcardcards) by a process calledresonant energy transfer.[1][2]Passive cards have three components which are sealed inside the plastic: an antenna consisting of a coil of wire, acapacitor, and anintegrated circuit(IC) which contains the user's ID number in specific formats and no other data. The reader has its own antenna, which continuously transmits a short range radio frequency field.
When the card is placed within range of the reader, the antenna coil and capacitor, which form atuned circuit, absorb and store energy from the field,resonatingat the frequency emitted by the reader. This energy isrectifiedtodirect currentwhich powers theintegrated circuit. The chip sends its ID number or other data to the antenna coil, which transmits it by radio frequency signals back to the reader unit. The reader checks whether the ID number from the card is correct, and then performs whatever function it has been programmed to do for that ID number. All the energy to power the card comes from the reader unit, so passive cards must be close to a reader to transmit their data.
An active card contains a flatlithium cellin addition to the above components to power it. The integrated circuit contains areceiverwhich uses the battery's power toamplifythe signal from the reader unit so it is stronger, allowing the card to detect the reader at a greater distance. The battery also powers atransmittercircuit in the chip which transmits a stronger return signal to cover the greater distance.
Proximity cards are all proprietary. This is also the case of the memory-based first generation ofcontactless smartcards. This means that there is no compatibility between the readers of a specific brand and the cards of another brand.
Contactless smartcardsare covered by theISO/IEC 14443and/or theISO/IEC 15693ORISO/IEC 18000standards. These standards define two types of card ("A" and "B", each with differentcommunications protocols) which typically have a range up to 10 cm (4 in). The relatedISO/IEC 15693(vicinity card) standard typically works up to a longer range of 100 centimetres (39 in).
The reality is thatISO/IEC 14443as well asISO/IEC 15693can only be fully implemented on microprocessor-based cards. The best way to check if a technology meets ISO standard is to ask the manufacturer if it can be emulated on other devices without any proprietary hardware.
The card readers communicate in various protocols, for example theWiegand protocolthat consists of a data 0 and a data 1 circuit (or binary or simple on/off (digital) type circuit). Other known protocols are mono directional Clock and Data or bidirectional OSDP (RS 485), RS 232 or UART. The earliest card formats were up to 64 bits long. As demand has increased, bit size has increased to continue to provide unique numbers. Often, the first several bits can be made identical; these are called facility or site codes. The idea is that company A has a facility code ofxnand a card set of 0001 through 1000 and company B has a facility code ofynand a card set also of 0001 through 1000. For smartcards, a numbering system is internationally harmonized and allocated by Netherlands-basedNEN (registration authority)according toISO/IEC 6523and ISO/IEC 15459 standards.
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Resonant inductive couplingormagnetic phase synchronous coupling[4][5]is a phenomenon withinductive couplingin which the coupling becomes stronger when the "secondary" (load-bearing) side of the loosely coupled coil resonates.[5]Aresonant transformerof this type is often used in analog circuitry as abandpass filter. Resonant inductive coupling is also used inwireless powersystems for portable computers, phones, and vehicles.
Various resonant coupling systems in use or are under development for short range (up to 2 meters)[6]wireless electricity systems to power laptops, tablets, smartphones,robot vacuums, implanted medical devices, and vehicles like electric cars,SCMaglevtrains[7]andautomated guided vehicles.[8]Specific technologies include:
Other applications include:
TheTesla coilis a resonant transformer circuit used to generate very high voltages, and is able to provide much higher current than high voltageelectrostatic machinessuch as theVan de Graaff generator.[10]However, this type of system radiates most of its energy into empty space, unlike modern wireless power systems which waste very little energy.
Resonant transformers are widely used inradiocircuits asbandpass filters, and in switching power supplies.
In 1894Nikola Teslaused resonant inductive coupling, also known as "electro-dynamic induction" to wirelessly light up phosphorescent and incandescent lamps at the 35 South Fifth Avenue laboratory, and later at the 46 E. Houston Street laboratory in New York City.[11][12][13]In 1897 he patented a device[14]called the high-voltage,resonant transformeror "Tesla coil." Transferring electrical energy from the primary coil to the secondary coil by resonant induction, a Tesla coil is capable of producingvery high voltagesathigh frequency. The improved design allowed for the safe production and utilization of high-potential electrical currents, "without serious liability of the destruction of the apparatus itself and danger to persons approaching or handling it."
In the early 1960s resonant inductive wireless energy transfer was used successfully in implantable medical devices[15]including such devices as pacemakers and artificial hearts. While the early systems used a resonant receiver coil, later systems[16]implemented resonant transmitter coils as well. These medical devices are designed for high efficiency using low power electronics while efficiently accommodating some misalignment and dynamic twisting of the coils. The separation between the coils in implantable applications is commonly less than 20 cm. Today resonant inductive energy transfer is regularly used for providing electric power in many commercially available medical implantable devices.[17]
Wireless electric energy transfer for experimentally powering electric automobiles and buses is a higher power application (>10 kW) of resonant inductive energy transfer. High power levels are required for rapid recharging and high energy transfer efficiency is required both for operational economy and to avoid negative environmental impact of the system. An experimental electrified roadway test track built circa 1990 achieved just above 60%energy efficiencywhile recharging the battery of a prototype bus at a specially equipped bus stop.[18][19]The bus could be outfitted with a retractable receiving coil for greater coil clearance when moving. The gap between the transmit and receive coils was designed to be less than 10 cm when powered. In addition to buses the use of wireless transfer has been investigated for recharging electric automobiles in parking spots and garages as well.
Some of these wireless resonant inductive devices operate at low milliwatt power levels and are battery powered. Others operate at higher kilowatt power levels. Current implantable medical and road electrification device designs achieve more than 75% transfer efficiency at an operating distance between the transmit and receive coils of less than 10 cm.[citation needed]
In 1993, Professor John Boys and Professor Grant Covic, of theUniversity of Aucklandin New Zealand, developed systems to transfer large amounts of energy across small air gaps.[4][5][20]It was putting into practical use as the moving crane and the AGV non-contact power supply in Japan.[8]In 1998, RFID tags were patented that were powered in this way.[21]
In November 2006,Marin Soljačićand other researchers at theMassachusetts Institute of Technologyapplied this near field behavior to wireless power transmission, based on strongly-coupled resonators.[22][23][24]In a theoretical analysis,[25]they demonstrate that, by designing electromagnetic resonators that suffer minimal loss due to radiation and absorption and have a near field with mid-range extent (namely a few times the resonator size), mid-range efficient wireless energy-transfer is possible. The reason is that, if two suchresonant circuitstuned to the same frequency are within a fraction of a wavelength, their near fields (consisting of 'evanescent waves') couple by means ofevanescent wave coupling. Oscillating waves develop between the inductors, which can allow the energy to transfer from one object to the other within times much shorter than all loss times, which were designed to be long, and thus with the maximum possible energy-transfer efficiency. Since the resonant wavelength is much larger than the resonators, the field can circumvent extraneous objects in the vicinity and thus this mid-range energy-transfer scheme does not require line-of-sight. By utilizing in particular the magnetic field to achieve the coupling, this method can be safe, since magnetic fields interact weakly with living organisms.
Apple Inc.applied for a patent on the technology in 2010, after WiPower did so in 2008.[26]
In the past, the power source used on the JR Tokai SCMaglev car was generating with a gas turbine generator. In 2011, they succeeded in powering while driving (CWD:charge while driving) across a large gap by the JR Tokai proprietary 9.8 kHz phase synchronization technology developed based on technology similar to AGV's wireless power scheme. And the Japanese Ministry of Land, Infrastructure and Transportation evaluated the technology as all the problems for practical use were cleared.[27]Construction of SCMaglev begin and commercial use will start in 2027.[28]
Non-resonantcoupled inductors, such as typicaltransformers, work on the principle of aprimary coilgenerating amagnetic fieldand a secondary coil subtending as much as possible of that field so that the power passing through the secondary is as close as possible to that of the primary. This requirement that the field be covered by the secondary results in very short range and usually requires amagnetic core. Over greater distances the non-resonant induction method is highly inefficient and wastes the vast majority of the energy in resistive losses of the primary coil.
Using resonance can help improve efficiency dramatically. If resonant coupling is used, the secondary coil is capacitive loaded so as to form a tuned LC circuit. If the primary coil is driven at the secondary side resonant frequency, it turns out that significant power may be transmitted between the coils over a range of a few times the coil diameters at reasonable efficiency.[29]
Compared to the costs associated with batteries, particularly non-rechargeable batteries, the costs of the batteries are hundreds of times higher. In situations where a source of power is available nearby, it can be a cheaper solution.[30]In addition, whereas batteries need periodic maintenance and replacement, resonant energy transfer can be used instead. Batteries additionally generate pollution during their construction and their disposal which is largely avoided.
Unlike mains-wired equipment, no direct electrical connection is needed and hence equipment can be sealed to minimize the possibility of electric shock.
Because the coupling is achieved using predominantly magnetic fields; the technology may be relatively safe. Safety standards and guidelines do exist in most countries for electromagnetic field exposures (e.g. ICNIRP[31][32]). Whether the system can meet the guidelines or the less stringent legal requirements depends on the delivered power and range from the transmitter. Maximum recommended B-field is a complicated function of frequency, the ICNIRP guidelines for example permit RMS fields of tens of microteslas below 100 kHz, falling with frequency to 200 nanoteslas in the VHF, and lower levels above 400 MHz, where body parts can sustain current loops comparable to a wavelength in diameter, and deep tissue energy absorption reaches a maximum.
Deployed systems already generate magnetic fields, for exampleinduction cookersin the tens of kHz where high fields are permitted, andcontactless smart cardreaders, where higher frequency is possible as the required energies are lower.
A study for the Swedish military found that 85 kHz systems for dynamic wireless power transfer for vehicles can cause electromagnetic interference at a radius of up to 300 kilometers.[33]
This process occurs in aresonant transformer, an electrical component which transformer consists of highQcoil wound on the same core with acapacitorconnected across a coil to make a coupledLC circuit.
The most basic resonant inductive coupling consists of one drive coil on the primary side and one resonance circuit on the secondary side.[34][5][2]In this case, when the resonant state on the secondary side is observed from the primary side, two resonances as a pair are observed.[35][5]One of them is called theantiresonantfrequency (parallel resonant frequency 1), and the other is called the resonant frequency (serial resonant frequency 1').[5]Theshort-circuit inductanceand resonant capacitor of the secondary coil are combined into a resonant circuit.[36][5]When the primary coil is driven with a resonant frequency (serial resonant frequency) of the secondary side, the phases of the magnetic fields of the primary coil and the secondary coil are synchronized.[5]As a result, the maximum voltage is generated on the secondary coil due to the increase of the mutual flux, and the copper loss of the primary coil is reduced, the heat generation is reduced, and the efficiency is relatively improved.[2]The resonant inductive coupling is thenear fieldwireless transmission of electrical energybetween magnetically coupled coils, which is part of aresonant circuittuned toresonateat the same frequency as the driving frequency.
In the transformer, only part of the flux generated by current through the primary coil is coupled to the secondary coil and vice versa. The part that couples is calledmutual fluxand the part that does not couple is calledleakage flux.[37]When the system is not in the resonance state, this leads to the open-circuit voltage appearing at the secondary being less than predicted by the turns ratio of the coils. The degree of coupling is captured by a parameter calledcoupling coefficient. The coupling coefficient,k, is defined as the ratio of transformer open-circuit voltage ratio to the ratio that would be obtained if all the flux coupled from one coil to the other. However, if it is not open circuit, the flux ratio will change. The value ofklies between 0 and ±1. Each coil inductance can be notionally divided into two parts in the proportionsk:(1−k). These are respectively an inductance producing the mutual flux and an inductance producing the leakage flux.
Coupling coefficient is a function of the geometry of the system. It is fixed by the positional relationship between the two coils. The coupling coefficient does not change between when the system is in the resonance state and when it is not in the resonance state, or even if the system is in resonance state and a secondary voltage larger than the turns ratio is generated. However, in the resonance case, the flux ratio changes and the mutual flux increases.
Resonant systems are said to be tightly coupled, loosely coupled, critically coupled or overcoupled. Tight coupling is when the coupling coefficient is around 1 as with conventional iron-core transformers. Overcoupling is when the secondary coil is so close and the formation of mutual flux is hindered by the effect of antiresonance, and critical coupling is when the transfer in the passband is optimal. Loose coupling is when the coils are distant from each other, so that most of the flux misses the secondary. In Tesla coils around 0.2 is used, and at greater distances, for example for inductive wireless power transmission, it may be lower than 0.01.
Generally the voltage gain of non resonantly coupled coils is directly proportional to the square root of the ratio of secondary and primary inductances.
However, if in the state of resonant coupling, higher voltage is generated. Theshort-circuit inductanceLsc2on the secondary side can be obtained by the following formula.
The short-circuit inductance Lsc2and the resonance capacitor Cr on the secondary side resonate. The resonance frequency ω2is as follows.
Assuming that the load resistance is Rl, the Q value of the secondary resonance circuit is as follows.
The voltage generated in the resonance capacitor Cr at the peak of the resonance frequency is proportional to the Q value. Therefore, the voltage gain Ar of the secondary coil with respect to the primary coil when the system is resonating,
In the case of the Type P-P, Q1 does not contribute to the voltage gain.
TheWiTricitytype magnetic resonance is characterized in that the resonant coils on the primary side and the resonant coils on the secondary side are paired. The primary resonant coil increases the primary driving coil current and increases the generated magnetic flux around the primary resonator. This is equivalent to driving the primary coil at high voltage. In the case of the type on the left figure, the general principle is that if a given oscillating amount of energy (for example a pulse or a series of pulses) is placed into a primary coil which is capacitively loaded, the coil will 'ring', and form an oscillating magnetic field.
Resonant transfer works by making a coilringwith an oscillating current. This generates an oscillatingmagnetic field. Because the coil is highly resonant, any energy placed in the coil dies away relatively slowly over very many cycles; but if a second coil is brought near it, the coil can pick up most of the energy before it is lost, even if it is some distance away. The fields used are predominantly non-radiative,near fields(sometimes calledevanescent waves), as all hardware is kept well within the 1/4 wavelength distance they radiate little energy from the transmitter to infinity.
The energy will transfer back and forth between the magnetic field in the inductor and the electric field across the capacitor at the resonant frequency. This oscillation will die away at a rate determined by the gain-bandwidth (Qfactor), mainly due to resistive and radiative losses. However, provided the secondary coil cuts enough of the field that it absorbs more energy than is lost in each cycle of the primary, then most of the energy can still be transferred.
Because theQfactor can be very high, (experimentally around a thousand has been demonstrated[38]with aircoredcoils) only a small percentage of the field has to be coupled from one coil to the other to achieve high efficiency, even though the field dies quickly with distance from a coil, the primary and secondary can be several diameters apart.
It can be shown that a figure of merit for the efficiency is:[39]
WhereQ1andQ2are the Q factors of the source and receiver coils respectively, andkis the coupling coefficient described above.
And the maximum achievable efficiency is:[39]
Because theQcan be very high, even when low power is fed into the transmitter coil, a relatively intense field builds up over multiple cycles, which increases the power that can be received—at resonance far more power is in the oscillating field than is being fed into the coil, and the receiver coil receives a percentage of that.
Unlike the multiple-layer secondary of a non-resonant transformer, coils for this purpose are often single layersolenoids(to minimiseskin effectand give improvedQ) in parallel with a suitablecapacitor. Alternative resonator geometries include wave-woundLitz wireandloop-gap resonators (LGRs). In the Litz wire-based resonators, insulation is either absent or lowpermittivityand low-loss materials such assilkare used to minimise dielectric losses.[citation needed]The LGR geometries have the advantage that electric fields outside the resonator structure are very weak which minimizes human exposure to electric fields and makes the power transfer efficiency insensitive to nearby dielectrics.[40]
To progressively feed energy into the primary coil with each cycle, different circuits can be used. One circuit employs aColpitts oscillator.[38]
In Tesla coils an intermittent switching system, a "circuit controller" or "break," is used to inject an impulsive signal into the primary coil; the secondary coil then rings and decays.[citation needed]
The secondary receiver coils are similar designs to the primary sending coils. Running the secondary at the same resonant frequency as the primary ensures that the secondary has a lowimpedanceat the transmitter's frequency and that the energy is optimally absorbed.
To remove energy from the secondary coil, different methods can be used, the AC can be used directly orrectifiedand a regulator circuit can be used to generate DC voltage.
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https://en.wikipedia.org/wiki/Resonant_inductive_coupling
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RFDumpis a software created by Lukas Grunwald and Christian Bottger for security auditingRFIDtags. It is periodically updated to support emerging RFID standards, such ase-passportandMifareencryption, that are currently found on many pay-as-you-go systems.
RFDump is a back-end GPL tool that interoperates directly with any RFID reader to make the contents stored on RFID tags accessible. The tool reads an RFID tag's meta information: tag ID, tag type, manufacturer, etc. The user data of a tag can be displayed and modified using either ahexor anASCIIeditor. Thecookiefeature demonstrates how simple it is to abuse RFID technology, such as companies using it to spy on consumers. RFDump works with the ACG Multi-Tag Reader or similar card reader hardware.
This security software article is astub. You can help Wikipedia byexpanding it.
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https://en.wikipedia.org/wiki/RFdump
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