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70-403: Unlike many other provably secure hash functions, the algorithm is quite fast, yielding a throughput of 40 Mbit/s on a 3.2 GHz Intel Pentium 4. Although SWIFFT satisfies many desirable cryptographic and statistical properties, it was not designed to be an "all-purpose" cryptographic hash function. For example, it is not a pseudorandom function , and would not be a suitable instantiation of

140-505: A ∈ F ( X − a ) . {\displaystyle X^{q}-X=\prod _{a\in F}(X-a).} It follows that GF( p ) contains a subfield isomorphic to GF( p ) if and only if m is a divisor of n ; in that case, this subfield is unique. In fact, the polynomial X − X divides X − X if and only if m is a divisor of n . Given a prime power q = p with p prime and n > 1 ,

210-473: A + b α + c α 2 ) = − a + ( − b ) α + ( − c ) α 2 (for  G F ( 8 ) , this operation is the identity) ( a + b α + c α 2 ) + ( d + e α + f α 2 ) = (

280-447: A + b α + c α 2 + d α 3 ) ( e + f α + g α 2 + h α 3 ) = ( a e + b h + c g + d f ) + ( a f + b e + b h + c g + d f + c h + d g ) α + (

350-442: A + d ) + ( b + e ) α + ( c + f ) α 2 ( a + b α + c α 2 ) ( d + e α + f α 2 ) = ( a d + b f + c e ) + ( a e + b d + b f + c e + c f ) α + (

420-796: A c + r b d ) + ( a d + b c ) α ( a + b α ) − 1 = a ( a 2 − r b 2 ) − 1 + ( − b ) ( a 2 − r b 2 ) − 1 α {\displaystyle {\begin{aligned}-(a+b\alpha )&=-a+(-b)\alpha \\(a+b\alpha )+(c+d\alpha )&=(a+c)+(b+d)\alpha \\(a+b\alpha )(c+d\alpha )&=(ac+rbd)+(ad+bc)\alpha \\(a+b\alpha )^{-1}&=a(a^{2}-rb^{2})^{-1}+(-b)(a^{2}-rb^{2})^{-1}\alpha \end{aligned}}} The polynomial X 3 − X − 1 {\displaystyle X^{3}-X-1}

490-598: A f + b e + c d + c f ) α 2 {\displaystyle {\begin{aligned}-(a+b\alpha +c\alpha ^{2})&=-a+(-b)\alpha +(-c)\alpha ^{2}\qquad {\text{(for }}\mathrm {GF} (8),{\text{this operation is the identity)}}\\(a+b\alpha +c\alpha ^{2})+(d+e\alpha +f\alpha ^{2})&=(a+d)+(b+e)\alpha +(c+f)\alpha ^{2}\\(a+b\alpha +c\alpha ^{2})(d+e\alpha +f\alpha ^{2})&=(ad+bf+ce)+(ae+bd+bf+ce+cf)\alpha +(af+be+cd+cf)\alpha ^{2}\end{aligned}}} The polynomial X 4 + X + 1 {\displaystyle X^{4}+X+1}

560-779: A g + b f + c e + c h + d g + d h ) α 2 + ( a h + b g + c f + d e + d h ) α 3 {\displaystyle {\begin{aligned}(a+b\alpha +c\alpha ^{2}+d\alpha ^{3})+(e+f\alpha +g\alpha ^{2}+h\alpha ^{3})&=(a+e)+(b+f)\alpha +(c+g)\alpha ^{2}+(d+h)\alpha ^{3}\\(a+b\alpha +c\alpha ^{2}+d\alpha ^{3})(e+f\alpha +g\alpha ^{2}+h\alpha ^{3})&=(ae+bh+cg+df)+(af+be+bh+cg+df+ch+dg)\alpha \;+\\&\quad \;(ag+bf+ce+ch+dg+dh)\alpha ^{2}+(ah+bg+cf+de+dh)\alpha ^{3}\end{aligned}}} The field GF(16) has eight primitive elements (the elements that have all nonzero elements of GF(16) as integer powers). These elements are

630-433: A GF( p ) - vector space . It follows that the number of elements of F is p for some integer n . The identity ( x + y ) p = x p + y p {\displaystyle (x+y)^{p}=x^{p}+y^{p}} (sometimes called the freshman's dream ) is true in a field of characteristic p . This follows from the binomial theorem , as each binomial coefficient of

700-404: A random oracle . The algorithm is less efficient than most traditional hash functions that do not give a proof of their collision-resistance. Therefore, its practical use would lie mostly in applications where the proof of collision-resistance is particularly valuable, such as digital signatures that must remain trustworthy for a long time. A modification of SWIFFT called SWIFFTX was proposed as

770-525: A candidate for SHA-3 function to the NIST hash function competition and was rejected in the first round. The algorithm is as follows: The FFT operation in step 4 is easy to invert, and is performed to achieve diffusion , that is, to mix the input bits. The linear combination in step 6 achieves confusion , since it compresses the input. This is just a high level description of what the algorithm does, some more advanced optimizations are used to finally yield

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840-429: A finite field. For any element x in F and any integer n , denote by n ⋅ x the sum of n copies of x . The least positive n such that n ⋅ 1 = 0 is the characteristic p of the field. This allows defining a multiplication ( k , x ) ↦ k ⋅ x of an element k of GF( p ) by an element x of F by choosing an integer representative for k . This multiplication makes F into

910-460: A function chosen randomly from the PRF family and a random oracle (a function whose outputs are fixed completely at random). Pseudorandom functions are vital tools in the construction of cryptographic primitives , especially secure encryption schemes . Pseudorandom functions are not to be confused with pseudorandom generators (PRGs). The guarantee of a PRG is that a single output appears random if

980-402: A high performing algorithm. Assuming the parameters ( n , m , p ) = (64, 16, 257) , any fixed compression function in the family takes a binary input of length mn = 1024 bits (128 bytes) to an output in the range ℤ p , which has size p = 257 . An output in ℤ p can easily be represented using 528 bits (66 bytes). The SWIFFT functions can be described as

1050-475: A lattice over ℤ p [α]/(α + 1) . At the moment, the fastest algorithms for finding short vectors are all exponential in n . Note that this ensures that there is no significant set of "weak instances" where the security of SWIFFT is weak. This guarantee is not given by most other provably secure hash functions. Known working attacks are the generalized birthday attack , which takes 2 operations, and inversion attacks which takes 2 operations for

1120-419: A lookup table filled with uniformly distributed random entries. However, in practice, a PRF is given an input string in the domain and a hidden random seed and runs multiple times with the same input string and seed, always returning the same value. Nonetheless, given an arbitrary input string, the output looks random if the seed is taken from a uniform distribution. A PRF is considered to be good if its behavior

1190-408: A non-zero element may be computed with the extended Euclidean algorithm; see Extended Euclidean algorithm § Simple algebraic field extensions . However, with this representation, elements of GF( q ) may be difficult to distinguish from the corresponding polynomials. Therefore, it is common to give a name, commonly α to the element of GF( q ) that corresponds to the polynomial X . So,

1260-481: A polynomial is given by Conway polynomials . They ensure a certain compatibility between the representation of a field and the representations of its subfields. In the next sections, we will show how the general construction method outlined above works for small finite fields. The smallest non-prime field is the field with four elements, which is commonly denoted GF(4) or F 4 . {\displaystyle \mathbb {F} _{4}.} It consists of

1330-507: A polynomial of degree < n having coefficients in ℤ p . A certain function in the SWIFFT family is specified by m fixed elements a 1 , …, a m ∈ R of the ring R {\displaystyle R} , that are called multipliers. The function corresponds to the following formula over R : The x 1 , …, x m ∈ R are polynomials with binary coefficients, and corresponding to

1400-537: A polynomial of degree < 2 n using an inverse fast Fourier transform. Instead of the normal Fourier transform, SWIFFT uses the number-theoretic transform . The number-theoretic transform uses roots of unity in ℤ p instead of complex roots of unity. In order for this to work, ℤ p needs to be a finite field and primitive 2 n roots of unity need to exist in this field. This can be done by taking p prime such that 2 n divides p − 1 . The parameters m , p , and n are subject to

1470-438: A prime number and k is a positive integer). In a field of order p , adding p copies of any element always results in zero; that is, the characteristic of the field is p . For q = p , all fields of order q are isomorphic (see § Existence and uniqueness below). Moreover, a field cannot contain two different finite subfields with the same order. One may therefore identify all finite fields with

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1540-432: A pseudorandom function is needed, they do not, in general, constitute a pseudorandom function family, as block ciphers such as AES are defined for only limited numbers of input and key sizes. A PRF is an efficient (i.e. computable in polynomial time), deterministic function that maps two distinct sets (domain and range) and looks like a truly random function. Essentially, a truly random function would just be composed of

1610-493: A simple algebraic expression over some polynomial ring R . A family of these functions depends on three main parameters: let n be a power of 2, let m > 0 be a small integer, and let p > 0 be a modulus (not necessarily prime , but is convenient to choose it prime). Define R to be the ring R = ℤ p [α]/(α + 1) , i.e., the ring of polynomials in α having integer coefficients, modulo p and α + 1 . An element of R can be written as

1680-410: A standard parameter choice. This is usually considered to be enough to render an attack by an adversary infeasible. Pseudorandom function In cryptography , a pseudorandom function family , abbreviated PRF , is a collection of efficiently-computable functions which emulate a random oracle in the following way: no efficient algorithm can distinguish (with significant advantage ) between

1750-485: A symbol that has the property α = r , in the same way that the complex number i is a symbolic square root of −1 . Then, the elements of GF( p ) are all the linear expressions a + b α , {\displaystyle a+b\alpha ,} with a and b in GF( p ) . The operations on GF( p ) are defined as follows (the operations between elements of GF( p ) represented by Latin letters are

1820-505: Is cyclic , so all non-zero elements can be expressed as powers of a single element called a primitive element of the field. (In general there will be several primitive elements for a given field.) The simplest examples of finite fields are the fields of prime order: for each prime number p , the prime field of order p may be constructed as the integers modulo p , Z / p Z {\displaystyle \mathbb {Z} /p\mathbb {Z} } . The elements of

1890-421: Is pseudorandom if the following conditions are satisfied: In an oblivious pseudorandom function, abbreviated OPRF, information is concealed from two parties that are involved in a PRF. That is, if Alice cryptographically hashes her secret value, cryptographically blinds the hash to produce the message she sends to Bob, and Bob mixes in his secret value and gives the result back to Alice, who unblinds it to get

1960-399: Is a finite set that is a field ; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms . The number of elements of a finite field is called its order or, sometimes, its size . A finite field of order q exists if and only if q is a prime power p (where p is

2030-407: Is a quadratic non-residue for p = 3, 5, 11, 13, ... , and 3 is a quadratic non-residue for p = 5, 7, 17, ... . If p ≡ 3 mod 4 , that is p = 3, 7, 11, 19, ... , one may choose −1 ≡ p − 1 as a quadratic non-residue, which allows us to have a very simple irreducible polynomial X + 1 . Having chosen a quadratic non-residue r , let α be a symbolic square root of r , that is,

2100-493: Is a symbol such that α 3 = α + 1. {\displaystyle \alpha ^{3}=\alpha +1.} The addition, additive inverse and multiplication on GF(8) and GF(27) may thus be defined as follows; in following formulas, the operations between elements of GF(2) or GF(3) , represented by Latin letters, are the operations in GF(2) or GF(3) , respectively: − (

2170-526: Is a symbol such that α 4 = α + 1 {\displaystyle \alpha ^{4}=\alpha +1} (that is, α is defined as a root of the given irreducible polynomial). As the characteristic of GF(2) is 2 , each element is its additive inverse in GF(16) . The addition and multiplication on GF(16) may be defined as follows; in following formulas, the operations between elements of GF(2) , represented by Latin letters are

SWIFFT - Misplaced Pages Continue

2240-442: Is allowed that the algorithm only works in a small but noticeable fraction of the SWIFFT family. Then we can find also an algorithm f 2 which can always find a short vector in any ideal lattice over the ring ℤ p [α]/(α + 1) in some feasible time T 2 , depending on T and p . This means that finding collisions in SWIFFT is at least as difficult as the worst-case scenario of finding short vectors in

2310-404: Is an odd prime, there are always irreducible polynomials of the form X − r , with r in GF( p ) . More precisely, the polynomial X − r is irreducible over GF( p ) if and only if r is a quadratic non-residue modulo p (this is almost the definition of a quadratic non-residue). There are ⁠ p − 1 / 2 ⁠ quadratic non-residues modulo p . For example, 2

2380-576: Is called the discrete logarithm of x to the base a . While a can be computed very quickly, for example using exponentiation by squaring , there is no known efficient algorithm for computing the inverse operation, the discrete logarithm. This has been used in various cryptographic protocols , see Discrete logarithm for details. When the nonzero elements of GF( q ) are represented by their discrete logarithms, multiplication and division are easy, as they reduce to addition and subtraction modulo q – 1 . However, addition amounts to computing

2450-420: Is indistinguishable from a truly random function. Therefore, given an output from either the truly random function or a PRF, there should be no efficient method to correctly determine whether the output was produced by the truly random function or the PRF. Pseudorandom functions take inputs x ∈ { 0 , 1 } ∗ {\displaystyle x\in \{0,1\}^{*}} . Both

2520-521: Is irreducible over GF(2) and GF(3) , that is, it is irreducible modulo 2 and 3 (to show this, it suffices to show that it has no root in GF(2) nor in GF(3) ). It follows that the elements of GF(8) and GF(27) may be represented by expressions a + b α + c α 2 , {\displaystyle a+b\alpha +c\alpha ^{2},} where a , b , c are elements of GF(2) or GF(3) (respectively), and α

2590-405: Is irreducible over GF(2) , that is, it is irreducible modulo 2 . It follows that the elements of GF(16) may be represented by expressions a + b α + c α 2 + d α 3 , {\displaystyle a+b\alpha +c\alpha ^{2}+d\alpha ^{3},} where a , b , c , d are either 0 or 1 (elements of GF(2) ), and α

2660-434: Is not unique. The number of primitive elements is φ ( q − 1) where φ is Euler's totient function . The result above implies that x = x for every x in GF( q ) . The particular case where q is prime is Fermat's little theorem . If a is a primitive element in GF( q ) , then for any non-zero element x in F , there is a unique integer n with 0 ≤ n ≤ q − 2 such that This integer n

2730-403: Is reducible, it is recommended to choose X + X + 1 with the lowest possible k that makes the polynomial irreducible. If all these trinomials are reducible, one chooses "pentanomials" X + X + X + X + 1 , as polynomials of degree greater than 1 , with an even number of terms, are never irreducible in characteristic 2 , having 1 as a root. A possible choice for such

2800-458: Is the non-trivial field automorphism, called the Frobenius automorphism , which sends α into the second root 1 + α of the above-mentioned irreducible polynomial X + X + 1 . For applying the above general construction of finite fields in the case of GF( p ) , one has to find an irreducible polynomial of degree 2. For p = 2 , this has been done in the preceding section. If p

2870-676: Is used in the Password Monitor functionality in Microsoft Edge . See the main article on Oblivious Pseudorandom Functions . PRFs can be used for: Finite field In mathematics , a finite field or Galois field (so-named in honor of Évariste Galois ) is a field that contains a finite number of elements . As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are

SWIFFT - Misplaced Pages Continue

2940-523: The integers mod p when p is a prime number . The order of a finite field is its number of elements, which is either a prime number or a prime power . For every prime number p and every positive integer k there are fields of order p , all of which are isomorphic . Finite fields are fundamental in a number of areas of mathematics and computer science , including number theory , algebraic geometry , Galois theory , finite geometry , cryptography and coding theory . A finite field

3010-480: The Euclidean division, one commonly chooses for P a polynomial of the form X n + a X + b , {\displaystyle X^{n}+aX+b,} which make the needed Euclidean divisions very efficient. However, for some fields, typically in characteristic 2 , irreducible polynomials of the form X + aX + b may not exist. In characteristic 2 , if the polynomial X + X + 1

3080-475: The binary input of length mn . To compute the above expression, the main problem is to compute the polynomial products a i ⋅ x i . A fast way to compute these products is given by the convolution theorem . This says that under certain restrictions, F { f ∗ g } = F { f } ⋅ F { g } , where F denotes the Fourier transform and ⋅ denotes the pointwise product. In

3150-463: The construction of the preceding section must involve this polynomial, and G F ( 4 ) = G F ( 2 ) [ X ] / ( X 2 + X + 1 ) . {\displaystyle \mathrm {GF} (4)=\mathrm {GF} (2)[X]/(X^{2}+X+1).} Let α denote a root of this polynomial in GF(4) . This implies that and that α and 1 + α are

3220-526: The discrete logarithm of a + a . The identity allows one to solve this problem by constructing the table of the discrete logarithms of a + 1 , called Zech's logarithms , for n = 0, ..., q − 2 (it is convenient to define the discrete logarithm of zero as being −∞ ). Zech's logarithms are useful for large computations, such as linear algebra over medium-sized fields, that is, fields that are sufficiently large for making natural algorithms inefficient, but not too large, as one has to pre-compute

3290-451: The elements of GF( q ) become polynomials in α , where P ( α ) = 0 , and, when one encounters a polynomial in α of degree greater or equal to n (for example after a multiplication), one knows that one has to use the relation P ( α ) = 0 to reduce its degree (it is what Euclidean division is doing). Except in the construction of GF(4) , there are several possible choices for P , which produce isomorphic results. To simplify

3360-400: The elements of GF(4) that are not in GF(2) . The tables of the operations in GF(4) result from this, and are as follows: A table for subtraction is not given, because subtraction is identical to addition, as is the case for every field of characteristic 2. In the third table, for the division of x by y , the values of x must be read in the left column, and the values of y in

3430-404: The equation x = 1 has at most k solutions in any field, q – 1 is the lowest possible value for k . The structure theorem of finite abelian groups implies that this multiplicative group is cyclic , that is, all non-zero elements are powers of a single element. In summary: Such an element a is called a primitive element of GF( q ) . Unless q = 2, 3 , the primitive element

3500-539: The expansion of ( x + y ) , except the first and the last, is a multiple of p . By Fermat's little theorem , if p is a prime number and x is in the field GF( p ) then x = x . This implies the equality X p − X = ∏ a ∈ G F ( p ) ( X − a ) {\displaystyle X^{p}-X=\prod _{a\in \mathrm {GF} (p)}(X-a)} for polynomials over GF( p ) . More generally, every element in GF( p ) satisfies

3570-415: The field GF( q ) may be explicitly constructed in the following way. One first chooses an irreducible polynomial P in GF( p )[ X ] of degree n (such an irreducible polynomial always exists). Then the quotient ring G F ( q ) = G F ( p ) [ X ] / ( P ) {\displaystyle \mathrm {GF} (q)=\mathrm {GF} (p)[X]/(P)} of

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3640-456: The final output, Bob is not able to see either Alice's secret value or the final output, and Alice is not able to see Bob's secret input, but Alice sees the final output which is a PRF of the two inputs -- a PRF of Alice's secret and Bob's secret. This enables transactions of sensitive cryptographic information to be secure even between untrusted parties. An OPRF is used in some implementations of password-authenticated key agreement . An OPRF

3710-457: The following classification theorem first proved in 1893 by E. H. Moore : The order of a finite field is a prime power. For every prime power q there are fields of order q , and they are all isomorphic. In these fields, every element satisfies x q = x , {\displaystyle x^{q}=x,} and the polynomial X − X factors as X q − X = ∏

3780-460: The following restrictions: A possible choice is ( n , m , p ) = (64,16,257) , which achieves a throughput of about 40 Mbit/s, security of about 2 operations for finding collisions, and a digest size of 512 bits. SWIFFT is an example of a provably secure cryptographic hash function . As with most security proofs, the security proof of SWIFFT relies on a reduction to a certain difficult-to-solve mathematical problem. Note that this means that

3850-545: The four elements 0, 1, α , 1 + α such that α = 1 + α , 1 ⋅ α = α ⋅ 1 = α , x + x = 0 , and x ⋅ 0 = 0 ⋅ x = 0 , for every x ∈ GF(4) , the other operation results being easily deduced from the distributive law . See below for the complete operation tables. This may be deduced as follows from the results of the preceding section. Over GF(2) , there is only one irreducible polynomial of degree 2 : X 2 + X + 1 {\displaystyle X^{2}+X+1} Therefore, for GF(4)

3920-477: The four roots of X + X + 1 and their multiplicative inverses . In particular, α is a primitive element, and the primitive elements are α with m less than and coprime with 15 (that is, 1, 2, 4, 7, 8, 11, 13, 14). The set of non-zero elements in GF( q ) is an abelian group under the multiplication, of order q – 1 . By Lagrange's theorem , there exists a divisor k of q – 1 such that x = 1 for every non-zero x in GF( q ) . As

3990-427: The general case of the convolution theorem, ∗ does not denote multiplication but convolution . It can, however, be shown that polynomial multiplication is a convolution. The fast Fourier transform is used to find the Fourier coefficients of each polynomial, which are then multiplied point-wise with the respective Fourier coefficients of the other polynomial. The resulting coefficients are then transformed to

4060-580: The input size I = | x | {\displaystyle I=|x|} and output size λ {\displaystyle \lambda } depend only on the index size n := | s | {\displaystyle n:=|s|} . A family of functions, f s : { 0 , 1 } I ( n ) → { 0 , 1 } λ ( n ) {\displaystyle f_{s}:\left\{0,1\right\}^{I(n)}\rightarrow \left\{0,1\right\}^{\lambda (n)}}

4130-484: The input was chosen at random. On the other hand, the guarantee of a PRF is that all its outputs appear random, regardless of how the corresponding inputs were chosen, as long as the function was drawn at random from the PRF family. A pseudorandom function family can be constructed from any pseudorandom generator, using, for example, the "GGM" construction given by Goldreich , Goldwasser , and Micali . While in practice, block ciphers are used in most instances where

4200-493: The multiplicative inverse of a root of P . In other words, the roots of P form a field of order q , which is equal to F by the minimality of the splitting field. The uniqueness up to isomorphism of splitting fields implies thus that all fields of order q are isomorphic. Also, if a field F has a field of order q = p as a subfield, its elements are the q roots of X − X , and F cannot contain another subfield of order q . In summary, we have

4270-444: The operations in GF( p ) ): − ( a + b α ) = − a + ( − b ) α ( a + b α ) + ( c + d α ) = ( a + c ) + ( b + d ) α ( a + b α ) ( c + d α ) = (

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4340-472: The operations in GF(2) . ( a + b α + c α 2 + d α 3 ) + ( e + f α + g α 2 + h α 3 ) = ( a + e ) + ( b + f ) α + ( c + g ) α 2 + ( d + h ) α 3 (

4410-441: The other axioms of a field, but whose multiplication is not required to be commutative, is called a division ring (or sometimes skew field ). By Wedderburn's little theorem , any finite division ring is commutative, and hence is a finite field. Let q = p be a prime power , and F be the splitting field of the polynomial P = X q − X {\displaystyle P=X^{q}-X} over

4480-407: The polynomial equation x − x = 0 . Any finite field extension of a finite field is separable and simple. That is, if E is a finite field and F is a subfield of E , then E is obtained from F by adjoining a single element whose minimal polynomial is separable . To use a piece of jargon, finite fields are perfect . A more general algebraic structure that satisfies all

4550-491: The polynomial ring GF( p )[ X ] by the ideal generated by P is a field of order q . More explicitly, the elements of GF( q ) are the polynomials over GF( p ) whose degree is strictly less than n . The addition and the subtraction are those of polynomials over GF( p ) . The product of two elements is the remainder of the Euclidean division by P of the product in GF( p )[ X ] . The multiplicative inverse of

4620-417: The prime field GF( p ) . This means that F is a finite field of lowest order, in which P has q distinct roots (the formal derivative of P is P ′ = −1 , implying that gcd( P , P ′ ) = 1 , which in general implies that the splitting field is a separable extension of the original). The above identity shows that the sum and the product of two roots of P are roots of P , as well as

4690-410: The prime field of order p may be represented by integers in the range 0, ..., p − 1 . The sum, the difference and the product are the remainder of the division by p of the result of the corresponding integer operation. The multiplicative inverse of an element may be computed by using the extended Euclidean algorithm (see Extended Euclidean algorithm § Modular integers ). Let F be

4760-407: The same order, and they are unambiguously denoted F q {\displaystyle \mathbb {F} _{q}} , F q or GF( q ) , where the letters GF stand for "Galois field". In a finite field of order q , the polynomial X − X has all q elements of the finite field as roots . The non-zero elements of a finite field form a multiplicative group . This group

4830-403: The security of SWIFFT relies strongly on the difficulty of this mathematical problem. The reduction in the case of SWIFFT is to the problem of finding short vectors in cyclic/ideal lattices. It can be proven that the following holds: Suppose we have an algorithm that, for a random version of SWIFFT given by f , can find collisions in f within some feasible time T , and with probability p . It

4900-471: The top row. (Because 0 ⋅ z = 0 for every z in every ring the division by 0 has to remain undefined.) From the tables, it can be seen that the additive structure of GF(4) is isomorphic to the Klein four-group , while the non-zero multiplicative structure is isomorphic to the group Z 3 . The map φ : x ↦ x 2 {\displaystyle \varphi :x\mapsto x^{2}}

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