Mathematics for Cryptography


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1 Mathematics for Cryptography Douglas R. Stinson David R. Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, N2L 3G1, Canada March 15, Groups and Modular Arithmetic 1.1 Modular Arithmetic 1.1 Definition (congruences). Suppose a and b are integers, and m is a positive integer. Then we write a b (mod m) if m divides b a. The phrase a b (mod m) is called a congruence, and it is read as a is congruent to b modulo m. The integer m is called the modulus. 1.2 Definition (modular reduction). Suppose we divide a and b by m, obtaining integer quotients and remainders, where the remainders are between 0 and m 1. That is, a = q 1 m + r 1 and b = q 2 m + r 2, where 0 r 1 m 1 and 0 r 2 m 1. Then it is not difficult to see that a b (mod m) if and only if r 1 = r 2. We will use the notation a mod m (without parentheses) to denote the remainder when a is divided by m, i.e., the value r 1 above. Thus a b (mod m) if and only if a mod m = b mod m. If we replace a by a mod m, we say that a is reduced modulo m. 1.3 Example. To compute 101 mod 7, we write 101 = Since 0 3 6, it follows that 101 mod 7 = 3. As another example, suppose we want to compute ( 101) mod 7. In this case, we write 101 = 7 ( 15) + 4. Since 0 4 6, it follows that ( 101) mod 7 = Remark. Some computer programming languages define a mod m to be the remainder in the range m + 1,..., m 1 having the same sign as a. For example, ( 101) mod 7 would be 3, rather than 4 as we defined it above. But for our purposes, it is much more convenient to define a mod m always to be nonnegative. 1.5 Definition (arithmetic modulo m). We now define arithmetic modulo m: Z m is the set {0,..., m 1}, equipped with two operations, + and. Addition and multiplication in Z m work exactly like real addition and multiplication, except that the results are reduced modulo m. 1.6 Example. Suppose we want to compute in Z 16. As integers, we have = 143. Then we reduce 143 modulo 16 as described above: 143 = , so 143 mod 16 = 15, and hence = 15 in Z 16. 1
2 1.2 Groups 1.7 Definition (group). A group is a pair G = (X, ), where X is a set and is a binary operation defined on X, that satisfies the following properties: The operation is associative, i.e., (a b) c = a (b c) for any a, b, c X. There is an element id X called the identity, such that a id = id a = a for any a X. For every a X, there exists an element b X called the inverse of a, such that a b = b a = id. 1.8 Definition. A group G = (X, ) is abelian if the the operation is commutative, i.e., a b = b a for any a, b X. 1.9 Definition. A group G = (X, ) is finite if X is a finite set Definition. The order of a finite group G = (X, ), denoted ord(g), is equal to X Remark. For notational convenience, most group operations are written as multiplication or addition. If the group operation is multiplication, then the identity is usually denoted by 1 and the inverse of a by a 1. If the group operation is addition, then the identity is usually denoted by 0 and the inverse of a by a Example (the additive group Z n ). Let n 2 be an integer. Then (Z n, +) is a finite abelian group of order n, where + denotes addition modulo n. The identity element is 0, and the inverse of a, usually denoted a, is ( a) mod n Example (the multiplicative group Z p ). Let p 2 be a prime. Define Z p = Z p \{0}. Then (Z p, ) is a finite abelian group of order p 1, where denotes multiplication modulo p. The identity element is 1, and the inverse of a, usually denoted a 1, is computed using the Extended Euclidean algorithm (see Theorem 1.52) Definition. For an integer n 2, φ(n) denotes the number of positive integers less than n that are relatively prime to n Theorem. φ(n) can be computed from the following formula: suppose that n has prime power factorization l e n = p i i (i.e., the p i s are distinct primes and e i 1 for 1 i l). Then φ(n) = i=1 l e p i 1 i (p i 1) = i=1 l ( e pi i e p i 1 ) i Example (the multiplicative group Z n ). This example generalizes Example Let n 2 be an integer. Define Z n = Z n \{d Z n : gcd(d, n) > 1}. Then (Z n, ) is a finite abelian group where denotes multiplication modulo n. The identity element is 1, and the inverse of a, usually denoted a 1, is computed using the Extended Euclidean algorithm (see Theorem 1.52). The order of (Z n, ) is equal to φ(n). 2 i=1
3 1.17 Example. The RSA Cryptosystem is constructed using the group Z n, where n = pq and p and q are distinct odd primes. For such an integer n, the order of (Z n, ) is equal to (p 1)(q 1) Example (matrices with nonzero determinant). Let n 2. The set of n n matrices with entries from Z p (where p is prime) having nonzero determinant is a multiplicative group. The identity is the n n matrix with 1s on the diagonal and 0s elsewhere. This is a nonabelian group, since matrix multiplication is not commutative Example (elliptic curves). Let p > 3 be prime. An elliptic curve is the set of solutions (x, y) Z p Z p to the congruence y 2 x 3 + ax + b (mod p), where a, b Z p are constants such that 4a 3 +27b 2 0 (mod p), together with a special point O called the point at infinity. Suppose we denote the set of points on the elliptic curve by E. It is possible to define an addition operation on E so that (E, +) is an abelian group. Addition is defined as follows (where all arithmetic operations are performed in Z p ): Suppose P = (x 1, y 1 ) and Q = (x 2, y 2 ) are points on E. If x 2 = x 1 and y 2 = y 1, then P + Q = O; otherwise P + Q = (x 3, y 3 ), where and x 3 = λ 2 x 1 x 2 and y 3 = λ(x 1 x 3 ) y 1, λ = Finally, define P + O = O + P = P for all P E. { (y 2 y 1 )(x 2 x 1 ) 1, if P Q (3x a)(2y 1 ) 1, if P = Q. 1.3 Orders of Group Elements 1.20 Definition (orders of group elements). For a finite group (X, ), define the order of an element a X (denoted ord(a)) to be the smallest positive integer m such that a} a {{ a} = id. m If the group operation is written multiplicatively, then a a a }{{} m is written as an exponentiation, a m. If the group operation is written additively, then the same expression is written as a multiplication, ma. The identity element is defined to have order Theorem. For a finite group (X, ), the order of any a X divides the order of the group, i.e., ord(a) ord(g) Theorem. For a finite group (X, ) and for any a X, the order of b = a i is ord(b) = ord(a) gcd(ord(a), i). (Here, for concreteness, we assume that the group operation is written multiplicatively.) 1.23 Example. If ord(a) = 100 and b = a 35, then ord(b) = 100 gcd(100, 35) = = Theorem. If ord(a) = i, then a 1 = a i 1. More generally, a i = a j if and only if i j (mod ord(a)). 3
4 1.4 Cyclic Groups and Primitive Elements 1.25 Definition (cyclic group). A finite abelian group (X, ) is a cyclic group if there exists an element a X having order equal to X. Such an element is called a generator of the group Example. Let n 2 be an integer. Then (Z n, +) is a cyclic group, and 1 is a generator. Further, an element a Z n is a generator of (Z n, +) if and only if gcd(a, n) = 1. The number of generators of (Z n, +) is φ(n) Example. Let p 2 be a prime. Then (Z p, ) is cyclic group, and a generator of this group is called a primitive element Theorem. (Z n, ) is cyclic group if and only if n = 2, 4, p k or 2p k, where p is an odd prime and k is a positive integer Theorem. α Z p is a primitive element if and only if for all primes q such that q (p 1). α (p 1)/q 1 (mod p) 1.30 Remark. Using Theorem 1.29, it is simple to test whether a given element α Z p is a primitive element (where p is an odd prime) provided that the factorization of p 1 is known Example. Suppose p = 13 and α = 2. The factorization of 12 into prime powers is 12 = Therefore, to verify that 2 is a primitive element modulo 13, it is sufficient to check that (mod 13) and (mod 13). This is much faster than checking all 12 powers of α Theorem. The number of generators of (Z p, ) is φ(p 1) = φ(φ(p)). 1.5 Subgroups and Cosets 1.33 Definition (subgroup). Suppose G = (X, ) is a finite group and Y X. We say that H = (Y, ) is a subgroup of G if H is also a (finite) group Theorem. Suppose G = (X, ) is a finite group and Y X. Then H = (Y, ) is a subgroup of G if and only if it is closed Definition (coset). Suppose H = (Y, ) is a subgroup of the group G = (X, ). For any a X, define the right coset Y a as follows: Also, define the left coset ay as follows: Y a = {y a : y Y }. ay = {a y : y Y } Theorem. Suppose H = (Y, ) is a subgroup of G = (X, ). Then, Y a = Y for all a. Furthermore, two right cosets Y a and Y a (or two left cosets ay and a Y ) are either identical or disjoint Corollary. A group X can be partitioned into right (or left) cosets of any subgroup Y. 4
5 1.38 Theorem (Lagrange s Theorem). Suppose H = (Y, ) is a subgroup of the finite group G = (X, ). Then ord(h) divides ord(g) Definition. Suppose that G = (X, ) is a finite group and y X. Define a = {a i : i 0} Remark. It is easy to see that ( a, ) is a cyclic subgroup of (X, ) and ord( a ) = ord(a). We say that ( a, ) is the subgroup generated by a. Lagrange s Theorem therefore shows that ord(a) ord(g), as stated previously in Theorem Group Isomorphisms and Homomorphisms 1.41 Definition. Two groups G = (X, ) and H = (Y, ) are isomorphic if there exists a bijection ϕ : X Y such that ϕ(a a ) = ϕ(a) ϕ(a ) for all a, a X Theorem. Any two cyclic groups of the same order n are isomorphic Corollary. If G = (X, ) is any finite group, and a X, then ( a, ) is isomorphic to (Z ord(a), +) Definition. A homomorphism from a group G = (X, ) to a group H = (Y, ) is a mapping ϕ : X Y such that ϕ(a a ) = ϕ(a) ϕ(a ) for all a, a X Remark. A homomorphism ϕ from a group G = (X, ) to a group H = (Y, ) is an isomorphism if and only if it is a bijection from X to Y. 1.7 Quadratic Residues 1.46 Definition (quadratic residue). Suppose p is an odd prime and a is an integer. a is defined to be a quadratic residue modulo p if a 0 (mod p) and the congruence y 2 a (mod p) has a solution y Z p. a is defined to be a quadratic nonresidue modulo p if a 0 (mod p) and a is not a quadratic residue modulo p Definition (Legendre symbol). Suppose p is an odd prime. For any integer a, define the Legendre symbol ( a p) as follows: ( ) a = p 0 if a 0 (mod p) 1 if a is a quadratic residue modulo p 1 if a is a quadratic nonresidue modulo p Theorem. Suppose p is an odd prime. Then ( ) a = a (p 1)/2 mod p. p 1.49 Remark. Suppose p is an odd prime. Then the mapping a ( a p) is a homomorphism from (Z p, ) to ({1, 1}, ) Theorem. Suppose p 3 (mod 4) is prime and suppose y is a quadratic residue modulo p. Then the two square roots of y modulo p are ±y (p+1)/4. 5
6 1.8 Euclidean Algorithm 1.51 Algorithm (Euclidean algorithm). The Euclidean algorithm computes the greatest common divisor of two positive integers, say a and b. The algorithm sets r 0 to be a and r 1 to be b, and performs the following sequence of divisions: r 0 = q 1 r 1 + r 2, 0 < r 2 < r 1 r 1 = q 2 r 2 + r 3, 0 < r 3 < r r m 2 = q m 1 r m 1 + r m, 0 < r m < r m 1 r m 1 = q m r m. The algorithm terminates when a division yields a remainder of 0. The last nonzero remainder, r m, is the greatest common divisor of a and b Algorithm (Extended Euclidean algorithm). Given two integers a and b, the Extended Euclidean algorithm computes integers s and t such that as + bt = gcd(a, b) Theorem (multiplicative inverses in Z n ). Let n 2. A multiplicative inverse a 1 mod n exists if and only if gcd(a, n) = 1. In this case, given inputs a and n, the Extended Euclidean algorithm will compute integers s and t such that as + nt = 1. Then a 1 s (mod n) Theorem (linear congruences mod n). Suppose gcd(a, n) = 1. Then the linear congruence ax c (mod n) has a unique solution modulo n, given by the formula x a 1 c (mod n) Theorem (linear congruences mod n). Suppose gcd(a, n) = d > 1. If c 0 (mod d), then the linear congruence ax c (mod n) has no solutions. If c 0 (mod d), then the linear congruence ax c (mod n) is equivalent to linear congruence a x c (mod n ), where a = a/d, c = c/d and n = n/d. This congruence has a unique solution modulo n by Theorem 1.54, say x = x 0 mod n. The original congruence has d solutions modulo n, namely, x = x 0 + in mod n, for 0 i d Definition. A finite continued fraction is an mtuple of nonnegative integers, say which is shorthand for the following expression: q 1 + [q 1,..., q m ], 1 1. q 2 + q qm 1.57 Remark. Continued fractions can be obtained from the sequence of quotients in the Euclidean algorithm Example. We compute the continued fraction expansion of 34/99. The Euclidean algorithm proceeds as follows: 34 = = = = = 3 1. The continued fraction expansion of 34/99 is [0, 2, 1, 10, 3], 6
7 1.59 Definition. The convergents of a finite continued fraction [q 1,..., q m ] are the values defined by the continued fractions [q 1,..., q i ], for 1 i m Example. The convergents of the continued fraction [0, 2, 1, 10, 3] are as follows: [0] = 0 [0, 2] = 1/2 [0, 2, 1] = 1/3 [0, 2, 1, 10] = 11/32, and [0, 2, 1, 10, 3] = 34/ Direct Products 1.61 Definition (direct product). Suppose that G = (X, ) and G = (X, ) are groups. The direct product G G is the group defined as follows: G G = (X X, ), where for all a, b X and all a, b X. (a, a ) (b, b ) = (a b, a b ) 1.62 Remark. Suppose (a, a ) G G. If the order of a is equal to d and the order of a is equal to d, then the order of (a, a ) is equal to the least common multiple of d and d Remark. Definition 1.61 can be extended in the obvious way to define a direct product of more than two groups Theorem (Fundamental Theorem of Abelian Groups). Every finite abelian group is isomorphic to a direct product of cyclic groups of prime power order Example. The factorization of 36 into prime powers is 36 = There are precisely four nonisomoprhic groups of order 36, namely, Z 4 Z 9, Z 2 Z 2 Z 9, Z 4 Z 3 Z 3 and Z 2 Z 2 Z 3 Z 3. 2 Rings and Fields 2.1 Rings 2.1 Definition (ring). A ring is a triple R = (X,, +), where X is a finite set and and + are a binary operations defined on X, that satisfies the following properties: (X, +) is an abelian group with identity 0. Multiplication is associative, i.e., for any a, b, c X, (ab)c = a(bc). The distributive property is satisfied, i.e., for any a, b, c X, (a + b)c = (ac) + (bc) and a(b + c) = (ab) + (ac). 2.2 Definition. A ring R = (X,, +) is a finite ring if X is a finite set. 2.3 Definition. A ring R = (X,, +) is a ring with identity if X contains a multiplicative identity, denoted by 1. 7
8 2.4 Definition. A ring R = (X,, +) is a commutative ring if multiplication is commutative. 2.5 Example. Some familiar examples of commutative rings include the integers, Z; the real numbers, R; and the complex numbers, C. These are all infinite rings. 2.6 Example. (Z m,, +) is a finite ring for any m Example (matrices). Let n 2. The set of n n matrices with entries from Z p is a ring, but not a commutative ring. 2.2 Fields 2.8 Definition (field). A ring R = (X,, +) is a field if it is a commutative ring with identity such that every nonzero element has a multiplicative inverse (i.e., (R\{0}, ) is an abelian group). 2.9 Example. (Z p,, +) is a finite field if p is prime Example (ring of polynomials). Suppose (A,, +) is a field and x is an indeterminate. Let A[x] denote the set of all polynomials with coefficients from A. Then (A[x],, +) is an infinite ring. 2.3 Direct Products and the Chinese Remainder Theorem 2.11 Definition (direct product). Suppose that R = (X,, +) and S = (Y,, +) are rings. The direct product R R is the ring defined as follows: R R = (X X,, +), where and for all a, b X and all a, b X. (a, a ) (b, b ) = (a b, a b ) (a, a ) + (b, b ) = (a + b, a + b ) 2.12 Remark. Definition 2.11 can be extended in the obvious way to define a direct product of more than two rings Remark. The direct product of two fields is not a field Definition. Two rings (or fields) R = (X,, +) and S = (Y,, +) are isomorphic if there exists a bijection ϕ : X Y such that ϕ(a a ) = ϕ(a) ϕ(a ) for all a, a X and ϕ(a+a ) = ϕ(a)+ϕ(a ) for all a, a X Theorem. Suppose M = m 1 m 2 m r, where gcd(m i, m j ) = 1 for all i j. Then the ring (Z M,, +) is isomorphic to the ring (Z m1 Z mr,, +) Remark. Define χ : Z M Z m1 Z mr, as follows: χ(a) = (a mod m 1,..., a mod m r ). Then χ can be shown to be an isomorphism of the two rings (Z M,, +) and (Z m1 Z mr,, +). 8
9 2.17 Remark. For 1 i r, define M i = M/m i and y i = M i 1 mod m i. Then the inverse function χ 1 : Z m1 Z mr Z M is χ 1 (a 1,..., a r ) = r a i M i y i mod M. i= Example. Suppose r = 3, m 1 = 7, m 2 = 11 and m 3 = 13. Then M = We compute M 1 = 143, M 2 = 91 and M 3 = 77, and then y 1 = 5, y 2 = 4 and y 3 = 12. Then the function χ 1 : Z 7 Z 11 Z 13 Z 1001 is the following: χ 1 (a 1, a 2, a 3 ) = (715a a a 3 ) mod Remark. The fact that the function χ 1 constitutes an isomorphism is an important result that is commonly known as the Chinese Remainder Theorem Theorem (Chinese remainder theorem). Suppose m 1,..., m r are pairwise relatively prime positive integers, and suppose a 1,..., a r are integers. Then the system of r congruences x a i (mod m i ) (1 i r) has a unique solution modulo M = m 1 m r, which is given by x = χ 1 (a 1,..., a m ). 2.4 Ideals and Quotient Rings 2.21 Definition (ideal). Suppose R = (X,, +) is a commutative ring. An ideal is a subset I X that satisfies the following properties: (I, +) is an abelian group, and ab I whenever a X and b I Definition (principal ideal). Suppose R = (X,, +) is a commutative ring and let c X. The principal ideal generated by c, which is denoted by (c), is the subset defined as follows: (c) = {ac : a X}. It is easy to see that a principal ideal is always an ideal Definition (quotient ring). Suppose R = (X,, +) is a commutative ring and I = (c) is a principal ideal. The quotient ring R/I is constructed as follows. R/I = (Y,, +), where Y consists of the (additive) cosets of I in (X, +). The sum of two cosets I + a and I + b is defined to be I + (a + b), for any a, b X, and the product of the two cosets I + a and I + b is defined to be I + ab Definition (principal ring). Suppose R = (X,, +) is a commutative ring. We say that R is a principal ring if every ideal is a principal ideal Example. Examples of principal rings include (Z,, +) as well as any polynomial ring (A[x],, +), where A is a field Example. Since (Z,, +) is a principal ring, it follows that any ideal I in this ring consists of all the multiples (positive and negative) of a positive integer c. The quotient ring Z/I is simply Z c. 9
10 2.5 Irreducible Polynomials and Finite Fields 2.27 Definition (irreducible polynomial). Then (A[x],, +) is a polynomial ring, where A is a field. A polynomial f(x) A[x] is irreducible if f(x) cannot be written as a product of two polynomials f 1 (x)f 2 (x), where f 1 (x) an f 2 (x) both have positive degree Example. In the ring Z 2 [x], we have that x = (x + 1)(x + 1), so x is reducible. Since x 2 + x = x(x + 1), this polynomial is also reducible. However, x 2 + x + 1 is irreducible Example. Suppose that A is any finite field and suppose n is a positive integer. Then there is at least one irreducible polynomial of degree n in (A[x],, +) Theorem. There exists a finite field of order n if and only if n = p k where p is prime and k 1 is an integer Definition. A finite field of order n = p k (where p is prime) is said to have characteristic p Theorem. Suppose p is prime and k 2. A finite field of order p k can be constructed as follows. Let f(x) Z p [x] be an irreducible polynomial of degree k. Then the quotient ring Z p [x]/(f(x)) is a finite field of order p k Remark. Multiplicative inverses in a finite field Z p [x]/(f(x)) can be computed using the Extended Euclidean Algorithm for polynomials Remark. For any polynomial f(x) Z p [x], the additive group (Z p [x]/(f(x)), +) is isomorphic to (Z p ) k Theorem. All finite fields of a given order n are isomorphic Remark. We denote the unique (up to isomorphism) finite field of order n by F n Example. The field F 8 can be constructed as either Z 2 [x]/(x 3 + x + 1) or Z 2 [x]/(x 3 + x 2 + 1), since both x 3 + x + 1 and x 3 + x are irreducible polynomials in Z 2 [x]. The two constructions yield isomorphic fields Theorem. The multiplicative group (F n \{0}, 0) is cyclic Definition. A generator of (F n \{0}, 0) is called a primitive element in F n. 10
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