# Computer Science A Cryptography and Data Security. Claude Crépeau

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1 Computer Science A Cryptography and Data Security Claude Crépeau

2 These notes are, largely, transcriptions by Anton Stiglic of class notes from the former course Cryptography and Data Security ( A) that was given by prof. Claude Crépeau at McGill University during the autumn of These notes are updated and revised by Claude Crépeau. 2

3 15 Digital signatures A digital signature scheme allows Alice to compute a signature s for a message m in a way that Bob, and others, can verify that s was in fact computed by Alice and no one else. Formally, a digital signature scheme is defined as follows: Definition 15.1 Let M be a finite set of messages and T a finite set of digital signatures such that for each (k a, k v ) K, there is a signing algorithm sig ka and a corresponding verification algorithm ver kv such that sig ka : M T and ver kv : M T {true, false} are polynomial-time computable functions and ver kv (m, y) = { true : if y = sigka (x) false : if y sig ka (x) A major difference between an authentication scheme and a signature scheme is that in an authentication scheme where Alice authenticates herself to Bob, Bob can fake Alice s authentication for any message. authentication M K a K v T verification 76

4 15.1 RSA signature scheme The RSA cryptographic scheme can be directly used as a signature scheme: the decryption function is used as the signature function and the verification function is obtained by comparing the message with the encryption of the signature ElGamal signature scheme We use the same keys as in the ElGamal encryption scheme, that is we have K = {(p, α, a, β) : β α a (mod p), α a generator of Z p }, p, α and β are public, a is kept secret. Unlike RSA, the functions for the ElGamal signature scheme are not identical to those of the ElGamal encryption scheme. The functions are constructed to try to make forgery difficult. Algorithm 15.1 ( ElGamal signature ) 1: Pick a random k such that 1 k p 2 and gcd(k,p 1) = 1. 2: γ α k mod p, δ (x aγ) k 1 mod p 1. 3: RETURN s = (γ,δ). Verification: V er K (x, γ, δ) = true β γ γ δ = α x mod p If the signature was constructed correctly, then the verification will succeed since 15.3 Bad usage β γ γ δ = β γ (α k ) δ mod p = β γ (α k ) (x aγ)k 1 mod p = (α a ) γ (α) (x aγ) mod p = α x mod p Revealing k or using the same k twice can cause forgery of chosen messages. If k is known, one can compute information on a from: aγ x δk mod p 1. 77

5 If the same k is used for two messages, we obtain the following Thus δ 1 = k 1 (x 1 aγ) mod p 1 δ 2 = k 1 (x 2 aγ) mod p 1 (δ 1 δ 2 )k = x 1 x 2 mod p 1. If δ 1 δ 2 0 mod p 1, we can compute d gcd(δ 1 δ 2, p 1). Since d δ 1 δ 2 and d p 1, we know that d (x 1 x 2 ). Thus we can write The equation becomes x := x 1 x 2 d δ := δ 2 δ 1 d p := p 1 d. x = kδ mod p Since gcd(δ, p ) = 1, we can compute (δ ) 1, then k = x (δ ) 1 mod p This yields d candidate values for k, we can choose the right k by verifying with the signature verification function. From k we can then deduce a Forgeries Some forgeries are now discussed by categories corresponding to the way Oscar forges a signature: Given x, set a γ and then try to find δ. The problem at hand would be to solve for δ given β γ γ δ = α x mod p, which is equivalent to solving for δ given γ δ = (α x )(β γ ) 1 mod p this is equivalent to the DLP modp. 78

6 Given x, set a δ, try to find γ. This reduces to trying to find γ given β γ γ δ = α x mod p. No efficient solution to this problem is known, this problem is not known to be related to any well-studied problem like DLP. Given x, try to simultaneously find δ and γ. There is no known way of doing this. Is it possible for Oscar to sign a random message? If Oscar chooses γ and δ and then tries to solve for x, he must compute log α (β γ γ δ ), yet another instance of the DLP. However, there is a way for Oscar to sign a random message by choosing γ, δ and x simultaneously, it is described by the following algorithm Algorithm 15.2 ( Forge ElGamal ) 1: Pick i and j such that 0 i,j p 2 and gcd(j,p 1) = 1. 2: γ α i β j mod p. 3: δ = γj 1 mod p 1, x γij 1 mod p 1. Theorem 15.2 The above algorithm gives a valid signature. Proof. β γ γ δ = β γ (α i β j ) γj 1 mod p = β γ (α γij 1 β γ ) mod p = α γij 1 mod p = α x mod p Note: in a variation of the ElGamal signature scheme, one uses h(x) instead of x, where h is a cryptographic hash function. Other than the fact that this enables signatures of data of arbitrary size, it also prevents the above forgery from being successful. 79

7 It is also possible for Oscar to forge some message given a previous message and signature (x, γ, δ). Algorithm 15.3 ( Forge From Previous ElGamal ) 1: Pick h,i,j such that 0 h,i,j p 2 and gcd(hγ jδ,p 1) = 1. 2: λ γ h α i β j mod p 3: µ δλ(hγ jδ) 1 mod p 1 4: x = λ(hx + iδ)(hγ jδ) 1 mod p 1. Theorem 15.3 The above algorithm gives (x, λ, µ) such that β λ λ µ = α x mod p 15.5 Digital Signature Standard The Digital Signature Standard (DSS) describes a Digital Signature Algorithm (DSA) in FIPS 186, it is a variation of the ElGamal system. DSS relieves the burden of oversized signatures (with ElGamal signing a 160-bit message using a 512 bit prime, for example, produces a signature that is 1024 bits long, DSS would produce a 320-bit signature). Algorithm 15.4 ( DSA key generation ) 1: Choose a 512-bit prime p 2: Pick a 160-bit prime q such that q p 1. 3: Choose α Z p a qth primitive root of 1 mod p. 4: Compute β α a mod p. 5: RETURN public p,q,α,β and private a Note: To pick α, you can start by picking α o a primitive element of Z p and then computing α α (p 1)/q o. 80

8 Algorithm 15.5 ( DSS signature ) 1: Pick a random k such that 1 k p 2. 2: γ (α k mod p) mod q 3: δ (x + aγ)k 1 mod q. 4: IF δ = 0, GOTO step 1 5: RETURN s = (γ,δ). Verification: e 1 xδ 1 mod q e 2 γδ 1 mod q V er K (x, γ, δ) = true (α e 1 β e 2 mod p) mod q = γ. If the signature was constructed correctly, then the verification will succeed since (α e 1 β e 2 mod p) mod q α e 1 α ae 2 α xδ 1 α aγδ 1 α δ 1 (x+aγ) α k = (γ mod p) mod q Undeniable signatures In this type of signature scheme, the verification protocol requires the cooperation of the signer. The scheme is composed of three components: a signing algorithm, a verification protocol and a disavowal protocol. A disavowal protocol enables one to determine whether the signer is attempting to disavow a valid signature or whether the signature was forged. 81

9 Chaum-Van Antwerpen s scheme The first undeniable signature scheme was introduced in [?]. Algorithm 15.6 ( Chaum-Van Antwerpen key generation ) 1: Select a random prime p = 2q + 1, where q is also prime. 2: Select α y 2 mod p, for a random y R {2,3,...,p 2}. 3: Select a random a R {1,2,...,q 1}, β α a mod p. 4: RETURN public (p, α, β) and private a. α is selected in such a way as to be a generator of the subgroup of order q in Z p. The scheme operates in Z p, however, we need to be able to compute in a multiplicative subgroup of Z p. Picking p = 2q + 1, p, q primes, enables us to do this, and in a large as possible subgroup. Algorithm 15.7 ( Chaum-Van Antwerpen signature ) 1: s x a mod p. 2: RETURN s Algorithm 15.8 ( Chaum-Van Antwerpen verification ) 1: Bob selects random secret integers e 1,e 2 R {1,2,...,q 1}. 2: Bob computes z s e 1 β e 2 mod p and sends z to Alice. 3: Alice computes w = (z) a 1 mod p and sends w to Bob. 4: Bob accepts w = x e 1 α e 2 mod p If Alice is honest, Bob will accept: w = (z) a 1 mod p = (s e 1 β e 2 ) a 1 mod p = (x ae 1 α ae 2 ) a 1 mod p = x e 1 α e 2 mod p Theorem 15.4 Suppose s x a mod p is a forged signature, the probability 82

10 that Bob will accept the signature in the above algorithm is 1/q. The following disavowal protocol allows Alice to convince Bob that a certain value is not a valid signature. However, Alice might attempt to disavow a valid signature. The following protocol essentially performs the verification protocol twice and checks that Alice is not cheating: Algorithm 15.9 ( Chaum-Van Antwerpen Disavowal ) 1: Bob randomly selects e 1,e 2 R {1,2,...,q 1} 2: Bob computes z s e 1 β e 2 mod p, sends z to Alice. 3: Alice computes w = (z) a 1 mod p and sends w to Bob. 4: IF w = x e 1 α e 2 mod p THEN RETURN valid /* Bob concludes that Alice is trying to disavow a valid sig */ 5: Bob selects random e 3,e 4 R {1,2,...,q 1} 6: Bob computes z s e 3 β e 4 mod p, sends z to Alice. 7: Alice computes w (z ) a 1 mod p and sends w to Bob. 8: IF w = x e 3 α e 4 mod p THEN RETURN valid /* Bob concludes that Alice is trying to disavow a valid sig */ 9: Bob computes c (wα e 2 ) e 3 mod p, c (w α e 4 ) e 1 mod p 10: IF c = c THEN RETURN forgery /* Bob concludes that the sig was a forgery */ 11: ELSE RETURN valid /* Bob concludes that Alice is trying to disavow a valid sig */ Theorem 15.5 The probability for Alice to successfully disavow a valid signature s = x a mod p, in the above algorithm, is 1/q. 83

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