Unformatted text preview: p) Alice can also disavow a signature, z, for a message, m. See  for details. Additional protocols for undeniable signatures can be found in [584,344]. Lein Harn and Shoubao Yang proposed a group undeniable signature scheme . Convertible Undeniable Signatures
An algorithm for a convertible undeniable signature, which can be verified, disavowed, and also converted to a conventional digital signature is given in . It’s based on the ElGamal digital signature algorithm. Like ElGamal, first choose two primes, p and q, such that q divides p -1. Now you have to create a number, g, less than q. First choose a random number, h, between 2 and p -1. Calculate g = h( p-1)/q mod p If g equals the 1, choose another random h. If it doesn’t, stick with the g you have. The private keys are two different random numbers, x and z, both less than q. The public keys are p, q, g, y, and u, where y = gx mod p u = gz mod p To compute the convertible undeniable signature of message m (which is actually the hash of a message), first choose a random number, t, between 1 and q -1. Then compute T = gt mod p and m' = Ttzm mod q. Now, compute the standard ElGamal signature on m'. Choose a random number, R, such that R is less than and relatively prime to p -1. Then compute r =gR mod p, and use the extended Euclidean algorithm to compute s, such that m' a rx + Rs (mod q) The signature is the ElGamal signature (r, s), and T. Here’s how Alice verifies her signature to Bob: (1) Bob generates two random numbers, a and b. He computes c = TTmagb mod p and sends that to Alice. (2) Alice generates a random number, k, and computes h1 = cgk mod p, and h2 = h1 z mod p, and sends both of those numbers to Bob. (3) Bob sends Alice a and b. (4) Alice verifies that c = TTmagb mod p. She sends k to Bob. (5) Bob verifies that h1 = TTmagb+k mod p, and that h2 = yrarsaub+k mod p. Alice can convert all of her undeniable signatures to normal signatures by publishing z. Now, anyone can verify her signature without her help. Undeniable signature sch...
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