Abstract

. This paper considers the computational complexity of the discrete logarithm and related problems in the context of "generic algorithms"---that is, algorithms which do not exploit any special properties of the encodings of group elements, other than the property that each group element is encoded as a unique binary string. Lower bounds on the complexity of these problems are proved that match the known upper bounds: any generic algorithm must perform\Omega (p 1=2 ) group operations, where p is the largest prime dividing the order of the group. Also, a new method for correcting a faulty Diffie-Hellman oracle is presented. 1 Introduction The discrete logarithm problem plays an important role in cryptography. The problem is this: given a generator g of a cyclic group G, and an element g x in G, determine x. A related problem is the Diffie-Hellman problem: given g x and g y , determine g xy . In this paper, we study the computational power of "generic algorithms"--- that is, ...

Citations

512	Efficient signature generation for smart cards - Schnorr - 1991
324	Fast Probabilistic Algorithms for Verification of Polynomial Identities - Schwartz - 1980
292	L.: A Hard-Core Predicate for all One-Way Functions - Goldreich, Levin - 1989
256	An improved algorithm for computing logarithms over GF (p) and its cryptographic significance - Pohlig, Hellman - 1978
193	Monte Carlo methods for index computation (mod p - Pollard - 1978
69	Algorithms for black box fields and their application to cryptography - Boneh, Lipton - 1996
64	Complexity of a determinate algorithm for the discrete logarithm - Nechaev
61	On the complexity of matrix group problems - Babai, Szemeredi - 1984
61	Towards the equivalence of breaking the Diffie-Hellman protocol and computing discrete logarithms - Maurer
30	Diffie-Hellman oracles - Maurer, Wolf