Towards the Equivalence of Breaking the Diffie-Hellman Protocol and Computing Discrete Logarithms

Let G be an arbitrary cyclic group with generator g and order |G| with known factorization. G could be the subgroup generated by g within a larger group H. Based on an assumption about the existence of smooth numbers in short intervals, we prove that breaking the Diffie-Hellman protocol for G and base g is equivalent to computing discrete logarithms in G to the base g when a certain side information string S of length 2 log |G| is given, where S depends only on |G| but not on the definition of G and appears to be of no help for computing discrete logarithms in G. If every prime factor p of |G| is such that one of a list of expressions in p, including p − 1 and p + 1, is smooth for an appropriate smoothness bound, then S can efficiently be constructed and therefore breaking the Diffie-Hellman protocol is equivalent to computing discrete logarithms.