Towards the Equivalence of Breaking

Let G be an arbitrary cyclic group with generator g and order jGj 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 Diie-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 jGj is given, where S depends only on jGj but not on the deenition of G and appears to be of no help for computing discrete logarithms in G. If every prime factor p of jGj 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 eeciently be constructed and therefore breaking the Diie-Hellman protocol is equivalent to computing discrete logarithms.