Diffie-hellman is as strong as discrete log for certain primes

Abstract

Diffie and Hellman proposed a key exchange scheme in 1976, which got their name in the literature afterwards. In the same epoch-making paper, they conjectured that breaking their scheme would be as hard as taking discrete logarithms. This problem has remained open for the multiplicative group modulo a prime P that they originally proposed. Here it is proven that both problems are (probabilisticly) polynomial-time equivalent if the totient of P-l has only small prime factors with respect to a (fixed) polynomial in 2logP. There is no algorithm known that solves the discrete log problem in probabilistic polynomial time for the this case, i.e., where the totient of P-l is smooth. Consequently, either there exists a (probabilistic) polynomial algorithm to solve the discrete log problem when the totient of P-l is smooth or there exist primes (satisfying this condition) for which Diffie-Hellman key exchange is secure.