DISCRETE LOGARITHMS, DIFFIE-HELLMAN, AND REDUCTIONS

Abstract. We consider the One-Prime-Not-p and All-Primes-But-p variants of the Discrete Logarithm (DL) problem in a group of prime order p. We give reductions to the Diffie-Hellman (DH) problem that do not depend on any unproved conjectures about smooth or prime numbers in short intervals. We show that the One-Prime-Not-p-DL problem reduces to DH in time roughly Lp(1/2); the All-Primes-But-p-DL problem reduces to DH in time roughly Lp(2/5); and the All-Primes-But-p-DL problem reduces to the DH plus Integer Factorization problems in polynomial time. We also prove that under the Riemann Hypothesis, with ε log p queries to a yes-or-no oracle one can reduce DL to DH in time roughly Lp(1/2); and under a conjecture about smooth numbers, with εlog p queries to a yes-or-no oracle one can reduce DL to DH in polynomial time. 1