On Certain Exponential Sums and the Distribution of Diffie-Hellman Triples

Abstract

Let g be a primitive root modulo a prime p. It is proved that the triples (gx, gy, gxy), x, y = 1, …, p−1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε>0 be fixed. Then ∑ x , y = 1 p - 1 exp ( 2 π i a g x + b g x + c g x p ) = O ( p 31 / 16 + ε ) uniformly for any integers a, b, c with gcd(a, b, c, p) = 1. Incomplete sums are estimated as well. The question is motivated by the assumption, often made in cryptography, that the triples (gx, gy, gxy) cannot be distinguished from totally random triples in feasible computation time. The results imply that this is in any case true for a constant fraction of the most significant bits, and for a constant fraction of the least significant bits.