On certain exponential sums and the distribution of Diffie-Hellman triples

Let g be a primitive root module a prime p. It is proved that the triples ( g(x), g(y), g(xy)), x,y + 1,...,p-1 are uniformly distributed module p in t he sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let epsilon > 0 be fixed. Then Sigma(x,y-1)(p-1) exp (2 pi i ag(x)+bg(y)+cg(xy)/p) = O(p(31/16+epsilon)) uniformly for any integers a, b,e with gcd(a, b, c,p)= 1. Incomplete sums a re estimated as well. The question is motivated by the assumption, often made in cryptography, th at the triples (g(x),g(y), g(xy)) cannot be distinguished from totally rand om 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.