Ran Canetti, Shai Halevi, et al.
Journal of Cryptology
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 Σp-1x,y-1 exp (2πiagx+bgy+cgxy/p) = O(p31/16+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.
Ran Canetti, Shai Halevi, et al.
Journal of Cryptology
Ran Canetti, Susan Hohenberger
CCS 2007
Ran Canetti, Ling Cheung, et al.
Dagstuhl Seminar Proceedings 2008
Ran Canetti, Yehuda Lindell, et al.
STOC 2002