On the Statistical Properties of Diffie–hellman Distributions

Let p be a large prime such that p−1 has some large prime factors, and let θ ∈ ZZp be an r-th power residue for all small factors of p− 1. The corresponding Diffie-Hellman (DH) distribution is (θx, θy, θxy) where x, y are randomly chosen from ZZp. A recently formulated assumption is that given p, θ of the above form it is infeasible to distinguish in reasonable time between DH distribution and triples of numbers chosen ∗IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA. [email protected] †Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada. [email protected] ‡Department of Mechanics and Mathematics, Moscow State University, Moscow, 119899, Russia. [email protected] §Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. [email protected] ¶Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. [email protected] ‖School of MPCE, Macquarie University, Sydney, NSW 2109, Australia. [email protected]