On the index of the Diffie–Hellman mapping

Abstract

Let γ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is the index of the largest subgroup U of G such that f (x)x −r is constant on each coset of U for some positive integer r. We determine the index of the univariate Diffie-Hellman mapping d(γ a) = γ a 2 , a = 0, 1,. .. , n − 1, and show that any mapping of small index coincides with d only on a small subset of G. Moreover, we prove similar results for the bivariate Diffie-Hellman mapping D(γ a , γ b) = γ ab , a, b = 0, 1,. .. , n − 1. In the special case that G is a subgroup of the multiplicative group of a finite field we present improvements.