On the index of the Diffie–Hellman mapping.

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. [ABSTRACT FROM AUTHOR]