On the Index of Diffie-Hellman Mapping (original) (raw)

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.