On the index of the Diffie-Hellman mapping

Let gamma 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(gamma(a))=gamma(a2), 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(gamma(a),gamma(b))=gamma(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.