On the Index of Diffie-Hellman Mapping

TL;DR

This paper analyzes the index of Diffie-Hellman mappings, determining their structure and uniqueness, and extends results to bivariate cases with improvements for subgroups of finite fields.

Contribution

It precisely determines the index of univariate and bivariate Diffie-Hellman mappings and shows their uniqueness among mappings of small index, with special improvements for finite field subgroups.

Findings

The index of the univariate Diffie-Hellman mapping is explicitly determined.

Mappings of small index are shown to coincide with the Diffie-Hellman mapping only on small subsets.

Results are extended to bivariate mappings and improved for subgroups of finite fields.

Abstract

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 $d(\gamma^a)=\gamma^{a^2}$, $a=0,1,\ldots,n-1$, 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 $D(\gamma^a,\gamma^b)=\gamma^{ab}$, $a,b=0,1,\ldots,n-1$. In the special case that $G$ is a subgroup of the multiplicative group of a finite field we present improvements.