On the additive index of the Diffie-Hellman mapping and the discrete logarithm

TL;DR

This paper investigates the additive index complexity measure for the Diffie-Hellman mapping and discrete logarithm, providing lower bounds under specific conditions, thus contributing to understanding their algebraic complexity.

Contribution

It establishes lower bounds on the additive index for the Diffie-Hellman mapping and discrete logarithm, advancing the analysis of their algebraic complexity in finite fields.

Findings

Lower bounds on the additive index are derived for the Diffie-Hellman mapping.

Lower bounds are established for the discrete logarithm as a self-mapping.

Results depend on certain algebraic conditions of the mappings.

Abstract

Several complexity measures such as degree, sparsity and multiplicative index for cryptographic functions including the Diffie-Hellman mapping and the discrete logarithm in a finite field have been studied in the literature. In 2022, Reis and Wang introduced another complexity measure, the additive index, of a self-mapping of a finite field. In this paper, under certain conditions, we determine lower bounds on the additive index of the univariate Diffie-Hellman mapping and a self-mapping of $\mathbb{F}_q$ which can be identified with the discrete logarithm in a finite field.