A provably quasi-polynomial algorithm for the discrete logarithm problem in finite fields of small characteristic

TL;DR

This paper presents a provably quasi-polynomial algorithm for computing discrete logarithms in finite fields of small characteristic, leveraging elliptic curve-based field presentations to improve upon previous heuristic methods.

Contribution

It introduces a new approach using elliptic curve-based presentations to achieve a provably quasi-polynomial time algorithm for discrete logarithms in small characteristic finite fields.

Findings

Algorithm is provably quasi-polynomial in complexity.

Uses elliptic curve-based field presentations for efficiency.

Improves upon previous heuristic algorithms.

Abstract

We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially follow the heuristically quasi-polynomial algorithm presented by Barbulescu, Gaudry, Joux and Thome'. The main difference is to use a presentation of the finite field based on elliptic curves: the abundance of elliptic curves ensures the existence of such a presentation.