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

TL;DR

This paper introduces a quasi-polynomial heuristic algorithm for solving the discrete logarithm problem in finite fields of small characteristic, significantly improving upon previous subexponential methods.

Contribution

The paper presents a new descent-based algorithm achieving quasi-polynomial complexity for discrete logarithms in small characteristic finite fields, advancing the state of the art.

Findings

Achieves quasi-polynomial heuristic complexity of $n^{O(\log n)}$

Improves asymptotic complexity compared to previous subexponential algorithms

Provides a new approach inspired by recent work of Joux

Abstract

In the present work, we present a new discrete logarithm algorithm, in the same vein as in recent works by Joux, using an asymptotically more efficient descent approach. The main result gives a quasi-polynomial heuristic complexity for the discrete logarithm problem in finite field of small characteristic. By quasi-polynomial, we mean a complexity of type $n^{O(\log n)}$ where $n$ is the bit-size of the cardinality of the finite field. Such a complexity is smaller than any $L(\varepsilon)$ for $\epsilon>0$. It remains super-polynomial in the size of the input, but offers a major asymptotic improvement compared to $L(1/4+o(1))$.