# Algorithmic aspects of elliptic bases in finite field discrete logarithm algorithms

## TL;DR

This paper explores the practical use of elliptic bases in finite field discrete logarithm algorithms, proposing a new model that could lead to efficient heuristic methods in small characteristic fields.

## Contribution

It introduces a new elliptic curve model for elliptic bases, enabling adaptation of existing Frobenius techniques for heuristic discrete logarithm algorithms.

## Key findings

- Preliminary experiments suggest comparable performance to current methods.
- The new model simplifies adaptation of Frobenius-based techniques.
- Potential for practical heuristic algorithms in small characteristic fields.

## Abstract

Elliptic bases, introduced by Couveignes and Lercier in 2009, give an elegant way of representing finite field extensions. A natural question which seems to have been considered independently by several groups is to use this representation as a starting point for small characteristic finite field discrete logarithm algorithms. This idea has been recently proposed by two groups working on it, in order to achieve provable quasi-polynomial time for discrete logarithms in small characteristic finite fields. In this paper, we don't try to achieve a provable algorithm but, instead, investigate the practicality of heuristic algorithms based on elliptic bases. Our key idea, is to use a different model of the elliptic curve used for the elliptic basis that allows for a relatively simple adaptation of the techniques used with former Frobenius representation algorithms. We haven't performed any record computation with this new method but our experiments with the field F 3 1345 indicate that switching to elliptic representations might be possible with performances comparable to the current best practical methods.