Algorithmic aspects of elliptic bases in finite field discrete logarithm algorithms

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 discrete logarithm algorithms in small characteristic finite fields. 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 do not 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 have not 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.