Cryptology ePrint Archive: Report 2019/782

Algorithmic aspects of elliptic bases in finite field discrete logarithm algorithms

Antoine Joux and Cecile Pierrot

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 donít 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 havenít performed any record computation with this new method but our experiments with the field GF(3^1345) indicate that switching to elliptic representations might be possible with performances comparable to the current best practical methods.

Category / Keywords: public-key cryptography / discrete logarithm problem, finite fields, elliptic representation

Date: received 4 Jul 2019

Contact author: cecile pierrot at inria fr,antoine joux@m4x org

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Version: 20190709:182334 (All versions of this report)

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