Algorithmic aspects of elliptic bases in finite field discrete logarithm algorithms

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.
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Submitted on : Thursday, July 4, 2019 - 4:10:22 PM
Last modification on : Friday, November 8, 2019 - 11:00:36 AM

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  • HAL Id : hal-02173688, version 1
  • ARXIV : 1907.02689

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Antoine Joux, Cécile Pierrot. Algorithmic aspects of elliptic bases in finite field discrete logarithm algorithms. 2019. ⟨hal-02173688⟩

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