Paper 2019/782

Algorithmic aspects of elliptic bases in finite field discrete logarithm algorithms

Antoine Joux
Cecile Pierrot

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 GF(3^1345) indicate that switching to elliptic representations might be possible with performances comparable to the current best practical methods.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Journal of Advances in Mathematics of Communications
discrete logarithm problem finite fields elliptic representation
Contact author(s)
antoine joux @ m4x org
cecile pierrot @ inria fr
2022-10-24: revised
2019-07-09: received
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      author = {Antoine Joux and Cecile Pierrot},
      title = {Algorithmic aspects of elliptic bases in finite field discrete logarithm algorithms},
      howpublished = {Cryptology ePrint Archive, Paper 2019/782},
      year = {2019},
      note = {\url{}},
      url = {}
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