A Heuristic Quasi-Polynomial Algorithm for Discrete Logarithm in Finite Fields of Small Characteristic

  • Razvan Barbulescu
  • Pierrick Gaudry
  • Antoine Joux
  • Emmanuel Thomé
Conference paper

DOI: 10.1007/978-3-642-55220-5_1

Part of the Lecture Notes in Computer Science book series (LNCS, volume 8441)
Cite this paper as:
Barbulescu R., Gaudry P., Joux A., Thomé E. (2014) A Heuristic Quasi-Polynomial Algorithm for Discrete Logarithm in Finite Fields of Small Characteristic. In: Nguyen P.Q., Oswald E. (eds) Advances in Cryptology – EUROCRYPT 2014. EUROCRYPT 2014. Lecture Notes in Computer Science, vol 8441. Springer, Berlin, Heidelberg

Abstract

The difficulty of computing discrete logarithms in fields \(\mathbb{F}_{q^k}\) depends on the relative sizes of k and q. Until recently all the cases had a sub-exponential complexity of type L(1/3), similar to the factorization problem. In 2013, Joux designed a new algorithm with a complexity of L(1/4 + ε) in small characteristic. In the same spirit, we propose in this article another heuristic algorithm that provides a quasi-polynomial complexity when q is of size at most comparable with k. By quasi-polynomial, we mean a runtime of nO(logn) where n is the bit-size of the input. For larger values of q that stay below the limit \(L_{q^k}(1/3)\), our algorithm loses its quasi-polynomial nature, but still surpasses the Function Field Sieve. Complexity results in this article rely on heuristics which have been checked experimentally.

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Copyright information

© International Association for Cryptologic Research 2014

Authors and Affiliations

  • Razvan Barbulescu
    • 1
  • Pierrick Gaudry
    • 1
  • Antoine Joux
    • 2
    • 3
  • Emmanuel Thomé
    • 1
  1. 1.Inria, CNRSUniversity of LorraineFrance
  2. 2.CryptoExpertsParisFrance
  3. 3.Chaire de Cryptologie de la Fondation UPMCSorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7606, LIP 6France

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