Advances in Cryptology – EUROCRYPT 2014

Volume 8441 of the series Lecture Notes in Computer Science pp 1-16

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

  • Razvan BarbulescuAffiliated withInria, CNRS, University of Lorraine
  • , Pierrick GaudryAffiliated withInria, CNRS, University of Lorraine
  • , Antoine JouxAffiliated withCryptoExpertsChaire de Cryptologie de la Fondation UPMC, Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7606, LIP 6
  • , Emmanuel ThoméAffiliated withInria, CNRS, University of Lorraine

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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 n O(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.