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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8441)


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.


Full Rank Discrete Logarithm Small Characteristic Discrete Logarithm Problem Linear Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© International Association for Cryptologic Research 2014

Authors and Affiliations

  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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