The difficulty of computing discrete logarithms in fields 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 , 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.
International Conference on the Theory and Application of Cryptographic Techniques (EuroCrypt)
2014
2026-06-08