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 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.
Keywords
- 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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References
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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. https://doi.org/10.1007/978-3-642-55220-5_1
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