Improving the Polynomial time Precomputation of Frobenius Representation Discrete Logarithm Algorithms - Simplified Setting for Small Characteristic Finite Fields

Antoine Joux 1 Cécile Pierrot 1
1 ALMASTY - ALgorithms for coMmunicAtion SecuriTY
LIP6 - Laboratoire d'Informatique de Paris 6
Abstract : In this paper, we revisit the recent small characteristic discrete logarithm algorithms. We show that a simplified description of the algorithm, together with some additional ideas, permits to obtain an improved complexity for the polynomial time precomputation that arises during the discrete logarithm computation. With our new improvements, this is reduced to $O(q^6)$, where $q$ is the cardinality of the basefield we are considering. This should be compared to the best currently documented complexity for this part, namely $O(q^7)$. With our simplified setting, the complexity of the precomputation in the general case becomes similar to the complexity known for Kummer (or twisted Kummer) extensions.
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Antoine Joux, Cécile Pierrot. Improving the Polynomial time Precomputation of Frobenius Representation Discrete Logarithm Algorithms - Simplified Setting for Small Characteristic Finite Fields. 20th International Conference on the Theory and Application of Cryptology and Information Security, Dec 2014, Kaoshiung, Taiwan. pp.378-397, ⟨10.1007/978-3-662-45611-8_20⟩. ⟨hal-01213649⟩

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