Asymptotic complexities of discrete logarithm algorithms in pairing-relevant finite fields

Gabrielle de Micheli (1, 2) , Pierrick Gaudry (1, 3, 2) , Cécile Pierrot (1, 2)

1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
2 LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
3 CNRS - Centre National de la Recherche Scientifique

We study the discrete logarithm problem at the boundary case between small and medium characteristic finite fields, which is precisely the area where finite fields used in pairing-based cryptosystems live. In order to evaluate the security of pairing-based protocols, we thoroughly analyze the complexity of all the algorithms that coexist at this boundary case: the Quasi-Polynomial algorithms, the Number Field Sieve and its many variants, and the Function Field Sieve. We adapt the latter to the particular case where the extension degree is composite, and show how to lower the complexity by working in a shifted function field. All this study finally allows us to give precise values for the characteristic asymptotically achieving the highest security level for pairings. Surprisingly enough, there exist special characteristics that are as secure as general ones.

Domaines: Cryptographie et sécurité [cs.CR]

Gabrielle de Micheli, Pierrick Gaudry, Cécile Pierrot. Asymptotic complexities of discrete logarithm algorithms in pairing-relevant finite fields. CRYPTO 2020 - 40th Annual International Cryptology Conference, Aug 2020, Santa Barbara / Virtual, United States. pp.32-61, ⟨10.1007/978-3-030-56880-1_2⟩. ⟨hal-02871839⟩