Computing discrete logarithms in $GF(p^6)$

Laurent Grémy (1) , Aurore Guillevic (1) , François Morain (2) , Emmanuel Thomé (1)

1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
2 GRACE - Geometry, arithmetic, algorithms, codes and encryption

The security of torus-based and pairing-based cryptography relies on the difficulty of computing discrete logarithms in small degree extensions of finite fields of large characteristic. It has already been shown that for degrees 2 and 3, the discrete logarithm problem is not as hard as once thought. We address the question of degree 6 and aim at providing real-life timings for such problems. We report on a record DL computation in a 132-bit subgroup of $GF(p^6)$ for a 22-decimal digit prime, with $p^6$ having 422 bits. The previous record was for a 79-bit subgroup in a 240-bit field. We used NFS-DL with a sieving phase over degree 2 polynomials, instead of the more classical degree 1 case. We show how to improve many parts of the NFS-DL algorithm to reach this target.

Domaines: Théorie de l'information et codage [math.IT]; Cryptographie et sécurité [cs.CR]

Laurent Grémy, Aurore Guillevic, François Morain, Emmanuel Thomé. Computing discrete logarithms in $GF(p^6)$. Selected Areas in Cryptography – SAC 2017, Aug 2017, Ottawa, Canada. pp.85-105, ⟨10.1007/978-3-319-72565-9_5⟩. ⟨hal-01624662⟩