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
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
2 GRACE - Geometry, arithmetic, algorithms, codes and encryption
Inria Saclay - Ile de France
Abstract : 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.
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Laurent Grémy, Aurore Guillevic, François Morain, Emmanuel Thomé. Computing discrete logarithms in $GF(p^6)$. 24th Annual Conference on Selected Areas in Cryptography, Aug 2017, Ottawa, Canada. pp.85-105, ⟨10.1007/978-3-319-72565-9_5⟩. ⟨hal-01624662⟩

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