Collecting relations for the number field sieve in $GF(p^6)$

Pierrick Gaudry 1 Laurent Grémy 1 Marion Videau 2, 1
1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in GF(p^6) with the Number Field Sieve. We provide new insights, improvements, and comparisons between different methods to select polynomials intended for a sieve in dimension 3 using a special-q strategy. We also take into account the Galois action to increase the relation productivity of the sieving phase. To validate our results, we ran several experiments and real computations for various selection methods and field sizes with our publicly available implementation of the sieve in dimension 3, with special-q and various enumeration strategies.
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Pierrick Gaudry, Laurent Grémy, Marion Videau. Collecting relations for the number field sieve in $GF(p^6)$. LMS Journal of Computation and Mathematics, London Mathematical Society, 2016, Special issue: Algorithmic Number Theory Symposium XII, 19, pp.332 - 350. ⟨10.1112/S1461157016000164⟩. ⟨hal-01273045v2⟩

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