A double large prime variation for small genus hyperelliptic index calculus

Pierrick Gaudry (1, 2, 3) , Emmanuel Thomé (1) , Nicolas Thériault (4) , Claus Diem (5)

1 SPACES - Solving problems through algebraic computation and efficient software
2 TANC - Algorithmic number theory for cryptology
3 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
4 CACR - Centre for Applied Cryptographic Research
5 Mathematischen Institut

In this article, we examine how the index calculus approach for computing discrete logarithms in small genus hyperelliptic curves can be improved by introducing a double large prime variation. Two algorithms are presented. The first algorithm is a rather natural adaptation of the double large prime variation to the intended context. On heuristic and experimental grounds, it seems to perform quite well but lacks a complete and precise analysis. Our second algorithm is a considerably simplified variant, which can be analyzed easily. The resulting complexity improves on the fastest known algorithms. Computer experiments show that for hyperelliptic curves of genus three, our first algorithm surpasses Pollard's Rho method even for rather small field sizes.

Mots clés: CRYPTOLOGY; ALGORITHMICS; HYPERELLIPTIC CURVES; DISCRETE LOGARITHM

Domaines: Autre [cs.OH]

Pierrick Gaudry, Emmanuel Thomé, Nicolas Thériault, Claus Diem. A double large prime variation for small genus hyperelliptic index calculus. [Research Report] RR-5764, INRIA. 2005. ⟨inria-00077334⟩