Counting points on genus-3 hyperelliptic curves with explicit real multiplication

Simon Abelard 1 Pierrick Gaudry 1 Pierre-Jean Spaenlehauer 1
1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field Fq, with explicit real multiplication by an order Z[η] in a totally real cubic field. Our main result states that this algorithm requires an expected number of O((log q) 6) bit-operations, where the constant in the O() depends on the ring Z[η] and on the degrees of polynomials representing the endomorphism η. As a proof-of-concept, we compute the zeta function of a curve defined over a 64-bit prime field, with explicit real multiplication by Z[2 cos(2π/7)].
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Communication dans un congrès
Thirteenth Algorithmic Number Theory Symposium ANTS-XIII, Jul 2018, Madison, United States
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Simon Abelard, Pierrick Gaudry, Pierre-Jean Spaenlehauer. Counting points on genus-3 hyperelliptic curves with explicit real multiplication. Thirteenth Algorithmic Number Theory Symposium ANTS-XIII, Jul 2018, Madison, United States. 〈hal-01816256v1〉

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