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arXiv · 1806.05834

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

Simon Abelard·Pierrick Gaudry·Pierre-Jean Spaenlehauer
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Abstract

We propose a Las Vegas probabilistic algorithm to compute the zeta function of a genus-3 hyperelliptic curve defined over a finite field $\mathbb F_q$, with explicit real multiplication by an order $\mathbb Z[η]$ in a totally real cubic field. Our main result states that this algorithm requires an expected number of $\widetilde O((\log q)^6)$ bit-operations, where the constant in the $\widetilde O()$ depends on the ring $\mathbb Z[η]$ 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 $\mathbb Z[2\cos(2π/7)]$.

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Simon Abelard, Pierrick Gaudry, Pierre-Jean Spaenlehauer. 2018-09-20. Counting points on genus-3 hyperelliptic curves with explicit real multiplication. https://doi.org/10.2140/obs.2019.2.1

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