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

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 $\mathbb Z[\eta]$ 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 $\mathbb Z[\eta]$ and on the degrees of polynomials representing the endomorphism $\eta$. 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 $\mathbb Z[2\cos(2\pi/7)]$.Comment: Proceedings of the ANTS-XIII conference (Thirteenth Algorithmic Number Theory Symposium

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

International audienceWe 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 $Z[2 cos(2π/7)$]

Counting points on hyperelliptic curves with explicit real multiplication in arbitrary genus

We present a probabilistic Las Vegas algorithm for computing the local zeta function of a genus-$g$ hyperelliptic curve defined over $\mathbb F_q$ with explicit real multiplication (RM) by an order $\Z[\eta]$ in a degree-$g$ totally real number field. It is based on the approaches by Schoof and Pila in a more favorable case where we can split the $\ell$-torsion into $g$ kernels of endomorphisms, as introduced by Gaudry, Kohel, and Smith in genus 2. To deal with these kernels in any genus, we adapt a technique that the author, Gaudry, and Spaenlehauer introduced to model the $\ell$-torsion by structured polynomial systems. Applying this technique to the kernels, the systems we obtain are much smaller and so is the complexity of solving them. Our main result is that there exists a constant $c>0$ such that, for any fixed $g$, this algorithm has expected time and space complexity $O((\log q)^{c})$ as $q$ grows and the characteristic is large enough. We prove that $c\le 9$ and we also conjecture that the result still holds for $c=7$.Comment: To appear in Journal of Complexity. arXiv admin note: text overlap with arXiv:1710.0344