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

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 $\mathbb{Z}[\eta ]$ in a totally real cubic field. Our main result states that this algorithm requires an expected number of $\tilde{O}((\log q{)}^6)$ bit-operations, where the constant in the $\tilde{O}()$ 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 $\mathbb{Z}[2\cos (2\pi /7)]$.

Proceedings of the ANTS-XIII conference (Thirteenth Algorithmic Number Theory Symposium)