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

TL;DR

This paper introduces a probabilistic algorithm for efficiently computing the zeta function of genus-3 hyperelliptic curves with explicit real multiplication over finite fields, significantly improving computational complexity.

Contribution

It presents a new Las Vegas algorithm that leverages explicit real multiplication to compute zeta functions with expected polylogarithmic complexity in the size of the finite field.

Findings

Algorithm requires expected tilde O((\u2113 q)^6) bit-operations.

Successfully computed zeta function for a curve over a 64-bit prime field.

Demonstrated practical feasibility with explicit real multiplication example.

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 $\widetilde O((\log q)^6)$ 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 $\mathbb Z[2\cos(2\pi/7)]$.