Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

TL;DR

This paper introduces a probabilistic algorithm that efficiently computes the local zeta function of hyperelliptic curves, improving complexity bounds and enabling faster calculations for fixed genus as the field size grows.

Contribution

It presents a new probabilistic algorithm combining existing methods with structured polynomial systems, achieving better complexity bounds for fixed genus hyperelliptic curves.

Findings

Expected time and space complexity is polynomial in log q for fixed genus

Algorithm outperforms previous methods in large characteristic fields

Complexity bound improved to O((log q)^{cg}) for some constant c

Abstract

We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over $\mathbb{F}_q$. It is based on the approaches by Schoof and Pila combined with a modeling of the $\ell$-torsion by structured polynomial systems. Our main result improves on previously known complexity bounds by showing that there exists a constant $c>0$ such that, for any fixed $g$, this algorithm has expected time and space complexity $O((\log q)^{cg})$ as $q$ grows and the characteristic is large enough.