Counting Points on Hyperelliptic Curves using Monsky-Washnitzer Cohomology

TL;DR

This paper presents an algorithm that efficiently counts points on hyperelliptic curves over finite fields by leveraging Monsky-Washnitzer cohomology to compute Frobenius polynomials, with favorable asymptotic complexity.

Contribution

It introduces a novel algorithm utilizing Monsky-Washnitzer cohomology for point counting on hyperelliptic curves over finite fields.

Findings

Asymptotic running time is O(g^{4+ε} n^{3+ε}) for fixed p.

Algorithm computes p-adic approximation to Frobenius characteristic polynomial.

Effective for arbitrary hyperelliptic curves over finite fields.

Abstract

We describe an algorithm for counting points on an arbitrary hyperelliptic curve over a finite field of odd characteristic, using Monsky-Washnitzer cohomology to compute a p-adic approximation to the characteristic polynomial of Frobenius. For fixed p, the asymptotic running time for a curve of genus g over the field of p^n elements is O(g^{4+\epsilon} n^{3+\epsilon}).