A point counting algorithm using cohomology with compact support

TL;DR

This paper presents a new point counting algorithm for hyperelliptic curves over finite fields using cohomology with compact support, offering a systematic cohomological approach with detailed complexity analysis.

Contribution

The paper introduces a cohomology-based algorithm for counting points on hyperelliptic curves, combining basis computation and Frobenius action representation with explicit complexity bounds.

Findings

Achieves $ ilde{O}(g^4 n^{3})$ time complexity

Uses Monsky-Washnitzer cohomology with compact support

Provides a detailed proof of correctness

Abstract

We describe an algorithm to count the number of rational points of an hyperelliptic curve defined over a finite field of odd characteristic which is based upon the computation of the action of the Frobenius morphism on a basis of the Monsky-Washnitzer cohomology with compact support. This algorithm follows the vein of a systematic exploration of potential applications of cohomology theories to point counting. Our algorithm decomposes in two steps. A first step which consists in the computation of a basis of the cohomology and then a second step to obtain a representation of the Frobenius morphism. We achieve a $\tilde{O}(g^4 n^{3})$ worst case time complexity and $O(g^3 n^3)$ memory complexity where $g$ is the genus of the curve and $n$ is the absolute degree of its base field. We give a detailed complexity analysis of the algorithm as well as a proof of correctness.