Fast point counting on genus two curves in characteristic three

TL;DR

This paper presents an efficient point counting algorithm for genus two curves over finite fields of characteristic three, with applications in cryptography and demonstrated effectiveness through computational experiments.

Contribution

It introduces a modified, more efficient version of a previous algorithm, optimized for large finite fields in cryptographic contexts.

Findings

Algorithm has quasi-quadratic complexity in field degree

Successfully applied to cryptographic-sized curves

Experimental Magma implementation confirms practicality

Abstract

In this article we give the details of an effective point counting algorithm for genus two curves over finite fields of characteristic three. The algorithm has an application in the context of curve based cryptography. One distinguished property of the algorithm is that its complexity depends quasi-quadratically on the degree of the finite base field. Our algorithm is a modified version of an earlier method that was developed in joint work with Lubicz. We explain how one can alter the original algorithm, on the basis of new theory, such that it can be used to efficiently count points on genus two curves over large finite fields. Examples of cryptographic size have been computed using an experimental Magma implementation of the algorithm which has been programmed by the author. Our computational results show that the quasi-quadratic algorithm of Lubicz and the author, with some improvements, is practical and relevant for cryptography.