Efficient Point-Counting Algorithms for Superelliptic Curves

TL;DR

This paper introduces efficient algorithms for counting points on superelliptic curves over finite fields, utilizing Hasse-Weil bounds, Hasse-Witt matrices, and extending existing methods for hyperelliptic curves.

Contribution

It presents new algorithms for point-counting on superelliptic curves, including a fast method for trinomial cases and a general approach extending hyperelliptic techniques.

Findings

Efficient point-counting algorithms for specific superelliptic curves.

Reduction of matrix entry simplification to quadratic Diophantine equations.

Extension of hyperelliptic curve methods to general superelliptic curves.

Abstract

In this paper, we present efficient algorithms for computing the number of points and the order of the Jacobian group of a superelliptic curve over finite fields of prime order p. Our method employs the Hasse-Weil bounds in conjunction with the Hasse-Witt matrix for superelliptic curves, whose entries we express in terms of multinomial coefficients. We present a fast algorithm for counting points on specific trinomial superelliptic curves and a slower, more general method for all superelliptic curves. For the first case, we reduce the problem of simplifying the entries of the Hasse-Witt matrix modulo p to a problem of solving quadratic Diophantine equations. For the second case, we extend Bostan et al.'s method for hyperelliptic curves to general superelliptic curves. We believe the methods we describe are asymptotically the most efficient known point-counting algorithms for certain families of trinomial superelliptic curves.