Computing in Jacobians of projective curves over finite fields

TL;DR

This paper presents efficient algorithms for computing with divisors and Jacobians of projective curves over finite fields, enabling various algebraic operations crucial for number theory and cryptography.

Contribution

It introduces algorithms based on Khuri-Makdisi's representation for performing key operations on divisors and Jacobians efficiently over finite fields.

Findings

Efficient decomposition of divisors into prime divisors

Fast computation of Frobenius and Kummer maps

Algorithms for generating random divisors and computing l-torsion bases

Abstract

We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations can be done efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite morphisms, and hence Picard and Albanese maps on Jacobians; generating uniformly random divisors and points on Jacobians; computing Frobenius maps and Kummer maps; and finding a basis for the $l$-torsion of the Picard group, where $l$ is a prime number different from the characteristic of the base field.