Improvements to Jacobian Arithmetic in Global Function Fields

TL;DR

This paper introduces two optimized algorithms for Jacobian arithmetic in global function fields, reducing computational steps and employing caching to enhance speed, supported by empirical results and a novel software implementation.

Contribution

It presents novel improvements to Jacobian arithmetic algorithms, including reduction step optimization and a memory-time trade-off, with the first software supporting unique divisor class representatives.

Findings

Algorithms are significantly faster in practice.

Empirical analysis confirms theoretical speedups.

First software supporting unique divisor class representatives.

Abstract

We present two improvements to arithmetic in the Jacobian of global function fields based on the approach of Hess. The first reduces the number of expensive reduction steps by optimizing for typical inputs rather than worst-case behavior, assuming the function field contains a degree-one place. The second introduces a memory-time trade-off that speeds up computations by caching frequently used intermediate results. Our asymptotic analysis and empirical experiments show that our improved algorithms are significantly faster in practice than previously published methods. To the best of our knowledge, our publicly-available software implementation of Jacobian arithmetic is the first to support unique representatives of divisor classes.