Fast Jacobian group operations for C_{3,4} curves over a large finite field

TL;DR

This paper presents optimized algorithms for group operations on Jacobians of C_{3,4} curves over large finite fields, achieving about 20% efficiency improvements over previous methods.

Contribution

It provides explicit formulas and demonstrates efficient addition and doubling algorithms for Jacobians of C_{3,4} curves of genus 3, improving computational performance.

Findings

Addition of two elements requires 117 multiplications and 2 inversions.

Doubling an element requires 129 multiplications and 2 inversions.

Algorithms are efficient even for low genus curves.

Abstract

Let C be an arbitrary smooth algebraic curve of genus g over a large finite field K. We revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi (math.NT/0409209, to appear in Math. Comp.). The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| >> g, and which asymptotically require O(g^{2.376}) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, we provide explicit formulae for performing the group law on Jacobians of C_{3,4} curves of genus 3. We show that, typically, the addition of two distinct elements in the Jacobian of a C_{3,4} curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.