Asymptotically fast group operations on Jacobians of general curves

TL;DR

This paper introduces probabilistic algorithms for efficiently performing group operations on Jacobians of general curves, significantly reducing computational complexity from previous methods by leveraging linear algebra techniques.

Contribution

It presents a novel probabilistic approach that, after a one-time precomputation, performs Jacobian group operations in nearly quadratic time using linear algebra.

Findings

Algorithms operate in $O(g^{3+psilon})$ field operations

Time complexity improved to $O(g^{2.376})$ with fast linear algebra

Achieves a significant reduction from the previous $O(g^4)$ complexity

Abstract

Let $C$ be a curve of genus $g$ over a field $k$. We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of $C$. After a precomputation, which is done only once for the curve $C$, the algorithms use only linear algebra in vector spaces of dimension at most $O(g \log g)$, and so take $O(g^{3 + \epsilon})$ field operations in $k$, using Gaussian elimination. Using fast algorithms for the linear algebra, one can improve this time to $O(g^{2.376})$. This represents a significant improvement over the previous record of $O(g^4)$ field operations (also after a precomputation) for general curves of genus $g$.