Efficient computation of Cantor's division polynomials of hyperelliptic curves over finite fields

TL;DR

This paper introduces a fast algorithm for computing Cantor's division polynomials of hyperelliptic curves over finite fields, utilizing p-adic differential equations and precision analysis to improve efficiency.

Contribution

It presents a novel algorithm that leverages p-adic methods and differential equations for explicit computation of Cantor's division polynomials.

Findings

Algorithm achieves faster computation of division polynomials.

Provides a sharp analysis of p-adic precision loss.

Enables explicit calculations over finite fields.

Abstract

Let $p$ be an odd prime number. We propose an algorithm for computing rational representations of isogenies between Jacobians of hyperelliptic curves via-adic differential equations with a sharp analysis of the loss of precision. Consequently, after having possibly lifted the problem in the $p$-adics, we derive fast algorithms for computing explicitly Cantor's division polynomials of hyperelliptic curves defined over finite fields.