Fast computation of hyperelliptic curve isogenies in odd characteristic

TL;DR

This paper introduces an efficient algorithm for explicitly computing isogenies between Jacobians of hyperelliptic curves over p-adic fields, leveraging differential equations with minimal precision loss.

Contribution

The work presents a novel algorithm for computing hyperelliptic curve isogenies with explicit rational representations, improving efficiency in odd characteristic.

Findings

Algorithm achieves explicit isogeny computation with logarithmic p-adic precision loss.

Method applies to Jacobians of hyperelliptic curves of genus g over p-adic extensions.

Provides a practical approach for isogeny-based cryptographic applications.

Abstract

Let p be an odd prime number and g $\ge$ 2 be an integer. We present an algorithm for computing explicit rational representations of isogenies between Jacobians of hyperelliptic curves of genus g over an extension K of the field of p-adic numbers Qp. It relies on an efficient resolution, with a logarithmic loss of p-adic precision, of a first order system of differential equations.