Explicit Kummer surface theory for arbitrary characteristic

TL;DR

This paper provides explicit equations and embeddings for Kummer surfaces associated with genus 2 Jacobians over any field, enhancing computational tools for cryptography and height calculations.

Contribution

It extends previous work by Flynn by explicitly describing Kummer surfaces for genus 2 Jacobians over arbitrary fields, including maps for arithmetic operations.

Findings

Explicit equations and embeddings for Kummer surfaces are derived.

Applications include improved cryptographic computations and height calculations.

The work generalizes earlier results to arbitrary characteristic fields.

Abstract

We explicitly find an equation and a projective embedding of the Kummer surface associated to the Jacobian of a curve of genus 2 given by an equation of the form y^2 + h(x)y = f(x) over an arbitrary ground field as well as several maps that can be used to perform arithmetic on it. This extends earlier work by Flynn and has applications, for instance, in computations of canonical heights for genus 2 Jacobians and in cryptography.