Explicit endomorphism of the Jacobian of a hyperelliptic function field of genus 2 using base field operations

TL;DR

This paper introduces an efficient explicit endomorphism for the Jacobian of genus 2 hyperelliptic curves, extending previous work to divisors with non-disjoint support and providing new formulas for divisor operations.

Contribution

It develops explicit formulas for endomorphisms on Jacobians of genus 2 hyperelliptic curves for divisors with non-disjoint support, expanding the computational tools available.

Findings

Derived explicit formulas for endomorphisms with non-disjoint support

Extended divisor doubling techniques with a new approach

Enhanced computational efficiency for genus 2 Jacobian operations

Abstract

We present an efficient endomorphism for the Jacobian of a curve $C$ of genus 2 (hyperelliptic) for divisors having a Non disjoint support. This extends the work of Costello and Lauter in [12] who calculated explicit formulae for divisor doubling and addition of divisors with disjoint support in $\mathbb{J}(C)$ using only base field operations. Explicit formulae is presented for this third case and a slightly different approach for divisor doubling.