Translating the discrete logarithm problem on Jacobians of genus 3 hyperelliptic curves with $(\ell,\ell,\ell)$-isogenies

TL;DR

This paper presents an algorithm to compute specific isogenies between Jacobians of genus 3 hyperelliptic and non-hyperelliptic curves, enabling the reduction of the discrete logarithm problem across these structures.

Contribution

It introduces a novel algorithm for $(ll,ll,ll)$-isogenies between Jacobians of genus 3 curves, facilitating cryptographic problem reduction.

Findings

Algorithm successfully computes the isogenies.

Reduces discrete logarithm problem from hyperelliptic to non-hyperelliptic Jacobians.

Potential cryptographic applications in curve-based systems.

Abstract

We give an algorithm to compute $(\ell,\ell,\ell)$-isogenies from the Jacobians of genus three hyperelliptic curves to the Jacobians of non-hyperelliptic curves. An important application is to reduce the discrete logarithm problem in the Jacobian of a hyperelliptic curve to the corresponding problem in the Jacobian of a non-hyperelliptic curve.