Pairing-based algorithms for jacobians of genus 2 curves with maximal endomorphism ring

TL;DR

This paper develops new pairing-based algorithms for genus 2 curve Jacobians with maximal endomorphism rings, enabling verification and construction of isogenies using Galois cohomology and Frobenius actions.

Contribution

It introduces criteria and methods for identifying maximal endomorphism rings and constructing horizontal isogenies for genus 2 Jacobians, advancing cryptographic applications.

Findings

Criterion for maximal endomorphism ring verification

Method for constructing horizontal (ell,ell)-isogenies

Application of Galois cohomology to pairing non-degeneracy

Abstract

Using Galois cohomology, Schmoyer characterizes cryptographic non-trivial self-pairings of the $\ell$-Tate pairing in terms of the action of the Frobenius on the $\ell$-torsion of the Jacobian of a genus 2 curve. We apply similar techniques to study the non-degeneracy of the $\ell$-Tate pairing restrained to subgroups of the $\ell$-torsion which are maximal isotropic with respect to the Weil pairing. First, we deduce a criterion to verify whether the jacobian of a genus 2 curve has maximal endomorphism ring. Secondly, we derive a method to construct horizontal $(\ell,\ell)$-isogenies starting from a jacobian with maximal endomorphism ring.