Pairing-based algorithms for jacobians of genus 2 curves with maximal endomorphism ring
This work addresses cryptographic applications in pairing-based algorithms for genus 2 curves, but it appears incremental as it builds on existing techniques.
The paper tackles the problem of studying non-degeneracy of the ℓ-Tate pairing on subgroups of ℓ-torsion for genus 2 curves, resulting in a criterion to verify maximal endomorphism ring and a method to construct horizontal (ℓ,ℓ)-isogenies.
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.