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

Using Galois cohomology, Schmoyer characterizes cryptographic non-trivial self-pairings of the ℓ-Tate pairing in terms of the action of the Frobenius on the ℓ-torsion of the jacobian of a genus 2 curve. We apply similar techniques to study the non-degeneracy of the ℓ-Tate pairing restrained to subgroups of the ℓ-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 (ℓ,ℓ)-isogenies starting from a jacobian with maximal endomorphism ring.