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

Sorina Ionica (1)

1 CARAMEL - Cryptology, Arithmetic: Hardware and Software

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.

Mots clés: abelian variety; Tate pairing; Galois cohomology

Domaines: Mathématiques générales [math.GM]

Sorina Ionica. Pairing-based algorithms for jacobians of genus 2 curves with maximal endomorphism ring. 2012. ⟨hal-00675045v3⟩