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 $\ell$-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.

💡 Research Summary

The paper investigates the relationship between the ℓ‑Tate pairing on the Jacobian of a genus‑2 curve and the action of the Frobenius endomorphism on its ℓ‑torsion subgroup, using Galois cohomology as the main theoretical tool. Building on Schmoyer’s characterization of non‑trivial self‑pairings of the ℓ‑Tate pairing, the authors extend the analysis to the restriction of the pairing to maximal isotropic subgroups with respect to the Weil pairing. The central insight is that the non‑degeneracy of the restricted ℓ‑Tate pairing can be decided by examining the Frobenius matrix on a chosen basis of J