Paper 2026/546

Hyperelliptic Gluing Isogeny Diffie–Hellman (HGIDH): A Genus-2 Gluing Isogeny Key-Exchange

Abstract

In this work, we propose Hyperelliptic Gluing Isogeny Diffie– Hellman (HGIDH), a key-exchange protocol built from gluing isogenies between the product of two supersingular elliptic curves and the Jacobian of a genus-2 hyperelliptic curve. The protocol leverages the Frey–Kani correspondence, using maximal isotropic subgroups of (E1 × E2)[N] to construct principally polarized abelian surfaces. Private keys are encoded as four scalars defining a non-cyclic, two-dimensional kernel, thereby avoiding the structural weaknesses exploited in SIDH-style attacks. We formalize the computational tasks underlying attacks on HGIDH through two intermediate problem formulations, which abstract the recovery of gluing kernels and the computation of genus-2 isogenies. We show that any efficient adversary solving these problems can be transformed into an algorithm solving standard supersingular isogeny problems, situating the security of HGIDH within the established hardness landscape. Furthermore, we analyze the resistance of the construction to known classical and quantum attacks, including torsion-point attacks and Costello–Smith-style meet-in-the-middle strategies.