Secret Keys in Genus-2 SIDH

We present a polynomial-time adaptive attack on the genus-2 variant of SIDH (G2SIDH) and describe an improvement to its secret selection procedure. G2SIDH is a generalisation of the Supersingular Isogeny Diffie-Hellman key exchange into the genus-2 setting and achieves the same security as SIDH while using fields a third of the size. We analyze the keyspace of G2SIDH and achieve an improvement to the secret selection by using symplectic bases for the torsion subgroups. This allows for the near uniform sampling of secrets without needing to solve multiple linear congruences as suggested by Flynn-Ti. More generally, using symplectic bases enables us to classify and enumerate isogeny kernel subgroups and thus simplify the secret sampling step for general genus-2 SIDH-style constructions. The proposed adaptive attack on G2SIDH is able to recover the secret when furnished with an oracle that returns a single bit of information. We ensure that the maliciously generated information provided by the attacker cannot be detected by implementing simple countermeasures, forcing the use of the Fujisaki-Okamoto transform for CCA2-security. We demonstrate this attack and show that it is able to recover the secret isogeny in all cases of G2SIDH using a symplectic basis before extending the strategy to arbitrary bases.