## Cryptology ePrint Archive: Report 2019/177

Genus Two Isogeny Cryptography

E.V. Flynn and Yan Bo Ti

Abstract: We study $(\ell,\ell)$-isogeny graphs of principally polarised supersingular abelian surfaces (PPSSAS). The $(\ell,\ell)$-isogeny graph has cycles of small length that can be used to break the collision resistance assumption of the genus two isogeny hash function suggested by Takashima. Algorithms for computing $(2,2)$-isogenies on the level of Jacobians and $(3,3)$-isogenies on the level of Kummers are used to develop a genus two version of the supersingular isogeny Diffie--Hellman protocol of Jao and de~Feo. The genus two isogeny Diffie--Hellman protocol achieves the same level of security as SIDH but uses a prime with a third of the bit length.

Category / Keywords: public-key cryptography / Post-quantum cryptography, Isogeny-based cryptography, Cryptanalysis, Key exchange, Hash function

Original Publication (in the same form): PQCrypto 2019