Threshold Schemes from Isogeny Assumptions

Abstract. We initiate the study of threshold schemes based on the Hard Homogeneous Spaces (HHS) framework of Couveignes. Quantum- resistant HHS based on supersingular isogeny graphs have recently be- come usable thanks to the record class group precomputation performed for the signature scheme CSI-FiSh. Using the HHS equivalent of the technique of Shamir’s secret sharing in the exponents, we adapt isogeny based schemes to the threshold setting. In particular we present threshold versions of the CSIDH public key encryption, and the CSI-FiSh signature schemes. The main highlight is a threshold version of CSI-FiSh which runs almost as fast as the original scheme, for message sizes as low as 1880 B, public key sizes as low as 128 B, and thresholds up to 56; other speed-size- threshold compromises are possible.