CSI-RAShi: Distributed Key Generation for CSIDH

We present an honest-majority Distributed Key Generation protocol (DKG) based on Shamir's (k, n)-threshold secret sharing in the setting of Very Hard Homogenous Spaces (VHHS). DKGs in the discrete logarithm setting use Pedersen commitments, for which there is no known analogue in the VHHS setting. As a replacement, we introduce a new primitive called piecewise verifiable proofs, which allow a prover to prove that a list of NP-statements is valid with respect to a common witness, and such that the different statements can be verified individually. Our protocol is robust and actively secure in the Quantum Random Oracle Model. For n participants, the total runtime of our protocol is 2 + lambda + n(1 + 4 lambda) group action evaluations, where lambda is the underlying security parameter, and is thus independent of the threshold k. When instantiated with CSIDH-512, this amounts to approximately 4.5 + 18n seconds.