Abstract. We present a variant of Regev’s cryptosystem first presented in [Reg05], but with a new choice of parameters. By a recent classical re-duction by Peikert we prove the scheme semantically secure based on the worst-case lattice problem GapSVP. From this we construct a threshold cryptosystem which has a very efficient and non-interactive decryption protocol. We prove the threshold cryptosystem secure against passive adversaries corrupting all but one of the players, and againts active ad-versaries corrupting less than one third of the players. We also describe how one can build a distributed key generation protocol. In the final part of the paper we show how one can, in zero-knowledge- prove knowledge of the plaintext contained in a given ciphertext from Regev’s original cryptosystem or our variant. The proof is of size only a constant times the size of the public key. 1
We extend a commitment scheme based on the learning with errors over rings (
RLWE ) problem, and present efficient companion zero-knowledge proofs of knowledge. Our scheme maps elements from the ring (or equivalently, n elements from
Fq ) to a small constant number of ring elements. We then construct
Σ -protocols for proving, in a zero-knowledge manner, knowledge of the message contained in a commitment. We are able to further extend our basic protocol to allow us to prove additive and multiplicative relations among committed values. Our protocols have a communication complexity of
O(Mnlogq) and achieve a negligible knowledge error in one run. Here M is the constant from a rejection sampling technique that we employ, and can be set close to 1 by adjusting other parameters. Previously known
Σ -protocols for LWE-related languages only achieved a noticeable or even constant knowledge error (thus requiring many repetitions of the protocol), or relied on “smudging” out the error (which necessitates working over large fields, resulting in poor efficiency).