RLWE-Based Zero-Knowledge Proofs for Linear and Multiplicative Relations

We present efficient Zero-Knowledge Proofs of Knowledge (ZKPoK) for linear and multiplicative relations among secret messages hidden as Ring Learning With Errors (RLWE) samples. Messages are polynomials in and our proposed protocols for a ZKPoK are based on the celebrated paper by Stern on identification schemes using coding problems (Crypto’93). Our 5-move protocol achieves a soundness error slightly above 1/2 and perfect Zero-Knowledge. As an application we present Zero-Knowledge Proofs of Knowledge of relations between committed messages. The resulting commitment scheme is perfectly binding with overwhelming probability over the choice of the public key, and computationally hiding under the RLWE assumption. Compared with previous Stern-based commitment scheme proofs we decrease computational complexity, improve the size of the parameters and reduce the soundness error of each round.