Zero Knowledge Proofs from Ring-LWE

Abstract

Zero-Knowledge proof is a very basic and important primitive, which allows a prover to prove some statement without revealing anything else. Very recently, Jain et al. proposed very efficient zero-knowledge proofs to prove any polynomial relations on bits, based on the Learning Parity with Noise (LPN) problem (Asiacrypt’12).

In this work, we extend analogous constructions whose security is based on the Ring Learning with Errors (RLWE) problem by adapting the techniques presented by Ling et al. (PKC’13). Specifically, we show a simple zero-knowledge proof of knowledge (Σ-protocol) for committed values, and prove any polynomial relations in the underlying ring. I.e. proving committed ring elements m, m ₁,...,m _t satisfying m = f(m ₁,...,m _t ) for any polynomial f. Comparing to other existing Σ-protocols, the extracted witness (error vector) has length only small constant times than the one possessed by the prover. When representing ring element as elements in ℤ _q , our protocol has amortized communication complexity $\tilde{O}(\lambda\cdot|f|)$ with exponentially small soundness in security parameter λ, where |f| is the size of the circuit in ℤ _q computing f.