Algebraic Techniques for Short(er) Exact Lattice-Based Zero-Knowledge Proofs

A key component of many lattice-based protocols is a zero-knowledge proof of knowledge of a vector \(\vec {s}\) with small coefficients satisfying \(A\vec {s}=\vec {u}\bmod \,q\). While there exist fairly efficient proofs for a relaxed version of this equation which prove the knowledge of \(\vec {s}'\) and c satisfying \(A\vec {s}'=\vec {u}c\) where \(\Vert \vec {s}'\Vert \gg \Vert \vec {s}\Vert \) and c is some small element in the ring over which the proof is performed, the proofs for the exact version of the equation are considerably less practical. The best such proof technique is an adaptation of Stern’s protocol (Crypto ’93), for proving knowledge of nearby codewords, to larger moduli. The scheme is a \(\varSigma \)-protocol, each of whose iterations has soundness error \(2{/}3\), and thus requires over 200 repetitions to obtain soundness error of \(2^{-128}\), which is the main culprit behind the large size of the proofs produced.