International Association
for Cryptologic Research

We introduce the first candidate lattice-based Designated Verifier (DV) ZK-SNARK protocol with \emph{quasi-optimal proof length} (quasi-linear in the security/privacy parameter), avoiding the use of the exponential smudging technique. Our ZK-SNARK also achieves significant improvements in proof length in practice, with proofs length below $6$ KB for 128-bit security/privacy level. Our main technical result is a new regularity theorem for `private' re-randomization of Module LWE (MLWE) samples using discrete Gaussian randomization vectors, also known as a lattice-based leftover hash lemma with leakage, which applies with a discrete Gaussian re-randomization parameter that is polynomial in the statistical privacy parameter. To obtain this result, we obtain bounds on the smoothing parameter of an intersection of a random $q$-ary SIS module lattice, Gadget SIS module lattice, and Gaussian orthogonal module lattice over standard power of 2 cyclotomic rings, and a bound on the minimum of module gadget lattices. We then introduce a new candidate \emph{linear-only} homomorphic encryption scheme called Module Half-GSW (HGSW), which is a variant of the GSW somewhat homomorphic encryption scheme over modules, and apply our regularity theorem to provide smudging-free circuit-private homomorphic linear operations for Module HGSW.