Lattice-Based Zero-Knowledge Proofs and Applications: Shorter, Simpler, and More General

We present a much-improved practical protocol, based on the hardness of Module-SIS and Module-LWE problems, for proving knowledge of a short vector s satisfying A s " t mod q. The currently mostefficient technique for constructing such a proof works by showing that the ℓ∞ norm of s is small. It creates a commitment to a polynomial vector m whose CRT coefficients are the coefficients of s and then shows that (1) A • CRT(m) " t mod q and (2) in the case that we want to prove that the ℓ∞ norm is at most 1, the polynomial product (m ´1) • m • (m `1) equals to 0. While these schemes are already quite practical, the requirement of using the CRT embedding and only being naturally adapted to proving the ℓ∞-norm, somewhat hinders the efficiency of this approach.

In this work, we show that there is a more direct and more efficient way to prove that the coefficients of s have a small ℓ2 norm which does not require an equivocation with the ℓ∞ norm, nor any conversion to the CRT representation. We observe that the inner product between two vectors r and s can be made to appear as a coefficient of a product (or sum of products) between polynomials which are functions of r and s. Thus, by using a polynomial product proof system and hiding all but one coefficient, we are able to prove knowledge of the inner product of two vectors (or of a vector with itself) modulo q. Using a cheap, "approximate range proof", one can then lift the proof to be over Z instead of Zq. Our protocols for proving short norms work over all (interesting) polynomial rings, but are particularly efficient for rings like Z[X](X n `1) in which the function relating the inner product of vectors and polynomial products happens to be a "nice" automorphism.

The new proof system can be plugged into constructions of various lattice-based privacy primitives in a black-box manner. As examples, we instantiate a verifiable encryption scheme and a group signature scheme which are more than twice as compact as the previously best solutions.