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

##### Abstract

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 $As=t\bmod q$. The currently most-efficient technique for constructing such a proof works by showing that the $\ell_\infty$ 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) $A\cdot \mathsf{CRT}(m)=t\bmod\,q$ and (2) in the case that we want to prove that the $\ell_\infty$ norm is at most $1$, the polynomial product $(m - 1)\cdot m\cdot(m+1)$ equals to $0$. While these schemes are already quite good for practical applications, the requirement of using the CRT embedding and only being naturally adapted to proving the $\ell_\infty$-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 $\ell_2$ norm which does not require an equivocation with the $\ell_\infty$ 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 modulo $q$. Using a cheap, approximate range proof, one can then lift the proof to be over $\mathbb{Z}$ instead of $\mathbb{Z}_q$. Our protocols for proving short norms work over all (interesting) polynomial rings, but are particularly efficient for rings like $\mathbb{Z}[X]/(X^n+1)$ 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.

Available format(s)
Category
Public-key cryptography
Publication info
A major revision of an IACR publication in Crypto 2022
Keywords
lattices lattice-based zero-knowledge proofs
Contact author(s)
nkn @ zurich ibm com
mpl @ zurich ibm com
History
2022-05-25: last of 3 revisions
See all versions
Short URL
https://ia.cr/2022/284

CC BY

BibTeX

@misc{cryptoeprint:2022/284,
author = {Vadim Lyubashevsky and Ngoc Khanh Nguyen and Maxime Plancon},
title = {Lattice-Based Zero-Knowledge Proofs and Applications: Shorter, Simpler, and More General},
howpublished = {Cryptology ePrint Archive, Paper 2022/284},
year = {2022},
note = {\url{https://eprint.iacr.org/2022/284}},
url = {https://eprint.iacr.org/2022/284}
}

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