Moz$\mathbb {Z}_{2^k}$arella: Efficient Vector-OLE and Zero-Knowledge Proofs over $\mathbb {Z}_{2^k}$

Abstract

Zero-knowledge proof systems are usually designed to support computations for circuits over $\mathbb {F}_2$ or $\mathbb {F}_p$ for large p, but not for computations over $\mathbb {Z}_{2^k}$, which all modern CPUs operate on. Although $\mathbb {Z}_{2^k}$-arithmetic can be emulated using prime moduli, this comes with an unavoidable overhead. Recently, Baum et al. (CCS 2021) suggested a candidate construction for a designated-verifier zero-knowledge proof system that natively runs over $\mathbb {Z}_{2^k}$. Unfortunately, their construction requires preprocessed random vector oblivious linear evaluation (VOLE) to be instantiated over $\mathbb {Z}_{2^k}$. Currently, it is not known how to efficiently generate such random VOLE in large quantities.

In this work, we present a maliciously secure, VOLE extension protocol that can turn a short seed-VOLE over $\mathbb {Z}_{2^k}$ into a much longer, pseudorandom VOLE over the same ring. Our construction borrows ideas from recent protocols over finite fields, which we non-trivially adapt to work over $\mathbb {Z}_{2^k}$. Moreover, we show that the approach taken by the QuickSilver zero-knowledge proof system (Yang et al. CCS 2021) can be generalized to support computations over $\mathbb {Z}_{2^k}$. This new VOLE-based proof system, which we call QuarkSilver, yields better efficiency than the previous zero-knowledge protocols suggested by Baum et al. Furthermore, we implement both our VOLE extension and our zero-knowledge proof system, and show that they can generate 13–50 million VOLEs per second for ${64}\,{\textrm{bit}}$ to ${256}\,{\textrm{bit}}$ rings, and evaluate ${1.3}\,\textrm{million}$ ${64}\,{\textrm{bit}}$ multiplications per second in zero-knowledge.