International Association
for Cryptologic Research

Suppose that a problem $\Pi$ has a statistical zero-knowledge (SZK) proof with communication complexity $m$. The question of batch verification for SZK asks whether one can prove that $k$ instances $x_1,\dots,x_k$ all belong to $\Pi$ with a statistical zero-knowledge proof whose communication complexity is better than $k \cdot m$ (which is the complexity of the trivial solution of executing the original protocol independently on each input). In a recent work, Kaslasi et al. (TCC, 2020) constructed such a batch verification protocol for any problem having a non-interactive SZK (NISZK) proof-system. Two drawbacks of their result are that their protocol is private-coin and is only zero-knowledge with respect to the honest verifier. In this work, we eliminate these two drawbacks by constructing a public-coin malicious-verifier SZK protocol for batch verification of NISZK. Similarly to the aforementioned prior work, the communication complexity of our protocol is $(k+poly(m)) \cdot polylog(k,m)$.

A statistical zero-knowledge proof (SZK) for a problem $\Pi$ enables a computationally unbounded prover to convince a polynomial-time verifier that $x \in \Pi$ without revealing any additional information about $x$ to the verifier, in a strong information-theoretic sense. Suppose, however, that the prover wishes to convince the verifier that $k$ separate inputs $x_1,\dots,x_k$ all belong to $\Pi$ (without revealing anything else). A naive way of doing so is to simply run the SZK protocol separately for each input. In this work we ask whether one can do better -- that is, is efficient batch verification possible for SZK? We give a partial positive answer to this question by constructing a batch verification protocol for a natural and important subclass of SZK -- all problems $\Pi$ that have a non-interactive SZK protocol (in the common random string model). More specifically, we show that, for every such problem $\Pi$, there exists an honest-verifier SZK protocol for batch verification of $k$ instances, with communication complexity $poly(n) + k \cdot poly(\log{n},\log{k})$, where $poly$ refers to a fixed polynomial that depends only on $\Pi$ (and not on $k$). This result should be contrasted with the naive solution, which has communication complexity $k \cdot poly(n)$. Our proof leverages a new NISZK-complete problem, called Approximate Injectivity, that we find to be of independent interest. The goal in this problem is to distinguish circuits that are nearly injective, from those that are non-injective on almost all inputs.