Compact NIZKs from Standard Assumptions on Bilinear Maps

Abstract

A non-interactive zero-knowledge (NIZK) protocol enables a prover to convince a verifier of the truth of a statement without leaking any other information by sending a single message. The main focus of this work is on exploring short pairing-based NIZKs for all \({{\textbf {NP}}}\) languages based on standard assumptions. In this regime, the seminal work of Groth, Ostrovsky, and Sahai (J.ACM’12) (GOS-NIZK) is still considered to be the state-of-the-art. Although fairly efficient, one drawback of GOS-NIZK is that the proof size is multiplicative in the circuit size computing the \({{\textbf {NP}}}\) relation. That is, the proof size grows by \(O(|C|\kappa )\), where C is the circuit for the \({{\textbf {NP}}}\) relation and \(\kappa \) is the security parameter. By now, there have been numerous follow-up works focusing on shortening the proof size of pairing-based NIZKs, however, thus far, all works come at the cost of relying either on a non-standard knowledge-type assumption or a non-static q-type assumption. Specifically, improving the proof size of the original GOS-NIZK under the same standard assumption has remained as an open problem. Our main result is a construction of a pairing-based NIZK for all of \({{\textbf {NP}}}\) whose proof size is additive in |C|, that is, the proof size only grows by \(|C| +\textsf{poly}(\kappa )\), based on the computational Diffie-Hellman assumption over specific pairing-free groups and decisional linear (DLIN) assumption. As by-products of our main result, we also obtain the following two results: (1) We construct a perfectly zero-knowledge NIZK (NIPZK) for \({{\textbf {NP}}}\) relations computable in \({{\textbf {NC}}}^1\) with proof size \(|w| \cdot \textsf{poly}(\kappa )\) where |w| is the witness length based on the DLIN assumption. This is the first pairing-based NIPZK for a non-trivial class of \({{\textbf {NP}}}\) languages whose proof size is independent of |C| based on a standard assumption. (2) We construct a universally composable (UC) NIZK for \({{\textbf {NP}}}\) relations computable in \({{\textbf {NC}}}^1\) in the erasure-free adaptive setting whose proof size is \(|w| \cdot \textsf{poly}(\kappa )\) from the DLIN assumption. This is an improvement over the recent result of Katsumata, Nishimaki, Yamada, and Yamakawa (CRYPTO’19), which gave a similar result based on a non-static q-type assumption. The main building block for all of our NIZKs is a constrained signature scheme with decomposable online-offline efficiency. This is a property which we newly introduce in this paper and construct from the DLIN assumption. We believe this construction is of an independent interest.