Exploring Constructions of Compact NIZKs from Various Assumptions

A non-interactive zero-knowledge (NIZK) protocol allows a prover to non-interactively convince a verifier of the truth of the statement without leaking any other information. In this study, we explore shorter NIZK proofs for all \(\mathbf{NP }\) languages. Our primary interest is NIZK proofs from falsifiable pairing/pairing-free group-based assumptions. Thus far, NIZKs in the common reference string model (CRS-NIZKs) for \(\mathbf{NP }\) based on falsifiable pairing-based assumptions all require a proof size at least as large as \(O(|C| \kappa )\), where C is a circuit computing the \(\mathbf{NP }\) relation and \(\kappa \) is the security parameter. This holds true even for the weaker designated-verifier NIZKs (DV-NIZKs). Notably, constructing a (CRS, DV)-NIZK with proof size achieving an additive-overhead \(O(|C|) + \mathsf {poly}(\kappa )\), rather than a multiplicative-overhead \(|C| \cdot \mathsf {poly}(\kappa )\), based on any falsifiable pairing-based assumptions is an open problem.

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