Paper 2020/1212

Triply Adaptive UC NIZK

Abstract

Non-interactive zero knowledge (NIZK) enables proving the validity of NP statement without leaking anything else. We study multi-instance NIZKs in the common reference string (CRS) model, against an adversary that adaptively corrupts parties and chooses statements to be proven. We construct the first such $\mathit{\text{triply adaptive}}$ NIZK that provides full adaptive soundness, as well as adaptive zero-knowledge, assuming either LWE or else LPN and DDH (previous constructions rely on non-falsifiable knowledge assumptions). In addition, our NIZKs are universally composable (UC). Along the way, we: - Formulate an ideal functionality, $$, which essentially captures $$ commitments, and show that it is realizable by existing protocols using standard assumptions. - Define and realize, under standard assumptions, Sigma protocols which satisfy triply adaptive security with access to $$. - Use the Fiat-Shamir transform, instantiated with correlation intractable hash functions, to compile a Sigma protocol with triply adaptive security with access to $$ into a triply adaptive UC-NIZK argument in the CRS model with access to $$, assuming LWE (or else LPN and DDH). - Use the UC theorem to obtain UC-NIZK in the CRS model.

Note: - Added notes in the non-interactive UC-commitment functionality (F_NICOM in Fig. 2) to denote the steps that capture the properties of binding, hiding, equivocation, and extraction.