Triply Adaptive UC NIZK

Non-interactive zero knowledge (NIZK) enables proving the validity of NP statement without leaking anything else. We study multiinstance 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 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 nonfalsifiable knowledge assumptions). In addition, our NIZKs are universally composable (UC). Along the way, we: - Formulate an ideal functionality, FNICOM, which essentially captures non-interactive 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 F NICOM. - Use the Fiat-Shamir transform, instantiated with correlation intractable hash functions, to compile a Sigma protocol with triply adaptive security with access to F NICOM into a triply adaptive UCNIZK argument in the CRS model with access to FNICOM, assuming LWE (or else LPN and DDH). - Use the UC theorem to obtain UC-NIZK in the CRS model.