New Techniques for Zero-Knowledge: Leveraging Inefficient Provers to Reduce Assumptions and Interaction

We present a transformation from NIZK with inefficient provers in the uniform random string (URS) model to ZAPs (two message witness indistinguishable proofs) with inefficient provers. While such a transformation was known for the case where the prover is efficient, the security proof breaks down if the prover is inefficient. Our transformation is obtained via new applications of Nisan-Wigderson designs, a combinatorial object originally introduced in the derandomization literature. We observe that our transformation is applicable both in the setting of super-polynomial provers/poly-time adversaries, as well as a new fine-grained setting, where the prover is polynomial time and the verifier/simulator/zero knowledge distinguisher are in a lower complexity class, such as $\mathsf{NC}^1$. We also present $\mathsf{NC}^1$-fine-grained NIZK in the URS model for all of $\mathsf{NP}$ from the worst-case assumption \oplus L/\mathsf{\poly} \not\subseteq \mathsf{NC}^1. Our techniques yield the following applications: 1. ZAPs for $\mathsf{AM}$ from Minicrypt assumptions (with super-polynomial time provers), 2. $\mathsf{NC}^1$-fine-grained ZAPs for $\mathsf{NP}$ from worst-case assumptions, 3. Protocols achieving an offline\u27\u27 notion of NIZK (oNIZK) in the standard (no-CRS) model with uniform soundness in both the super-polynomial setting (from Minicrypt assumptions) and the $\mathsf{NC}^1$-fine-grained setting (from worst-case assumptions). The oNIZK notion is sufficient for use in indistinguishability-based proofs

All-But-Many Encryption: A New Framework for Fully-Equipped UC Commitments

We present a general framework for constructing non-interactive universally composable (UC) commitment schemes that are secure against adaptive adversaries in the non-erasure model under a re-usable common reference string. Previously, such ``fully-equipped\u27\u27 UC commitment schemes have been known only in [CF01,CLOS02], with strict expansion factor O(k); meaning that to commit L bits, communication strictly requires O(Lk)$ bits, where k denotes the security parameter. Efficient construction of a fully-equipped UC commitment scheme is a long-standing open problem. We introduce new abstraction, called all-but-many encryption (ABME), and prove that it captures fully-equipped UC commitment schemes. We propose the first fully-equipped UC commitment scheme with optimal expansion factor O(1) from our ABME scheme related to the DCR assumption. We also provide an all-but-many lossy trapdoor function (ABM-LTF)[Hof12] from our DCR-based ABME scheme, with a better lossy rate than [Hof12]