Quasi-Optimal SNARGs via Linear Multi-Prover Interactive Proofs

Succinct non-interactive arguments (SNARGs) enable verifying$$\mathsf {NP} $$ computations with significantly less complexity than that required for classical$$\mathsf {NP} $$ verification. In this work, we focus on simultaneously minimizing the proof size and the prover complexity of SNARGs. Concretely, for a security parameter$$\lambda $$, we measure the asymptotic cost of achieving soundness error$$2^{-\lambda }$$ against provers of size$$2^\lambda $$. We say a SNARG is quasi-optimally succinct if its proof length is$$\widetilde{O}(\lambda )$$, and that it is quasi-optimal, if moreover, its prover complexity is only polylogarithmically greater than the running time of the classical$$\mathsf {NP} $$ prover. We show that this definition is the best we could hope for assuming that$$\mathsf {NP} $$ does not have succinct proofs. Our definition strictly strengthens the previous notion of quasi-optimality introduced in the work of Boneh et al. (Eurocrypt 2017). This work gives the first quasi-optimal SNARG for Boolean circuit satisfiability from a concrete cryptographic assumption. Our construction takes a two-step approach. The first is an information-theoretic construction of a quasi-optimal linear multi-prover interactive proof (linear MIP) for circuit satisfiability. Then, we describe a generic cryptographic compiler that transforms our quasi-optimal linear MIP into a quasi-optimal SNARG by relying on the notion of linear-only vector encryption over rings introduced by Boneh et al. Combining these two primitives yields the first quasi-optimal SNARG based on linear-only vector encryption. Moreover, our linear MIP construction leverages a new robust circuit decomposition primitive that allows us to decompose a circuit satisfiability instance into several smaller circuit satisfiability instances. This primitive may be of independent interest. Finally, we consider (designated-verifier) SNARGs that provide optimal succinctness for a non-negligible soundness error. Concretely, we put forward the notion of “1-bit SNARGs” that achieve soundness error$$1\text {/}2$$ with only one bit of proof. We first show how to build 1-bit SNARGs from indistinguishability obfuscation, and then show that 1-bit SNARGs also suffice for realizing a form of witness encryption. The latter result highlights a two-way connection between the soundness of very succinct argument systems and powerful forms of encryption.