Shorter Non-interactive Zero-Knowledge Arguments and ZAPs for Algebraic Languages

We put forth a new framework for building pairing-based non-interactive zero- knowledge (NIZK) arguments for a wide class of algebraic languages, which are an extension of linear languages, containing disjunctions of linear languages and more. Our approach differs from the Groth-Sahai methodology, in that we rely on pairings to compile a $\Sigma$-protocol into a NIZK. Our framework enjoys a number of interesting features: – conceptual simplicity, parameters derive from the $\Sigma$-protocol; – proofs as short as resulting from the Fiat-Shamir heuristic applied to the underlying $\Sigma$-protocol; – fully adaptive soundness and perfect zero-knowledge in the common random string model with a single random group element as CRS; – yields simple and efficient two-round, public coin, publicly-verifiable perfect witness-indistinguishable (WI) arguments (ZAPs) in the plain model. To our knowledge, this is the first construction of two-rounds statistical witness-indistinguishable arguments from pairing assumptions. Our proof system relies on a new (static, falsifiable) assumption over pairing groups which generalizes the standard kernel Diffie-Hellman assumption in a natural way and holds in the generic group model (GGM) and in the algebraic group model (AGM). Replacing Groth-Sahai NIZKs with our new proof system allows to improve several important cryptographic primitives. In particular, we obtain the shortest tightly-secure structure-preserving signature scheme (which are a core component in anonymous credentials), the shortest tightly-secure quasi-adaptive NIZK with unbounded simulation soundness (which in turns implies the shortest tightly-mCCA-secure cryptosystem), and shorter ring signatures.