The knowledge complexity of quadratic residuosity languages

AbstractNoninteractive perfect zero-knowledge (ZK) proofs are very elusive objects. In fact, since the introduction of the noninteractive model of Blum . (1988), the only perfect zero-knowledge proof known was the one for quadratic nonresiduosity of Blum . (1991). The situation is no better in the interactive case where perfect zero-knowledge proofs are known only for a handful of particular languages.In this work, we show that a large class of languages related to quadratic residuosity admits noninteractive perfect zero-knowledge proofs. More precisely, we give a protocol for the language of thresholds of quadratic residuosity.Moreover, we develop a new technique for converting noninteractive zero-knowledge proofs into round-optimal zero-knowledge proofs for an even wider class of languages. The transformation preserves perfect zero knowledge in the sense that, if the noninteractive proof we started with is a perfect zero-knowledge proof, then we obtain a round-optimal perfect zero-knowledge proof. The noninteractive perfect zero-knowledge proofs presented in this work can be transformed into 4-round (which is optimal) interactive perfect zero-knowledge proofs. Until now, the only known 4-round perfect ZK proof systems were the ones for quadratic nonresiduosity (Goldwasser et al., 1989) and for graph nonisomorphism (Goldreich et al., 1986) and no 4-round perfect zero-knowledge proof system was known for the simple case of the language of quadratic residues

Increasing the Power of the Dealer in Non-interactive ZeroKnowledge Proof Systems

Abstract. We introduce weaker models for non-interactive zero knowledge, in which the dealer is not restricted to deal a truly random string and may also have access to the input to the protocol (i.e. the statement to prove). We show in these models a non-interactive statistical zero-knowledge proof for every language that has (interactive) statistical zero-knowledge proof, and a computational zero-knowledge proof for every language in NP. We also show how to change the latter proof system to fit the model of non-interactive computational zero-knowledge with preprocessingto improve existingresults in term of the number of bit commitments that are required for the protocol to work.