Weak Zero-Knowledge Beyond the Black-Box Barrier

Abstract

The round complexity of zero-knowledge protocols is a long-standing open question, yet to be settled under standard assumptions. So far, the question has appeared equally challenging for relaxations such as weak zero-knowledge and witness hiding. Protocols satisfying these relaxed notions under standard assumptions have at least four messages, just like full-fledged zero knowledge. The difficulty in improving round complexity stems from a fundamental barrier: none of these notions can be achieved in three messages via reductions (or simulators) that treat the verifier as a black box. We introduce a new non-black-box technique and use it to obtain the first protocols that cross this barrier under standard assumptions. Our main results are: \begin{itemize} \item Weak zero-knowledge for $NP$in two messages, assuming quasipolynomially-secure fully-homomorphic encryption and other standard primitives (known from quasipolynomial hardness of Learning with Errors), as well as subexponentially-secure one-way functions. \item Weak zero-knowledge for $NP$ in three messages under standard polynomial assumptions (following for example from fully-homomorphic encryption and factoring). \end{itemize} We also give, under polynomial assumptions, a two-message witness-hiding protocol for any language $L \in NP$ that has a witness encryption scheme. This protocol is also publicly verifiable. Our technique is based on a new {\em homomorphic trapdoor paradigm}, which can be seen as a non-black-box analog of the classic Feige-Lapidot-Shamir trapdoor paradigm