Concurrent Zero Knowledge Without Complexity Assumptions

Abstract

Abstract. We provide unconditional constructions of concurrent statistical zero-knowledge proofs for a variety of non-trivial problems (not known to have probabilistic polynomial-time algorithms). The problems include Graph Isomorphism, Graph Nonisomorphism, Quadratic Residuosity, Quadratic Nonresiduosity, a restricted version of Statistical Difference, and approximate versions of the (coNP forms of the) Shortest Vector Problem and Closest Vector Problem in lattices.For some of the problems, such as Graph Isomorphism and Quadratic Residuosity, the proof systems have provers that can be implemented in polynomial time (given an NP witness) and haveÕ(log n) rounds, which is known to be essentially optimal for black-box simulation.To the best of our knowledge, these are the first constructions of concurrent zero-knowledge proofs in the plain, asynchronous model (i.e., without setup or timing assumptions) that do not require complexity assumptions (such as the existence of one-way functions).

A Characterization of Non-interactive Instance-Dependent Commitment-Schemes (NIC)

“…Our framework strengthens and simplifies the result of Micciancio, Ong, Sahai, and Vadhan [18], who showed that a NIC with reversed properties 3 can replace the bit commitment-scheme in the protocol of [23]. Unlike [18], since we already have a characterization result, we do not need to construct such a NIC for specific problems (e.g., GRAPH-NONISOMORPHISM) or to be familiar with their definition (e.g., the lattice problems of [19]). Also, our framework shows that such NIC are closed under monotone boolean formulae.…”

“…For example, applying this framework to the result of Micciancio et al [18] (who showed that some problems, including GRAPH-NONISOMORPHISM and QUADRATIC-RESIDUOUSITY, unconditionally have a concurrent zero-knowledge proof) we easily get that arbitrary, monotone boolean formulae over a large class of problems (which contains, e.g., the complement of any random self-reducible problem) unconditionally have a concurrent zero-knowledge proof.…”

Concurrent Statistical Zero-Knowledge Arguments for NP from One Way Functions

“…In [MOSV06], Micciancio et al show how to build concurrent statistical zero-knowledge proofs for a variety of problems unconditionally, that is, without making any unproven complexity assumptions. However since these were statistical zero-knowledge proofs, their results could not include proofs for all languages in NP (unless NP is in AM∩coAM and the polynomial hierarchy collapses).…”

“…We will follow [MOSV06] in formalizing the properties of the PRS preamble we need. Without loss of generality, assume that there are Q concurrent sessions.…”

“…Lemma 1 (implicit in [PRS02], adapted from [MOSV06]). There exists a PPT concurrently-extractable simulator CEC-Sim with a fixed strategy SIMULATE such that for COM and all concurrent adversaries V * , for settings of parameters σ=poly(n), k =Õ(log n), and Q =poly(n), we have the ensembles…”