Fair-Zero Knowledge.

We introduce Fair Zero-Knowledge , a multi-verifier ZK system where every proof is guaranteed to be “zero-knowledge for all verifiers.” That is, if an honest verifier accepts a fair zero-knowledge proof, then he is assured that all other verifiers also learn nothing more than the verity of the statement in question, even if they maliciously collude with a cheating prover. We construct Fair Zero-Knowledge systems based on standard complexity assumptions (specifically, the quadratic residuosity assumption) and an initial, one-time use of a physically secure communication channel (specifically, each verifier sends the prover a private message in an envelope). A ll other communication occurs (and must occur) on a broadcast channel. The main technical challenge of our construction consists of provably removing any possibility of using steganography in a ZK proof. To overcome this technical difficulty, we introduce tools —such as Unique Zero Knowledge— that may be of independent interest.