One-message statistical Zero-Knowledge Proofs and space-bounded verifier
Traditional mathematical proofs are communicated by a paper or a book as a single message. The reader has to understand the proof and be persuaded by it without any interaction or coordination with the writer of the proof. Statements whose proofs can be written down as a single message and then verified efficiently constitute the class NP.
In modern communication settings, such as distributed and cryptographic scenarios, hiding some or all of the details of a proof while convincing the verifier of the validity of a statement is an essential operation. This is captured by the important concept of Zero-Knowledge Proofs which deals with characterizations and techniques of how to convince a polynomial-time machines that a statement is true without giving any clue about the proof.
All NP statements have a zero-knowledge proof but, unlike written proofs, these proofs require either interaction between the verifier and the prover , or agreement between the verifier and the prover on some initial truly random string independent of the theorem [1, 2] and thus are not “one-message” proofs. In fact, it has been shown  that only languages in BPP have “one-message proofs” and thus, if NP had one-message proofs, the seemingly very unlikely conclusion would be that \(NP \subseteq BPP\). However, since the traditional one-message proofs minimize coordination and interaction, it is important to characterize when it is still possible to have meaningful one-message zero-knowledge proofs for NP.
In this work we show that when a bound on the space of the verifier is known (which is applicable in various computational settings), then all NP statements have one-message zero-knowledge proofs. Furthermore, our one-message proofs are perfect zero-knowledge, which without the space bound would imply the unlikely event that PH=Σ2 [3, 11].
KeywordsTuring Machine Proof System Input Tape Information Block Random Tape
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