Non-interactive zero-knowledge proof

A non-interactive zero-knowledge proof allows a prover to send one proof message that a verifier can check without an ensuing challenge-response conversation, while satisfying an appropriate zero-knowledge proof definition. Non-interactive describes the proof exchange; it does not mean that setup assumptions disappear.^[ref]

In a common reference string model, both parties use public setup data $\sigma$. The prover constructs a proof $\pi$ for a public statement $x$, and the verifier evaluatesThe setup's distribution and the adversary's knowledge of its generation must be specified. In the tutorial's basic common-random-string model, the zero-knowledge simulator produces a reference-string/proof pair with the required computational indistinguishability from a real pair.^[ref]

An original toy relation separates a statement from a cryptographic witness. Let the public values be $n=15$ and $y=4$, and ask whether there is a value $w$ withThe value $w=2$ satisfies the relation because $2^2\equiv 4(\mathrm{mod}15)$. Sending $w$ directly verifies that example but reveals that witness; it does not demonstrate a zero-knowledge property. A non-interactive proof would instead require a defined proof construction, verifier and simulation argument.

The tiny modulus is chosen only for transparent modular arithmetic; it is not a secure cryptographic parameter. Likewise, a single short message is not automatically a non-interactive zero-knowledge proof: efficient verification, soundness and information disclosure remain separate obligations.