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Cryptographic setup is the generation and distribution of initial parameters or other resources required by a cryptographic protocol. Its specification includes the input cryptographic security parameter, the sampling procedure, the public output and any information that must remain secret. Setup is part of the protocol's security model, rather than an arbitrary source of well-formatted constants.

A common reference string gives one example. Parties to a non-interactive zero-knowledge proof may share a public string $\sigma$ sampled by the prescribed setup. The definition describes both its distribution and how that setup participates in real and simulated proof experiments.^[ref]

An original distribution example makes the generation rule explicit. Let a toy setup sample two independent uniform bits $a,b$ and publishIts support is $\{000,011,101,110\}$, each with probability $1/4$. A uniformly sampled three-bit string instead has eight possible values, each with probability $1/8$. The two generators publish strings of the same length, but the second satisfies the parity equation only half the time.

This toy setup is not a secure proof-system construction. It demonstrates why replacing a specified generator by a different random-looking source changes the experiment. Public availability of setup data and correctness of its generation are separate properties.

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.