Prover in an interactive proof ($P$)

A prover in an interactive proof is the participant that sends messages intended to convince a verifier in an interactive proof of a public statement. Formally it is a strategy: its next response can depend on the public input, the preceding conversation and any private information or randomness it has. The basic interactive proof system model does not impose the verifier's polynomial time bound on this strategy.^[ref]

An honest prover follows a designated strategy supporting completeness of an interactive proof. A cheating prover may depart from it; soundness of an interactive proof must still hold for every allowed cheating strategy. Calling a participant a prover therefore does not mean that its assertions are trusted.

A cryptographic witness can make an otherwise difficult response easy to produce. For an original composition example, let $\varphi :G_0\to G_1$ be a known graph isomorphism. If the prover sends $H=\pi (G_0)$ for a chosen permutation $\pi$, it can supply $\pi$ as an isomorphism from $G_0$ to $H$, or $\pi \circ {\varphi }^{-1}$ as an isomorphism from $G_1$ to $H$. Neither response requires solving a new isomorphism search problem from scratch.

Thus a theoretical allowance of unlimited prover computation is different from a claim about implementation cost. Some protocols have efficient honest provers when the witness is supplied; others are studied primarily for the verification power they demonstrate.