Interactive proof system ($(P,V)$)

An interactive proof system is a protocol in which a prover in an interactive proof exchanges messages with a verifier in an interactive proof to establish a public statement. The verifier is a randomized algorithm running in polynomial time; the prover need not have that resource bound. Completeness of an interactive proof and soundness of an interactive proof specify acceptance behavior on true and false statements.^[ref]

For a decision problem represented by a set $L$, write the acceptance bounds asThe first condition supplies an honest proving strategy; the second constrains every cheating strategy, including adaptive answers. The bounds need a useful separation between $c$ and $s$.

Graph non-isomorphism gives a concrete protocol. Given two graphs $G_0,G_1$ on the same number of vertices, the verifier secretly chooses a uniform bit $b$, randomly relabels $G_b$ using a permutation, and sends the result $H$. The prover guesses which input graph was chosen; the verifier accepts exactly when the guess equals $b$.^[ref]

If the graphs are non-isomorphic, an unrestricted prover can identify the correct isomorphism class and always answer correctly. If they are isomorphic, the two ways of generating $H$ have identical probability distributions, so $H$ gives no information about $b$ and any guess succeeds with probability $1/2$. The verifier's secret random choice creates the test. This soundness argument alone makes no claim that the protocol is a zero-knowledge proof.

A verifier in an interactive proof is the efficient participant that evaluates a public statement using messages from a prover in an interactive proof and its own random choices. Its algorithm specifies which challenge to send, how to process the response and when to accept or reject. In the standard model it runs in probabilistic polynomial time in the public input length.^[ref]

The verifier's guarantee concerns more than one successful conversation. Completeness of an interactive proof requires a suitable honest prover to succeed on true statements. Soundness of an interactive proof bounds success of every cheating strategy on false statements. A verifier that accepts whenever it receives any message meets neither requirement merely by being fast.

For an original checking example, suppose a response claims a graph isomorphism between two20-vertex graphs encoded by adjacency matrices. Checking that the vertex mapping is a bijection and that all $20^2=400$ adjacency entries agree with their mapped entries verifies that response. It does not require searching through all $20!$ vertex permutations.

An interactive proof system can use such inexpensive checks in a randomized conversation. The verifier's random choices must follow the specified protocol: exposing a challenge too early, reusing randomness in an unjustified way or changing the acceptance rule can alter its mathematical guarantees.

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.

Soundness of an interactive proof bounds how often a verifier in an interactive proof can be induced to accept a false statement. For every false input $x$ and every cheating prover in an interactive proof strategy $P^{\ast }$, an acceptance bound $s(|x|)$ requiresIn the usual interactive proof system definition, the quantifier includes computationally unrestricted cheating strategies. Bounding only one tested strategy is insufficient.^[ref]

An original repetition example shows why conditional bounds matter. Suppose a single fair bit $B$ is reused in two tests, and both accept exactly when $B=0$. Each test separately accepts with probability $1/2$, but both accept with probability $1/2$, not $1/4$. Merely multiplying marginal probabilities would be wrong.

Instead, if every next test has acceptance probability at most $s$ conditional on every preceding accepting history, the chain rule for probabilities giveswith the first term unconditional and zero-probability histories handled trivially. For $s=1/2$ and $k=8$, the bound is $1/256$. The relevant protocol must justify those conditional bounds, rather than assuming that repetition automatically makes every transcript independent.

Soundness protects the false-statement side of verification. It neither guarantees acceptance of true statements nor establishes that a successful interaction hides a cryptographic witness; those are separate completeness and zero-knowledge proof questions.

Completeness of an interactive proof guarantees that true statements can convince the verifier in an interactive proof when the designated prover in an interactive proof follows the protocol. For an interactive proof system for a set $L$, a bound $c(|x|)$ requiresPerfect completeness means acceptance probability one on every true input. This is different from soundness of an interactive proof, which concerns every cheating strategy on false inputs.^[ref]

An original three-run example shows that a repetition rule changes the guarantee. Suppose three independent runs on a true statement each accept with probability $c=0.9$. Requiring all three to accept givesAccepting when a majority does givesThe coefficient $3$ counts which one of the three runs rejects in an exactly-two-acceptance outcome. These stipulated independent trials illustrate the aggregation rule; the soundness of the majority rule must be assessed separately.

Thus lowering the chance of false acceptance is not the only design objective. A verifier that always rejects has excellent false-statement behavior but no useful completeness. A successful proof system needs both sides of its acceptance guarantee.