 Proof of knowledge

A proof of knowledge is a proof protocol whose success is linked to an efficient procedure for obtaining a cryptographic witness from the prover's behavior. The procedure is a knowledge extractor, with access to the prover specified by the definition. The ordinary verifier in an interactive proof need not receive the witness during the real exchange.^[ref]

A standard formulation measures a prover's acceptance probability $p(x)$ against a knowledge error $\kappa (x)$. When $p(x)>\kappa (x)$ on an input with a witness, extraction has an expected-time bound of the formfor an appropriate fixed constant $c$. The relation, oracle access and false-input guarantees must also be specified; this expression alone is not the complete definition.^[ref]

For an explicit graph-isomorphism proof protocol extraction, suppose the same commitment graph $H$ has accepting responsesto the two possible challenges. The compositionis a graph isomorphism, hence a witness for the public claim. Rewinding in cryptographic proofs can provide the extractor with differently challenged continuations of a common earlier state.

The formula explains a knowledge claim without invoking a human notion of understanding. Soundness of an interactive proof and zero-knowledge proof privacy are separate requirements: excluding false statements and hiding information do not by themselves establish extraction.

Proof of knowledge is a concept in cryptography and computer science that pertains to demonstrating that a party (the prover) possesses certain information (usually a secret or private key) without revealing the information itself. It is often associated with zero-knowledge proofs, which are protocols that allow one party to prove to another that they know a value (like a password or cryptographic key) without disclosing the value itself.