Proof of knowledge

In cryptography, a proof of knowledge is an interactive proof in which the prover succeeds in 'convincing' a verifier that the prover knows something. What it means for a machine to 'know something' is defined in terms of computation. A machine 'knows something', if this something can be computed, given the machine as an input. As the program of the prover does not necessarily spit out the knowledge itself (as is the case for zero-knowledge proofs) a machine with a different program, called the knowledge extractor is introduced to capture this idea. We are mostly interested in what can be proven by polynomial time bounded machines. In this case the set of knowledge elements is limited to a set of witnesses of some language in NP.

Let ${\displaystyle x}$ be a language element of language ${\displaystyle L}$ in NP, and ${\displaystyle W(x)}$ the set of witnesses for x that should be accepted in the proof. This allows us to define the following relation: ${\displaystyle R=\{(x,w):x\in L,w\in W(x)\}}$.

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