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I'm trying to study the difference between non-interactive zero knowledge (NIZK) proof of knowledge and proof of membership. I have known that in NIZK proof of knowledge the prover wants to convince the verifier that he knows the proof of a given theorem without revealing any additional information about the witness. In NIZK proof of membership the prover just convinces the verifier only of the veridicity of the statement. However, I cannot find good materials describing a formal definition of NIZK proof of membership. I have known that there exists an PPT extractor for NIZK proof of knowledge that can extract the witness, but what about proof of membership? What properties should it satisfy?

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