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Group Signature schemes are used to provide anonymity of the signers. During signature generation, signers provide a NIZK proof to prove that they are certified group members. Verifiers check whether the proof is valid during verification of signature. Can we use non-adaptive NIZK proof to prove signer is a certified group member?

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This question is hard to answer in general, without specifying a concrete group signature scheme and a security model in which you want the group signature scheme to be secure.

However, we may observe some things: first you will need a non-interactive proof/argument system as signatures are non-interactive by definition. Second, you will most likely be required to include the common reference string (CRS) of the proof/argument system in the group public key. Third, you will probably (in any reasonably strong model I can currently think of) have to simulate proofs (resp. signatures) after handing the group signing key over to the adversary. In these cases you will definitely require adaptive zero-knowledge proofs/arguments, i.e., ones that allow the adversary to choose the targeted statements after seeing the CRS.

Nevertheless, it can not be excluded that one can define a (weaker, maybe also non-adaptive) security model as well as a construction in this security model which does not have this requirement. The question is whether such a model would still provide the security guarantees one would hope for.

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