Proofs of membership vs. proofs of knowledge

We investigate the relationship between interactive proofs of membership and interactive proofs of knowledge. Previous results in this area show that many proofs of membership for some languages are also proofs of knowledge of an associated relation, raising the question of whether all proofs of membership are proofs of knowledge. In this paper we clarify the relationship between these two notions of proofs. It turns out that a precise relationship depends on the kind of relation considered. Clearly, any proof of membership is a proof of knowledge for some easy to compute relation. On the other hand, we define a notion of tight relations, referring to relations that capture the computational advantage communicated by a prover to a poly-time verifier in an interactive protocol