# ZK proof that I possess a ZK proof for membership in $L$?

A zero-knowledge proof system for a language $L$ is an interactive proof system where a prover $P$ (a Turing machine) tries to convince a verifier $V$ (a polynomially bounded Turing machine) in a sequence of message exchanges that $x \in L$ so that $V$ learns nothing from the interaction except whether $x \in L$ or not.

Question. Is it known how iterate this, i.e.: how to produce a ZK proof that I possess a ZK proof for membership in $L$?

• What would it mean to "possess" such a proof? ​ (possessing the transcript of such a proof?) ​ ​ ​ ​ – user12859 Sep 23 '15 at 13:42
• @RickyDemer I'm not sure how to define this properly. It's part of the question. A ZK proof system consists of a couple of TMs with certain properties. Can I demonstrate the possession of said TMs without leaking any other information? – Martin Berger Sep 23 '15 at 13:54
• Do you want it to be for the proof system or just for a particular x? ​ – user12859 Sep 23 '15 at 13:59
• @RickyDemer For the proof system. But I assume that the two problems are related. – Martin Berger Sep 23 '15 at 14:02
• In that case, something like MIP* would be needed, since the property of L having a zero-knowledge proof system is undecidable by reduction from the halting problem. ​ – user12859 Sep 23 '15 at 14:13

Here's a better instance of what you're looking for. Look up non-interactive zero knowledge proofs. Suppose I know a string $p$ that is a valid non-interactive zero knowledge (NIZK) proof of some statement $s$. Moreover, there is a polynomial-time algorithm that allows verifying whether $p$ is a valid NIZK proof of $s$. Therefore, there's a way I can use a ZK proof to prove to Alice in zero-knowledge that I know a string $p$ that is a valid NIZK proof of $s$, without revealing $p$ to Alice (because this is proving a NP statement).
That said, this probably isn't useful. What would be the point? I don't think this lets Alice conclude anything more than if I just proved in ZK to her that $s$ is true.