# What do we know about the relationship between IP, ZKP?

The question is fairly simple and is actually stated in the title although I will elaborate a bit further on it. Also, I'm not really sure if this belongs here or it partly belongs to cstheory.stackexchange.com.

Currently I'm trying to understand ZKP as a class of problems. I have read that IP=PSPACE and that ZKP exists under the assumption of OWF, and it basically is IP with the extra property of zero knowledge. I also have read that any NP problem has a ZKP, so NP $$\subseteq$$ ZKP. What do we know about IP and ZKP?

Edit : I haven't seen any classification diagrams with ZKP mentioned. Is there any resource available for this?

It is known that $$\mathsf{IP} = \mathsf{PSPACE} = \mathsf{CZK}$$ assuming the existence of one-way functions, see here.

The existence of one-way functions is necessary, since it is already known that one-way functions are (essentially - there are minor caveats) necessary to prove that $$\mathsf{CZK}$$ contains $$\mathsf{NP}$$. See here and here.

Of course, the first equality is for $$\mathsf{CZK}$$ with an all powerful prover. In cryptography, we are typically interested in zero-knowledge proofs where the prover is efficient given a witness. Having a witness for the language implies that the language is in $$\mathsf{NP}$$; it is known that, assuming one-way functions, all languages in $$\mathsf{NP}$$ have a zero-knowledge proofs with an efficient prover (see here), and as I said above, one-way functions are necessary for this.

Not sure, but I think I found the answer, I would really like to hear if this is actually correct.

The site complexityzoo.net mentions the following :

Assuming the existence of one-way functions, CZK contains NP [GMW91], and actually equals IP=PSPACE [BGG+90]. However, none of these implications of one-way functions relativize (Impagliazzo, unpublished).

I'm not really sure I can understand the last sentence, IP=ZK under the assumption of OWF.