One thing I do not understand and would like to learn more about is zero-knowledge proofs is that their soundness seems to rest on the data-generating mechanism. Suppose you had an adversary Alice who claims to possess a constructive proof that graph-isomorphism [1] is in P. Secretly, she has proven that graph-isomorphism is NP-complete, but only for a vanishingly small family of graphs, and almost all random graphs…
You may be interested in Mahaney's Theorem ( https://en.m.wikipedia.org/wiki/Mahaney%27s_theorem ), which 'answers' a special case of this: If by vanishingly small you mean polynomial size, and you also assume Alice has an algorithm that can distinguish the two subclasses of graph isomorphism, then P=NP if Alice has these facts. By 'distinguish', I specifically mean that Alice has a polynomial time algorithm that wil…
Even if challenge instances are superpolynomial in the worst case, empirically, we know from SAT solving competitions that a very small fraction of randomly sampled k-SAT instances are truly hard and most are in P. It is nontrivial to design a challenge whose average case is superpolynomial, and there are many open questions in the field of average-case complexity [1].
I would be very skeptical that the challenge generator is not somehow poisoned. Even if the prover did not collude with the protocol designer to poison it directly, if she can infer any information about the internal state of the verifier from the instances he proposes, she may be able to solve future challenges much more easily than would be possible by random chance.