I am not an expert in Zero-knowledge proofs and I am interested in its applicability to general mathematical proofs.
Suppose we're willing to assume that NP-complete problems are intractable and require exponential time algorithms.
Can we construct a zero-knowledge proof of the intractability of Hamiltonian cycle problem, under this assumption? Is it possible without assuming the existence of one-way functions?
"A zero-knowledge proof of the intractability of the Hamiltonian cycle problem" doesn't make sense. You should refer to the formal definition of a zero-knowledge proof system. When we talk about a zero-knowledge proof system, it is a zero-knowledge proof system for a family of statements. For the problem to be interesting, the family needs to be infinite.
"The intractability of the Hamiltonian cycle problem" is not a family of statements; it is a single statement. It doesn't make sense to talk about a zero-knowledge proof in that sense -- you end up with something trivial and degenerate and uninteresting.
In particular, that single statement ("the Hamiltonian cycle problem is intractable") is either true, or it is false. If it's true, there is a trivial zero-knowledge proof system for it: the prover sends nothing, and the verifier outputs "yes, the statement is true". If it's false, there is also a trivial zero-knowledge proof system for it: the prover sends nothing, and the verifier outputs "no, the statement is false". Either way, there exists a zero-knowledge proof system that satisfies the definition. But this is not very interesting or meaningful.
From a formal mathematical perspective, in the phrase "zero-knowledge proof", you shouldn't think about "zero-knowledge" as a modifier to the word "proof". Rather, you should think about the phrase "zero-knowledge proof system" as an indivisible term with a specific mathematical definition. A "zero-knowledge proof system" is not even in the same category as a "proof"; they are like apples and oranges.
Ostrovsky and Wigderson showed that one-way functions are necessary for zero-knowledge proofs for languages beyond BPP. To the extent that your question makes sense at all, this should show that you do need one-way functions.
