(You are not referring to the P != NP conjecture I suppose? It is just as uncertain, but it has absolutely no relevance to those two assumptions in the bitcoin protocol.)
I am. And it has. At least to 'what the protocol can handle in principle being thrown at it', not 'what the protocol is and does right now'. As long as there
are problems that are hard to solve but easy to check, the protocol can be adapted to the threat that a particular problem turned out to be easy to solve after all.
First, let me say that I was just reacting to that "pure incorruptible mathematics" bullshit by the reporter. I am
not arguing that the lack of mathematical proof iis a flaw of bitcoin. My skepticism does not come from that "flaw" (or from any other strictly technical issue), but from political/social/practical problems, for which my probabilities are obviously different from those of most people in this thread.
As for the P != NP conjecture, I do not need to tell you that it makes sense only if the problem instance size n is allowed to go to infinity, since the difference between polynomial and superpolynomial appears only in the limit, when n is "just about to get there". The two classes are indistinguishable if one assums that n is bounded, say n < 10^500. But the bitcoin mining problem and the signature forging problem have a fixed n, threfore only a finite number of instances; in that case, complexity theory says that both problems are trivial and can be solved in constant time (e.g. by a simple table lookup). Obviously this theoretical conclusion is irrelevant to the practical question.
The sad fact is that theoretical computer science does not have any tools yet that can provide useful lower bounds for problems of fixed size. Even assuming P != NP, there is no way to prove that the current protocol, with specific key and hash sizes, is safe. It may even be the case that forging a bitcoin transaction signature is easier than checking it, or that finding the right nonce for a block is easier than computing SHA256 twice -- independently of whether P = NP or not. We simply do not know.
If someday one were to find fast solutions for those puzzles, then complexity theory indeed says that there must exist safe replacements for them -- assuming P != NP, of course. But the theory cannot identify those replacements, not even tell how many bits one needs to use in keys and hashes to ensure that inverting those functions is hard enough. Conversely, even if P = NP, the current protocol may be safe, or there may exist safe alternatives for those two puzzles. (As you surely know, "polynomial" does not mean "computable in practice", and "superpolynomial" does not mean "impossible to compute in practice".)
The difficulty of those two puzzles, for the currently chosen key and hash sizes, has been verified only experimentally, by tests that have ruled out some weaknesses that people have thought of checking. (For a trivial example, someone surely must have checked that the leading bits of the block hash are not linear functions of the nonce bits, not even approximately.) However, a thorough check would require testing all possible algorithms (or boolean circuits) up to some large size, for all possible inputs -- which is totally outside the realm of practicality.
With no
known shortcuts, exhaustive enumeration is pretty much the best way we
now have to solve those puzzles. That is all one can say about the issue.