At its core, to do public-key cryptography, you just need a difficult instance of an NP-but-non-P problem. Quantum computers put factorization and discrete logarithms into P, but there are still plenty of harder NP problems that have not been tapped. If quantum computing starts to become a significant threat, we'll probably see renewed interest in those.
Maybe I'm misunderstanding but if you had one of these you would have answered a fairly important question in CS theory. Perhaps you mean to say "NP-complete", but even then I'm not sure the proposition is correct.