We don't even know if difficult problems exist; at least in a way that is useful to cryptography (e.g. become easy once you know a small shared secret). We hav

We don't even know if difficult problems exist; at least in a way that is useful to cryptography (e.g. become easy once you know a small shared secret). We have some problems (factorization, and discrete log) that no one has found a way to efficiently solve on a Turing machine; but we don't really have a solid reason to think these problems are difficult. All we know is that our best mathamaticians have been stuck on them for generations. The bigger problem is that even these reductions to assumed difficult problems represent only a small portion of cryptography. We prove things like AES and SHA512 are secure by publishing them and waiting a few years to see if anyone publishes a break. No meaningful reduction to well researched problems, just a hope that we wont discover we missed something after making it a standard.