Cryptography based on really hard problems

Abstract

Up until now we have looked at basing cryptography on problems which are believed to be hard, e.g. that AES is a PRF, that factoring a product of large primes is hard, that finding discrete logarithms is hard. But there is no underlying reason why these problems should be hard. Computer Science gives us a whole theory of categorizing hard problems, called complexity theory. Yet none of our hard problems appear to be what a complexity theorist would call hard. Indeed, in comparison to what complexity theorists discuss, factoring and discrete logarithms are comparatively easy.