Perfect Structure on the Edge of Chaos

We construct trapdoor permutations based on (sub-exponential) indistinguishability obfuscation and oneway functions, thereby providing the first candidate that is not based on the hardness of factoring. Our construction shows that even highly structured primitives, such as trapdoor permutations, can be potentially based on hardness assumptions with noisy structures such as those used in candidate constructions of indistinguishability obfuscation. It also suggest a possible way to construct trapdoor permutations that resist quantum attacks, and that their hardness may be based on problems outside the complexity class SZK — indeed, while factoring-based candidates do not possess such security, future constructions of indistinguishability obfuscation might. As a corollary, we eliminate the need to assume trapdoor permutations and injective one-way function in many recent constructions based on indistinguishability obfuscation.