Computational Pseudo-Randomness

One of the most important and fundamental discoveries of Theoretical Computer Science is the surprising connection between the computational power of randomness, and computational lower bounds on explicit functions. The currently strongest result of this form states [l] : Theorem 1 If EXP has no subexponentially small circuits then BPP had determinastic, pseudo-polynomial time algorithms. The key mechanism behind this connection is called a pseudo-random generator. There are two different constructions known the “classical” one of [2, 191, which uses the difficulty of computing functions whose inverse is easy, and the more recent one of [14, 151, which can use essentially any hard function. The talk will motivate and define the notions above. Then it will survey the main ideas behind the constructions of both generators, the proofs that they are pseudo-random, and the theorem above. This will lead to several natural open problems and conjectures, of which the most important (and I believe, solvable with present technology) is Conjecture 1 EXP # BPP There will be no time to discuss the related (and equally interesting) topic of pseudo-random generators for restricted models, such as constant-depth circuits and log-space Turing machines.