Randomness vs. Time: De-randomization under a uniform assumption Russell Impagliazzo∗ Avi Wigdersony Department of Computer Science Institute of Computer Science University of California Hebrew University San Diego, CA 91097-0114 Jerusalem, Israel 91904
[email protected] [email protected] Abstract 1 Introduction, History, and Intuition We prove that if BPP = EXP, then every prob- lem in BPP can be solved6 deterministically in 1.1 Motivation subexponential time on almost every input ( The introduction of randomization into efficient on every samplable ensemble for infinitely many computation has been one of the most fertile and input sizes). This is the first derandomiza- useful ideas in computer science. In cryptogra- tion result for BP P based on uniform, non- phy and asynchronous computing, randomization cryptographic hardness assumptions. It implies makes possible tasks that are impossible to per- the following gap in the average-instance com- form deterministically. Even for function com- plexities of problems in BP P : either these com- putation, many examples are known in which plexities are always sub-exponential or they con- randomization allows considerable savings in re- tain arbitrarily large exponential functions. sources like space and time over deterministic al- We use a construction of a small \pseudo- gorithms, or even \only" simplifies them. random" set of strings from a \hard function" But to what extent is this seeming power of in EXP which is identical to that used in the randomness over determinism real? The most analogous non-uniform results of [21, 3]. How- famous concrete version of this question regards ever, previous proofs of correctness assume the the power of BP P , the class of problems solv- \hard function" is not in P=poly.