Pseudorandom Generators, Typically-Correct Derandomization, and Circuit Lower Bounds

• Published in: Computational complexity 2012-03, Vol.21 (1), p.3-61
• Main Authors: Kinne, Jeff, van Melkebeek, Dieter, Shaltiel, Ronen
• Format: Article
• Language: English
• Subjects: Algorithm Analysis and Problem Complexity; Computational Mathematics and Numerical Analysis; Computer Science
• ISSN: 1016-3328; 1420-8954
• DOI: 10.1007/s00037-011-0019-z

The area of derandomization attempts to provide efficient deterministic simulations of randomized algorithms in various algorithmic settings. Goldreich and Wigderson introduced a notion of “typically-correct” deterministic simulations, which are allowed to err on few inputs. In this paper, we further the study of typically-correct derandomization in two ways. First, we develop a generic approach for constructing typically-correct derandomizations based on seed-extending pseudorandom generators, which are pseudorandom generators that reveal their seed. We use our approach to obtain both conditional and unconditional typically-correct derandomization results in various algorithmic settings. We show that our technique strictly generalizes an earlier approach by Shaltiel based on randomness extractors and simplifies the proofs of some known results. We also demonstrate that our approach is applicable in algorithmic settings where earlier work did not apply. For example, we present a typically-correct polynomial-time simulation for every language in BPP based on a hardness assumption that is (seemingly) weaker than the ones used in earlier work. Second, we investigate whether typically-correct derandomization of BPP implies circuit lower bounds. Extending the work of Kabanets and Impagliazzo for the zero-error case, we establish a positive answer for error rates in the range considered by Goldreich and Wigderson. In doing so, we provide a simpler proof of the zero-error result. Our proof scales better than the original one and does not rely on the result by Impagliazzo, Kabanets, and Wigderson that NEXP having polynomialsize circuits implies that NEXP coincides with EXP.