Pseudorandom Generators without the XOR Lemma

摘要

R. Impagliazzo and A. Wigderson (1997, in “Proceedings of the twenty-ninth Annual ACM Symposium on Theory of Computing,” pp. 220–229) have recently shown that if there exists a decision problem solvable in time 2O(n) and having circuit complexity 2Ω(n) (for all but finitely many n) then P=BPP. This result is a culmination of a series of works showing connections between the existence of hard predicates and the existence of good pseudorandom generators. The construction of Impagliazzo and Wigderson goes through three phases of “hardness amplification” (a multivariate polynomial encoding, a first derandomized XOR Lemma, and a second derandomized XOR Lemma) that are composed with a pseudorandom generator construction of N. Nisan and A. Wigderson (1999, J. Comput. System Sci.49, 149–167). In this paper we present two different approaches to proving the main result of Impagliazzo and Wigderson. In developing each approach, we introduce new techniques and prove new results that could be useful in future improvements and/or applications of hardness-randomness trade-offs. Our first result is that when (a modified version of) the Nisan–Wigderson generator construction is applied with a “mildly” hard predicate, the result is a generator that produces a distribution indistinguishable from having large min-entropy. An extractor can then be used to produce a distribution computationally indistinguishable from uniform. This is the first construction of a pseudorandom generator that works with a mildly hard predicate without doing hardness amplification. We then show that in the Impagliazzo–Wigderson construction only the first hardness-amplification phase (encoding with multivariate polynomial) is necessary, since it already gives the required average-case hardness. We prove this result by (i) establishing a connection between the hardness-amplification problem and a list-decoding problem for error-correcting codes; and (ii) presenting a list-decoding algorithm for error-correcting codes based on multivariate polynomials that improves and simplifies a previous one by S. Arora and M. Sudan (1997, in “Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing,” pp. 485–495).