Impagliazzo and Wigderson [IW97] have recently shown that if there exists a decision problem solvable in time 2 and having circuit complexity 2 (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 the Nisan–Wigderson [NW94] generator. 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 (encodThe full version of this paper appears as [STV98] and an extended abstract appears as [STV99]. Laboratory for Computer Science, 545 Technology Square, MIT, Cambridge, MA 02141. E-mail: email@example.com. zDepartment of Computer Science, Columbia University, 500W 120th St., New York, NY 10027. Email: firstname.lastname@example.org. Work done at MIT. Laboratory for Computer Science, 545 Technology Square, MIT, Cambridge, MA 02141. E-mail: email@example.com. URL: http://theory.lcs.mit.edu/ ̃salil. Supported by a DOD/NDSEG graduate fellowship and partially by DARPA grant DABT63-96-C-0018. ing 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 based on multivariate polynomials; and (ii) presenting a list-decoding algorithm that improves and simplifies a previous one by Arora and Sudan [AS97].
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