Pseudorandom generators without the XOR Lemma (extended abstract)

Impagliazzo and Wigderso” [lW97] have recently show” that if there exists a decision problem solvable in time Z”cn) and having circuit complexity Z”(“) (for all but tinitely 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 pseudorandomgenerators. The construction of lmpagliazzo and Wigderson goes through three phases of “hardness amplitication” (a multivatiafe polynomial encoding, a first derandomized XOR Lemma. and a second derandomized XOR Lemma) that are composed with the NisanWigderson [NW941 generator. In this paper we present two different approaches to proving the main result of lmpagliazzo and Wigderso”. 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 ofJ the NisanWigderson generator construction is applied with a “mildly” hard predicate, the result is a generator that produces a distribution indistinguishable fmm having large min.entropy. An extmctor can the” be used to produce a distribution computationally indistinguishable from uniform. This is the first constructionof a pseudorandomgenerator that works with a mildly hard predicate without doing hardness amplification. We then show that in the lmpagliazzo-Wigderso” co”stmction only the first hardness-amplification phase (encoding with multivariate polynomial) is necessary, since it already gives the required average-case hardness. We pmve this result by (i) establishing a connection between the hardness-amplification problem and a listdecoding 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].