Pseudorandom Generators without the XOR Lemma

Impagliazzo andWigderson IW have recently shown that if there exists a decision problem solvable in time O n and having circuit complexity n for all but nitely 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 andWigderson goes through three phases of hardness ampli cation a multivariate polynomial encoding a rst derandomized XOR Lemma and a second derandomized XOR Lemma that are composed with the Nisan Wigderson NW generator In this paper we present two di erent 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 o s Our rst result is that when a modi ed version of the Nisan Wigderson generator construction is applied with a mildly hard predicate the result is a generator that produces a distribution indis tinguishable from having large min entropy An extractor can then be used to produce a distribution computationally indistinguishable from uniform This is the rst construction of a pseudorandom gen erator that works with a mildly hard predicate without doing hardness ampli cation We then show that in the Impagliazzo Wigderson construction only the rst hardness ampli cation 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 ampli cation problem and a list decoding problem for error correcting codes and ii presenting a list decoding algo rithm for error correcting codes based on multivariate polynomials that improves and simpli es a previous one by Arora and Sudan AS