Pseudorandom generators without the XOR Lemma

Impagliazzo and Wigderson IW97] have recently shown that if there exists a decision problem solvable in time 2O(n) and having circuit complexity 2 (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 and Wigderson 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 NW94] 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...

• Type: Journal
• Year: 1998
• Where: ECCC
• Authors: Madhu Sudan, Luca Trevisan, Salil P. Vadhan
• Keywords: Ampli Cation, ECCC 1998, Multivariate Polynomials, Xor Lemma