## Abstract

A "randomness extractor" is an algorithm that given a sample from a distribution with sufficiently high min-entropy and a short random seed produces an output that is statistically indistinguishable from uniform. (Min-entropy is a measure of the amount of randomness in a distribution.) We present a simple, self-contained extractor construction that produces good extractors for all minentropies. Our construction is algebraic and builds on a new polynomial-based approach introduced by Ta-Shma et al. [2001b]. Using our improvements, we obtain, for example, an extractor with output length m = k/(log n) ^{O(1/α)} and seed length (1 + α)log n for an arbitrary 0 < α ≤ 1, where n is the input length, and k is the min-entropy of the input distribution. A "pseudorandom generator" is an algorithm that given a short random seed produces a long output that is computationally indistinguishable from uniform. Our technique also gives a new way to construct pseudorandom generators from functions that require large circuits. Our pseudorandom generator construction is not based on the Nisan-Wigderson generator [Nisan and Wigderson 1994], and turns worst-case hardness directly into pseudorandomness. The parameters of our generator match those in Impagliazzo and Wigderson [1997] and Sudan et al. [2001] and in particular are strong enough to obtain a new proof that P = BPP if E requires exponential size circuits. Our construction also gives the following improvements over previous work: -We construct an optimal "hitting set generator" that stretches O(log n) random bits into S ^{Ω(1)} pseudorandom bits when given a function on log n bits that requires circuits of size s. This yields a quantitatively optimal hardness versus randomness tradeoff for both RP and BPP and solves an open problem raised in Impagliazzo et al. [1999]. -We give the first construction of pseudorandom generators that fool nondeterministic circuits when given a function that requires large nondeterministic circuits. This technique also give a quantitatively optimal hardness versus randomness tradeoff for AM and the first hardness amplification result for nondeterministic circuits.

Original language | English |
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Pages (from-to) | 172-216 |

Number of pages | 45 |

Journal | Journal of the ACM |

Volume | 52 |

Issue number | 2 |

DOIs | |

State | Published - 2005 |

## Keywords

- Hardness versus randomness
- Pseudorandom generator
- Randomness extractor

## ASJC Scopus subject areas

- Software
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence