Deterministic Extraction from Weak Random Sources - Preamble

Preface Roughly speaking, a deterministic extractor is a function that 'extracts' almost perfect random bits from a 'weak random source'-a distribution that contains some entropy but is far from being truly random. In this book we explicitly construct deterministic extractors and related objects for various types of sources. A basic theme in this book is a methodology of recycling randomness that enables increasing the output length of deterministic ex-tractors to near-optimal length. Our results are as follows. Deterministic Extractors for Bit-Fixing Sources An (n, k)-bit-fixing source is a distribution X over {0, 1} n such that there is a subset of k variables in X 1 ,. .. , X n that are uniformly distributed and independent of each other, and the remaining n − k variables are fixed in advance to some (unknown) constants. We give constructions of deterministic bit-fixing source extractors that extract (1−o(1))k bits whenever k > (log n) c for some universal constant c > 0. Thus, our constructions extract almost all the randomness from bit-fixing sources and work even when k is small. Our technique gives a general method to transform deterministic bit-fixing source extractors that extract few bits into extractors which extract almost all the bits. Deterministic Extractors for Affine Sources over Large Fields An (n, k)-affine source over a finite field F is a random variable X = (X 1 , ..., X n) ∈ F n , that is uniformly distributed over an (unknown) k-dimensional affine sub-space of F n. There has been much interest lately in extractors for affine sources over F 2. It can be shown that a random function D : {0, 1} n → {0, 1} is with high probability an extractor for (n, k)-affine sources over F 2 whenever k ≥ 3 · log n. The best explicit construction due to Bourgain [10] works when k = δ · n for constant δ. We focus on the case of a large field, specifically, a field of size n c for constant c > 0, i.e., a field size that is polynomially large in the dimension of the space. When working with a field of size larger than n 20 we show how to deterministically extract practically all the randomness from an (n, k)-affine source for any k ≥ 2. Extractors and Rank Extractors for Polynomial Sources We construct explicit deterministic extractors from polynomial sources, namely from distributions …