Improved Constructions of Three Source Extractors

Abstract

We study the problem of constructing extractors for independent weak random sources. The probabilistic method shows that there exists an extractor for two independent weak random sources on n bits with only logarithmic min-entropy. However, previously the best known explicit two source extractor only achieves min-entropy 0.499n, and the best known three source extractor only achieves min-entropy n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0.9</sup> . It is a long standing open problem to construct extractors that work for smaller min-entropy. In this paper we construct an extractor for three independent weak random sources on n bits with min-entropy n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2+δ</sup> , for any constant 0 <; δ <; 1/2. This improves the previous best result of. In addition, we consider the problem of constructing extractors for three independent weak sources, such that one of the sources is significantly shorter than the min-entropy of the other two, in the spirit of. We give an extractor that works in the case where the longer, n-bit sources only have min-entropy polylog(n), and the shorter source also has min-entropy polylog(n). This improves the result of. We also study the problem of constructing extractors for afflne sources over GF(2). Previously the best known deterministic extractor for n-bit affine sources in this case achieves entropy n/√(log log n). In this paper we construct an extractor for two independent affine sources with entropy n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2+δ</sup> , for any constant 0 <; δ <; 1/2. Our constructions mainly use the extractors for some where random sources in and the lossless condenser in.