Error reduction for extractors

Abstract

An extractor is a function which extracts (almost) truly random bits from a weak random source, using a small number of additional random bits as a catalyst. We present a general method to reduce the error of any extractor. Our method works particularly well in the case that the original extractor extracts up to a constant function of the source min-entropy and achieves a polynomially small error. In that case, we are able to reduce the error to (almost) any /spl epsiv/, using only O(log(1//spl epsiv/)) additional truly random bits (while keeping the other parameters of the original extractor more or less the same). In other cases (e.g. when the original extractor extracts all the min-entropy or achieves only a constant error), our method is not optimal but it is still quite efficient and leads to improved constructions of extractors. Using our method, we are able to improve almost all known extractors in the case where the error required is relatively small (e.g. less than a polynomially small error). In particular, we apply our method to the new extractors of L. Trevisan (1999) and R. Raz et al. (1999) to obtain improved constructions in almost all cases. Specifically, we obtain extractors that work for sources of any min-entropy on strings of length n which (a) extract any 1/n/sup /spl gamma// fraction of the min-entropy using O[log n+log(1//spl epsiv/)] truly random bits (for any /spl gamma/>0), (b) extract any constant fraction of the min-entropy using O[log/sup 2/n+log(1//spl epsiv/)] truly random bits, and (c) extract all the min-entropy using O[log/sup 3/n+log n/spl middot/log(1//spl epsiv/)] truly random bits.