Error Reduction for Extractors

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 fraction of the source min-entropy and achieves a polynomially small error. In that case, we are able to reduce the error to (almost) any ", using only O(log(1=")) 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 polynomially small error). In particular, we apply our method to the new extractors of [Tre99, RRV99] to get 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 fraction of the min-entropy using O(log n + log(1=")) truly random bits (for any > 0), (b) extract any constant fraction of the min-entropy using O(log2 n+ log(1=")) truly random bits, and (c) extract all the min-entropy using O(log3 n + logn log(1=")) truly random bits. Department of Applied Mathematics and Computer Science, Weizmann Institute, Rehovot, 76100 Israel. E-mail: [email protected] Work supported by an American-Israeli BSF grant 95-00238. yDepartment of Applied Mathematics and Computer Science, Weizmann Institute, Rehovot, 76100 Israel. E-mail: [email protected] Research supported by an Eshkol Fellowship of the Israeli Ministry of Science and by ESPRIT working group RAND2. zMIT Laboratory for Computer Science. 545 Technology Square. Cambridge, MA 02139. E-mail: [email protected]. URL: http://theory.lcs.mit.edu/ ̃salil. Supported by a DOD/NDSEG fellowship and partially by DARPA grant DABT63-96-C0018.