Summary: Extractors for Three Uneven­Length Sources
Anup Rao #,1 and David Zuckerman ##,2
1 School of Mathematics, Institute for Advanced Study
arao@ias.edu
2 Department of Computer Science, University of Texas at Austin
diz@cs.utexas.edu
Abstract. We construct an e#cient 3­source extractor that requires one of the sources to be signif­
icantly shorter than the min­entropy of the other two sources. Our extractors work even when the
longer, n­bit sources have min­entropy
n# (1) and the shorter source has min­entropy log 10 n. Previous
constructions for independent sources with min­entropy n # required #(1/#) sources [Rao06]. Our con­
struction relies on lossless condensers [GUV07] based on Parvaresh­Vardy codes [PV05], as well as on
a 2­source extractor for a block source and general source [BRSW06].
1 Introduction
Motivated by the widespread use of randomness in computer science, researchers have sought algorithms
to extract randomness from a distribution that is only weakly random. A general weak source is one with
some min­entropy: a distribution has min­entropy k if all strings have probability at most 2 -k . We would
like to extract randomness from a weak source knowing only k and not the exact distribution. However, this
is impossible, even for more restricted sources [SV86].
Therefore, Santha and Vazirani showed how to extract randomness from two independent restricted weak

Source: Anderson, Richard - Department of Computer Science and Engineering, University of Washington at Seattle

Collections: Computer Technologies and Information Sciences