 
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
