期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2021
卷号：21
语种：English
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：We consider the problem of extracting randomness from \textit{sumset sources}, a general class of weak sources introduced by Chattopadhyay and Li (STOC, 2016). An (nkC) -sumset source X is a distribution on 01n of the form X1+X2++XC, where Xi's are independent sources on n bits with min-entropy at least k. Prior extractors either required the number of sources C to be a large constant or the min-entropy k to be at least 051n. As our main result, we construct an explicit extractor for sumset sources in the setting of C=2 for min-entropy poly(logn) and polynomially small error. We can further improve the min-entropy requirement to (logn)(loglogn)1+o(1) at the expense of worse error parameter of our extractor. We find applications of our sumset extractor for extracting randomness from other well-studied models of weak sources such as affine sources, small-space sources, and interleaved sources. Interestingly, it is unknown if a random function is an extractor for sumset sources. We use techniques from additive combinatorics to show that it is a disperser, and further prove that an affine extractor works for an interesting subclass of sumset sources which informally corresponds to the ``low doubling" case (i.e., the support of X1+X2 is not much larger than 2k).
