2021/10/25 by Chattopadhyay, Eshan, Liao, Jyun-Jie
#Computational Complexity (cs.CC) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2110.12652
We consider the problem of extracting randomness from sumset sources, a general class of weak sources introduced by Chattopadhyay and Li (STOC, 2016). An (n,k,C)-sumset source X is a distribution on \0,1\n 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 0.51 n. As our main result, we construct an explicit extractor for sumset sources in the setting of C=2 for min-entropy poly(log n) and polynomially small error. We can further improve the min-entropy requirement to (log n) ⋅ (log log n)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).