Explicit two-source extractors and resilient functions
Explicit two-source extractors and resilient functions
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DOI：
10.4007/annals.2019.189.3.1
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发表时间：
2019-05-01
影响因子：
4.9
通讯作者：
Zuckerman, David
中科院分区：
文献类型：
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作者：
Chattopadhyay, Eshan;Zuckerman, David
我们明确地在n位上明确构建了两个独立源的提取器，每个源具有最小的log（c）n的最小log（c）n。最好的先前提取器，由波尔加恩（Bourgain），要求每个来源具有最小的entropy .499n.499n.a构造中的关键成分是在n位上明确的单调，几乎平衡的布尔功能，这对n位均具有弹性，这对n尺寸n（尺寸）（ 1-delta）对于任何三角洲> 0。实际上，我们的构造更强，因为它为非合理的概括提供了明确的提取器n位上的位固定源，其中一些未知的n-q位几乎独立地选择了几乎是polygog（n），其余的q = n（1-delta）位由对手选择作为n的任意函数。 -Q位。 Viola的最佳先前结构实现了Q = n（1/2-delta）。您的显式两源提取器直接意味着在n个顶点上的2（（log log log n）o（1）O（log n）O（1）O（log log n）O（1））Ramsey图，改善Barak等人获得的界限。并匹配科恩的独立作品。
We explicitly construct an extractor for two independent sources on n bits, each with min-entropy at least log(C) n for a large enough constant C. Our extractor outputs one bit and has error n(-Omega(1)). The best previous extractor, by Bourgain, required each source to have min-entropy .499n.A key ingredient in our construction is an explicit construction of a monotone, almost-balanced Boolean function on n bits that is resilient to coalitions of size n(1-delta) for any delta > 0. In fact, our construction is stronger in that it gives an explicit extractor for a generalization of non-oblivious bit-fixing sources on n bits, where some unknown n - q bits are chosen almost polylog(n)-wise independently, and the remaining q = n(1-delta) bits are chosen by an adversary as an arbitrary function of the n - q bits. The best previous construction, by Viola, achieved q = n(1/2-delta).Our explicit two-source extractor directly implies an explicit construction of a 2((log log N)O(1)) Ramsey graph over N vertices, improving bounds obtained by Barak et al. and matching an independent work by Cohen.