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An Efficient Reduction from Two-Source to Nonmalleable Extractors: Achieving Near-Logarithmic Min-Entropy
SIAM Journal on Computing ( IF 1.2 ) Pub Date : 2019-11-05 , DOI: 10.1137/17m1133245 Avraham Ben-Aroya , Dean Doron , Amnon Ta-Shma
SIAM Journal on Computing ( IF 1.2 ) Pub Date : 2019-11-05 , DOI: 10.1137/17m1133245 Avraham Ben-Aroya , Dean Doron , Amnon Ta-Shma
SIAM Journal on Computing, Ahead of Print.
The breakthrough result of Chattopadhyay and Zuckerman [Explicit two-source extractors and resilient functions, in Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC), ACM, 2016, pp. 670--683] gives a reduction from the construction of explicit two-source extractors to the construction of explicit nonmalleable extractors. However, even assuming the existence of optimal explicit nonmalleable extractors, we only obtain a two-source extractor for $\mathrm{poly}(\log n)$ entropy, rather than the optimal $O(\log n)$. In this paper we modify the construction to solve the above barrier. Using the currently best explicit nonmalleable extractors, we get explicit bipartite Ramsey graphs for sets of size $2^k$ for $k=O(\log n \frac{\log\log n}{\log\log\log n})$. Any further improvement in the construction of nonmalleable extractors would immediately yield a corresponding two-source extractor. Intuitively, Chattopadhyay and Zuckerman use an extractor as a sampler, and we observe that we could use a weaker object---a somewhere-random condenser with a small entropy gap and a very short seed. We also show how to explicitly construct this weaker object using the error reduction technique of Raz, Reingold, and Vadhan [Error reduction for extractors, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE, 1999, pp. 191--201], and the constant-degree dispersers of Zuckerman [Linear degree extractors and the inapproximability of max clique and chromatic number, in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing (STOC), ACM, 2006, pp. 681--690] that also work against extremely small tests.
中文翻译：
从两源到不可破坏提取器的有效降低：实现近对数最小熵
《 SIAM计算杂志》，预印本。
Chattopadhyay和Zuckerman的突破性结果[明确的两源提取器和弹性函数，在第48届ACM SIGACT计算理论研讨会（STOC），2016年，ACM，第670--683页]中进行了讨论。构造显式两源提取器，以构造显式不可恶意提取器。但是，即使假设存在最佳的显式不可修改的提取器，我们也只能获得$ \ mathrm {poly}（\ log n）$熵的双源提取器，而不是最优的$ O（\ log n）$熵。在本文中，我们修改了构造以解决上述障碍。使用当前最佳的显式不可微提取器，对于$ k = O（\ log n \ frac {\ log \ log n} {\ log \ log \ log n}），我们获得了大小为$ 2 ^ k $的集合的显式二分Ramsey图。 $。不可捏造萃取器的结构上的任何进一步改进将立即产生相应的两源萃取器。直观地讲，Chattopadhyay和Zuckerman使用提取器作为采样器，并且我们观察到可以使用较弱的对象-某个具有较小熵间隙和非常短种子的随机聚光器。我们还展示了如何使用Raz，Reingold和Vadhan的错误减少技术[提取器的错误减少，[第40届IEEE计算机科学基础年度研讨会（FOCS），IEEE，1999，pp 191--201]和Zuckerman的恒度分散器[线性度提取器和最大团和色数的不可逼近性，在第38届ACM计算理论年度学术会议论文集（STOC）中，ACM，2006年，第
更新日期：2019-11-05
The breakthrough result of Chattopadhyay and Zuckerman [Explicit two-source extractors and resilient functions, in Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC), ACM, 2016, pp. 670--683] gives a reduction from the construction of explicit two-source extractors to the construction of explicit nonmalleable extractors. However, even assuming the existence of optimal explicit nonmalleable extractors, we only obtain a two-source extractor for $\mathrm{poly}(\log n)$ entropy, rather than the optimal $O(\log n)$. In this paper we modify the construction to solve the above barrier. Using the currently best explicit nonmalleable extractors, we get explicit bipartite Ramsey graphs for sets of size $2^k$ for $k=O(\log n \frac{\log\log n}{\log\log\log n})$. Any further improvement in the construction of nonmalleable extractors would immediately yield a corresponding two-source extractor. Intuitively, Chattopadhyay and Zuckerman use an extractor as a sampler, and we observe that we could use a weaker object---a somewhere-random condenser with a small entropy gap and a very short seed. We also show how to explicitly construct this weaker object using the error reduction technique of Raz, Reingold, and Vadhan [Error reduction for extractors, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE, 1999, pp. 191--201], and the constant-degree dispersers of Zuckerman [Linear degree extractors and the inapproximability of max clique and chromatic number, in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing (STOC), ACM, 2006, pp. 681--690] that also work against extremely small tests.
中文翻译：
从两源到不可破坏提取器的有效降低：实现近对数最小熵
《 SIAM计算杂志》，预印本。
Chattopadhyay和Zuckerman的突破性结果[明确的两源提取器和弹性函数，在第48届ACM SIGACT计算理论研讨会（STOC），2016年，ACM，第670--683页]中进行了讨论。构造显式两源提取器，以构造显式不可恶意提取器。但是，即使假设存在最佳的显式不可修改的提取器，我们也只能获得$ \ mathrm {poly}（\ log n）$熵的双源提取器，而不是最优的$ O（\ log n）$熵。在本文中，我们修改了构造以解决上述障碍。使用当前最佳的显式不可微提取器，对于$ k = O（\ log n \ frac {\ log \ log n} {\ log \ log \ log n}），我们获得了大小为$ 2 ^ k $的集合的显式二分Ramsey图。 $。不可捏造萃取器的结构上的任何进一步改进将立即产生相应的两源萃取器。直观地讲，Chattopadhyay和Zuckerman使用提取器作为采样器，并且我们观察到可以使用较弱的对象-某个具有较小熵间隙和非常短种子的随机聚光器。我们还展示了如何使用Raz，Reingold和Vadhan的错误减少技术[提取器的错误减少，[第40届IEEE计算机科学基础年度研讨会（FOCS），IEEE，1999，pp 191--201]和Zuckerman的恒度分散器[线性度提取器和最大团和色数的不可逼近性，在第38届ACM计算理论年度学术会议论文集（STOC）中，ACM，2006年，第
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