利用局部单调性改进无序分支程序的伪随机性

Eshan Chattopadhyay, P. Hatami, Omer Reingold, Avishay Tal
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引用次数: 30

摘要

我们提出了一个显式伪随机生成器,其种子长度为Õ((logn)w+1),用于读取一次,无关的,宽度为w的分支程序,可以以任何顺序读取其输入位。这改进了Impagliazzo, Meka和Zuckerman (FOCS ' 12)的工作,他们要求种子长度为n1/2+o(1)。我们工作中的一个核心成分是我们在分支程序的傅立叶谱上证明的下一个界。对于任意宽度w读一次,无关分支程序B:{0,1}n→{0,1},任意k∈{1,…,n},[复公式未显示],这解决了Reingold, Steinke和Vadhan (RANDOM ' 13)提出的一个猜想。我们的分析关键是在分支程序的边缘标记上使用了局部单调性的概念。我们在局部单调假设下给出了证明的关键部分,并给出了如何推导出无限制分支规划的结果。
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Improved pseudorandomness for unordered branching programs through local monotonicity
We present an explicit pseudorandom generator with seed length Õ((logn)w+1) for read-once, oblivious, width w branching programs that can read their input bits in any order. This improves upon the work of Impagliazzo, Meka and Zuckerman (FOCS’12) where they required seed length n1/2+o(1). A central ingredient in our work is the following bound that we prove on the Fourier spectrum of branching programs. For any width w read-once, oblivious branching program B:{0,1}n→ {0,1}, any k ∈ {1,…,n}, [complex formula not displayed] This settles a conjecture posed by Reingold, Steinke and Vadhan (RANDOM’13). Our analysis crucially uses a notion of local monotonicity on the edge labeling of the branching program. We carry critical parts of our proof under the assumption of local monotonicity and show how to deduce our results for unrestricted branching programs.
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