正则分支程序伪随机性的Fourier分析
Pseudorandomness for Regular Branching Programs via Fourier Analysis
摘要：
我们提出了一个显式的伪随机发生器的遗忘，只读一次，置换分支程序的恒定宽度，可以读取其输入位在任何顺序。种子长度是$O(\log^2 n)$，其中$n$是分支程序的长度。这个模型之前已知的最佳种子长度是$n^{1/2+o(1)}$，它是Impagliazzo，Meka和Zuckerman（FOCS 2012）的生成器的特殊情况（它为大小为$s$的任意分支程序提供了$s^{1/2+o(1)}$的种子长度）。我们的技术还为宽度为$2^{n^{o(1)}}$的一般不经意的只读分支程序提供了种子长度$n^{1/2+o(1)}$，这与Impagliazzo等人的结果无法比拟。我们的伪随机发生器类似于Gopalan等人（FOCS 2012）用于只读CNF的伪随机发生器，但分析完全不同;我们的分析基于分支程序的傅立叶分析。特别是，我们表明，一个不经意的，只读一次，定期分支程序的宽度$w$具有傅立叶质量最多$(2w^2)^k$在$k$级，独立的长度的程序。
英文摘要：
We present an explicit pseudorandom generator for oblivious, read-once, permutation branching programs of constant width that can read their input bits in any order. The seed length is $O(\log^2 n)$, where $n$ is the length of the branching program. The previous best seed length known for this model was $n^{1/2+o(1)}$, which follows as a special case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of $s^{1/2+o(1)}$ for arbitrary branching programs of size $s$). Our techniques also give seed length $n^{1/2+o(1)}$ for general oblivious, read-once branching programs of width $2^{n^{o(1)}}$, which is incomparable to the results of Impagliazzo et al.Our pseudorandom generator is similar to the one used by Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite different; ours is based on Fourier analysis of branching programs. In particular, we show that an oblivious, read-once, regular branching program of width $w$ has Fourier mass at most $(2w^2)^k$ at level $k$, independent of the length of the program.
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