ECCC-Report TR14-076https://eccc.weizmann.ac.il/report/2014/076Comments and Revisions published for TR14-076en-usTue, 27 May 2014 22:01:07 +0300
Revision 1
| Pseudorandomness and Fourier Growth Bounds for Width 3 Branching Programs |
Thomas Steinke,
Salil Vadhan,
Andrew Wan
https://eccc.weizmann.ac.il/report/2014/076#revision1We present an explicit pseudorandom generator for oblivious, read-once, width-$3$ branching programs, which can read their input bits in any order. The generator has seed length $\tilde{O}( \log^3 n ).$ The previously best known seed length for this model is $n^{1/2+o(1)}$ due to Impagliazzo, Meka, and Zuckerman (FOCS '12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM '13) for \textit{permutation} branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any $f:\{0,1\}^n\rightarrow \{0,1\}$ computed by such a branching program, and $k\in [n],$ $$\sum_{s\subseteq [n]: |s|=k} \left| \hat{f}[s] \right| \leq n^2 \cdot (O(\log n))^k,$$ where $\widehat{f}[s] = \mathbb{E}\left[f[U] \cdot (-1)^{s \cdot U}\right]$ is the standard Fourier transform over $\mathbb{Z}_2^n$. The base $O(\log n)$ of the Fourier growth is tight up to a factor of $\log \log n$.Tue, 27 May 2014 22:01:07 +0300https://eccc.weizmann.ac.il/report/2014/076#revision1
Paper TR14-076
| Pseudorandomness and Fourier Growth Bounds for Width 3 Branching Programs |
Thomas Steinke
https://eccc.weizmann.ac.il/report/2014/076We present an explicit pseudorandom generator for oblivious, read-once, width-$3$ branching programs, which can read their input bits in any order. The generator has seed length $\tilde{O}( \log^3 n ).$ The previously best known seed length for this model is $n^{1/2+o(1)}$ due to Impagliazzo, Meka, and Zuckerman (FOCS '12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM '13) for \textit{permutation} branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any $f:\{0,1\}^n\rightarrow \{0,1\}$ computed by such a branching program, and $k\in [n],$ $$\sum_{s\subseteq [n]: |s|=k} \left| \hat{f}[s] \right| \leq n^2 \cdot (O(\log n))^k,$$ where $\widehat{f}[s] = \mathbb{E}\left[f[U] \cdot (-1)^{s \cdot U}\right]$ is the standard Fourier transform over $\mathbb{Z}_2^n$. The base $O(\log n)$ of the Fourier growth is tight up to a factor of $\log \log n$.Tue, 27 May 2014 21:53:34 +0300https://eccc.weizmann.ac.il/report/2014/076