ECCC-Report TR17-171https://eccc.weizmann.ac.il/report/2017/171Comments and Revisions published for TR17-171en-usThu, 09 Nov 2017 00:43:30 +0200
Revision 1
| Improved Pseudorandomness for Unordered Branching Programs through Local Monotonicity |
Avishay Tal,
Eshan Chattopadhyay,
Omer Reingold,
Pooya Hatami
https://eccc.weizmann.ac.il/report/2017/171#revision1We present an explicit pseudorandom generator with seed length $\tilde{O}((\log n)^{w+1})$ for read-once, oblivious, width $w$ branching programs that can read their input bits in any order. This improves upon the work of Impaggliazzo, Meka and Zuckerman (FOCS'12) where they required seed length $n^{1/2+o(1)}$.
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 $B:\{0,1\}^n\rightarrow \{0,1\}$, any $k \in \{1,\ldots,n\}$, $$\sum_{S\subseteq[n]: |S|=k}|\widehat{B}(S)| \le O(\log n)^{wk}.$$ 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.
Thu, 09 Nov 2017 00:43:30 +0200https://eccc.weizmann.ac.il/report/2017/171#revision1
Paper TR17-171
| Improved Pseudorandomness for Unordered Branching Programs through Local Monotonicity |
Avishay Tal,
Eshan Chattopadhyay,
Omer Reingold,
Pooya Hatami
https://eccc.weizmann.ac.il/report/2017/171We present an explicit pseudorandom generator with seed length $\tilde{O}((\log n)^{w+1})$ for read-once, oblivious, width $w$ branching programs that can read their input bits in any order. This improves upon the work of Impaggliazzo, Meka and Zuckerman (FOCS'12) where they required seed length $n^{1/2+o(1)}$.
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 $B:\{0,1\}^n\rightarrow \{0,1\}$, any $k \in \{1,\ldots,n\}$, $$\sum_{S\subseteq[n]: |S|=k}|\widehat{B}(S)| \le O(\log n)^{wk}.$$ 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.
Mon, 06 Nov 2017 15:07:40 +0200https://eccc.weizmann.ac.il/report/2017/171