Revision #1 Authors: Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett

Accepted on: 30th April 2018 16:29

Downloads: 1586

Keywords:

We propose a new framework for constructing pseudorandom generators for $n$-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in $[-1,1]^n$. Next, we use a fractional pseudorandom generator as steps of a random walk in $[-1,1]^n$ that converges to $\{-1,1\}^n$. We prove that this random walk converges fast (in time logarithmic in $n$) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity $s$, whose seed length is polynomial in $s$. Other examples include functions computed by branching programs of various sorts or by bounded depth circuits.

TR18-015 Authors: Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett

Publication: 25th January 2018 22:18

Downloads: 2099

Keywords:

We propose a new framework for constructing pseudorandom generators for $n$-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in $[-1,1]^n$. Next, we use a fractional pseudorandom generator as steps of a random walk in $[-1,1]^n$ that converges to $\{-1,1\}^n$. We prove that this random walk converges fast (in time logarithmic in $n$) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity $s$, whose seed length is polynomial in $s$. Other examples include functions computed by branching programs of various sorts or by bounded depth circuits.

The answer to Open problem 1.8 is no. See https://eccc.weizmann.ac.il/report/2015/182/