Revision #1 Authors: Eshan Chattopadhyay, Jyun-Jie Liao

Accepted on: 3rd June 2020 03:57

Downloads: 27

Keywords:

In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom

generator for length $n$ and width $w$ read-once branching programs with seed

length $O(\log n\cdot \log(nw)+\log n\cdot\log(1/\varepsilon))$ and error

$\varepsilon$. It remains a central question to reduce the seed length to

$O(\log (nw/\varepsilon))$, which would prove that $\mathbf{BPL}=\mathbf{L}$.

However, there has been no improvement on Nisan's construction for the case

$n=w$, which is most relevant to space-bounded derandomization.

Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced

the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit

construction of a PRPD with seed length $\tilde{O}(\log n\cdot

\log(nw)+\log(1/\varepsilon))$. A PRPD is a relaxation of a pseudorandom

generator, which suffices for derandomizing $\mathbf{BPL}$ and also implies a

hitting set. Unfortunately, their construction is quite involved and

complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler

hitting set generator with seed length $O(\log n\cdot

\log(nw)+\log(1/\varepsilon))$, but their techniques are restricted to hitting

sets.

In this work, we construct a PRPD with seed length $$O(\log n\cdot \log

(nw)\cdot \log\log(nw)+\log(1/\varepsilon)).$$ This improves upon the

construction in [BCG18] by a $O(\log\log(1/\varepsilon))$ factor, and is

optimal in the small error regime. In addition, we believe our construction and

analysis to be simpler than the work of Braverman, Cohen and Garg.

added details about using the Saks-Zhou scheme with PRPDs (Appendix A). Other minor changes.

TR20-069 Authors: Eshan Chattopadhyay, Jyun-Jie Liao

Publication: 4th May 2020 01:01

Downloads: 132

Keywords:

In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length $n$ and width $w$ read-once branching programs with seed length $O(\log n\cdot \log(nw)+\log n\cdot\log(1/\varepsilon))$ and error $\varepsilon$. It remains a central question to reduce the seed length to $O(\log (nw/\varepsilon))$, which would prove that $\mathbf{BPL}=\mathbf{L}$. However, there has been no improvement on Nisan's construction for the case $n=w$, which is most relevant to space-bounded derandomization.

Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced the notion of a \emph{pseudorandom pseudo-distribution} (PRPD) and gave an explicit construction of a PRPD with seed length $\tilde{O}(\log n\cdot \log(nw)+\log(1/\varepsilon))$. A PRPD is a relaxation of a pseudorandom generator, which suffices for derandomizing $\mathbf{BPL}$ and also implies a hitting set. Unfortunately, their construction is quite involved and complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler hitting set generator with seed length $O(\log n\cdot \log(nw)+\log(1/\varepsilon))$, but their techniques are restricted to hitting sets.

In this work, we construct a PRPD with seed length

$$O(\log n\cdot \log (nw)\cdot \log\log(nw)+\log(1/\varepsilon)).$$

This improves upon the construction in \cite{BCG18} by a $O(\log\log(1/\varepsilon))$ factor, and is optimal in the small error regime. In addition, we believe our construction and analysis to be simpler than the work of Braverman, Cohen and Garg.