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Revision #1 to TR12-116 | 14th September 2012 18:54

A Derandomized Switching Lemma and an Improved Derandomization of AC0

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Revision #1
Authors: Luca Trevisan, TongKe Xue
Accepted on: 14th September 2012 18:54
Downloads: 1931
Keywords: 


Abstract:

We describe a new pseudorandom generator for AC0. Our generator $\epsilon$-fools circuits of depth $d$ and size $M$ and uses a seed of length $\tilde O( \log^{d+4} M/\epsilon)$. The previous best construction for $d \geq 3$ was due to Nisan, and had seed length $O(\log^{2d+6} M/\epsilon)$.
A seed length of $O(\log^{2d + \Omega(1)} M)$ is best possible given Nisan-type generators and the current state of circuit lower bounds; Seed length $\Omega(\log^d M/\epsilon)$ is a barrier for any pseudorandom generator construction given the current state of circuit lower bounds. For $d=2$, a pseudorandom generator of seed length $\tilde O(\log^2 M/\epsilon)$ was known.

Our generator is based on a ``pseudorandom restriction'' generator which outputs restrictions that satisfy the conclusions of the Hastad Switching Lemma and that uses a seed of polylogarithmic length.


Paper:

TR12-116 | 13th September 2012 03:28

A Derandomized Switching Lemma and an Improved Derandomization of AC0





TR12-116
Authors: Luca Trevisan
Publication: 13th September 2012 03:28
Downloads: 1627
Keywords: 


Abstract:

We describe a new pseudorandom generator for AC0. Our generator $\epsilon$-fools circuits of depth $d$ and size $M$ and uses a seed of length $\tilde O( \log^{d+4} M/\epsilon)$. The previous best construction for $d \geq 3$ was due to Nisan, and had seed length $O(\log^{2d+6} M/\epsilon)$.
A seed length of $O(\log^{2d + \Omega(1)} M)$ is best possible given Nisan-type generators and the current state of circuit lower bounds; Seed length $\Omega(\log^d M/\epsilon)$ is a barrier for any pseudorandom generator construction given the current state of circuit lower bounds. For $d=2$, a pseudorandom generator of seed length $\tilde O(\log^2 M/\epsilon)$ was known.

Our generator is based on a ``pseudorandom restriction'' generator which outputs restrictions that satisfy the conclusions of the Hastad Switching Lemma and that uses a seed of polylogarithmic length.



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