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Revision #2 to TR12-057 | 26th September 2012 03:37

Pseudorandomness from Shrinkage

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Revision #2
Authors: Russell Impagliazzo, Raghu Meka, David Zuckerman
Accepted on: 26th September 2012 03:37
Downloads: 649
Keywords: 


Abstract:

One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer a quantitative loss in parameters, and hence do not give nontrivial implications for models where we only know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can indeed construct PRGs which are essentially best possible without in turn improving the lower bounds.

More specifically, say that a circuit family has shrinkage exponent Gamma if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of $p^{\Gamma}$. Our PRG uses a seed of length roughly $s^{1/(\Gamma + 1)}$ to fool circuits in the family of size $s$. By instantiating this generic construction, we get PRGs for the following classes:
1) de Morgan formulas of size $s$, seed length $s^{1/3+o(1)}$.
2) Formulas over an arbitrary basis of size $s$, seed length $s^{1/2+o(1)}$.
3) Read-once formulas, seed length $s^{.234...}$.
4) Branching programs of size $s$, seed length $s^{1/2 + o(1)}$.

The previous best PRGs known for these classes used seeds of length bigger than $n/2$ to output $n$ bits, and worked only when the size $s=O(n)$.



Changes to previous version:

Fixes some typos and minor notational changes.


Revision #1 to TR12-057 | 22nd May 2012 02:47

Pseudorandomness from Shrinkage


Abstract:

One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer a quantitative loss in parameters, and hence do not give nontrivial implications for models where we only know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can indeed construct PRGs which are essentially best possible without in turn improving the lower bounds.

More specifically, say that a circuit family has shrinkage exponent Gamma if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of $p^{\Gamma}$. Our PRG uses a seed of length roughly $s^{1/(\Gamma + 1)}$ to fool circuits in the family of size $s$. By instantiating this generic construction, we get PRGs for the following classes:
1) de Morgan formulas of size $s$, seed length $s^{1/3+o(1)}$.
2) Formulas over an arbitrary basis of size $s$, seed length $s^{1/2+o(1)}$.
3) Read-once formulas, seed length $s^{.234...}$.
4) Branching programs of size $s$, seed length $s^{1/2 + o(1)}$.

The previous best PRGs known for these classes used seeds of length bigger than $n/2$ to output $n$ bits, and worked only when the size $s=O(n)$.



Changes to previous version:

Reference to recent work.


Paper:

TR12-057 | 7th May 2012 15:53

Pseudorandomness from Shrinkage


Abstract:

One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer a quantitative loss in parameters, and hence do not give nontrivial implications for models where we only know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can indeed construct PRGs which are essentially best possible without in turn improving the lower bounds.

More specifically, say that a circuit family has shrinkage exponent Gamma if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of $p^{\Gamma}$. Our PRG uses a seed of length roughly $s^{1/(\Gamma + 1)}$ to fool circuits in the family of size $s$. By instantiating this generic construction, we get PRGs for the following classes:
1) de Morgan formulas of size $s$, seed length $s^{1/3+o(1)}$.
2) Formulas over an arbitrary basis of size $s$, seed length $s^{1/2+o(1)}$.
3) Read-once formulas, seed length $s^{.234...}$.
4) Branching programs of size $s$, seed length $s^{1/2 + o(1)}$.

The previous best PRGs known for these classes used seeds of length bigger than $n/2$ to output $n$ bits, and worked only when the size $s=O(n)$.



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