"Pseudorandomness from Shrinkage"

Russell Impagliazzo

UCSD

Monday, June 11, 2012, 2:00 pm

EBU3B, Room 4140

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 pGamma. Our PRG uses a seed of length roughly s1/(Gamma+1) to fool circuits in the family of size s. By instantiating this generic construction, we get PRGs for the following classes:

de Morgan formulas of size s, seed length s1/3+o(1).

Formulas over an arbitrary basis of size s, seed length s1/2+o(1).

Read-once formulas, seed length s0.234....

Branching programs of size s, seed length s1/2+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).

The talk will not assume any familiarity with formula lower bounds or pseudo-random generators.

Joint work with Raghu Meka and David Zuckerman.