TR08-045 Authors: Omer Reingold, Luca Trevisan, Madhur Tulsiani, Salil Vadhan

Publication: 24th April 2008 01:47

Downloads: 2536

Keywords:

A theorem of Green, Tao, and Ziegler can be stated (roughly)

as follows: if R is a pseudorandom set, and D is a dense subset of R,

then D may

be modeled by a set M that is dense in the entire domain such that D and

M are indistinguishable. (The precise statement refers to``measures'' or

distributions rather than sets.) The proof of this theorem is very

general, and it applies to notions of pseudorandomness and

indistinguishability defined in terms of any family of distinguishers

with some mild closure properties.

The proof proceeds via iterative partitioning and an energy increment

argument, in the spirit of the proof of the weak Szemeredi regularity

lemma. The ``reduction'' involved in the proof has exponential

complexity in the distinguishing probability.

We present a new proof inspired by Nisan's proof of Impagliazzo's

hardcore set theorem. The reduction in our proof has polynomial

complexity in the distinguishing probability and provides a new

characterization of the notion of ``pseudoentropy'' of a distribution.

We also follow the connection between the two theorems and obtain a new

proof of Impagliazzo's hardcore set theorem via iterative partitioning

and energy increment. While our reduction has exponential complexity in

some parameters, it has the advantage that the hardcore set is

efficiently recognizable.