Simple extractors via constructions of cryptographic pseudo-random generators

TL;DR

This paper introduces simple, efficient extractors derived from cryptographic pseudo-random generator constructions, notably from one-way permutations, improving simplicity and efficiency over previous methods.

Contribution

It demonstrates that pseudo-random generators from one-way permutations can be used to build simple, efficient extractors without designs or polynomial error-correcting codes.

Findings

Extractors operate in $O( ext{log}^2 n)$ time per output bit.

They work for sources with min entropy $ heta n$, for any constant $ heta > 0$.

Output length is approximately $n^{ heta/3}$.

Abstract

Trevisan has shown that constructions of pseudo-random generators from hard functions (the Nisan-Wigderson approach) also produce extractors. We show that constructions of pseudo-random generators from one-way permutations (the Blum-Micali-Yao approach) can be used for building extractors as well. Using this new technique we build extractors that do not use designs and polynomial-based error-correcting codes and that are very simple and efficient. For example, one extractor produces each output bit separately in $O(\log^2 n)$ time. These extractors work for weak sources with min entropy $\lambda n$, for arbitrary constant $\lambda > 0$, have seed length $O(\log^2 n)$, and their output length is $\approx n^{\lambda/3}$.