An explicit two-source extractor with min-entropy rate near 4/9

04/15/2018
by   Mark Lewko, et al.
0

In 2005 Bourgain gave the first explicit construction of a two-source extractor family with min-entropy rate less than 1/2. His approach combined Fourier analysis with innovative but inefficient tools from arithmetic combinatorics and yielded an unspecified min-entropy rate which was greater than .499. This remained essentially the state of the art until a 2015 breakthrough of Chattopadhyay and Zuckerman in which they gave an alternative approach which produced extractors with arbitrarily small min-entropy rate. In the current work, we revisit the Fourier analytic approach. We give an improved analysis of one of Bourgain's extractors which shows that it in fact extracts from sources with min-entropy rate near 21/44 =.477..., moreover we construct a variant of this extractor which we show extracts from sources with min-entropy rate near 4/9 = .444.... While this min-entropy rate is inferior to Chattopadhyay and Zuckerman's construction, our extractors have the advantage of exponential small error which is important in some applications. The key ingredient in these arguments is recent progress connected to the restriction theory of the finite field paraboloid by Rudnev and Shkredov. This in turn relies on a Rudnev's point-plane incidence estimate, which in turn relies on Kollár's generalization of the Guth-Katz incidence theorem.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
04/15/2018

An explicit two-source extractor with min-entropy near 4/9

In 2005 Bourgain gave the first explicit construction of a two-source ex...
research
11/07/2021

Extractors: Low Entropy Requirements Colliding With Non-Malleability

The known constructions of negligible error (non-malleable) two-source e...
research
09/20/2023

Extractors for Polynomial Sources over 𝔽_2

We explicitly construct the first nontrivial extractors for degree d ≥ 2...
research
11/26/2022

Extractors for Images of Varieties

We construct explicit deterministic extractors for polynomial images of ...
research
01/10/2018

An Entropy Lower Bound for Non-Malleable Extractors

A (k,ε)-non-malleable extractor is a function nmExt : {0,1}^n ×{0,1}^d ...
research
04/16/2019

Extractors for small zero-fixing sources

A random variable X is an (n,k)-zero-fixing source if for some subset V⊆...
research
04/11/2023

Optimizing Linear Correctors: A Tight Output Min-Entropy Bound and Selection Technique

Post-processing of the raw bits produced by a true random number generat...

Please sign up or login with your details

Forgot password? Click here to reset