Deterministic Extraction from Weak Random Sources [electronic resource] /
Gabizon, Ariel.author.
European Association for Theoretical Computer Science.
A deterministic extractor is a function that extracts almost perfect random bits from a weak random source. In this research monograph the author constructs deterministic extractors for several types of sources. A basic theme in this work is a methodology of recycling randomness which enables increasing the output length of deterministic extractors to near optimal length. The author's main work examines deterministic extractors for bit-fixing sources, deterministic extractors for affine sources and polynomial sources over large fields, and increasing the output length of zero-error dispersers. This work will be of interest to researchers and graduate students in combinatorics and theoretical computer science.
Description based upon print version of record.
Preface; Contents; 1 Introduction; 1.1 Organization of This Book; 1.2 The Classical Story; 1.2.1 Seeded Extractors; 1.2.2 Deterministic Extraction for Restricted Classes; 1.3 Other Motivations; 1.4 Techniques --- the Recycling Paradigm; 1.4.1 A Simple Example; 1.4.2 The General Principle and the Applicationfor Affine Sources; 1.4.3 The Recycling Paradigm in Bit-Fixing Sources; 1.4.4 The Recycling Paradigm for Zero-Error Dispersers; 1.4.5 What Else Is There in This Book?; 2 Deterministic Extractors for Bit-Fixing Sources by Obtaining an Independent Seed; 2.1 Introduction
2.1.1 Bit-Fixing Sources2.1.2 Our Results; 2.1.3 Overview of Techniques; 2.1.4 Outline; 2.2 Preliminaries; 2.2.1 Averaging Samplers; 2.2.2 Probability Distributions; 2.3 Obtaining an Independent Seed; 2.3.1 Seed Obtainers and Their Application; 2.3.2 Constructing Seed Obtainers; 2.4 Extracting a Few Bits for Any k; 2.5 Sampling and Partitioning with a Short Seed; 2.6 A Seeded Bit-Fixing Source Extractor with a Short Seed; 2.7 Deterministic Extractors for Bit-Fixing Sources; 2.7.1 An Extractor for Large k (Proof of Theorem 2.1); 2.7.2 An Extractor for Small k (Proof of Theorem 2.2)
2.8 Discussion and Open Problems3 Deterministic Extractors for Affine Sources over Large Fields; 3.1 Introduction; 3.1.1 Affine Source Extractors; 3.1.2 Our Results; 3.1.3 Previous Work; 3.2 Overview of Techniques; 3.2.1 Extracting Many Bits from Lines; 3.2.2 Linear Seeded Affine Source Extractors; 3.2.3 Using the Correlated Randomness as a Seed; 3.3 Preliminaries; 3.3.1 Probability Distributions; 3.3.2 Characters of Finite Fields; 3.4 Extracting One Bit from Lines; 3.5 Extracting Many Bits from Lines; 3.6 A Linear Seeded Extractor for Affine Sources; 3.7 Composing Extractors
3.8 Putting It All Together4 Extractors and Rank Extractors for Polynomial Sources; 4.1 Introduction; 4.1.1 Rank Extractors; 4.1.2 Extractors and Condensers for Polynomial Sources; 4.1.3 Rank Versus Entropy --- Weak Polynomial Sources; 4.1.4 Organization; 4.2 General Preliminaries; 4.2.1 Probability Distributions; 4.2.2 Polynomials over Finite Fields; 4.2.3 The Number of Solutions to a System of Polynomial Equations; 4.3 Algebraic Independence and Rank; 4.4 An Explicit Rank Extractor; 4.4.1 Preliminaries for the Proof of Theorem 4.4; 4.4.2 Proof of Theorem 4.4
4.5 Extractors for Polynomial Sources4.5.1 Preliminaries for the Proof of Theorem 4.5; 4.5.2 Proof of Theorem 4.5; 4.6 Improving the Output Length; 4.7 Extractors for Weak Polynomial Sources; 4.7.1 Proof of Theorem 4.9; 4.7.2 The Entropy of a Polynomial Mapping; 4.8 Rank Extractors over the Complex Numbers; 4.9 Discussion and Open Problems; 5 Increasing the Output Length of Zero-Error Dispersers; 5.1 Introduction; 5.1.1 Randomness Extractors and Dispersers; 5.1.2 Zero-Error Dispersers; 5.1.3 Increasing the Output Length of Zero-Error Dispersers; 5.1.4 Applications; 5.1.5 Outline
5.2 Preliminaries
Includes bibliographical references.
A deterministic extractor is a function that extracts almost perfect random bits from a weak random source. In this research monograph the author constructs deterministic extractors for several types of sources. A basic theme in this work is a methodology of recycling randomness which enables increasing the output length of deterministic extractors to near optimal length. The author's main work examines deterministic extractors for bit-fixing sources, deterministic extractors for affine sources and polynomial sources over large fields, and increasing the output length of zero-error dispersers. This work will be of interest to researchers and graduate students in combinatorics and theoretical computer science.
Computers.
Computer science—Mathematics.
Algebraic geometry.
Combinatorics.
Theory of Computation.
Mathematics of Computing.
Algebraic Geometry.
Combinatorics.
Monographs in Theoretical Computer Science. An EATCS Series