Finite Fields and Their Applications: Character Sums and Polynomials

• Publisher: De Gruyter
• Year: 2013
• Tongue: English
• Leaves: 288
• Series: Radon Series on Computational and Applied Mathematics; 11
• Category: Library

✦ Synopsis

This book is based on the invited talks of the "RICAM-Workshop on Finite Fields and Their Applications: Character Sums and Polynomials" held at the Federal Institute for Adult Education (BIfEB) in Strobl, Austria, from September 2-7, 2012.

Finite fields play important roles in many application areas such as coding theory, cryptography, Monte Carlo and quasi-Monte Carlo methods, pseudorandom number generation, quantum computing, and wireless communication. In this book we will focus on sequences, character sums, and polynomials over finite fields in view of the above mentioned application areas:

• Chapters 1 and 2 deal with sequences mainly constructed via characters and analyzed using bounds on character sums.
• Chapters 3, 5, and 6 deal with polynomials over finite fields.
• Chapters 4 and 9 consider problems related to coding theory studied via finite geometry and additive combinatorics, respectively.
• Chapter 7 deals with quasirandom points in view of applications to numerical integration using quasi-Monte Carlo methods and simulation.
• Chapter 8 studies aspects of iterations of rational functions from which pseudorandom numbers for Monte Carlo methods can be derived.

The goal of this book is giving an overview of several recent research directions as well as stimulating research in sequences and polynomials under the unified framework of character theory.

✦ Table of Contents

Preface
Character Sums and Polyphase Sequence Families with Low Correlation, Discrete Fourier Transform (DFT), and Ambiguity
1 Introduction
2 Basic Definitions and Concepts
2.1 Notations
2.2 Polynomial Functions over Fq
2.3 Characters of Finite Fields
2.4 The Weil Bounds on Character Sums
3 Correlation, DFT, and Ambiguity Functions
3.1 Operators on Sequences
3.2 Correlation Functions
3.3 Ambiguity Functions
3.4 Convolution and Correlation
3.5 Optimal Correlation, DFT, and Ambiguity
4 Polyphase Sequences for Three Metrics
4.1 Sequences from the Additive Group of ZN and the Additive Group of Zp
4.1.1 Frank-Zadoff-Chu (FZC) Sequences
4.1.2 Another Class for Zn
4.1.3 Sequences from Fp Additive Characters
4.2 Sequences from Fp Multiplicative Characters
4.3 Sequences from Fq Additive Characters
4.4 Sequences from Fq Multiplicative Characters
4.5 Sequences Defined by Indexing Field Elements Alternatively
5 Sequences with Low Degree Polynomials
5.1 Methods for Generating Signal Sets from a Single Sequence
5.2 Sequences with Low Odd Degree Polynomials
5.2.1 Fq Additive Sequences with Low Odd Degree Polynomials
5.2.2 Fq Multiplicative Sequences with Low Odd Degree Polynomials
5.3 Sequences from Power Residue and Sidel’nikov Sequences
5.3.1 Interleaved Structure of Sidel’nikov Sequences
5.3.2 Sequences from Linear and/or Quadratic/Inverse Polynomials
5.4 Sequences from Hybrid Characters
5.4.1 Sequences Using Weil Representation and Their Generalizations
5.4.2 Generalization to Fq Hybrid Sequences
5.5 A New Construction
6 Two-Level Autocorrelation Sequences and Double Exponential Sums
6.1 Prime Two-Level Autocorrelation Sequences
6.2 Hadamard Transform, Second-Order Decimation-Hadamard Transform, and Hadamard Equivalence
6.3 Conjectures on Ternary 2-Level Autocorrelation Sequences
7 Some Open Problems
7.1 Current Status of the Conjectures on Ternary 2-Level Autocorrelation
7.2 Possibility of Multiplicative Sequences with Low Autocorrelation
7.3 Problems in Four Alternative Classes of Sequences and the General Hybrid Construction
8 Conclusions
Measures of Pseudorandomness
1 Introduction
2 Definition of the Pseudorandom Measures
3 Typical Values of Pseudorandom Measures
4 Minimum Values of Pseudorandom Measures
5 Connection between Pseudorandom Measures
6 Constructions
7 Family Measures
8 Linear Complexity
9 Multidimensional Theory
10 Extensions
Existence Results for Finite Field Polynomials with Specified Properties
1 Introduction
2 A Survey of Known Results
2.1 Normal Bases
2.2 Primitive Normal Bases
2.3 Prescribed Coefficients
2.4 Primitive Polynomials: Prescribed Coefficients
2.5 Primitive Normal Polynomials: Prescribed Coefficients
3 A Survey of Methodology and Techniques
3.1 Basic Approach
3.2 A p-adic Approach to Coefficient Constraints
3.3 The Sieving Technique
4 Conclusion
Incidence Structures, Codes, and Galois Geometries
1 Introduction
2 Galois Closed Codes
3 Extension Codes of Simplex and First-Order Reed-Muller Codes
4 Simple Incidence Structures and Their Codes
5 Embedding Theorems
6 Designs with Classical Parameters
7 Two-Weight Codes
8 Steiner Systems
9 Configurations
10 Conclusion and Open Problems
Special Mappings of Finite Fields
1 Introduction
2 Different Notions for Optimal Non-linearity
2.1 Almost Perfect Nonlinear (APN) Mappings
2.2 Bent and Almost Bent (AB) Mappings
3 Functions with a Linear Structure
4 Crooked Mappings
5 Planar Mappings
6 Switching Construction
7 Products of Linearized Polynomials
On The Classification of Perfect Nonlinear (PN) and Almost Perfect Nonlinear (APN) Monomial Functions
1 Introduction
2 Background and Motivation
2.1 PN and Planar Functions
2.2 APN Functions
3 Outline of APN Functions Classification Proof
3.1 Singularities in APN case
3.2 A Warm-Up Case
4 PN Functions Classification Proof: Analysis of Singularities
4.1 Singular Points in Case (b.1)
4.2 Singular Points at Infinity
4.3 The Multiplicities
4.4 Further Analysis
4.5 Type(i)
4.6 Type(iii)
4.7 Type(ii)
5 Case (b.1): Assuming Bt(x,y) Irreducible over Fp
6 Case (b.1): Assuming Bt (x, y) not Irreducible over Fp
Finite Fields and Quasirandom Points
1 Introduction
2 General Background
3 General Construction Principles
4 The Combinatorics of Nets
5 Duality Theory
6 Special Constructions of Nets
6.1 Polynomial Lattices
6.2 Hyperplane Nets
6.3 Nets Obtained from Global Function Fields
7 Special Constructions of (T,s)-Sequences
7.1 Faure Sequences and Niederreiter Sequences-c
7.2 Sequences Obtained from Global Function Fields
7.3 Sequences with Finite-Row Generating Matrices
Iterations of Rational Functions: Some Algebraic and Arithmetic Aspects
1 Introduction
1.1 Background
1.2 Notation
1.3 Iterations
2 Distribution of Elements, Degree Growth and Representation
2.1 Exponential Sums and Linear Combinations of Iterates
2.2 Generic Multivariate Polynomials
2.3 Systems with Slow Degree Growth
2.4 Exponential Degree Growth, but Sparse Representation
2.5 Representation of Iterates
2.6 Deligne and Dwork-Regular Polynomials
2.7 Distribution in Prime and Polynomial Times
3 Structure of Rational Function Maps
3.1 Trajectory Length and Periodic Structure
3.2 Graph of Rational Function Maps
3.3 Common Composites and Intersection of Orbits
4 Geometric Properties of Orbits
4.1 Diameter of Orbits
4.2 Convex Hull of Trajectories
5 Stability, Absolute Irreducibility and Coprimality
5.1 Motivation
5.2 Stable Univariate Polynomials
5.3 On the Growth of the Number of Irreducible Factors
5.4 Stable Multivariate Polynomials
5.5 Coprimality of Iterates
6 More Problems
6.1 Multiplicative Independence
6.2 Complete Polynomials
Additive Combinatorics over Finite Fields: New Results and Applications
1 Introduction
2 Notation
3 Estimates from Arithmetic Combinatorics
3.1 Classical Sum-Product Problem
3.2 Multifold Sum-Product Problem
3.3 Sum-Inversion Estimates
3.4 Equations over Finite Fields with Variables from Arbitrary Sets
3.5 Incidence Bounds
3.6 Polynomial and Other Nonlinear Functions on Sets
3.7 Structured Sets
3.8 Elliptic Curve Analogues
3.9 Matrix Analogues
4 Applications
4.1 Exponential and Character Sums
4.2 Waring, Erdős-Graham and Other Additive Problems in Finite Fields
4.3 Intersections of Almost Arithmetic and Geometric Progressions
4.4 Exponential Congruence
4.5 Hidden Shifted Power Problem
Sum-Product Estimates and Multiplicative Orders of γ and γ + γ-1 in Finite Fields
4.7 Expansion of Dynamical Systems
Index