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.