Bent functions: Results and Applications to Cryptography

Natalia Tokareva

Publisher:

Academic Press

Publication Date:

2016

Number of Pages:

202

Format:

Paperback

Price:

74.95

ISBN:

9780128023181

Category:

Monograph

MAA Review
Table of Contents

We do not plan to review this book.

Foreword
Preface
Notation
- Notation
Chapter 1: Boolean Functions
- Abstract
- Introduction
- 1.1 Definitions
- 1.2 Algebraic Normal Form
- 1.3 Boolean Cube and Hamming Distance
- 1.4 Extended Affinely Equivalent Functions
- 1.5 Walsh-Hadamard Transform
- 1.6 Finite Field and Boolean Functions
- 1.7 Trace Function
- 1.8 Polynomial Representation of a Boolean Function
- 1.9 Trace Representation of a Boolean Function
- 1.10 Monomial Boolean Functions
Chapter 2: Bent Functions: An Introduction
- Abstract
- Introduction
- 2.1 Definition of a Nonlinearity
- 2.2 Nonlinearity of a Random Boolean Function
- 2.3 Definition of a Bent Function
- 2.4 If n Is Odd?
- 2.5 Open Problems
- 2.6 Surveys
Chapter 3: History of Bent Functions
- Abstract
- Introduction
- 3.1 Oscar Rothaus
- 3.2 V.A. Eliseev and O.P. Stepchenkov
- 3.3 From the 1970s to the Present
Chapter 4: Applications of Bent Functions
- Abstract
- Introduction
- 4.1 Cryptography: Linear Cryptanalysis and Boolean Functions
- 4.2 Cryptography: One Historical Example
- 4.3 Cryptography: Bent Functions in CAST
- 4.4 Cryptography: Bent Functions in Grain
- 4.5 Cryptography: Bent Functions in HAVAL
- 4.6 Hadamard Matrices and Graphs
- 4.7 Links to Coding Theory
- 4.8 Bent Sequences
- 4.9 Mobile Networks, CDMA
- 4.10 Remarks
Chapter 5: Properties of Bent Functions
- Abstract
- Introduction
- 5.1 Degree of a Bent Function
- 5.2 Affine Transformations of Bent Functions
- 5.3 Rank of a Bent Function
- 5.4 Dual Bent Functions
- 5.5 Other Properties
Chapter 6: Equivalent Representations of Bent Functions
- Abstract
- Introduction
- 6.1 Hadamard Matrices
- 6.2 Difference Sets
- 6.3 Designs
- 6.4 Linear Spreads
- 6.5 Sets of Subspaces
- 6.6 Strongly Regular Graphs
- 6.7 Bent Rectangles
Chapter 7: Bent Functions with a Small Number of Variables
- Abstract
- Introduction
- 7.1 Two and Four Variables
- 7.2 Six Variables
- 7.3 Eight Variables
- 7.4 Ten and More Variables
- 7.5 Algorithms for Generation of Bent Functions
- 7.6 Concluding Remarks
Chapter 8: Combinatorial Constructions of Bent Functions
- Abstract
- Introduction
- 8.1 Rothaus’s Iterative Construction
- 8.2 Maiorana-McFarland Class
- 8.3 Partial Spreads: PS⁺, PS^-
- 8.4 Dillon’s Bent Functions: PS_ap
- 8.5 Dobbertin’s Construction
- 8.6 More Iterative Constructions
- 8.7 Minterm Iterative Constructions
- 8.8 Bent Iterative Functions: BI
- 8.9 Other Constructions
Chapter 9: Algebraic Constructions of Bent Functions
- Abstract
- Introduction
- 9.1 An Algebraic Approach
- 9.2 Bent Exponents: General Properties
- 9.3 Gold Bent Functions
- 9.4 Dillon Exponent
- 9.5 Kasami Bent Functions
- 9.6 Canteaut-Leander Bent Functions (MF-1)
- 9.7 Canteaut-Charpin-Kuyreghyan Bent Functions (MF-2)
- 9.8 Niho Exponents
- 9.9 General Algebraic Approach
- 9.10 Other Constructions
Chapter 10: Bent Functions and Other Cryptographic Properties
- Abstract
- Introduction
- 10.1 Cryptographic Criteria
- 10.2 High Degree and Balancedness
- 10.3 Correlation Immunity and Resiliency
- 10.4 Algebraic Immunity
- 10.5 Vectorial Bent Functions, Almost Bent Functions, and Almost Perfect Nonlinear Functions
Chapter 11: Distances Between Bent Functions
- Abstract
- Introduction
- 11.1 The Minimal Possible Distance Between Bent Functions
- 11.2 Classification of Bent Functions at the Minimal Distance from the Quadratic Bent Function
- 11.3 Upper Bound for the Number of Bent Functions at the Minimal Distance from an Arbitrary Bent Function
- 11.4 Bent Functions at the Minimal Distance from a McFarland Bent Function
- 11.5 Locally Metrically Equivalent Bent Functions
- 11.6 The Graph of Minimal Distances of Bent Functions
Chapter 12: Automorphisms of the Set of Bent Functions
- Abstract
- Introduction
- 12.1 Preliminaries
- 12.2 Shifts of the Class of Bent Functions
- 12.3 Duality Between Definitions of Bent and Affine Functions
- 12.4 Automorphisms of the Set of Bent Functions
- 12.5 Metrically Regular Sets
Chapter 13: Bounds on the Number of Bent Functions
- Abstract
- Introduction
- 13.1 Preliminaries
- 13.2 The Number of Bent Functions for Small n
- 13.3 Upper Bounds
- 13.4 Direct Lower Bounds
- 13.5 Iterative Lower Bounds
- 13.6 Lower Bound from the Bent Iterative Functions
- 13.7 Testing of the Lower Bound for Small n
- 13.8 Asymptotic Problem and Hypotheses
Chapter 14: Bent Decomposition Problem
- Abstract
- Introduction
- 14.1 Preliminaries
- 14.2 Partial Results
- 14.3 Boolean Function as the Sum of a Constant Number of Bent Functions
- 14.4 Any Cubic Boolean Function in Eight Variables is the Sum of at Most Four Bent Functions
- 14.5 Decomposition of Dual Bent Functions
Chapter 15: Algebraic Generalizations of Bent Functions
- Abstract
- Introduction
- 15.1 Preliminaries
- 15.2 The q-Valued Bent Functions
- 15.3 The p-ary Bent Functions
- 15.4 Bent Functions Over a Finite Field
- 15.5 Bent Functions Over Quasi-Frobenius Local Rings
- 15.6 Generalized Boolean Bent Functions (of Schmidt)
- 15.7 Bent Functions from a Finite Abelian Group into the Set of Complex Numbers on the Unit Circle
- 15.8 Bent Functions from a Finite Abelian Group into a Finite Abelian Group
- 15.9 Non-Abelian Bent Functions
- 15.10 Vectorial G-Bent Functions
- 15.11 Multidimensional Bent Functions on a Finite Abelian Group
Chapter 16: Combinatorial Generalizations of Bent Functions
- Abstract
- Introduction
- 16.1 Symmetric Bent Functions
- 16.2 Homogeneous Bent Functions
- 16.3 Rotation-Symmetric Bent Functions
- 16.4 Normal Bent Functions
- 16.5 Self-Dual and Anti-Self-Dual Bent Functions
- 16.6 Partially Defined Bent Functions
- 16.7 Plateaued Functions
- 16.8 Z-Bent Functions
- 16.9 Negabent Functions, Bent₄-Functions, and I-Bent Functions
Chapter 17: Cryptographic Generalizations of Bent Functions
- Abstract
- Introduction
- 17.1 Semibent Functions (Near-Bent Functions)
- 17.2 Balanced (Semi-) Bent Functions
- 17.3 Partially Bent Functions
- 17.4 Hyperbent Functions
- 17.5 Bent Functions of Higher Order
- 17.6 k-Bent Functions
References
Index