# Bent Functions

### Results and Applications to Cryptography

1st Edition - August 6, 2015

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• Author: Natalia Tokareva
• eBook ISBN: 9780128025550
• Paperback ISBN: 9780128023181

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## Description

Bent Functions: Results and Applications to Cryptography offers a unique survey of the objects of discrete mathematics known as Boolean bent functions. As these maximal, nonlinear Boolean functions and their generalizations have many theoretical and practical applications in combinatorics, coding theory, and cryptography, the text provides a detailed survey of their main results, presenting a systematic overview of their generalizations and applications, and considering open problems in classification and systematization of bent functions. The text is appropriate for novices and advanced researchers, discussing proofs of several results, including the automorphism group of bent functions, the lower bound for the number of bent functions, and more.

## Key Features

• Provides a detailed survey of bent functions and their main results, presenting a systematic overview of their generalizations and applications
• Presents a systematic and detailed survey of hundreds of results in the area of highly nonlinear Boolean functions in cryptography
• Appropriate coverage for students from advanced specialists in cryptography, mathematics, and creators of ciphers

Researchers in discrete math and cryptography; students and professors of math and IT departments

• 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
• 1.6 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
• 6.2 Difference Sets
• 6.3 Designs
• 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: PSap
• 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

## Product details

• No. of pages: 220
• Language: English
• Published: August 6, 2015
• eBook ISBN: 9780128025550
• Paperback ISBN: 9780128023181

### Natalia Tokareva

Dr. Natalia Tokareva is a senior researcher at the Laboratory of Discrete Analysis in the Sobolev Institute of Mathematics and she teaches courses in cryptology in the Department of Mathematics and Mechanics at Novosibirsk State University. She has studied bent functions and their applications for several years, publishing one monograph (in Russian) and more than 12 articles. She has been a participant of many international conferences and seminars and presentations in the area of bent functions, particularly with applications in cryptography. Her research interests include Boolean functions in cryptography, bent functions, block and stream ciphers, cryptanalysis, coding theory, combinatorics, and algebra. She is chief of the seminar "Cryptography and Cryptanalysis" at the Sobolev Institute of Mathematics and she supervises BS, MS, and PhD students in discrete mathematics and cryptology.

#### Affiliations and Expertise

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation Department of Mathematics and Mechanics, Novosibirsk State University, Novosibirsk, Russian Federation

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