Bent functions. Fundamentals and results.

*(English)*Zbl 1364.94008
Cham: Springer (ISBN 978-3-319-32593-4/hbk; 978-3-319-32595-8/ebook). xxvi, 544 p. (2016).

Bent functions have important applications in cryptography and coding theory (being Boolean functions with maximal Hamming distance from the set of affine functions), and many interesting relations to further topics in mathematics, like combinatorics, design theory, graph theory. For instance they correspond to Hadamard difference sets in the elementary abelian \(2\)-group or to relative difference sets in the elementary abelian group. Since their introduction in the 1960s, there has been a lot of research on this class of functions in various directions. Many main results on bent functions can be found in [C. Carlet and S. Mesnager, Des. Codes Cryptography 78, No. 1, 5–50 (2016; Zbl 1378.94028)].

This book is a comprehensive and very detailed collection of results on bent functions obtained in the decades after their introduction. Motivated by applications in cryptography and coding, it first of all focuses on Boolean (bent) functions. Particular emphasis is put on the construction and representation of bent functions in the framework of finite fields. Many results are presented in considerable detail. For instance quite in detail representations of bent functions (and their duals) as univariate monomials, binomials, …are treated.

The book is rather self-contained, besides from references to the original literature (every Chapter finishes with a reference list), for many of the presented results also the proofs are given. As prerequisite some background in linear algebra and in the basics of finite fields is required. First of all, the book is addressed to researchers in discrete mathematics and their applications in cryptography or coding theory. It can serve as a comprehensive reference book on topics on (Boolean) bent functions. Furthermore it is interesting for theoretical computer scientists or engineers.

After some basics on Boolean functions and some mathematical foundations in Chapters 1 and 2, in Chapter 3 the main cryptographic criteria for Boolean functions are discussed. Being the functions with maximal nonlinearity, this Chapter provides the motivation for investigating bent functions in connection with cryptography. Several equivalent definitions of (Boolean) bent functions are presented in Chapter 4, the enumeration and classification problem is touched and it is pointed to applications of bent functions in cryptography and coding. Chapters 5-7 deal with constructions of bent functions (primary and secondary). Besides from the classical constructions, the Maiorana-McFarland construction and the partial spread construction, families of polynomials over finite fields which represent bent functions (in trace form) are presented. An own Chapter, Chapter 8, analyses Dillon’s \(\mathcal{H}\) class of bent functions, which are bent functions that are affine on the elements of the Desarguesian spread. Their connection to hyperovals and o-polynomials is pointed out. Classes of bent functions based on partial spreads and spreads (partial spread bent functions and Dillon’s \(\mathcal{H}\) class) are then revisited in Chapter 14. Spreads are explained in more detail, besides from the Desarguesian, also other spreads like AndrĂ©’s spread or semifield spreads are suggested for the construction of bent functions. Chapters 9-11 present results on so-called hyperbent functions, a subclass of bent functions with maximal distance from monomial permutations (which is meaningful only in the framework of finite fields). In Chapter 12, vectorial bent functions are introduced. Whereas the book in general focuses on Boolean bent functions (motivated by applications in coding and cryptography), Chapter 13 summarizes some results on bent functions in odd characteristic. In Chapter 15, some generalizations respectively special classes of bent functions are collected, like partially bent functions, rotation symmetric and idempotent bent functions, homogeneous bent functions, \(Z\)-bent functions, negabent functions. More comprehensively, plateaued functions are treated in Chapter 16, and the subclass of semi-bent functions is analysed in detail in Chapter 17. Finally Chapter 18 is devoted to linear error-correcting codes with few weights constructed from bent functions.

This book is a comprehensive and very detailed collection of results on bent functions obtained in the decades after their introduction. Motivated by applications in cryptography and coding, it first of all focuses on Boolean (bent) functions. Particular emphasis is put on the construction and representation of bent functions in the framework of finite fields. Many results are presented in considerable detail. For instance quite in detail representations of bent functions (and their duals) as univariate monomials, binomials, …are treated.

The book is rather self-contained, besides from references to the original literature (every Chapter finishes with a reference list), for many of the presented results also the proofs are given. As prerequisite some background in linear algebra and in the basics of finite fields is required. First of all, the book is addressed to researchers in discrete mathematics and their applications in cryptography or coding theory. It can serve as a comprehensive reference book on topics on (Boolean) bent functions. Furthermore it is interesting for theoretical computer scientists or engineers.

After some basics on Boolean functions and some mathematical foundations in Chapters 1 and 2, in Chapter 3 the main cryptographic criteria for Boolean functions are discussed. Being the functions with maximal nonlinearity, this Chapter provides the motivation for investigating bent functions in connection with cryptography. Several equivalent definitions of (Boolean) bent functions are presented in Chapter 4, the enumeration and classification problem is touched and it is pointed to applications of bent functions in cryptography and coding. Chapters 5-7 deal with constructions of bent functions (primary and secondary). Besides from the classical constructions, the Maiorana-McFarland construction and the partial spread construction, families of polynomials over finite fields which represent bent functions (in trace form) are presented. An own Chapter, Chapter 8, analyses Dillon’s \(\mathcal{H}\) class of bent functions, which are bent functions that are affine on the elements of the Desarguesian spread. Their connection to hyperovals and o-polynomials is pointed out. Classes of bent functions based on partial spreads and spreads (partial spread bent functions and Dillon’s \(\mathcal{H}\) class) are then revisited in Chapter 14. Spreads are explained in more detail, besides from the Desarguesian, also other spreads like AndrĂ©’s spread or semifield spreads are suggested for the construction of bent functions. Chapters 9-11 present results on so-called hyperbent functions, a subclass of bent functions with maximal distance from monomial permutations (which is meaningful only in the framework of finite fields). In Chapter 12, vectorial bent functions are introduced. Whereas the book in general focuses on Boolean bent functions (motivated by applications in coding and cryptography), Chapter 13 summarizes some results on bent functions in odd characteristic. In Chapter 15, some generalizations respectively special classes of bent functions are collected, like partially bent functions, rotation symmetric and idempotent bent functions, homogeneous bent functions, \(Z\)-bent functions, negabent functions. More comprehensively, plateaued functions are treated in Chapter 16, and the subclass of semi-bent functions is analysed in detail in Chapter 17. Finally Chapter 18 is devoted to linear error-correcting codes with few weights constructed from bent functions.

Reviewer: Wilfried Meidl (Linz)

##### MSC:

94-02 | Research exposition (monographs, survey articles) pertaining to information and communication theory |

94A60 | Cryptography |

94C10 | Switching theory, application of Boolean algebra; Boolean functions (MSC2010) |

94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |

94B05 | Linear codes, general |

06E30 | Boolean functions |