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Let m be an even positive integer. A Boolean bent function f on ${\mathbb F}_{ 2^{\text {m}-1}} \times {\mathbb F}_{ 2^{}}$ is called a cyclic bent function if for any $\text {a}\neq \text {b}\in {\mathbb F}_{ 2^{\text {m}-1}}$ and $\epsilon \in {\mathbb F}_{ 2^{}}$ , $\text {f}({\it\text { ax}}_{1},\text {x}_{2})+\text {f}({\it\text { bx}}_{1},\text {x}_{2}+\epsilon)$ is always bent, where $\text {x}_{1}\in {\mathbb F}_{ 2^{\text {m}-1}}, \text {x}_{2} \in {\mathbb F}_{ 2^{}}$ . Cyclic bent functions look extremely rare. This paper focuses on cyclic bent functions on ${\mathbb F}_{ 2^{\text {m}-1}} \times {\mathbb F}_{ 2^{}}$ and their applications. The first objective of this paper is to establish a link between quadratic cyclic bent functions and a special type of prequasifields, and construct a class of quadratic cyclic bent functions from the Kantor-Williams prequasifields. The second objective is to use cyclic bent functions to construct families of optimal sequences. The results of this paper show that cyclic bent functions have nice applications in several fields such as coding theory, symmetric cryptography, and CDMA communication.

In this paper, we first present two systematic constructions of bent functions by modifying the truth tables of Rothaus’s bent function and Maiorana-McFarland’s bent function respectively. The number of the newly constructed bent functions by modifying the truth table of Rothaus’s bent function is also determined. The methods of constructing self-dual bent functions are then given after the dual functions of these bent functions being determined. Finally, as an application, two constructions of 2-rotation symmetric bent functions are presented in this paper.

This paper is composed of two main parts related to the nonlinearity of vectorial functions. The first part is devoted to maximally nonlinear $(n,m)$ functions (the so-called bent vectorial functions), which contribute to an optimal resistance to both linear and differential attacks on symmetric cryptosystems. They can be used in block ciphers at the cost of additional diffusion/compression/expansion layers, or as building blocks for the construction of substitution boxes (S-boxes), and they are also useful for constructing robust codes and algebraic manipulation detection codes. A main issue on bent vectorial functions is to characterize bent monomial functions $Tr_{m}^{n} (\lambda x^{d})$ from $\mathbb {F}_{2^{n}}$ to $\mathbb {F}_{2^{m}}$ (where $m$ is a divisor of $n$ ) leading to a classification of those bent monomials. We also treat the case of functions with multiple trace terms involving general results and explicit constructions. Furthermore, we investigate some open problems raised by Pasalic et al. and Muratovic-Ribic et al. in a series of papers on vectorial functions. The second part is devoted to the nonlinearity of $(n,m)$ -functions. No tight upper bound is known when $\frac {n}2 . The covering radius bound is the only known upper bound in this range (the Sidelnikov–Chabaud–Vaudenay bound coincides with it when $m=n-1$ and it has no sense when $m ). Finding better bounds is an open problem since the 1990s. Moreover, no bound has been found during the last 23 years, which improve upon the covering radius bound for a large part of $(n,m)$ -functions. We derive such upper bounds for functions, which are sufficiently unbalanced or which satisfy some conditions. These upper bounds imply some necessary conditions for vectorial functions to have large nonlinearity.

A classical method of constructing a linear code over $ {\mathrm {GF}}(q)$ with a $t$ -design is to use the incidence matrix of the $t$ -design as a generator matrix over $ {\mathrm {GF}}(q)$ of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 2-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterization of highly nonlinear Boolean functions is presented.

As a class of optimal combinatorial objects, bent functions have important applications in cryptography, sequence design, and coding theory. Bent idempotents are a subclass of bent functions and of great interest, since they can be stored in less space and allow faster computation of the Walsh-Hadamard transform. The objective of this paper is to present a generic construction of bent functions from known ones. It includes the previous constructions of bent functions by Mesnager and Xu et al. as special cases, and produces new bent functions, which cannot be produced by earlier ones. In particular, it also generates infinite families of bent idempotents over ${F}_{2^{2m}}$ of any algebraic degree between 2 and $m$ . This together with a recent construction by Su and Tang gives a positive answer to an open problem on bent idempotents proposed by Carlet. In addition, an infinite family of anti-self-dual bent functions is obtained in which the sum of any three distinct functions is again an anti-self-dual bent function in this family. This solves an open problem recently proposed by Mesnager.

In the mid 1960s, Rothaus proposed the so-called “most general form” of constructing new bent functions by using three (initial) bent functions whose sum is again bent. In this paper, we utilize a special case of Rothaus construction when two of these three bent functions differ by a suitably chosen characteristic function of an $n/2$ -dimensional subspace. This simplification allows us to treat the induced bent conditions more easily, also implying the possibility to specify the initial functions in the partial spread class and most notably to identify several instances of the so-called non-normal bent functions. Affine inequivalent bent functions within this class are then identified using a suitable selection of initial bent functions within the partial spread class (stemming from the complete Desarguesian spread). It is also shown that when the initial bent functions belong to the class $\mathcal {D}$ , then, under certain conditions, the constructed functions provably do not belong to the completed Maiorana–McFarland class. We conjecture that our method potentially generates an infinite class of non-normal bent functions (all tested ten-variable functions are non-normal but unfortunately they are weakly normal) though there are no efficient computational tools for confirming this.

In this paper, we investigate properties of generalized bent Boolean functions and $2^{k}$ -bent (i.e., negabent, octabent, hexadecabent, et al. ) Boolean functions in a uniform framework. From the Hadamard matrices, Hodžic and Pasalic presented sufficient conditions for generalized bent functions. Using cyclotomic fields and the decomposition of generalized bent functions, we generalize their results, prove that Hodžic and Pasalic’s conditions of generalized bent functions are not only sufficient but also necessary, and completely characterize generalized bent functions in terms of their component functions. Furthermore, we present a secondary construction of bent functions or semi-bent functions from generalized bent functions. Finally, we give the relations of generalized bent functions and $2^{k}$ -bent functions, demonstrate that $2^{k}$ -bent functions are actually a special class of generalized bent functions, and completely characterize $2^{k}$ -bent functions.

Rotation symmetric bent functions and their generation two-rotation symmetric bent functions are two classes of cryptographically significant Boolean functions. However, few constructions have been presented in the literature, which either have restriction on integer $n$ or have algebraic degree no more than 4. In this paper, for any even integer $n\ge 4$ , three classes of bent functions are presented respectively. Most notably, the proposed $n$ -variable rotation symmetric bent functions and two-rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to $n/2$ . Besides, we obtain bent idempotent functions with the maximal algebraic degree $n/2$ .

For any bent function, it is very interesting to determine its dual function, because the dual function is also bent in certain cases. For $k$ odd and $\gcd (n, k)=1$ , it is known that the Coulter–Matthews bent function $f(x)=Tr\left({ax^{\frac {3^{k}+1}{2}}}\right)$ is weakly regular bent over $\mathbb {F}_{3^{n}}$ , where $a\in \mathbb {F}_{3^{n}}^{*}$ , and $Tr(\cdot ):\mathbb {F}_{3^{n}}\rightarrow \mathbb {F}_{3}$ is the trace function. In this paper, we investigate the dual function of $f(x)$ , aiming to determine a universal formula. In particular, for two cases, we determine the formula explicitly: for the case of $n=3t+1$ and $k=2t+1$ with $t\geq 2$ , the dual function is given by \begin{equation*}Tr\left ({-\frac {x^{3^{2t+1}+3^{t+1}+2}}{a^{3^{2t+1}+3^{t+1}+1}}-\frac {x^{3^{2t}+1}}{a^{-3^{2t}+3^{t}+1}}+\frac {x^{2}}{a^{-3^{2t+1}+3^{t+1}+1}}}\right );\end{equation*} and for the case of $n=3t+2$ and $k=2t+1$ with $t\geq 2$ , the dual function is given by \begin{equation*}Tr\left ({-\frac {x^{3^{2t+2}+1}}{a^{3^{2t+2}-3^{t+1}+3}}-\frac {x^{2\cdot 3^{2t+1}+3^{t+1}+1}}{a^{3^{2t+2}+3^{t+1}+1}}+\frac {x^{2}}{a^{-3^{2t+2}+3^{t+1}+3}}}\right ).\end{equation*} As a byproduct, we find two new classes of ternary bent functions with only three terms. Moreover, we also prove that in certain cases $f(x)$ is regular bent.

In this paper, we introduce a sequence of discrete Fourier transforms and define new versions of bent functions, which we shall call (weak and strong) octa/hexadeca and, in general, $2^{k}$ -bent functions. We investigate relationships between these classes and completely characterize the octabent and hexadecabent functions in terms of bent functions. We further find relative difference sets based upon these functions.

Nonlinearity is one of the most challenging combinatorial property in the domain of Boolean function research. Obtaining nonlinearity greater than the bent concatenation bound for odd number of variables continues to be one of the most sought after combinatorial research problems. The pioneering result in this direction has been discovered by Patterson and Wiedemann in 1983 (IEEE-IT), which considered Boolean functions on $5 \times 3 = 15$ variables that are invariant under the actions of the cyclic group ${GF(2^{5})}^\ast \,{\cdot }\, {GF(2^{3})}^\ast $ as well as the group of Frobenius automorphisms. Some of these Boolean functions possess nonlinearity greater than the bent concatenation bound. The next possible option for exploring such functions is on $7 \times 3 = 21$ variables. However, obtaining such functions remained elusive for more than three decades even after substantial efforts as evident in the literature. In this paper, we exploit combinatorial arguments together with heuristic search to demonstrate such functions for the first time.

The problem of constructing bent-negabent functions that do not belong to the completed Maiorana-McFarland class emerges implicitly through a series of construction methods proposed recently. These approaches manage to optimize the algebraic degree of bent-negabent functions, but all of the constructed bent-negabent functions belong to the completed Maiorana-McFarland class. In this paper, we use the indirect sum construction (proposed by Carlet in 2004) for constructing the bent-negabent functions that are not provably contained in this class, which is the first significant attempt in this direction. To achieve this, we first provide a class of bent functions with certain desirable properties that does not belong to this class and demonstrate the existence of the class members. Then, embedding these functions in the framework of the indirect sum construction, we are able to specify sufficient conditions for bent-negabent functions not being contained in the completed Maiorana-McFarland class.

New quadratic bent functions in polynomial form are constructed in this paper. The constructions give new Boolean bent, generalized Boolean bent and p-ary bent functions. Based on Z 4 -valued quadratic forms, a simple method provides several new constructions of generalized Boolean bent functions. From these generalized Boolean bent functions a method is presented to transform them into Boolean bent and semi-bent functions. Moreover, many new p-ary bent functions can also be obtained by applying similar methods.