Related papers: Bent functions satisfying the dual bent condition …
In this article, we study bent functions on $\mathbb{F}_2^{2m}$ of the form $f(x,y) = x \cdot \phi(y) + h(y)$, where $x \in \mathbb{F}_2^{m-1} $ and $ y \in \mathbb{F}_2^{m+1}$, which form the generalized Maiorana-McFarland class (denoted…
Every Boolean bent function $f$ can be written either as a concatenation $f=f_1||f_2$ of two complementary semi-bent functions $f_1,f_2$; or as a concatenation $f=f_1||f_2||f_3||f_4$ of four Boolean functions $f_1,f_2,f_3,f_4$, all of which…
In this article, we provide the first systematic analysis of bent functions $f$ on $\mathbb{F}_2^{n}$ in the Maiorana-McFarland class $\mathcal{MM}$ regarding the origin and cardinality of their $\mathcal{M}$-subspaces, i.e., vector…
Bent functions of the form $\mathbb{F}_2^n\rightarrow\mathbb{Z}_q$, where $q\geqslant2$ is a positive integer, are known as generalized bent (gbent) functions. Gbent functions for which it is possible to define a dual gbent function are…
An important classification of permutations over $\mathbb{F}_2^m$, suitable for constructing Maiorana-McFarland bent functions on $\mathbb{F}_2^m \times \mathbb{F}_2^m$ with the unique $M$-subspace of maximal dimension, was recently…
Two classes of ternary bent functions of degree four with two and three terms in the univariate representation that belong to the completed Maiorana-McFarland class are found. Binomials are mappings $\F_{3^{4k}}\mapsto\fthree$ given by…
Bent functions are balanced by restricting their domains to vectors with either even or odd Hamming weights, which ensures an equal number of pre-images for both, 0 and 1. Using the previous fact, we can construct bent functions on two…
In this paper, we consider the characterization of the bentness of quadratic Boolean functions of the form $f(x)=\sum_{i=1}^{\frac{m}{2}-1} Tr^n_1(c_ix^{1+2^{ei}})+ Tr_1^{n/2}(c_{m/2}x^{1+2^{n/2}}) ,$ where $n=me$, $m$ is even and $c_i\in…
We give a new simple construction for known binary quadratic symmetric bent and almost bent functions. In particular, for even number of variables, they are self-dual and anti-self-dual quadratic bent functions, respectively, which are not…
Generalized bent (gbent) functions is a class of functions $f: \mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$, where $q \geq 2$ is a positive integer, that generalizes a concept of classical bent functions through their co-domain extension. A lot…
For any positive integers $n=2k$ and $m$ such that $m\geq k$, in this paper we show the maximal number of bent components of any $(n,m)$-functions is equal to $2^{m}-2^{m-k}$, and for those attaining the equality, their algebraic degree is…
Bent functions can be classified into regular bent functions, weakly regular but not regular bent functions, and non-weakly regular bent functions. Regular and weakly regular bent functions always appear in pairs since their duals are also…
Bent functions are Boolean functions in an even number of variables that are indicators of Hadamard difference sets in elementary abelian 2-groups. A bent function in m variables is said to be normal if it is constant on an affine space of…
The necessary and sufficient conditions for a class of functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$, where $q \geq 2$ is an even positive integer, have been recently identified for $q=4$ and $q=8$. In this article we give an…
A one to one correspondence between regular generalized bent functions from $\F_2^n$ to $\Z_{2^m},$ and $m-$tuples of Boolean bent functions is established. This correspondence maps self-dual (resp. anti-self-dual) generalized bent…
Let $n=2m$. In the present paper, we study the binomial Boolean functions of the form $$f_{a,b}(x) = \mathrm{Tr}_1^{n}(a x^{2^m-1 }) +\mathrm{Tr}_1^{2}(bx^{\frac{2^n-1}{3} }), $$ where $m$ is an even positive integer, $a\in…
Vectorial dual-bent functions have recently attracted some researchers' interest as they play a significant role in constructing partial difference sets, association schemes, bent partitions and linear codes. In this paper, we further study…
Bent functions as optimal combinatorial objects are difficult to characterize and construct. In the literature, bent idempotents are a special class of bent functions and few constructions have been presented, which are restricted by the…
A function $F:\mathbb{F}_2^n\rightarrow \mathbb{F}_2^n$, $n=2m$, can have at most $2^n-2^m$ bent component functions. Trivial examples are obtained as $F(x) = (f_1(x),\ldots,f_m(x),a_1(x),\ldots, a_m(x))$, where…
In 2017, Zhang et al. proposed a question (not open problem) and two open problems in [IEEE TIT 63 (8): 5336--5349, 2017] about constructing bent functions by using Rothaus' construction. In this note, we prove that the sufficient…