Related papers: On More Bent Functions From Dillon Exponents
Dillon-like Boolean functions are known, in the literature, to be those trace polynomial functions from $\mathbb{F}_{2^{2n}}$ to $\mathbb{F}_{2}$, with all the exponents being multiples of $2^n-1$ often called Dillon-like exponents. This…
Let $p\ge 7$, $q=p^m$. $K_q(a)=\sum_{x\in \mathbb{F}_{p^m}} \zeta^{\mathrm{Tr}^m_1(x^{p^m-2}+ax)}$ is the Kloosterman sum of $a$ on $\mathbb{F}_{p^m}$, where $\zeta=e^{\frac{2\pi\sqrt{-1}}{p}}$. The value $1-\frac{2}{\zeta+\zeta^{-1}}$ of…
Inspired by a recent work of Mesnager, we present several new infinite families of quadratic ternary bent, near-bent and 2-plateaued functions from some known quadratic ternary bent functions. Meanwhile, the distribution of the Walsh…
The characterization and construction of bent functions are challenging problems. The paper generalizes the constructions of Boolean bent functions by Mesnager \cite{M2014}, Xu et al. \cite{XCX2015} and $p$-ary bent functions by Xu et al.…
In this paper, we discover that any univariate Niho bent function is a sum of functions having the form of Leander-Kholosha bent functions with extra coefficients of the power terms. This allows immediately, knowing the terms of an…
In this article a technique for constructing $p$-ary bent functions from near-bent functions is presented. Two classes of quadratic $p$-ary functions are shown to be near-bent. Applying the construction of bent functions to these classes of…
In this note, the polar decomposition of binary fields of even extension degree is used to reduce the evaluation of the Walsh transform of binomial Boolean functions to that of Gauss sums. In the case of extensions of degree four times an…
In this paper, a new construction of quaternary bent functions from quaternary quadratic forms over Galois rings of characteristic 4 is proposed. Based on this construction, several new classes of quaternary bent functions are obtained, and…
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…
The $p$-ary function $f(x)$ mapping $\mathrm{GF}(p^{4k})$ to $\mathrm{GF}(p)$ given by $f(x)={\rm Tr}_{4k}\big(x^{p^{3k}+p^{2k}-p^k+1}+x^2\big)$ is proven to be a weakly regular bent function and the exact values of its Walsh transform…
The $p$-ary function $f(x)$ mapping $\mathrm{GF}(p^{4k})$ to $\mathrm{GF}(p)$ and given by $f(x)={\rm Tr}_{4k}\big(ax^d+bx^2\big)$ with $a,b\in\mathrm{GF}(p^{4k})$ and $d=p^{3k}+p^{2k}-p^k+1$ is studied with the respect to its exponential…
In this work we study $p$-adic continuous functions in several variables taking values on $\mathbb{Z}_p$. We describe the orthonormal van der Put base of these functions and study various Lipschitz conditions in several variables,…
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…
Let $K_{q^n}(a)$ be a Kloosterman sum over the finite field $\F_{q^n}$ of characteristic $p$. In this note so called subfield conjecture is proved in case $p>3$: if $a\ne0$ belongs to the proper subfield $\F_q$ of $\F_{q^n}$, then…
We study a new method of constructing Boolean bent functions from cyclotomic mappings. Three generic constructions are obtained by considering different branch functions such as Dillon functions, Niho functions and Kasami functions over…
The number of $n$-ary bent functions is less than $2^{3\cdot2^{n-3}(1+o(1))}$ as $n$ is even and $n\rightarrow\infty$. Keywords: Boolean function, bent function, upper bound
We obtain an explicit criterion for $p$-ary functions to produce association schemes in terms of their Walsh spectrum. Employing this characterization, we explicitly find a correlation between $p$-ary bent functions and association schemes;…
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…
This note presents generalizations of the partial spread bent functions introduced by Dillon, as well as the corresponding relative difference sets in nonabelian groups.
We return to some past studies of hyperkloosterman sums ([9,10]) via $p$-adic cohomology with an aim to improve earlier results. In particular, we work here with Dwork's $\theta_\infty$-splitting function and a better choice of basis for…