# On constructing bent functions from cyclotomic mappings

Xi Xie, Nian Li, Qiang Wang and Xiangyong Zeng X. Xie and X. Zeng are with the
Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and
Statistics, Hubei University, Wuhan 430062, China. N. Li is with the Hubei Key
Laboratory of Applied Mathematics, School of Cyber Science and Technology,
Hubei University, Wuhan 430062, China. Q. Wang is with the School of
Mathematics and Statistics, Carleton University, Ottawa, K1S 5B6, Canada.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
> Abstract: 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 multiplicative cosets and additive cosets
> respectively. As a result, several new explicit infinite families of bent
> functions and their duals are derived. We demonstrate that some previous
> constructions are special cases of our simple constructions. In addition, by
> studying their polynomial forms, we observe that the last construction
> provides some examples which are EA-inequivalent to five classes of
> monomials, Dillon type and Niho type polynomials.
>
> Keywords: Bent functions, Walsh transform, Multiplicative cyclotomic
> mappings, Additive cyclotomic mappings.

##  1 Introduction

Boolean bent functions were first introduced by Rothaus in 1976 [35] as an
interesting combinatorial object with maximum Hamming distance to the set of
all affine functions. Over the last four decades, bent functions have
attracted a lot of research interest due to their important applications in
cryptography [8], sequences [34] and coding theory [13, 15]. Kumar, Scholtz
and Welch in [23] generalized the notion of Boolean bent functions to the case
of functions over an arbitrary finite field. Then a lot of research has been
devoted to the construction of bent functions, which we refer the readers to
[11], [31], and Chapter 6 of [10]. The construction methods can be divided
into two categories: primary and secondary constructions. Primary
constructions build bent functions from scratch, see, e.g., the papers [4, 14,
16, 17, 24, 25, 29, 35]. In contrast, secondary constructions provide bent
functions from the known ones, and a non-exhaustive list of references is [5,
6, 7, 9, 14, 27, 30, 36, 40]. Though there are many works on bent functions,
the classification and the general structure of bent functions are still not
clear. Therefore it remains attractive to continue studying bent functions.

Cyclotomic mappings of the first order were first introduced by Evans [18] and
Niederreiter and Winterhof [33]. It was further generalized by Wang [38, 39].
Let $\mathbb{F}_{p^{n}}$ be the finite field with $p^{n}$ elements. The so-
called index $d$ generalized cyclotomic mappings of $\mathbb{F}_{p^{n}}$ are
functions $\mathbb{F}_{p^{n}}\rightarrow\mathbb{F}_{p^{n}}$ that agree with a
suitable monomial function $x\rightarrow ax^{r}$ (for a fixed
$a\in\mathbb{F}_{p^{n}}$ and non-negative integer $r$) on each cyclotomic
coset of the index $d$ subgroup of $\mathbb{F}_{p^{n}}^{*}$. It is called a
cyclotomic mapping if all exponents of monomials over all cosets are the same
exponent. It turns out that every polynomial fixing $0$ can be represented by
a cyclotomic mapping uniquely according to its index [2]. Cyclotomic mappings
are used extensively to study the permutation behaviour of polynomials over
finite fields [3, 38, 39]. Recently it was also used to construct linear codes
with few weights [19]. In this paper, we explore the new application of
cyclotomic mappings in the construction of bent functions.

Let $q=2^{m}$ and $m$ be a positive integer. In the first and second generic
constructions (see Theorems 2 and 4), we use Dillon functions and Niho
functions as the branch functions over index $q+1$ cyclotomic cosets of
${\mathbb{F}}_{q^{2}}$, respectively. Some explicit classes of bent functions
are provided and their duals are determined as well. Using Magma we also
demonstrate that many examples can be derived from these classes. For the
third construction, we introduce a new variation of cyclotomic mappings.
Namely, we partition the finite field ${\mathbb{F}}_{q^{2}}$ as a union of
additive cosets and then use Kasami functions as branch functions over these
additive cosets. As a result, we obtain another generic construction of bent
functions (see Theorem 8). Because the conditions in our construction are easy
to meet, we demonstrate our construction by obtaining some explicit classes of
non-quadratic bent functions (see Corollaries 3 and 4). Finally, switching
between the cyclotomic form and polynomial form, we illustrate that our
constructions can produce new infinite classes of bent polynomials, as well as
several previous known classes. In fact, these generic constructions produce
bent polynomials belonging to the Partial Spread class, the class
$\mathcal{H}$ and the Maiorana-McFarland class respectively.

The rest of this paper is organized as follows. Some preliminaries are given
in Section 2. Section 3 and Section 4 construct bent functions using Dillon
functions, Niho functions and Kasami functions as the branch functions over
multiplicative cyclotomic cosets and additive cyclotomic cosets respectively.
In Section 5, we study the polynomial forms of our newly constructed bent
functions and briefly discuss the EA-equivalence between our functions and
known ones. Finally, Section 6 concludes this paper.

##  2 Preliminaries

Throughout this paper, let $\mathbb{Z}_{d}$ denote the set
$\\{0,1,\cdots,d-1\\}$. In addition, let $\mathbb{F}_{p^{n}}$ be the finite
field with $p^{n}$ elements and $\mu_{e}=\\{x\in\mathbb{F}_{p^{n}}:x^{e}=1\\}$
be the set of $e$-th roots of unity in $\mathbb{F}_{p^{n}}$ for
$e\,|\,p^{n}-1$, where $p$ is a prime and $n$ is a positive integer. The
(absolute) trace function ${\rm
Tr}_{1}^{n}:\mathbb{F}_{p^{n}}\longrightarrow\mathbb{F}_{p}$ is defined by
${\rm Tr}_{1}^{n}(x)=\sum_{i=0}^{n-1}x^{p^{i}}$ for all
$x\in\mathbb{F}_{p^{n}}$.

Given a function $f(x)$ mapping from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p}$,
the Walsh transform of $f(x)$ is defined by

$\widehat{f}(b)=\sum\nolimits_{x\in\mathbb{F}_{p^{n}}}{\xi_{p}}^{f(x)-{\rm
Tr}_{1}^{n}(bx)},\,b\in\mathbb{F}_{p^{n}},$

where $\xi_{p}=e^{\frac{2\pi\sqrt{-1}}{p}}$ is a complex primitive $p$-th root
of unity. Then $f(x)$ is called a $p$-ary bent function if all its Walsh
coefficients satisfy $|\widehat{f}(b)|=p^{n/2}$ [35, 23]. A $p$-ary bent
function $f(x)$ is called regular if
$\widehat{f}(b)=p^{n/2}\omega^{\widetilde{f}(b)}$ holds for some function
$\widetilde{f}(x)$ mapping $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_{p}$, and it is
called weakly regular if there exists a complex $\mu$ having unit magnitude
such that $\widehat{f}(b)=\mu^{-1}p^{n/2}\omega^{\widetilde{f}(b)}$ for all
$b\in\mathbb{F}_{p^{n}}$. The function $\widetilde{f}(x)$ is called the dual
of $f(x)$ and it is also bent.

The following lemmas are helpful for the subsequent sections.

######  Lemma 1.

([21]) Let $n=2m$ and $a\in{\mathbb{F}}_{q^{2}}^{*}$, where $q=p^{m}$ and $p$
is a prime. Then

$\sum\nolimits_{x\in\mu_{q+1}}\xi_{p}^{{\rm
Tr}_{1}^{n}(ax)}=1-K_{m}(a^{q+1}).$

Here $K_{m}(a^{q+1}):=\sum_{x\in{\mathbb{F}}_{q}}\xi_{p}^{{\rm
Tr}_{1}^{m}(a^{q+1}x+x^{q-2})}$ is the Kloosterman sum over
${\mathbb{F}}_{q}$.

######  Lemma 2.

([37]) Let $n=2m$ be an even positive integer and
$a,b\in\mathbb{F}_{2^{n}}^{*}$ satisfying ${\rm Tr}_{1}^{n}(b/a^{2})=0$. Then
the quadratic equation $x^{2}+ax+b=0$ has

(1) both two solutions in $\mu_{2^{m}+1}$ if and only if $b=a^{1-2^{m}}$ and
${\rm Tr}_{1}^{m}(b/a^{2})=1$.

(2) exactly one solution in $\mu_{2^{m}+1}$ if and only if $b\neq a^{1-2^{m}}$
and

$(1+b^{2^{m}+1})(1+a^{2^{m}+1}+b^{2^{m}+1})+a^{2}b^{2^{m}}+a^{2^{m}}b=0.$

##  3 Bent functions from multiplicative cyclotomic mappings

Let $p$ be a prime, $d,\,n$ be positive integers such that $d\mid p^{n}-1$,
and $\omega$ be a primitive element of $\mathbb{F}_{p^{n}}$. Let $C$ be the
(unique) index $d$ subgroup of $\mathbb{F}_{p^{n}}^{*}$. Then the cosets of
$C$ in $\mathbb{F}_{p^{n}}^{*}$ are of the form $C_{i}:=\omega^{i}C$ for
$i\in\mathbb{Z}_{d}$. It can be seen that
$\mathbb{F}_{p^{n}}=(\bigcup_{i=0}^{d-1}C_{i})\bigcup\\{0\\}$ and
$C_{i}\bigcap C_{j}=\emptyset$ for $i\neq j$. Let
$(a_{0},a_{1},\cdots,a_{d-1})\in\mathbb{F}_{p^{n}}^{d}$ and
$r_{0},r_{1},\cdots,r_{d-1}$ be $d$ non-negative integers. A generalized
cyclotomic mapping [3, 39] of $\mathbb{F}_{p^{n}}$ of index $d$ is defined as
follows:

$F(x)=\left\\{\begin{array}[]{ll}0,&{\rm if}\,\,x=0,\\\ a_{i}x^{r_{i}},&{\rm
if}\,\,x\in C_{i},\,i\in\mathbb{Z}_{d}.\end{array}\right.$ (1)

Because the finite field is partitioned into the union of $0$ and the
multiplicative cosets, we call these cyclotomic mappings as multiplicative
cyclotomic mappings. In this paper, we investigate the bentness of the
function $f(x)={\rm Tr}_{1}^{n}(F(x))$. To study the Walsh transform of $f(x)$
at point $b\in\mathbb{F}_{p^{n}}$, we first define

$S_{i}(b)=\sum\nolimits_{x\in C_{i}}\xi_{p}^{{\rm
Tr}_{1}^{n}(a_{i}x^{r_{i}})-{\rm Tr}_{1}^{n}(bx)}$ (2)

for $i\in\mathbb{Z}_{d}$. This gives

$\displaystyle\widehat{f}(b)$ $\displaystyle=$
$\displaystyle\sum\nolimits_{x\in\mathbb{F}_{p^{n}}}\xi_{p}^{{\rm
Tr}_{1}^{n}(F(x))-{\rm Tr}_{1}^{n}(bx)}$ (3) $\displaystyle=$ $\displaystyle
1+\sum_{i\in\mathbb{Z}_{d}}\sum_{x\in C_{i}}\xi_{p}^{{\rm
Tr}_{1}^{n}(a_{i}x^{r_{i}})-{\rm Tr}_{1}^{n}(bx)}$ $\displaystyle=$
$\displaystyle 1+\sum\nolimits_{i\in\mathbb{Z}_{d}}S_{i}(b).$

Therefore, to determine $\widehat{f}(b)$, it suffices to calculate $S_{i}(b)$
for all $i\in\mathbb{Z}_{d}$. In this section, we focus on the construction of
bent functions from multiplicative cyclotomic mappings of index $q+1$ over
$\mathbb{F}_{q^{2}}$, where $q=p^{m}$ and $n=2m$. In this case,
$C={\mathbb{F}}_{q}^{*}$ and $C_{i}=\omega^{i}{\mathbb{F}}_{q}^{*}$. Note that
$\omega^{i}{\mathbb{F}}_{q}^{*}=\\{\omega^{i+k(q+1)}:k\in\mathbb{Z}_{q-1}\\}=\\{\omega^{i(q-1)(2^{n-1}-1)+(2^{n-1}i+k)(q+1)}:k\in\mathbb{Z}_{q-1}\\}=u^{i}{\mathbb{F}}_{q}^{*}$
with the notation $u=\omega^{(q-1)(2^{n-1}-1)}$. Then $S_{i}(b)$ defined by
(2) becomes

$\displaystyle S_{i}(b)$ $\displaystyle=\sum\nolimits_{x\in
u^{i}{\mathbb{F}}_{q}^{*}}\xi_{p}^{{\rm Tr}_{1}^{n}(a_{i}x^{r_{i}})-{\rm
Tr}_{1}^{n}(bx)}$
$\displaystyle=\sum\nolimits_{y\in{\mathbb{F}}_{q}^{*}}\xi_{p}^{{\rm
Tr}_{1}^{n}(a_{i}(u^{i}y)^{r_{i}})-{\rm Tr}_{1}^{n}(bu^{i}y)}$
$\displaystyle=\sum\nolimits_{y\in{\mathbb{F}}_{q}^{*}}\xi_{p}^{{\rm
Tr}_{1}^{m}(\alpha_{i}y^{r_{i}})-{\rm Tr}_{1}^{m}((bu^{i}+b^{q}u^{-i})y)}$

with $\alpha_{i}:=a_{i}u^{ir_{i}}+a_{i}^{q}u^{-ir_{i}}$ due to
$u\in\mu_{q+1}$. The general case is difficult to calculate, we shall study
$S_{i}(b)$ for the following two cases:

If $r_{i}\equiv 0\,({\rm mod}\,q-1)$, then

$\displaystyle S_{i}(b)$ $\displaystyle=$ $\displaystyle\xi_{p}^{{\rm
Tr}_{1}^{m}(\alpha_{i})}\sum\nolimits_{y\in{\mathbb{F}}_{q}^{*}}\xi_{p}^{-{\rm
Tr}_{1}^{m}((bu^{i}+b^{q}u^{-i})y)}$ (6) $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}(q-1)\xi_{p}^{{\rm
Tr}_{1}^{m}(\alpha_{i})},&{\rm if}\,\,bu^{i}+b^{q}u^{-i}=0,\\\ -\xi_{p}^{{\rm
Tr}_{1}^{m}(\alpha_{i})},&{\rm otherwise}.\end{array}\right.$

If $r_{i}\equiv p^{t_{i}}\,({\rm mod}\,q-1)$ for some integer $t_{i}$, then

$\displaystyle S_{i}(b)$
$\displaystyle=\sum\nolimits_{y\in{\mathbb{F}}_{q}^{*}}\xi_{p}^{{\rm
Tr}_{1}^{m}((\alpha_{i}^{p^{-t_{i}}}-(bu^{i}+b^{q}u^{-i}))y)}$
$\displaystyle=\left\\{\begin{array}[]{ll}q-1,&{\rm
if}\,\,bu^{i}+b^{q}u^{-i}=\alpha_{i}^{p^{-t_{i}}},\\\ -1,&{\rm
otherwise},\end{array}\right.$

which implies that $S_{i}(b)=qT_{i}(b)-1$ with the notation

$T_{i}(b)=\left\\{\begin{array}[]{ll}1,&{\rm
if}\,\,bu^{i}+b^{q}u^{-i}=\alpha_{i}^{p^{-t_{i}}},\\\ 0,&{\rm
otherwise}.\end{array}\right.$ (7)

In the sequel, we always assume that $n=2m$, $q=2^{m}$ and
$u=\omega^{(q-1)(2^{n-1}-1)}$ is a primitive element of $\mu_{q+1}$ with
$u^{\infty}=0$. Define $R_{0}:=\\{i\in\mathbb{Z}_{q+1}:r_{i}\equiv 0\,({\rm
mod}\,q-1)\\}$ and $R_{1}:=\\{i\in\mathbb{Z}_{q+1}:r_{i}\equiv
p^{t_{i}}\,({\rm mod}\,q-1)\\}$. Then we characterize the Walsh transform of a
Boolean function $f(x)$ in the case of $\\#R_{0}+\\#R_{1}=q+1$.

######  Theorem 1.

Let $r_{i}$ be $q+1$ non-negative integers satisfying $\\#R_{0}+\\#R_{1}=q+1$
and $a_{i}\in{\mathbb{F}}_{q^{2}}$ for $i\in\mathbb{Z}_{q+1}$. Define

$f(x)={\rm Tr}_{1}^{n}(a_{i}x^{r_{i}}),\,\,{\rm if}\,\,x\in
u^{i}\mathbb{F}_{q}^{*},\,i\in\\{\infty\\}\cup\mathbb{Z}_{q+1}.$ (8)

Then for any $b\in{\mathbb{F}}_{q^{2}}$,

$\widehat{f}(b)=\left\\{\begin{array}[]{lll}q(M_{0}+\sum_{i\in
R_{1}}T_{i}(b))-(M_{0}+\\#R_{1}-1),&{\rm if}\,\,b=0,\\\ q((-1)^{{\rm
Tr}_{1}^{m}(\alpha_{t})}+\sum_{i\in R_{1}}T_{i}(b))-(M_{0}+\\#R_{1}-1),&{\rm
if}\,\,b^{q-1}=u^{2t},\,t\in R_{0},\\\ q\sum_{i\in
R_{1}}T_{i}(b)-(M_{0}+\\#R_{1}-1),&{\rm if}\,\,b^{q-1}=u^{2t},\,t\in R_{1},\\\
\end{array}\right.$

where $M_{0}:=\sum_{i\in R_{0}}(-1)^{{\rm Tr}_{1}^{m}(\alpha_{i})}$ with
$\alpha_{i}:=a_{i}u^{ir_{i}}+a_{i}^{q}u^{-ir_{i}}$ and $T_{i}(b)$ is defined
by (7).

###### Proof.

In this case, for any $b\in{\mathbb{F}}_{q^{2}}$, (3) turns into

$\displaystyle\widehat{f}(b)$ $\displaystyle=$ $\displaystyle
1+\sum_{i\in\mathbb{Z}_{q+1}}S_{i}(b)=1+\sum_{i\in R_{0}}S_{i}(b)+\sum_{i\in
R_{1}}S_{i}(b)$ (9) $\displaystyle=$ $\displaystyle 1+\sum_{i\in
R_{0}}S_{i}(b)+\sum_{i\in R_{1}}(qT_{i}(b)-1)$ $\displaystyle=$
$\displaystyle\sum_{i\in R_{0}}S_{i}(b)+q\sum_{i\in
R_{1}}T_{i}(b)-\\#R_{1}+1.$

Next, we calculate $\sum_{i\in R_{0}}S_{i}(b)$ for $b\in{\mathbb{F}}_{q^{2}}$
by considering the following three cases.

Case 1: $b=0$. In this case, (6) gives $S_{i}(b)=(q-1)(-1)^{{\rm
Tr}_{1}^{m}(\alpha_{i})}$ for all $i\in R_{0}$. This means $\sum_{i\in
R_{0}}S_{i}(b)=(q-1)M_{0}$ with the notation $M_{0}=\sum_{i\in
R_{0}}(-1)^{{\rm Tr}_{1}^{m}(\alpha_{i})}$.

Case 2: $b^{q-1}=u^{2t}$ for some $t\in R_{0}$. If this case happens, then
$b^{q}u^{-t}+bu^{t}=0$. Further, (6) implies $S_{t}(b)=(q-1)(-1)^{{\rm
Tr}_{1}^{m}(\alpha_{t})}$. Next we claim that $b^{q}u^{-i}+bu^{i}\neq 0$ for
any $i\in R_{0}\backslash\\{t\\}$. Suppose that there exists $t^{\prime}\in
R_{0}\backslash\\{t\\}$ such that $b^{q}u^{-t^{\prime}}+bu^{t^{\prime}}=0$.
Then $u^{2t^{\prime}}=b^{q-1}=u^{2t}$, i.e., $u^{2(t-t^{\prime})}=1$, which is
impossible due to $\gcd(2,\,q+1)=1$ and $t\neq t^{\prime}$. Thus
$b^{q}u^{-i}+bu^{i}\neq 0$ for any $i\in R_{0}\backslash\\{t\\}$. From (6) we
can know that $S_{i}(b)=-(-1)^{{\rm Tr}_{1}^{m}(\alpha_{i})}$ for $i\in
R_{0}\backslash\\{t\\}$. Hence one concludes

$\sum_{i\in R_{0}}S_{i}(b)=(q-1)(-1)^{{\rm Tr}_{1}^{m}(\alpha_{t})}-\sum_{i\in
R_{0}\backslash\\{t\\}}(-1)^{{\rm Tr}_{1}^{m}(\alpha_{i})}=q(-1)^{{\rm
Tr}_{1}^{m}(\alpha_{t})}-M_{0}.$

Case 3: $b^{q-1}=u^{2t}$ for some $t\in R_{1}$. If this case happens,
$b^{q-1}\neq u^{2i}$, i.e., $b^{q}u^{-i}+bu^{i}\neq 0$ for all $i\in R_{0}$.
Then (6) yields $S_{i}(b)=-(-1)^{{\rm Tr}_{1}^{m}(\alpha_{i})}$ for $i\in
R_{0}$. Therefore we have $\sum_{i\in R_{0}}S_{i}(b)=-M_{0}$.

Substituting the values of $\sum_{i\in R_{0}}S_{i}(b)$ in Cases 1-3 into (9)
gives the desired result. ∎

Next we consider the construction of bent functions from two classes of
multiplicative cyclotomic mappings of index $q+1$ over $\mathbb{F}_{q^{2}}$.

###  3.1 Bent functions from Dillon functions

First of all, we focus on the branch functions with Dillon exponents. In this
case, we consider $R_{0}=\mathbb{Z}_{q+1}$ and $R_{1}=\emptyset$. And we
assume $q>2$ in this subsection. Then we state our first result on bent
functions.

######  Theorem 2.

Let $a_{i}\in{\mathbb{F}}_{q^{2}}$ and $l_{i}$ be non-negative integers for
$i\in\mathbb{Z}_{q+1}$. Then

$f(x)={\rm Tr}_{1}^{n}(a_{i}x^{l_{i}(q-1)}),\,\,{\rm if}\,\,x\in
u^{i}\mathbb{F}_{q}^{*},\,i\in\\{\infty\\}\cup\mathbb{Z}_{q+1}$

is bent if and only if

$\sum\nolimits_{i\in\mathbb{Z}_{q+1}}(-1)^{{\rm
Tr}_{1}^{n}(a_{i}u^{-2il_{i}})}=1.$

Moreover, the dual function of $f(x)$ is

$\widetilde{f}(x)={\rm Tr}_{1}^{n}(a_{i}x^{l_{i}(1-q)}),\,\,{\rm
if}\,\,x^{q-1}=u^{2i},\,i\in\\{\infty\\}\cup\mathbb{Z}_{q+1}.$

###### Proof.

From Theorem 1, for any $b\in{\mathbb{F}}_{q^{2}}$, one has

$\widehat{f}(b)=\left\\{\begin{array}[]{lll}qM_{0}-(M_{0}-1),&{\rm
if}\,\,b=0,\\\ q(-1)^{{\rm Tr}_{1}^{m}(\alpha_{t})}-(M_{0}-1),&{\rm
if}\,\,b^{q-1}=u^{2t},\,t\in\mathbb{Z}_{q+1}\\\ \end{array}\right.$ (10)

due to that $\\#R_{1}=0$ and $\sum_{i\in R_{1}}T_{i}(b)=0$. Here
$\alpha_{i}=a_{i}u^{il_{i}(q-1)}+a_{i}^{q}u^{-il_{i}(q-1)}=a_{i}u^{-2il_{i}}+a_{i}^{q}u^{2il_{i}}$
and $M_{0}=\sum_{i\in\mathbb{Z}_{q+1}}(-1)^{{\rm
Tr}_{1}^{m}(\alpha_{i})}=\sum_{i\in\mathbb{Z}_{q+1}}(-1)^{{\rm
Tr}_{1}^{n}(a_{i}u^{-2il_{i}})}$. Note that when $i$ runs over
$\mathbb{Z}_{q+1}$, $u^{2i}$ runs over $\mu_{q+1}$, which implies
$\\{0\\}\bigcup\\{b\in{\mathbb{F}}_{q^{2}}:b^{q-1}=u^{2i},i\in\mathbb{Z}_{q+1}\\}={\mathbb{F}}_{q^{2}}$.
Then we claim that $|\widehat{f}(b)|=q$ for any $b\in{\mathbb{F}}_{q^{2}}$ if
and only if $M_{0}=1$. Obviously, if $M_{0}=1$, then (10) yields
$|\widehat{f}(b)|=q$ for any $b\in{\mathbb{F}}_{q^{2}}$. On the other hand, if
$|\widehat{f}(b)|=q$ for any $b\in{\mathbb{F}}_{q^{2}}$, then
$|\widehat{f}(0)|=q$ and then (10) indicates
$\widehat{f}(0)=qM_{0}-(M_{0}-1)=(q-1)M_{0}+1=\pm q$, which leads to $M_{0}=1$
since $q>2$. Thus $|\widehat{f}(b)|=q$ for any $b\in{\mathbb{F}}_{q^{2}}$ if
and only if $M_{0}=1$. More precisely, (10) gives $\widehat{f}(0)=q$, and if
$b^{q-1}=u^{2t}$ for some $t\in\mathbb{Z}_{q+1}$, then

$\widehat{f}(b)=q(-1)^{{\rm Tr}_{1}^{m}(\alpha_{t})}=q(-1)^{{\rm
Tr}_{1}^{n}(a_{t}u^{-2tl_{t}})}=q(-1)^{{\rm
Tr}_{1}^{n}(a_{t}b^{l_{t}(1-q)})}.$

This completes the proof. ∎

A slight modification of the condition given in Theorem 2 such that $f(x)$ is
bent allows us to construct bent functions explicitly. Note that

$\sum_{i\in\mathbb{Z}_{q+1}}(-1)^{{\rm
Tr}_{1}^{n}(a_{i}u^{-2il_{i}})}=\sum_{i\in\mathbb{Z}_{q}}((-1)^{{\rm
Tr}_{1}^{n}(a_{i}u^{-2il_{i}})}-(-1)^{{\rm
Tr}_{1}^{n}(a_{q}u^{-2il_{q}})})+\sum_{i\in\mathbb{Z}_{q+1}}(-1)^{{\rm
Tr}_{1}^{n}(a_{q}u^{-2il_{q}})}$

and when $\gcd(l_{q},\,q+1)=1$ and $a_{q}\neq 0$,

$\sum_{i\in\mathbb{Z}_{q+1}}(-1)^{{\rm
Tr}_{1}^{n}(a_{q}u^{-2il_{q}})}=\sum_{z\in\mu_{q+1}}(-1)^{{\rm
Tr}_{1}^{n}(a_{q}z)}=1-K_{m}(a_{q}^{q+1})$

due to Lemma 1. That means $f(x)$ in Theorem 2 is bent if and only if

$\sum\nolimits_{i\in\mathbb{Z}_{q}}((-1)^{{\rm
Tr}_{1}^{n}(a_{i}u^{-2il_{i}})}-(-1)^{{\rm
Tr}_{1}^{n}(a_{q}u^{-2il_{q}})})=K_{m}(a_{q}^{q+1}).$

By using this observation, we obtain the following construction of bent
functions for the case $\\#\\{a_{i}:i\in\mathbb{Z}_{q+1}\\}\leq 2$ directly.

######  Theorem 3.

Let $a_{1}\in{\mathbb{F}}_{q^{2}},a_{2}\in{\mathbb{F}}_{q^{2}}^{*}$,
$l_{1},\,l_{2}$ be non-negative integers and $\gcd(l_{2},\,q+1)=1$. Denote
$N:=\\{u^{i}{\mathbb{F}}_{q}^{*}:i\in\mathbb{Z}\\}$, where $\mathbb{Z}$ is a
subset of $\mathbb{Z}_{q}$. Then

$f(x)=\left\\{\begin{array}[]{ll}{\rm Tr}_{1}^{n}(a_{1}x^{l_{1}(q-1)}),&{\rm
if}\,\,x\in N,\\\ {\rm Tr}_{1}^{n}(a_{2}x^{l_{2}(q-1)}),&{\rm
otherwise}\end{array}\right.$

is a bent function if and only if

$\sum\nolimits_{i\in\mathbb{Z}}((-1)^{{\rm
Tr}_{1}^{n}(a_{1}u^{-2il_{1}})}-(-1)^{{\rm
Tr}_{1}^{n}(a_{2}u^{-2il_{2}})})=K_{m}(a_{2}^{q+1}).$ (11)

Moreover, the dual function of $f(x)$ is

$\widetilde{f}(x)=\left\\{\begin{array}[]{ll}{\rm
Tr}_{1}^{n}(a_{1}x^{l_{1}(1-q)}),&{\rm
if}\,\,x^{q-1}=u^{2i},\,i\in\mathbb{Z},\\\ {\rm
Tr}_{1}^{n}(a_{2}x^{l_{2}(1-q)}),&{\rm otherwise}.\end{array}\right.$

It is easy to construct bent functions by selecting suitable set $N$ and
parameters $a_{1},a_{2}$ when $\gcd(l_{1},\,q+1)=\gcd(l_{2},\,q+1)=1$. In the
case of $N=\mu_{r(q-1)}$ with $r\mid(q+1)$, one has
$\mathbb{Z}=\\{i\in\mathbb{Z}_{q}:(q+1)/r\mid i\\}$. Then (11) becomes

$\sum\nolimits_{i\in\mathbb{Z}_{q},(q+1)/r|i}((-1)^{{\rm
Tr}_{1}^{n}(a_{1}u^{-2il_{1}})}-(-1)^{{\rm
Tr}_{1}^{n}(a_{2}u^{-2il_{2}})})=K_{m}(a_{2}^{q+1}),$

that is,

$\sum\nolimits_{i\in\mathbb{Z}_{r}}((-1)^{{\rm
Tr}_{1}^{n}(a_{1}\varepsilon^{i})}-(-1)^{{\rm
Tr}_{1}^{n}(a_{2}\varepsilon^{i})})=K_{m}(a_{2}^{q+1})$ (12)

due to $\gcd(l_{i},\,q+1)=1$ for $i=1,\,2$, where $\varepsilon$ is a primitive
element of $\mu_{r}$. Firstly, set $a_{1}=\varepsilon^{j}a_{2}$ for
$j\in\mathbb{Z}_{r}$, then

$\sum\nolimits_{i\in\mathbb{Z}_{r}}(-1)^{{\rm
Tr}_{1}^{n}(a_{1}\varepsilon^{i})}=\sum\nolimits_{i\in\mathbb{Z}_{r}}(-1)^{{\rm
Tr}_{1}^{n}(a_{2}\varepsilon^{i+j})}=\sum\nolimits_{i\in\mathbb{Z}_{r}}(-1)^{{\rm
Tr}_{1}^{n}(a_{2}\varepsilon^{i})}.$

Then the following result can be obtained from Theorem 3 and (12) directly.

######  Corollary 1.

Let $c\in{\mathbb{F}}_{q^{2}}^{*}$, $\epsilon\in\mu_{r}$, $r$ and $l_{i}$ be
positive integers satisfying $r|(q+1)$ and $\gcd(l_{i},\,q+1)=1$ for
$i=1,\,2$. Define

$f(x)=\left\\{\begin{array}[]{ll}{\rm Tr}_{1}^{n}(\epsilon
cx^{l_{1}(q-1)}),&{\rm if}\,\,x\in\mu_{r(q-1)},\\\ {\rm
Tr}_{1}^{n}(cx^{l_{2}(q-1)}),&{\rm otherwise}.\end{array}\right.$

Then $f(x)$ is a bent function if and only if $K_{m}(c^{q+1})=0$. Moreover,
the dual function of $f(x)$ is

$\widetilde{f}(x)=\left\\{\begin{array}[]{ll}{\rm Tr}_{1}^{n}(\epsilon
cx^{l_{1}(1-q)}),&{\rm if}\,\,x\in\mu_{r(q-1)},\\\ {\rm
Tr}_{1}^{n}(cx^{l_{2}(1-q)}),&{\rm otherwise}.\end{array}\right.$

######  Remark 1.

If we use $l_{1}=l_{2}$ and $\epsilon=1$ in Corollary 1, then $f(x)$ is
reduced to the monomial case, and Corollary 1 gives ${\rm
Tr}_{1}^{n}(cx^{l_{1}(q-1)})$ is bent if and only if $K_{m}(c^{q+1})=0$, which
is consistent with the results presented by Dillon [14] for $l_{1}=1$, Leander
[24] and Charpin and Gong [12] for $l_{1}$ with $\gcd(l_{1},\,q+1)=1$. By the
way, $f(x)$ in Corollary 1 is bent if and only if the function ${\rm
Tr}_{1}^{n}(cx^{l_{2}(q-1)})$ is bent.

######  Example 1.

Let $q=2^{6}$, $l_{1}=l_{2}=1$, $r=5$. According to Magma, there are 3118
pairs $(\epsilon,\,c)$ with $\epsilon\neq 1$ such that $f(x)$ in Corollary 1
is bent over $\mathbb{F}_{2^{12}}$. Take $\epsilon=\omega^{819}$ and
$c=\omega^{5}$, where $\omega$ is a primitive element of $\mathbb{F}_{2^{6}}$.
Then $\epsilon\in\mu_{5}$ and it can be checked that $K_{6}(\omega^{65})=0$.
Corollary 1 now establishes that

$f(x)=\left\\{\begin{array}[]{ll}{\rm Tr}_{1}^{12}(\omega^{824}x^{63}),&{\rm
if}\,\,x\in\mu_{315},\\\ {\rm Tr}_{1}^{12}(\omega^{5}x^{63}),&{\rm
otherwise}\end{array}\right.$

is a bent function over $\mathbb{F}_{2^{12}}$ and its dual is

$\widetilde{f}(x)=\left\\{\begin{array}[]{ll}{\rm
Tr}_{1}^{12}(\omega^{824}x^{-63}),&{\rm if}\,\,x\in\mu_{315},\\\ {\rm
Tr}_{1}^{12}(\omega^{5}x^{-63}),&{\rm otherwise}.\end{array}\right.$

Secondly, by setting $r=3$, we give the following result.

######  Corollary 2.

Let $q=2^{m}$ for an odd integer $m>1$, $a_{1}\in{\mathbb{F}}_{q^{2}}$,
$a_{2}\in{\mathbb{F}}_{q^{2}}^{*}$ and $l_{i}$ be positive integers satisfying
$\gcd(l_{i},\,q+1)=1$ for $i=1,\,2$. Then

$f(x)=\left\\{\begin{array}[]{ll}{\rm Tr}_{1}^{n}(a_{1}x^{l_{1}(q-1)}),&{\rm
if}\,\,x\in\mu_{3(q-1)},\\\ {\rm Tr}_{1}^{n}(a_{2}x^{l_{2}(q-1)}),&{\rm
otherwise}\end{array}\right.$

is a bent function if and only if

$K_{m}(a_{2}^{q+1})=4\big{(}(1-{\rm Tr}_{1}^{n}(a_{1}))(1-{\rm
Tr}_{1}^{n}(a_{1}\theta))-(1-{\rm Tr}_{1}^{n}(a_{2}))(1-{\rm
Tr}_{1}^{n}(a_{2}\theta))\big{)},$

where $\theta$ is a 3-rd root of unity. Moreover, the dual function of $f(x)$
is

$\widetilde{f}(x)=\left\\{\begin{array}[]{ll}{\rm
Tr}_{1}^{n}(a_{1}x^{l_{1}(1-q)}),&{\rm if}\,\,x\in\mu_{3(q-1)},\\\ {\rm
Tr}_{1}^{n}(a_{2}x^{l_{2}(1-q)}),&{\rm otherwise}.\end{array}\right.$

###### Proof.

According to Theorem 3 and (12), $f(x)$ is bent if and only if

$\sum_{i=0}^{2}((-1)^{{\rm Tr}_{1}^{n}(a_{1}\theta^{i})}-(-1)^{{\rm
Tr}_{1}^{n}(a_{2}\theta^{i})})=K_{m}(a_{2}^{q+1}),$ (13)

where $\theta$ is a $3$-rd root of unity. Note that $\theta^{2}+\theta+1=0$,
which means ${\rm Tr}_{1}^{n}(a_{j}\theta^{2})={\rm Tr}_{1}^{n}(a_{j})+{\rm
Tr}_{1}^{n}(a_{j}\theta)$ for $j=1,\,2$. Observe that for two Boolean
functions $f_{1}(x)$ and $f_{2}(x)$,

$(-1)^{f_{1}(x)}+(-1)^{f_{2}(x)}+(-1)^{f_{1}(x)+f_{2}(x)}=4(1-f_{1}(x))(1-f_{2}(x))-1.$

This implies

$\sum_{i=0}^{2}(-1)^{{\rm Tr}_{1}^{n}(a_{j}\theta^{i})}=4(1-{\rm
Tr}_{1}^{n}(a_{j}))(1-{\rm Tr}_{1}^{n}(a_{j}\theta))-1$

for $j=1,\,2$. The desired result then follows from (13). This completes the
proof. ∎

######  Example 2.

Let $q=2^{5}$, $l_{1}=l_{2}=1$. According to Magma, there are 218715 pairs
$(a_{1},\,a_{2})$ with $a_{1}\neq a_{2}$ such that $f(x)$ in Corollary 2 is
bent over $\mathbb{F}_{2^{10}}$. For example, take $a_{1}=1$ and
$a_{2}=\omega^{9}$, where $\omega$ is a primitive element of
$\mathbb{F}_{2^{10}}$. Then $\omega^{341}$ is a 3-rd root of unity. It can be
verified that $K_{5}(\omega^{33})=-4$, ${\rm Tr}_{1}^{n}(1)=0$, ${\rm
Tr}_{1}^{n}(\omega^{341})=1$, ${\rm Tr}_{1}^{n}(\omega^{9})={\rm
Tr}_{1}^{n}(\omega^{350})=0$. Corollary 2 now establishes that

$f(x)=\left\\{\begin{array}[]{ll}{\rm Tr}_{1}^{10}(x^{31}),&{\rm
if}\,\,x\in\mu_{93},\\\ {\rm Tr}_{1}^{10}(\omega^{9}x^{31}),&{\rm
otherwise}\end{array}\right.$

is a bent function over $\mathbb{F}_{2^{10}}$ and its dual is

$\widetilde{f}(x)=\left\\{\begin{array}[]{ll}{\rm Tr}_{1}^{10}(x^{-31}),&{\rm
if}\,\,x\in\mu_{93},\\\ {\rm Tr}_{1}^{10}(\omega^{9}x^{-31}),&{\rm
otherwise}.\end{array}\right.$

###  3.2 Bent functions from Niho functions

Secondly, we focus on the branch functions with Niho exponents. Without loss
of generality, we assume that $r_{i}=s_{i}(q-1)+1$ for all
$i\in\mathbb{Z}_{q+1}$. In this case, we consider $R_{0}=\emptyset$ and
$R_{1}=\mathbb{Z}_{q+1}$. Then the second main result on bent functions
follows from Theorem 1.

######  Theorem 4.

Let $a_{i}\in{\mathbb{F}}_{q^{2}}$ and $s_{i}$ be integers with $0\leq
s_{i}\leq q$ for $i\in\mathbb{Z}_{q+1}$. Define

$f(x)={\rm Tr}_{1}^{n}(a_{i}x^{s_{i}(q-1)+1}),\,\,{\rm if}\,\,x\in
u^{i}\mathbb{F}_{q}^{*},\,i\in\\{\infty\\}\cup\mathbb{Z}_{q+1}.$

Then for any $b\in{\mathbb{F}}_{q^{2}}$,
$\widehat{f}(b)=q(\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)-1)$, where

$T_{i}(b)=\left\\{\begin{array}[]{ll}1,&{\rm
if}\,\,bu^{i}+b^{q}u^{-i}=\alpha_{i},\\\ 0,&{\rm
otherwise}.\end{array}\right.$ (14)

with $\alpha_{i}=a_{i}u^{i(1-2s_{i})}+a_{i}^{q}u^{i(2s_{i}-1)}$. Moreover,
$f(x)$ is bent if and only if $\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)=0$ or $2$.

We first give a simple case of above construction.

######  Theorem 5.

Let $a_{i}\in{\mathbb{F}}_{q^{2}}$ and $s_{i}$ be integers with $0\leq
s_{i}\leq q$ satisfying $a_{i}u^{i(1-2s_{i})}+a_{i}^{q}u^{i(2s_{i}-1)}=c$ for
any $i\in\mathbb{Z}_{q+1}$, where $c$ is a fixed element in
${\mathbb{F}}_{q}^{*}$. Then $f(x)$ defined as in Theorem 4 is bent and its
dual is ${\rm Tr}_{1}^{m}(c^{-2}x^{q+1})+1$.

###### Proof.

In this case, one knows $\alpha_{i}=c$ for any $i\in\mathbb{Z}_{q+1}$, and
then for $b\in{\mathbb{F}}_{q^{2}}$, $T_{i}(b)$ given by (14) turns into

$T_{i}(b)=\left\\{\begin{array}[]{ll}1,&{\rm if}\,\,bu^{i}+b^{q}u^{-i}=c,\\\
0,&{\rm otherwise}.\end{array}\right.$

Next we determine $T_{i}(b)$ for any $b\in{\mathbb{F}}_{q^{2}}$. It can be
verified that $T_{i}(0)=0$ for any $i\in\mathbb{Z}_{q+1}$ due to $c\neq 0$.
Hence $\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(0)=0$ and then $\widehat{f}(0)=-q$
according to Theorem 4. For $b\in{\mathbb{F}}_{q^{2}}^{*}$, $T_{i}(b)=1$ if
and only if $u^{2i}+b^{-1}cu^{i}+b^{q-1}=0$. Note that the equation

$z^{2}+b^{-1}cz+b^{q-1}=0$

has 2 (resp. 0) solutions in $\mu_{q+1}$ if ${\rm
Tr}_{1}^{m}(\frac{b^{q-1}}{(b^{-1}c)^{2}})={\rm Tr}_{1}^{m}(b^{q+1}c^{-2})=1$
(resp. ${\rm Tr}_{1}^{m}(b^{q+1}c^{-2})=0$); indeed, this follows from Lemma 2
because ${\rm Tr}_{1}^{n}(\frac{b^{q-1}}{(b^{-1}c)^{2}})=0$ and
$b^{q-1}=(b^{-1}c)^{1-q}$ due to $c\in{\mathbb{F}}_{q}^{*}$. From this fact we
can derive that there are 2 or 0 $T_{i}$’s equal to 1 and others $T_{i}=0$. By
Theorem 4, $\widehat{f}(b)=q(\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)-1)=q$ if
${\rm Tr}_{1}^{m}(b^{q+1}c^{-2})=1$, or
$\widehat{f}(b)=q(\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)-1)=-q$ if ${\rm
Tr}_{1}^{m}(b^{q+1}c^{-2})=0$. Together with $\widehat{f}(0)=-q$, we conclude
$\widehat{f}(b)=q(-1)^{{\rm Tr}_{1}^{m}(b^{q+1}c^{-2})+1}$ for any
$b\in{\mathbb{F}}_{q^{2}}$. This completes the proof. ∎

######  Remark 2.

It can be seen that the bent function $f(x)$ given in Theorem 5 is a dual of
the Kasami bent function. On the other hand, set $a_{i}=a$ with
$a\in{\mathbb{F}}_{q^{2}}\backslash{\mathbb{F}}_{q}$ and $s_{i}=2^{m-1}+1$ for
all $z\in\mathbb{Z}_{q+1}$, then
$a_{i}u^{i(1-2s_{i})}+a_{i}^{q}u^{i(2s_{i}-1)}=a+a^{q}=c\in{\mathbb{F}}_{q}^{*}$.
In this case, $f(x)={\rm Tr}_{1}^{n}(ax^{2^{-1}(q+1)})={\rm
Tr}_{1}^{m}(c^{2}x^{q+1})$ and Theorem 5 implies that $f(x)$ is bent and its
dual is ${\rm Tr}_{1}^{m}(c^{-2}x^{q+1})+1$. This coincides with the result
obtained by Mesnager in [30].

In other cases, we obtain a general result.

######  Theorem 6.

Let $a_{i}\in{\mathbb{F}}_{q^{2}}$, $s_{i}$ be integers with $0\leq s_{i}\leq
q$ and denote $\alpha_{i}=a_{i}u^{(1-2s_{i})i}+a_{i}^{q}u^{(2s_{i}-1)i}$,
where $i\in\mathbb{Z}_{q+1}$. Then $f(x)$ defined as in Theorem 4 is bent if
and only if one of the following conditions is satisfied:

(1) $\alpha_{i}\neq 0$ for all $i\in\mathbb{Z}_{q+1}$, and for any
$t\in\mathbb{Z}_{q+1}$, each element in the multiset
$\\{\\{\frac{\alpha_{i}}{u^{i}+u^{2t-i}}:i\in\mathbb{Z}_{q+1}\backslash\\{t\\}\\}\\}$
has multiplicity 2.

(2) $\alpha_{i_{1}}=\alpha_{i_{2}}=0$ for two distinct integers
$i_{1},i_{2}\in\mathbb{Z}_{q+1}$ and $\alpha_{i}\neq 0$ for
$i\in\mathbb{Z}_{q+1}\backslash\\{i_{1},\,i_{2}\\}$; all elements in the set
$\\{\frac{\alpha_{i}}{u^{i}+u^{2t-i}}:i\in\mathbb{Z}_{q+1}\backslash\\{i_{1},\,i_{2}\\}\\}$
are distinct for $t=i_{1},i_{2}$; and each element in the multiset
$\\{\\{\frac{\alpha_{i}}{u^{i}+u^{2t-i}}:i\in\mathbb{Z}_{q+1}\backslash\\{t,\,i_{1},\,i_{2}\\}\\}\\}$
has multiplicity 2 for any
$t\in\mathbb{Z}_{q+1}\backslash\\{i_{1},\,i_{2}\\}$.

###### Proof.

From Theorem 4, to prove $f(x)$ is bent, it suffices to prove
$\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)=0$ or $2$ for any
$b\in{\mathbb{F}}_{q^{2}}$, where $T_{i}(b)$ is defined by (14). We shall
distinguish the cases $b=0$ and $b\neq 0$ as follows.

If $b=0$, then (14) yields $T_{i}(0)=1$ if and only if $\alpha_{i}=0$. This
indicates
$\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(0)=\\#\\{i\in\mathbb{Z}_{q+1}:\alpha_{i}=0\\}.$
Thus $f(x)$ is bent only if there are exactly 0 or 2 $\alpha_{i}$’s equal 0.

If $b^{q-1}=u^{2t}$ for some $t\in\mathbb{Z}_{q+1}$, then $T_{i}(b)$ given by
(14) turns into

$T_{i}(b)=\left\\{\begin{array}[]{ll}1,&{\rm
if}\,\,b(u^{i}+u^{2t-i})=\alpha_{i},\\\ 0,&{\rm otherwise}.\end{array}\right.$
(15)

Note that $u^{i}+u^{2t-i}=0$ if and only if $i=t$. We consider the value of
$\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)$ as follows:

(1) $\alpha_{i}\neq 0$ for any $i\in\mathbb{Z}_{q+1}$. If this case happens,
one readily knows $T_{t}(b)=0$ and

$T_{i}(b)=\left\\{\begin{array}[]{ll}1,&{\rm
if}\,\,b=\frac{\alpha_{i}}{u^{i}+u^{2t-i}},\\\ 0,&{\rm
otherwise}\end{array}\right.$ (16)

for $i\in\mathbb{Z}_{q+1}\backslash\\{t\\}$ according to (15). Then we can
deduce that
$\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)=\\#\\{i\in\mathbb{Z}_{q+1}\backslash\\{t\\}:\frac{\alpha_{i}}{u^{i}+u^{2t-i}}=b\\}$,
which implies that $\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)=0$ or $2$ if and only
if each element in the multiset
$\\{\\{\frac{\alpha_{i}}{u^{i}+u^{2t-i}}:i\in\mathbb{Z}_{q+1}\backslash\\{t\\}\\}\\}$
has multiplicity 2.

(2) There are two distinct integers $i_{1},\,i_{2}\in\mathbb{Z}_{q+1}$ such
that $\alpha_{i_{1}}=\alpha_{i_{2}}=0$. If $t=i_{1}$, then (15) yields
$T_{i_{1}}(b)=1$, $T_{i_{2}}(b)=0$, and other $T_{i}(b)$’s are given by (16)
due to $u^{i}+u^{2t-i}\neq 0$ for $i\neq t$. Hence
$\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)=1+\\#\\{i\in\mathbb{Z}_{q+1}\backslash\\{i_{1},\,i_{2}\\}:\frac{\alpha_{i}}{u^{i}+u^{2t-i}}=b\\}>0$.
This implies
$\\#\\{i\in\mathbb{Z}_{q+1}\backslash\\{i_{1},\,i_{2}\\}:\frac{\alpha_{i}}{u^{i}+u^{2t-i}}=b\\}=1$.
It can be verified that $(\frac{\alpha_{i}}{u^{i}+u^{2t-i}})^{q-1}=u^{2t}$ and
there are $q-1$ $b$’s such that $b^{q-1}=u^{2t}$ for any
$b\in{\mathbb{F}}_{q^{2}}^{*}$. Thus
$\\#\\{i\in\mathbb{Z}_{q+1}\backslash\\{i_{1},\,i_{2}\\}:\frac{\alpha_{i}}{u^{i}+u^{2t-i}}=b\\}=1$
for any $b^{q-1}=u^{2t}$ if and only if all elements in the set
$\\{\frac{\alpha_{i}}{u^{i}+u^{2i_{1}-i}}:i\in\mathbb{Z}_{q+1}\backslash\\{i_{1},\,i_{2}\\}\\}$
are distinct. The case $t=i_{2}$ is similar.

If $t\neq i_{1}$ and $t\neq i_{2}$, then (15) indicates
$T_{t}(b)=T_{i_{1}}(b)=T_{i_{2}}(b)=0$ and other $T_{i}(b)$’s are given by
(16). Thus
$\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)=\\#\\{i\in\mathbb{Z}_{q+1}\backslash\\{t,\,i_{1},\,i_{2}\\}:\frac{\alpha_{i}}{u^{i}+u^{2t-i}}=b\\}$,
which implies that $\sum_{i\in\mathbb{Z}_{q+1}}T_{i}(b)=0$ or $2$ if and only
if each element in the multiset
$\\{\\{\frac{\alpha_{i}}{u^{i}+u^{2t-i}}:i\in\mathbb{Z}_{q+1}\backslash\\{t,\,i_{1},\,i_{2}\\}\\}\\}$
has multiplicity 2. This completes the proof. ∎

We don’t have an efficient way to construct the explicit classes of such bent
functions. However, we can search for many bent examples by Magma, and some of
them are given below.

######  Example 3.

Let $q=2^{3}$. According to Magma, it can be verified that $s_{i}=4$ for all
$i\in\mathbb{Z}_{9}$, $a_{i}=1$ for $i=0,2,4$, $a_{7}=\omega^{5}$ and others
$a_{i}=\omega^{2}$; $s_{i}=2$ for all $i\in\mathbb{Z}_{9}$, $a_{i}=1$ for
$i=2,4,7,8$, $a_{7}=\omega^{6}$ for $i=1,3$ and others $a_{i}=\omega^{2}$
satisfy the conditions (1) and (2) in Theorem 6 respectively. Then Theorem 6
establishes that

$f(x)=\left\\{\begin{array}[]{lll}{\rm Tr}_{1}^{6}(x^{29}),&{\rm if}\,\,x\in
w^{7i}{\mathbb{F}}_{q}^{*},\,\,i\in\\{0,2,4\\},\\\ {\rm
Tr}_{1}^{6}(\omega^{5}x^{29}),&{\rm if}\,\,x\in
w^{7i}{\mathbb{F}}_{q}^{*},\,\,i\in\\{7\\},\\\ {\rm
Tr}_{1}^{6}(\omega^{2}x^{29}),&{\rm otherwise}\end{array}\right.$

and

$f(x)=\left\\{\begin{array}[]{lll}{\rm Tr}_{1}^{6}(x^{15}),&{\rm if}\,\,x\in
w^{7i}{\mathbb{F}}_{q}^{*},\,\,i\in\\{2,4,7,8\\},\\\ {\rm
Tr}_{1}^{6}(\omega^{6}x^{15}),&{\rm if}\,\,x\in
w^{7i}{\mathbb{F}}_{q}^{*},\,\,i\in\\{1,3\\},\\\ {\rm
Tr}_{1}^{6}(\omega^{2}x^{15}),&{\rm otherwise}\end{array}\right.$

are bent functions over $\mathbb{F}_{2^{6}}$.

##  4 Bent functions from additive cyclotomic mappings

In this section, we construct bent functions from additive cyclotomic
mappings. Let $p$ be a prime and $d,\,n$ be positive integers such that $d\mid
p^{n}$. Let $C$ be the index $d$ subgroup of the additive group of
$\mathbb{F}_{p^{n}}$. Then the left cosets of $\mathbb{F}_{p^{n}}$ modulo $C$
are of the form $C_{i}:=\upsilon_{i}+C$ for $i\in\mathbb{Z}_{d}$, where
$\upsilon_{i}$ is a representative of $C_{i}$. It can be seen that
$\mathbb{F}_{p^{n}}=\bigcup_{i=0}^{d-1}C_{i}$ and $C_{i}\bigcap
C_{j}=\emptyset$ for $i\neq j$. Let
$(a_{0},a_{1},\cdots,a_{d-1})\in\mathbb{F}_{p^{n}}^{d}$ and
$r_{0},r_{1},\cdots,r_{d-1}$ be $d$ non-negative integers. Define a cyclotomic
mapping of $\mathbb{F}_{p^{n}}$ of index $d$ as follows:

$F(x)=a_{i}x^{r_{i}},{\rm if}\,\,x\in C_{i},\,i\in\mathbb{Z}_{d},$ (17)

which is with respect to the additive cosets and we call it an additive
cyclotomic mapping. In this paper, we investigate the bentness of functions
$f(x)={\rm Tr}_{1}^{n}(F(x))$. For any $b\in\mathbb{F}_{p^{n}}$, a calculation
gives

$\displaystyle\widehat{f}(b)$
$\displaystyle=\sum\nolimits_{x\in\mathbb{F}_{p^{n}}}\xi_{p}^{{\rm
Tr}_{1}^{n}(F(x))-{\rm Tr}_{1}^{n}(bx)}$
$\displaystyle=\sum_{i\in\mathbb{Z}_{d}}\sum_{x\in C_{i}}\xi_{p}^{{\rm
Tr}_{1}^{n}(a_{i}x^{r_{i}})-{\rm Tr}_{1}^{n}(bx)}$
$\displaystyle=\sum\nolimits_{i\in\mathbb{Z}_{d}}S_{i}(b),$

where $S_{i}(b)$ is defined as in (2). Therefore, to calculate
$\widehat{f}(b)$, it suffices to determine $S_{i}(b)$ for all
$i\in\mathbb{Z}_{d}$. In the case of $C=\mathbb{F}_{2^{k}}$ and
$r_{i}=2^{t_{i}}+1$ with $k|t_{i}$ and $k|n$, one knows $d=2^{n-k}$ and
$S_{i}(b)$ defined by (2) becomes

$\displaystyle S_{i}(b)$ $\displaystyle=\sum\nolimits_{x\in
v_{i}+\mathbb{F}_{2^{k}}}(-1)^{{\rm Tr}_{1}^{n}(a_{i}x^{2^{t_{i}}+1})+{\rm
Tr}_{1}^{n}(bx)}$
$\displaystyle=\sum\nolimits_{y\in\mathbb{F}_{2^{k}}}(-1)^{{\rm
Tr}_{1}^{n}\big{(}a_{i}(v_{i}+y)^{2^{t_{i}}+1}\big{)}+{\rm
Tr}_{1}^{n}(b(v_{i}+y))}$ $\displaystyle=(-1)^{{\rm
Tr}_{1}^{n}(a_{i}v_{i}^{2^{t_{i}}+1}+bv_{i})}\sum\nolimits_{y\in\mathbb{F}_{2^{k}}}(-1)^{{\rm
Tr}_{1}^{k}({\rm
Tr}_{k}^{n}(a_{i}v_{i}^{2^{t_{i}}}+a_{i}v_{i}+a_{i}^{2^{-1}}+b)y)},$

which gives

$S_{i}(b)=\left\\{\begin{array}[]{ll}2^{k}(-1)^{{\rm
Tr}_{1}^{n}(a_{i}v_{i}^{2^{t_{i}}+1}+bv_{i})},&{\rm if}\,\,{\rm
Tr}_{k}^{n}(a_{i}(v_{i}^{2^{t_{i}}}+v_{i})+a_{i}^{2^{-1}}+b)=0,\\\ 0,&{\rm
otherwise}.\end{array}\right.$

Then we give the Walsh transform of $f(x)$ by the following theorem.

######  Theorem 7.

Let $n$, $k$ and $t_{i}$ be positive integers satisfying $k|n$ and $k|t_{i}$
for $i\in\mathbb{Z}_{2^{n-k}}$. Let

$f(x)={\rm Tr}_{1}^{n}(a_{i}x^{2^{t_{i}}+1}),\,\,{\rm if}\,\,x\in
v_{i}+\mathbb{F}_{2^{k}},\,i\in\mathbb{Z}_{2^{n-k}},$

where $a_{i}\in\mathbb{F}_{2^{n}}$ and $\upsilon_{i}$ are the representatives
of cosets of $\mathbb{F}_{2^{n}}$ modulo $\mathbb{F}_{2^{k}}$. Then for any
$b\in\mathbb{F}_{2^{n}}$,

$\widehat{f}(b)=2^{k}\sum_{i\in E(b)}(-1)^{{\rm
Tr}_{1}^{n}(a_{i}v_{i}^{2^{t_{i}}+1}+bv_{i})}$

and $E(b)$ is defined by

$E(b):=\\{i\in\mathbb{Z}_{2^{n-k}}:{\rm Tr}_{k}^{n}(b)={\rm
Tr}_{k}^{n}(a_{i}(v_{i}^{2^{t_{i}}}+v_{i})+a_{i}^{2^{-1}})\\}.$ (18)

By using Kasami functions, we generate bent functions from a class of additive
cyclotomic mappings of index $q$ over $\mathbb{F}_{q^{2}}$.

######  Theorem 8.

Let $\xi$ be a primitive element of $\mathbb{F}_{q}$ and define
$\xi^{\infty}=0$, where $q=2^{m}$ and $m$ is a positive integer. Define

$f(x)={\rm Tr}_{1}^{m}(\alpha_{i}x^{q+1}),\,\,{\rm if}\,\,x\in
N_{i},\,i\in\\{\infty\\}\cup\mathbb{Z}_{q-1},$

where $\alpha_{i}\in{\mathbb{F}}_{q}$ and
$N_{i}=\\{x\in\mathbb{F}_{q^{2}}:x^{q}+x=\xi^{i}\\}$ for
$i\in\\{\infty\\}\cup\mathbb{Z}_{q-1}$. If

$\\{\alpha_{i}^{2}\xi^{2i}+\alpha_{i}:i\in\\{\infty,0,\cdots,q-2\\}\\}={\mathbb{F}}_{q},$

then $f(x)$ is a bent function over ${\mathbb{F}}_{q^{2}}$. Moreover, for any
$b\in{\mathbb{F}}_{q^{2}}$, the Walsh transform of $f(x)$ is

$\widehat{f}(b)=q(-1)^{\varphi_{t}(b)},\,\,{\rm
if}\,\,b^{q}+b=\alpha_{t}\xi^{t}+\alpha_{t}^{2^{-1}},\,\,t\in\\{\infty\\}\cup\mathbb{Z}_{q-1}$

with

$\varphi_{t}(b)=\left\\{\begin{array}[]{ll}{\rm Tr}_{1}^{m}(\xi^{t}b),&{\rm
if}\,\,\alpha_{t}=0,\\\ {\rm Tr}_{1}^{m}(\alpha_{t}^{-1}b^{q+1})+1,&{\rm
if}\,\,\alpha_{t}\neq 0.\end{array}\right.$ (19)

###### Proof.

In this case, $k=t_{i}=m$, $\alpha_{i}=a_{i}+a_{i}^{q}$ for
$i\in\mathbb{Z}_{q-1}$ and $\alpha_{\infty}=a_{q-1}+a_{q-1}^{q}$. Then we
calculate the values of $v_{i}$ for $i\in\mathbb{Z}_{q}$ in Theorem 8. Suppose
that $x_{0}$ is a solution of the equation $x^{q}+x=1$. It can be verified
that $\xi^{i}x_{0}$ is a solution of the equation $x^{q}+x=\xi^{i}$. This
allows us to write
$N_{i}=\\{x\in\mathbb{F}_{q^{2}}:x^{q}+x=\xi^{i}\\}=\xi^{i}x_{0}+\mathbb{F}_{q}$
for each $i\in\\{\infty\\}\cup\mathbb{Z}_{q-1}$. That means
$v_{i}=\xi^{i}x_{0}$ for $i\in\mathbb{Z}_{q-1}$ and
$v_{q-1}=\xi^{\infty}x_{0}=0$. Combining with the values of $v_{i}$, for any
$b\in{\mathbb{F}}_{q^{2}}$, $E(b)$ given by (18) turns into

$E(b)=\\{i\in\mathbb{Z}_{q}:b^{q}+b=\beta_{i}\\},$ (20)

with

$\displaystyle\beta_{i}$ $\displaystyle={\rm
Tr}_{m}^{n}(a_{i}(v_{i}^{q}+v_{i})+a_{i}^{2^{-1}})$ $\displaystyle={\rm
Tr}_{m}^{n}(a_{i}((\xi^{i}x_{0})^{q}+\xi^{i}x_{0}))+(a_{i}+a_{i}^{q})^{2^{-1}}$
$\displaystyle={\rm
Tr}_{m}^{n}(a_{i}(x_{0}^{q}+x_{0}))\xi^{i}+\alpha_{i}^{2^{-1}}=\alpha_{i}\xi^{i}+\alpha_{i}^{2^{-1}}$

for $i\in\mathbb{Z}_{q-1}$ and

$\beta_{q-1}={\rm
Tr}_{m}^{n}(a_{q-1}(v_{q-1}^{q}+v_{q-1})+a_{q-1}^{2^{-1}})=(a_{q-1}+a_{q-1}^{q})^{2^{-1}}=\alpha_{\infty}\xi^{\infty}+\alpha_{\infty}^{2^{-1}}$

due to $\alpha_{i}=a_{i}+a_{i}^{q}$ for $i\in\mathbb{Z}_{q-1}$ and
$\alpha_{\infty}=a_{q-1}+a_{q-1}^{q}$. Therefore
$\\{\beta_{i}:i\in\mathbb{Z}_{q}\\}=\\{\alpha_{i}\xi^{i}+\alpha_{i}^{2^{-1}}:i\in\\{\infty,0,\cdots,q-2\\}\\}={\mathbb{F}}_{q}$,
which implies there exist a unique $t\in\\{\infty,0,\cdots,q-2\\}$ such that
$b^{q}+b=\alpha_{t}\xi^{t}+\alpha_{t}^{2^{-1}}$ for any
$b\in{\mathbb{F}}_{q^{2}}$. Moreover, in the case of
$b^{q}+b=\alpha_{t}\xi^{t}+\alpha_{t}^{2^{-1}}$, $E(b)=\\{t\\}$ and from
Theorem 7, one derives

$\displaystyle\widehat{f}(b)=$ $\displaystyle q(-1)^{{\rm
Tr}_{1}^{n}(a_{t}(\xi^{t}x_{0})^{q+1}+b\xi^{t}x_{0})}$ $\displaystyle=$
$\displaystyle q(-1)^{{\rm
Tr}_{1}^{m}(\alpha_{t}(\xi^{t}x_{0})^{q+1}+(bx_{0}+(bx_{0})^{q})\xi^{t})}$
$\displaystyle=$ $\displaystyle q(-1)^{{\rm
Tr}_{1}^{m}(\alpha_{t}\xi^{2t}x_{0}^{q+1}+((b+b^{q})x_{0}+b^{q})\xi^{t})}$

due to $x_{0}^{q}+x_{0}=1$ and $\xi\in\mathbb{F}_{q}$. Denote
$\varphi_{t}(b):={\rm
Tr}_{1}^{m}(\alpha_{t}\xi^{2t}x_{0}^{q+1}+((b+b^{q})x_{0}+b^{q})\xi^{t})$.
Then $\widehat{f}(b)=q(-1)^{\varphi_{t}(b)}$, which implies $f(x)$ is bent.
Furthermore, we claim that $\varphi_{t}(b)$ is given by (19). If
$\alpha_{t}=0$, then $b^{q}+b=0$ and one readily gets $\varphi_{t}(b)={\rm
Tr}_{1}^{m}(\xi^{t}b)$. For $\alpha_{t}\neq 0$, combining with the facts
$b^{q}+b=\alpha_{t}\xi^{t}+\alpha_{t}^{2^{-1}}$ and $x_{0}^{q}+x_{0}=1$, a
calculation gives

$\displaystyle\varphi_{t}(b)=$ $\displaystyle{\rm
Tr}_{1}^{m}\big{(}\alpha_{t}\xi^{2t}x_{0}(x_{0}+1)+\xi^{t}\big{(}(\alpha_{t}\xi^{t}+\alpha_{t}^{2^{-1}})x_{0}+(\alpha_{t}\xi^{t}+\alpha_{t}^{2^{-1}}+b)\big{)}\big{)}$
$\displaystyle=$ $\displaystyle{\rm
Tr}_{1}^{m}(\alpha_{t}\xi^{2t}x_{0}^{2}+\alpha_{t}^{2^{-1}}\xi^{t}x_{0}+\alpha_{t}\xi^{2t}+\alpha_{t}^{2^{-1}}\xi^{t}+b\xi^{t})$
$\displaystyle=$
$\displaystyle\alpha_{t}^{2^{-1}}\xi^{t}(x_{0}^{q}+x_{0})+\alpha_{t}^{2^{-1}}(\xi^{t}+\xi^{t})+\sum\nolimits_{i=0}^{m-1}(b\xi^{t})^{2^{i}}$
$\displaystyle=$
$\displaystyle\alpha_{t}^{2^{-1}}\xi^{t}+\sum\nolimits_{i=0}^{m-1}(b\xi^{t})^{2^{i}}$

On the other hand, we have

$\displaystyle{\rm Tr}_{1}^{m}(\alpha_{t}^{-1}b^{q+1})=$ $\displaystyle{\rm
Tr}_{1}^{m}(\alpha_{t}^{-1}b(b+\alpha_{t}\xi^{t}+\alpha_{t}^{2^{-1}}))$
$\displaystyle=$ $\displaystyle{\rm
Tr}_{1}^{m}(\alpha_{t}^{-1}b^{2}+\alpha_{t}^{-2^{-1}}b+b\xi^{t})$
$\displaystyle=$
$\displaystyle\alpha_{t}^{-2^{-1}}(b^{q}+b)+\sum\nolimits_{i=0}^{m-1}(b\xi^{t})^{2^{i}}$
$\displaystyle=$
$\displaystyle\alpha_{t}^{-2^{-1}}(\alpha_{t}\xi^{t}+\alpha_{t}^{2^{-1}})+\sum\nolimits_{i=0}^{m-1}(b\xi^{t})^{2^{i}}$
$\displaystyle=$
$\displaystyle\alpha_{t}^{2^{-1}}\xi^{t}+\sum\nolimits_{i=0}^{m-1}(b\xi^{t})^{2^{i}}+1.$

Therefore, $\varphi_{t}(b)={\rm Tr}_{1}^{m}(\alpha_{t}^{-1}b^{q+1})+1$. This
completes the proof. ∎

######  Remark 3.

If one takes
$\alpha_{\infty}=\alpha_{0}=\cdots=\alpha_{q-2}=a\in{\mathbb{F}}_{q}^{*}$,
then $f(x)$ in Theorem 8 is reduced to the monomial case, that is $f(x)={\rm
Tr}_{1}^{m}(ax^{q+1})$. It can be verified that $a^{2}\xi^{2i}+a$ are $q$
distinct elements in ${\mathbb{F}}_{q}$ when $i$ runs over
$\\{\infty,0,\cdots,q-2\\}$. Otherwise, assume that there are distinct
$i,j\in\\{\infty,0,\cdots,q-2\\}$ such that $a^{2}\xi^{2i}+a=a^{2}\xi^{2j}+a$,
then $\xi^{2(i-j)}=1$, which is impossible since $\gcd(2,\,q-1)=1$ and
$i-j\neq 0$. Thus
$\\{a^{2}\xi^{2i}+a:i\in\\{\infty,0,\cdots,q-2\\}\\}={\mathbb{F}}_{q}.$
Theorem 8 gives that $f(x)$ is bent with the dual function ${\rm
Tr}_{1}^{m}(a^{-1}x^{q+1})+1$, which is consistent with the result obtained by
Mesnager in [30].

Note that it is easy to find parameters satisfying the condition given in
Theorem 8 such that $f(x)$ is bent. To show that, we provide an equivalent
condition which help us to explicitly construct bent functions. Without of
loss of generality, assume that $\alpha_{\infty}\neq 0$. It can be verified
that the condition
$\\{\alpha_{i}^{2}\xi^{2i}+\alpha_{i}:i\in\mathbb{Z}_{q-1}\\}\bigcup\\{\alpha_{\infty}\\}={\mathbb{F}}_{q}$
is equivalent to

$\\{\alpha_{\infty}^{2}\xi^{2i}+\alpha_{\infty}:i\in\mathbb{Z}_{q-1}\\}=\\{\alpha_{i}^{2}\xi^{2i}+\alpha_{i}:i\in\mathbb{Z}_{q-1}\\}.$
(21)

By using it, we give the following construction of bent functions for the case
$\\#\\{\alpha_{i}:i\in\\{\infty\\}\cup\mathbb{Z}_{q-1}\\}\leq 2$. Precisely,
let $\alpha_{i}=a$ for $i\in\mathbb{Z}\subseteq\mathbb{Z}_{q-1}$ and
$\alpha_{i}=c\neq 0$ for others $i$. Then (21) becomes

$\\{c^{2}\xi^{2i}+c:i\in\mathbb{Z}\\}=\\{a^{2}\xi^{2i}+a:i\in\mathbb{Z}\\}.$
(22)

We shall present the bent functions for two cases: $a=0$ and $a\neq 0$.

In the case of $a=0$, (22) is equivalent to $c^{2}\xi^{2i}+c=0$, that is,
$\xi^{i}=c^{2^{m-1}-1}$ for a unique $i\in\mathbb{Z}_{q-1}$. This means
$\alpha_{i}=0$ for such $i$ and $\alpha_{i}=c\neq 0$ for others $i$. Then the
following corollary can be obtained from Theorem 8.

######  Corollary 3.

Let $q=2^{m}$ and $c\in\mathbb{F}_{q}^{*}$, where $m$ is a positive integer.
Then

$f(x)=\left\\{\begin{array}[]{ll}0,&{\rm if}\,\,x^{q}+x=c^{2^{m-1}-1},\\\ {\rm
Tr}_{1}^{m}(cx^{q+1}),&{\rm otherwise}\end{array}\right.$

is a bent function over $\mathbb{F}_{q^{2}}$, and its dual is

$\widetilde{f}(x)=\left\\{\begin{array}[]{ll}{\rm
Tr}_{1}^{m}(c^{2^{m-1}-1}x),&{\rm if}\,\,x\in{\mathbb{F}}_{q},\\\ {\rm
Tr}_{1}^{m}(c^{-1}x^{q+1})+1,&{\rm otherwise}.\end{array}\right.$

If $a\neq 0$, then Theorem 8 and (22) give the following corollary directly.

######  Corollary 4.

Let $q=2^{m}$ and $a,\,c\in\mathbb{F}_{q}^{*}$ satisfying (22). Denote
$N:=\\{\xi^{i}:i\in\mathbb{Z}\\}$, where $\xi$ is a primitive element of
$\mathbb{F}_{q}$ and $\mathbb{Z}$ is a subset of $\mathbb{Z}_{q-1}$. Then

$f(x)=\left\\{\begin{array}[]{ll}{\rm Tr}_{1}^{m}(ax^{q+1}),&{\rm
if}\,\,x^{q}+x\in N,\\\ {\rm Tr}_{1}^{m}(cx^{q+1}),&{\rm
otherwise},\end{array}\right.$

is a bent function over $\mathbb{F}_{q^{2}}$ and its dual is

$\widetilde{f}(x)=\left\\{\begin{array}[]{ll}{\rm
Tr}_{1}^{m}(a^{-1}x^{q+1})+1,&{\rm
if}\,\,x^{q}+x=a\xi^{i}+a^{2^{-1}},\,i\in\mathbb{Z}\\\ {\rm
Tr}_{1}^{m}(c^{-1}x^{q+1})+1,&{\rm otherwise}.\end{array}\right.$

It is clear that any $c\in{\mathbb{F}}_{q}^{*}$ gives a bent function as in
Corollary 3. Then we only give an example for the construction in Corollary 4.

######  Example 4.

Let $q=2^{4}$ and $N=\\{\xi^{2}\\}$, where $\xi$ is a primitive element of
$\mathbb{F}_{2^{4}}$. According to Magma, there are 14 pairs $(a_{1},\,a_{2})$
with $a_{1}\neq a_{2}$ such that $f(x)$ in Corollary 4 is bent over
$\mathbb{F}_{2^{8}}$. For example, take $a=\xi^{9}$ and $c=\xi^{2}$. It can be
verified that $a^{2}\xi^{4}+a=c^{2}\xi^{4}+c=1$. Corollary 4 now establishes
that

$f(x)=\left\\{\begin{array}[]{ll}{\rm Tr}_{1}^{4}(\xi^{9}x^{17}),&{\rm
if}\,\,x^{q}+x=\xi^{2},\\\ {\rm Tr}_{1}^{4}(\xi^{2}x^{17}),&{\rm
otherwise}\end{array}\right.$

is a bent function over $\mathbb{F}_{2^{8}}$, and its dual is

$\widetilde{f}(b)=\left\\{\begin{array}[]{ll}{\rm
Tr}_{1}^{4}(\xi^{6}x^{17})+1,&{\rm if}\,\,x^{q}+x=1,\\\ {\rm
Tr}_{1}^{4}(\xi^{13}x^{17})+1,&{\rm otherwise}.\end{array}\right.$

##  5 Switching between cyclotomic form and polynomial form of bent functions

Boolean functions $f,\,f^{\prime}:\mathbb{F}_{2^{n}}\rightarrow\mathbb{F}_{2}$
are extended-affine equivalent (EA-equivalent) if there exist an affine
permutation $L$ of $\mathbb{F}_{2^{n}}$ and an affine function
$l:\mathbb{F}_{2^{n}}\rightarrow\mathbb{F}_{2}$ such that
$f^{\prime}(x)=(f\circ L)(x)+l(x)$. A class of bent functions is called
complete if it is globally invariant under EA-equivalence and the completed
version of a class is the set of all functions EA-equivalent to the functions
in the class. In this section, we first investigate the polynomial form of
those bent functions proposed in Section 3 and Section 4, and then study the
EA-equivalence between proposed bent functions and known ones.

The switching between multiplicative cyclotomic form and polynomial form is
characterized as below.

######  Lemma 3.

([3, 39]) Let $F(x)$ be an index $d$ generalized cyclotomic mapping defined as
in (1). Then the polynomial form of $F(x)$ is

$F(x)=\frac{1}{d}\sum_{i,j=0}^{d-1}\omega^{-ij\cdot\frac{p^{n}-1}{d}}a_{i}x^{j\cdot\frac{p^{n}-1}{d}+r_{i}}.$

Recall that $u=\omega^{(q-1)(2^{n-1}-1)}$, i.e., $\omega^{q-1}=u^{-2}$. From
Lemma 3, we give the polynomial forms of bent functions proposed in Section 3.

(1) Dillon case: First of all, the polynomial form of $f(x)$ proposed in
Theorem 2 is

$f(x)=\sum_{i,j=0}^{q}{\rm Tr}_{1}^{n}(u^{2ij}a_{i}x^{(q-1)(j+l_{i})}),$

which is a Dillon type polynomial. Note that such $f(x)$ restricted to the
cosets $u{\mathbb{F}}_{q}^{*}$ are constant where $u$ ranges over $\mu_{q+1}$,
and thus belong to $\mathcal{PS}_{ap}$ class [14]. In fact, Dillon in his
thesis [14] shows more precisely that, a Boolean function over
${\mathbb{F}}_{q^{2}}$ with the form $g(x^{q-1})$ and $g(0)=0$, is bent if and
only if $g(h)=1$ for exactly $2^{m-1}$ elements in $\mu_{q+1}$. Obviously, in
Theorem 2, $f(x)=g(x^{q-1})$ with $g(x)=\sum_{i,j=0}^{q}{\rm
Tr}_{1}^{n}(u^{2ij}a_{i}x^{j+l_{i}})$ and the condition
$\sum\nolimits_{i\in\mathbb{Z}_{q+1}}(-1)^{{\rm
Tr}_{1}^{n}(a_{i}u^{-2il_{i}})}=1$ implies that there are exactly $2^{m-1}$
$t$’s in $\mathbb{Z}_{q+1}$ such that $g(u^{t})=1$ if $u^{tl_{t}}$ runs over
$\mu_{q+1}$ when $t$ runs over $\mathbb{Z}_{q+1}$. Thus our construction gives
a subclass of bent functions of Dillon’s construction. We remark that some
generalizations of Dillon’s construction were given later on, for example, see
[22, 28, 32]. In fact, our construction can also explain some previous
infinite classes of Dillon type bent polynomials. For example, from Algorithm
1 given in [3], the cyclotomic form of a general Dillon type polynomial
$f(x)={\rm Tr}_{1}^{n}(\sum_{t=0}^{q}\gamma_{t}x^{t(q-1)})$ is

$f(x)={\rm Tr}_{1}^{n}((\sum\nolimits_{t=0}^{q}\gamma_{t}u^{-2it})),\,\,{\rm
if}\,\,x\in u^{i}\mathbb{F}_{q}^{*},\,i\in\\{\infty\\}\cup\mathbb{Z}_{q+1}.$

Then Theorem 2 yields $f(x)$ is bent if and only if

$\sum\nolimits_{i\in\mathbb{Z}_{q+1}}(-1)^{{\rm
Tr}_{1}^{n}(\sum\nolimits_{t=0}^{q}\gamma_{t}u^{-2it})}=\sum\nolimits_{z\in\mu_{q+1}}(-1)^{{\rm
Tr}_{1}^{n}(\sum_{t=0}^{q}\gamma_{t}z^{t})}=1,$

which is consistent with the result presented by Li et al. in [26]. Although
our first construction coincides with some known ones in some sense, the
choice of parameters are more flexible in our construction. This can help to
construct new explicit infinite classes of Dillon type polynomials. For
instance, set $r=3$, then for odd $m$ and $K_{m}(c^{q+1})=0$, Corollary 1
generates the bent function with the polynomial form

$f(x)={\rm Tr}_{1}^{n}\big{(}\sum\nolimits_{i=0}^{(q+1)/3-1}\big{(}c\epsilon
x^{(3i+l_{1})(q-1)}+cx^{(3i+l_{2})(q-1)}\big{)}+cx^{l_{2}(q-1)}\big{)}.$

For $n=4$ and $n=6,\,8$, the $2$-th power coset representatives of $k(q-1)$,
$k=1,\cdots,q$ modulo $q+1$ are $1$ and $1,3$ respectively, which means all
Dillon type bent ponomials are of the forms ${\rm Tr}_{1}^{n}(ax^{q-1})$ over
$\mathbb{F}_{2^{4}}$ and ${\rm Tr}_{1}^{n}(ax^{q-1}+bx^{3(q-1)})$ over
$\mathbb{F}_{2^{6}}$ or $\mathbb{F}_{2^{8}}$. Li et al. [26] has characterized
the bentness for such two type functions. For $n\geq 10$, due to the shortage
of computer memory, we cannot verify the EA-equivalence between two bent
functions. Thus the equivalence between the newly proposed Dillon type bent
functions and the previously known ones is not clear, and we leave it to
interested readers.

(2) Niho case: Secondly, the polynomial form of $f(x)$ proposed in Theorem 4
is

$f(x)=\sum_{i,j=0}^{q}{\rm Tr}_{1}^{n}(u^{2ij}a_{i}x^{(q-1)(j+s_{i})+1}),$

which is a Niho type polynomial. Such $f(x)$ is linear over each element of
the Desarguesian spread and thus belongs to the class $\mathcal{H}$ [14]. From
the polynomial perspective, our construction concludes some previous infinite
classes of Niho type bent polynomials. For example, from Algorithm 1 given in
[3], the cyclotomic form of a general Niho type polynomial $f(x)={\rm
Tr}_{1}^{n}(\sum_{t=1}^{k}\gamma_{t}x^{s_{t}(q-1)+1})$ is

$f(x)={\rm
Tr}_{1}^{n}((\sum\nolimits_{t=1}^{k}\gamma_{r}u^{-2s_{t}i})x),\,\,{\rm
if}\,\,x\in u^{i}\mathbb{F}_{q}^{*},\,i\in\\{\infty\\}\cup\mathbb{Z}_{q+1}.$

Then Theorem 4 yields $f(x)$ is bent if and only if
$bz+b^{q}z^{-1}+\sum_{t=1}^{k}(\gamma_{t}z^{1-2s_{t}}+\gamma_{t}^{q}z^{2s_{t}-1})=0$
has $0$ or $2$ solutions in $\mu_{q+1}$ for any $b\in{\mathbb{F}}_{q^{2}}$,
which coincides with the result presented by Leander and Kholosha in [25]. In
terms of the equivalence, Abdukhalikov [1] has determined all the equivalence
classes of Niho type bent functions for $m\leq 6$. For instance, Example 3 is
equivalent to ${\rm Tr}_{1}^{3}(x^{36})+{\rm Tr}_{1}^{6}(x^{22})$ over
$\mathbb{F}_{2^{6}}$. However, for $m\geq 7$, we have not yet found an
efficient way to select appropriate parameters to search for new bent
functions. We encourage interested readers to construct specific infinite
classes of Niho type bent functions from cyclotomic mappings.

As for the additive case, the polynomial form of bent functions from additive
cyclotomic mappings can be obtained using Lagrange interpolation.

(3) Kasami case: The polynomial form of $f(x)$ proposed in Theorem 8 is

$f(x)=\sum_{i=0}^{q-2}{\rm
Tr}_{1}^{m}(a_{i}x^{q+1})((x^{q}+x+\xi^{i})^{q-1}+1)+{\rm
Tr}_{1}^{m}(a_{\infty}x^{q+1})((x^{q}+x)^{q-1}+1).$ (23)

According to the equivalent condition for a bent function belonging to the
Maiorana-McFarland class $\mathcal{MM}$ [14, 29] given by [14], it can be
checked that $f(x)$ in (23) belongs to the $\mathcal{MM}$ class. Note that
Fernando and Hou [20] also characterised the polynomial form of such type
functions.

######  Lemma 4.

([20]) Let $\xi$ be an element of order $k$ with $k|(p^{n}-1)$, and define
$\xi^{\infty}=0$. Let
$f_{\infty},\,f_{0},\cdots,f_{k-1}\in\mathbb{F}_{p^{n}}[x]$ and
$\phi:\mathbb{F}_{p^{n}}\rightarrow\\{\xi^{i}:i=\infty,0,\cdots,k-1\\}$. Then
the polynomial form of

$H(x)=f_{i}(x)\,\,{\rm if}\,\,\phi(x)=\xi^{i},\,i\in\\{\infty,0,\cdots,k-1\\}$

is

$H(x)=f_{\infty}(x)(1-\phi(x)^{p^{n}-1})+\frac{1}{k}\sum_{i,j=0}^{k-1}\xi^{-ij}f_{j}(x)\phi(x)^{i}.$

Taking $k=q-1$, $\phi(x)=x^{q}+x$ and $f_{i}(x)={\rm
Tr}_{1}^{m}(a_{i}x^{q+1})$ for $i=\infty,0,\cdots,k-1$, then $f(x)$ in Theorem
8 coincides with $H(x)$, which can be rewritten as

$f(x)={\rm
Tr}_{1}^{m}(a_{\infty}x^{q+1})(1+(x^{q}+x)^{p^{n}-1})+\sum_{i,j=0}^{q-2}\xi^{-ij}{\rm
Tr}_{1}^{m}(a_{j}x^{q+1})(x^{q}+x)^{i}.$

In fact, this is consistent with the expansion of (23). From a polynomial
point of view, our construction produce infinite classes of bent functions
with algebraic degrees higher than 2. For instance, $f(x)$ in Corollary 3 is
of the polynomial form

$f(x)={\rm Tr}_{1}^{m}(cx^{q+1})(x^{q}+x+c^{2^{m-1}-1})^{q-1}.$ (24)

It can be verified that the algebraic degree of $f(x)$ is $m$ for $2\leq m\leq
10$ when $c=1$, which achieves the optimal algebraic degree. Next, we study
the EA-equivalence between $f(x)$ in (24) and known bent polynomials. Firstly,
it can be verified by Magma that $f(x)$ is EA-inequivalent to the five classes
of bent monomials. Recall from (1) and (2), when $m=3$, all known Dillon type
bent polynomials are of the form ${\rm Tr}_{1}^{6}(ax^{q-1}+bx^{3(q-1)})$ and
there are only two Niho type bent polynomials, ${\rm Tr}_{1}^{3}(x^{36})$ and
${\rm Tr}_{1}^{3}(x^{36})+{\rm Tr}_{1}^{6}(x^{22})$, up to equivalence. Magma
shows $f(x)$ are EA-inequivalent to these three classes of bent functions. We
also note that there are so many bent functions based on the second
construction and thus we decide not to consider the equivalence with all known
ones. Therefore we leave this problem for future study.

##  6 Concluding remarks

In this paper, we investigated the construction of Boolean bent functions from
cyclotomic mappings. Firstly, using Dillon functions and Niho functions as the
branch functions over index $q+1$ multiplicative cyclotomic cosets of
${\mathbb{F}}_{q^{2}}$ respectively, we obtained two generic constructions of
bent functions and then derived several new explicit infinite families of bent
functions. Secondly, a generic construction was presented by using Kasami
functions as branch functions over index $q$ additive cyclotomic cosets of
${\mathbb{F}}_{q^{2}}$, from which we got some explicit constructions of bent
functions. Finally, switching between the cyclotomic form and polynomial form,
we showed these three classes of bent functions belong to the $\mathcal{PS}$
class, the class $\mathcal{H}$ and the $\mathcal{MM}$ class respectively. EA-
equivalence of these bent functions has been briefly discussed and it is worth
pursuing a further study.

## Acknowledgements

This work was supported by the National Key Research and Development Program
of China (No. 2021YFA1000600), the National Natural Science Foundation of
China (No. 62072162), the Natural Science Foundation of Hubei Province of
China (No. 2021CFA079), the Knowledge Innovation Program of Wuhan-Basic
Research (No. 2022010801010319), the Innovation Group Project of the Natural
Science Foundation of Hubei Province of China (No. 2003AFA021), and Natural
Sciences and Engineering Research Council of Canada (RGPIN- 2023-04673).

## References

  * [1] Abdukhalikov, K.: Equivalence classes of Niho bent functions, Des. Codes Cryptogr. 89, 1509-1534 (2021).
  * [2] Akbary A., Ghioca D., Wang Q.: On permutation polynomials of prescribed shape. Finite Fields Appl. 15 (2), 195-206 (2009).
  * [3] Bors A., Wang Q.: Generalized cyclotomic mappings: Switching between polynomial, cyclotomic, and wreath product form, Commun. Math. Res. (2021).
  * [4] Canteaut A., Charpin P., Kyureghyan G.: A new class of monomial bent functions, Finite Fields Appl. 14(1), 221-241 (2008).
  * [5] Carlet C.: Two new classes of bent functions, In: Helleseth T. (eds) Advances in EUROCRYPT 1993. LNCS 765, Springer, Berlin (1994).
  * [6] Carlet C.: On the secondary constructions of resilient and bent functions, In: Feng K., Niederreiter H., Xing C. (eds.) Proceedings of the Workshop on Coding, Cryptography and Combinatorics 2003, pp. 3-28. Birkhäuser Verlag (2004).
  * [7] Carlet C.: On bent and highly nonlinear balanced/resilient functions and their algebraic immunities, In: Fossorier M., Imai H., Lin S., Poli A. (eds.) Proceedings of AAECC 2006, LNCS 3857, pp. 1-28 (2006).
  * [8] Carlet C.: Boolean functions for cryptography and error correcting codes, In: Crama Y., Hammer P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, 1st edn, pp. 257-397. Cambridge University Press, New York (2010).
  * [9] Carlet C., Zhang, F., Hu, Y.: Secondary constructions of bent functions and their enforcement, Adv. Math. Commun. 6(3), 305-314 (2012).
  * [10] Carlet C.: Boolean functions for cryptography and coding theory, Cambridge, U.K.: Cambridge Univ. Press (2021).
  * [11] Carlet C., Mesnager S.: Four decades of research on bent functions, Des. Codes Cryptogr. 78(1), 5-50 (2016).
  * [12] Charpin P., Gong G.: Hyperbent functions, Kloosterman sums and Dickson polynomials, IEEE Trans. Inf. Theory 9(54), 4230-4238 (2008).
  * [13] Cohen G., Honkala I., Litsyn S., Lobstein A.: Covering codes, North-Holland Mathematical Library 54, North-Holland, Amsterdam (1997).
  * [14] Dillon J. F.: Elementary Hadamard difference sets, Ph.D. dissertation, Univ. Maryland, Collage Park, (1974).
  * [15] Ding C., Fan C., Zhou Z.: The dimension and minimum distance of two classes of primitive BCH codes, Finite Fields Appl. 45, 237-263 (2017).
  * [16] Dobbertin H.: Construction of bent functions and balanced Boolean functions with high nonlinearity, in Fast Software Encryption, vol. 1008. Berlin, Germany: Springer, pp. 61-74 (1995).
  * [17] Dobbertin H., Leander G., Canteaut A., Carlet C., Felke P., Gaborit P.: Construction of bent functions via Niho power functions, J. Combin. Theory Ser. A 113(5), 779-798 (2006).
  * [18] Evans A. B: Orthomorphism graphs of groups. Lecture Notes in Mathematics, 1535. Springer-Verlag, Berlin, 1992.
  * [19] Fang J., Sun Y., Wang L., Wang Q: Two-weight or three-weight binary linear codes from cyclotomic mappings, Finite Fields Appl. 85, 102114 (2023).
  * [20] Fernando N., Hou X.: A piecewise construction of permutation polynomials over finite fields, Finite Fields Appl. 18, 1184-1194 (2012).
  * [21] Helleseth T., Kholosha A.: Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory 52(5), 2018-2032 (2006).
  * [22] Hou X.: $q$-ary bent functions constructed from chain rings, Finite Fields Appl. 4, 55-61 (1998).
  * [23] Kumar P. V., Scholtz R. A., Welch L. R.: Generalized bent functions and their properties, J. Combin. Theory Ser. A 40(1), 90-107 (1985).
  * [24] Leander N. G.: Monomial bent functions, IEEE Trans. Inf. Theory 52(2), 738-743 (2006).
  * [25] Leander N. G., Kholosha A.: Bent functions with $2^{r}$ Niho exponents, IEEE Trans. Inf. Theory 52(12), 5529-5532 (2006).
  * [26] Li N., Helleseth T., Tang X., Kholosha A.: Several new classes of bent functions from Dillon exponents, IEEE Trans. Inf. Theory 59(3), 1818-1831 (2013).
  * [27] Li Y., Kan H., Mesnager S., Peng J., Tan C. H., Zheng L.: Generic constructions of (Boolean and vectorial) bent functions and their consequences, IEEE Trans. Inf. Theory 68(4), 2735-2751 (2022).
  * [28] Lisoněk P., Lu H. Y.: Bent functions on partial spreads, Des. Codes Cryptogr. 73(1), 209-216 (2014).
  * [29] McFarland R. L.: A family of noncyclic difference sets, J. Combin. Theory Ser. A 15, 1-10 (1973).
  * [30] Mesnager S.: Several new infinite families of bent functions and their duals, IEEE Trans. Inf. Theory 60(7), 4397-4407 (2014).
  * [31] Mesnager S.: Bent functions-fundamentals and results, Cham, Switzerland: Springer, pp. 1-544 (2016).
  * [32] Nyberg N.: Perfect nonlinear S-boxes, In: Davies, D.W. (eds) Advances in EUROCRYPT 1991, LNCS 547, Springer, Berlin, pp. 378-386, (1992).
  * [33] Niederreiter H., Winterhof A.: Cyclotomic $\mathcal{R}$-orthomorphisms of finite fields, Discrete Math. 295, 161-171 (2005).
  * [34] Olsen J., Scholtz R., Welch L.: Bent-function sequences, IEEE Trans. Inf. Theory 28(6), 858-864 (1982).
  * [35] Rothaus O. S.: On bent functions, J. Combin. Theory Ser. A 20(3), 300-305 (1976).
  * [36] Tang C., Zhou Z., Qi Y., Zhang X., Fan C., Helleseth T.: Generic construction of bent functions and bent idempotents with any possible algebraic degrees. IEEE Trans. Inf. Theory 63(10), 6149-6157 (2017).
  * [37] Tu Z., Zeng X., Li C., Helleseth T.: A class of new permutation trinomials, Finite Fields Appl. 50, 178-195 (2018).
  * [38] Wang Q.: Cyclotomic mapping permutation polynomials over finite fields, in: Golomb, S.W., Gong, G., Helleseth, T., Song, HY. (Eds.), Sequences, Subsequences, and Consequences, LNCS 4893, Springer, Berlin, pp. 119-128 (2007).
  * [39] Wang Q.: Cyclotomy and permutation polynomials of large indices, Finite Fields Appl. 22, 57-69 (2013).
  * [40] Xie X., Li N., Zeng X., Tang X., Yao Y.: Several classes of bent functions over finite fields, Des. Codes Cryptogr. 91, 309-332 (2023).