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[1]

[1]This work was supported by the National Natural Science Foundation of China
under Grant No.12361001.

[style=chinese, orcid=0000-0002-4753-7206]

1] organization=School of Mathematics,addressline=Aba Teachers
University,city=Wenchuan,state=Sichuan,postcode=623000,country=China

[style=chinese] [1]

2] organization=School of Mathematical and Sciences,addressline=Sichuan Normal
University,city=Chengdu,state=Sichuan,postcode=610066,country=China

# The characterization of hyper-bent function with multiple trace terms in the
extension field

Peng Han<EMAIL_ADDRESS>[  Keli Pu<EMAIL_ADDRESS>[

###### Abstract

Bent functions are maximally nonlinear Boolean functions with an even number
of variables, which include a subclass of functions, the so-called hyper-bent
functions whose properties are stronger than bent functions and a complete
classification of hyper-bent functions is elusive and inavailable. In this
paper, we solve an open problem of Mesnager that describes hyper-bentness of
hyper-bent functions with multiple trace terms via Dillon-like exponents with
coefficients in the extension field $\mathbb{F}_{2^{2m}}$ of this field
$\mathbb{F}_{2^{m}}$. By applying Möbius transformation and the theorems of
hyperelliptic curves, hyper-bentness of these functions are successfully
characterized in this field $\mathbb{F}_{2^{2m}}$ with $m$ odd integer.

###### keywords:

Möbius transformation Bent function Hyper-bent function Hyperelliptic curve

##  1 Introduction

Boolean functions play a vital role in cryptographic applications in terms of
its nonlinearity based criterion, such as design of substitution box in block
cipher, generation of pseudo-random key stream in stream cipher. Let $n=2m$ be
an even positive integer number. Let $\mathbb{F}_{2^{n}}$ be the finite field
with $2^{n}$ elements and $\mathbb{F}^{\ast}_{2^{n}}$ be the multiplication
group of $\mathbb{F}_{2^{n}}$, where items are not including 0 element over
the finite field $\mathbb{F}_{2^{n}}$. Let $f$ be a Boolean function over
$\mathbb{F}_{2^{n}}$, $\chi_{f}(x)=(-1)^{f(x)}$ is its sign function. For
$x,y\in\mathbb{F}_{2^{n}}$, the inner product is written as $x\cdot
y=\mathrm{Tr}^{n}_{1}(xy)$. The Walsh Hadamard transform of $f$ is the
discrete Fourier transform of $\chi_{f}(x)$, i.e.,

$\displaystyle\centering\widehat{\chi}_{f}(\omega)=\sum\limits_{x\in\mathbb{F}_{2^{n}}}(-1)^{f(x)+\mathrm{Tr}^{n}_{1}(\omega
x)},~{}\forall\omega\in\mathbb{F}_{2^{n}}.\@add@centering$

In addition, the Walsh Hadamard transform satisfies Parseval relation

$\sum\limits_{\omega\in\mathbb{F}_{2^{n}}}{\widehat{\chi}_{f}}^{2}(\omega)=2^{2n}.$

Bent functions in 1976 introduced by Rothaus [1] have attracted more and more
attention with achieving the highest possible nonlinearity and only exist for
the Boolean functions of even numbers of variables by Parseval equation and
have been extensively applied in the fields of coding theory [2, 3],
cryptography [4], sequence designs [5], graph theory [6, 7], and association
scheme [8]. The Walsh Hadamard transform is a powerful tool to determine
bentness of Boolean function. A Boolean function in $\mathbb{F}_{2^{n}}$ is
said to be bent if and only if $\widehat{\chi}_{f}(\omega)=\pm
2^{\frac{n}{2}}$. Up to now, a complete classification of bent functions is
still vague. However, a number of interesting results on bent functions have
been found via primary constructions and secondary constructions [9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Hyper-bent function is a subclass
of bent functions, which possesses even stronger properties than bent
functions were introduced by Youssef and Gong [23] in 2001. In fact, the
initial definition of hyper-bent functions was proposed by Golomb and Gong
[24] in 1999. The extended Walsh Hadamard transform of $f$ is defined by the
following form:

$\widehat{\chi}_{f}(\omega,k)=\sum\limits_{x\in\mathbb{F}_{2^{n}}}(-1)^{f(x)+\mathrm{Tr}^{n}_{1}(\omega
x^{k})},~{}\forall\omega\in\mathbb{F}_{2^{n}},$

where $gcd(k,2^{n}-1)=1$. By employing the extended Walsh Hadamard transform,
the concept of hyper-bent functions can be precisely defined, i.e., a bent
function $f$ is said to be the hyper-bent function if and only if
$\widehat{\chi}_{f}(\omega,k)=\pm 2^{\frac{n}{2}}$. The algebraic degree of
bent function is at most $n/2$. Specificly, hyper-bent function in
$\mathbb{F}_{2^{n}}$ is exact $n/2$. The classifications of hyper-bent
functions seem fuzzy. Monomial hyper-bent functions in $\mathbb{F}_{2^{n}}$
are famous bent functions discovered by Dillon [13]. Charpin and Gong [25]
have characterized a large class of hyper-bent functions involving Dickson
polynomials, which includes the well-known monomial functions with Dillon
exponent $r(2^{m}-1)$, where $r$ is a positive integer. By applying
Kloosterman sums, we can determine a bent function whether it is a hyper-bent
function. Let $a$ be an element over the finite field
$\mathbb{F}^{\ast}_{2^{m}}$. The class of monomial functions
$\mathrm{Tr}^{n}_{1}(ax^{r(2^{m}-1)})$ is hyper-bent if and only if the value
of Kloosterman sums satisfies $K_{m}(a)=0$ in [26, 27, 28]. A family of
binomial functions is captured with
$f^{(r)}_{a,b}(x)=\mathrm{Tr}^{n}_{1}(ax^{r(2^{m}-1)})+\mathrm{Tr}^{2}_{1}(bx^{\frac{2^{n}-1}{3}})$,
where $a\in\mathbb{F}^{\ast}_{2^{n}}$ and $b\in\mathbb{F}^{\ast}_{4}$.
According to Mesnager’s [29], when $r=1$, it is a hyper-bent function if and
only if the value of Kloosterman sums on coefficient $a$ is 4. At the same
time, Mesnager professor in [30] also presents the fact that the value of
Kloosterman sums satisfies $K_{m}(a)=4$ if $f^{(1)}_{a,b}(x)$ is a bent
functions, where $m$ is an even positive integer. There is an open problem
[31] left by Mesnager in [29] for the binomial bent functions with the case
$m$ even. Recently, this problem is resolved by Tang et al. [32] applying
computing tools, which include Walsh transform of Boolean functions by means
of multiplicative characters, Gauss sums and relative properties of graph
theory, i.e., if $K_{m}(a^{2^{m}+1})=4$, then $f^{(1)}_{a,b}(x)$ is bent
function. Let $D_{r}(x)$ be $r$-th order Dickson polynomial. Charpin and Gong
in [25] studied the hyper-bentness of Boolean functions of this form
$f_{a_{r}}(x)=\sum\limits_{r\in R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})$,
where $R$ consists of a set of representatives of the cyclotomic coset modulo
$2^{m}+1$ and the coefficient $a_{r}$ lies in this field $\mathbb{F}_{2^{m}}$.
Charpin and Gong prove the fact if all of $a_{r}$ belong to subfield
$\mathbb{F}_{2^{m}}$, then $f_{a_{r}}(x)$ is a hyper-bent function if and only
if
$\sum\limits_{x\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}^{m}_{1}(x^{-1})+g(x)}=2^{m}-2wt(g)$,
where $g(x)=\sum\limits_{r\in R}\mathrm{Tr}^{m}_{1}(a_{r}D_{r}(x))$.
Mesnager [33, 34] has characterized another class of hyper-bent functions,
which is distinct from that of Charpin and Gong by means of Dickson
polynomials. This function $f_{a_{r},b}(x)=\sum\limits_{r\in
R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})+\mathrm{Tr}^{2}_{1}(bx^{\frac{2^{n}-1}{3}})$
is extended by Mesnager to deal with Charpin-Gong like functions, where the
same restriction coefficient $a_{r}$ lives in $\mathbb{F}_{2^{m}}$ and $b$ is
in $\mathbb{F}_{4}$. In addition, the coefficient $b$ is in $\mathbb{F}_{16}$
and $m\equiv 2(mod\,4)$, the hyper-bentness of this function
$f_{a_{r},b}(x)=\sum\limits_{r\in
R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})+\mathrm{Tr}^{4}_{1}(bx^{\frac{2^{n}-1}{5}})$
was discussed in [35]. We generalize the results in [34] of the hyper-bentness
of Charpin-Gong like functions with an additional trace term over the finite
field $\mathbb{F}_{4}$ by the form $f^{(r)}_{a_{r},b}(x)=\sum\limits_{r\in
R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})+\mathrm{Tr}^{2}_{1}(bx^{\frac{2^{n}-1}{3}})$,
where the coefficient $a_{r}$ lies in the extension field $\mathbb{F}_{2^{n}}$
rather than in $\mathbb{F}_{2^{m}}$.
The rest parts of this paper are organized as follows. In Section 2, some
elementary notations and preliminaries are recalled. In Section 3, the hyper-
bentness of $f^{(r)}_{a_{r},b}(x)$ is presented with coefficient $a_{r}$
living in $\mathbb{F}_{2^{n}}$. In Section 4, we conclude the relative results
in this paper.

##  2 Preliminaries

Throughout this paper, let $n=2m$ be an even positive integer, where $m$ odd.
Let $\mathbb{F}_{q}$ be a finite field with $q$ elements. While working over
the finite field $\mathbb{F}_{2^{n}}$, we denote the notation
$\frac{1}{0}=0^{2^{n}-2}=0$. The set $U_{2^{m}+1}$ is always the group of
$(2^{m}+1)$-th roots of unity that is
$U_{2^{m}+1}=\\{u\in\mathbb{F}^{\ast}_{2^{n}}|u^{2^{m}+1}=1\\}$.

###  2.1 Trace functions and Hyper-bent functions

Trace function plays a crucial role in cryptography and coding theory. For a
positive integer $k$ dividing $n$. Trace function is a map from
$\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{k}}$ denoted by
$\mathrm{Tr}^{n}_{k}(\cdot)$. It is defined as

$\mathrm{Tr}^{n}_{k}(x)=x+x^{2^{k}}+\cdots+x^{2^{n-k}},\forall
x\in\mathbb{F}_{2^{n}}.$

In particular, $\mathrm{Tr}^{n}_{1}(x)$ is called the absolute trace function
and has the following transitivity property:

$\mathrm{Tr}^{n}_{1}(x)=\mathrm{Tr}^{k}_{1}(\mathrm{Tr}^{n}_{k}(x)).$

Let $\Gamma_{n}$ be the set of integers obtained by selecting an element in
each cyclotomic coset modulo $2^{n}-1$. $o(j)$ is the smallest positive
integer containing $j$ such that $j2^{o(j)}\equiv j(2^{n}-1)$ holds. Every
non-zero Boolean function in $\mathbb{F}_{2^{n}}$ can be expressed as the
following form:

$f(x)=\sum\limits_{j\in\Gamma_{n}}\mathrm{Tr}^{o(j)}_{1}(a_{j}x^{j})+\varepsilon(1+x^{2^{n}-1}),$

where $a_{j}\in\mathbb{F}_{2^{o(j)}}$, $\varepsilon=wt(f)(mod\,2)$ and $wt(f)$
is the cardinality of support set of $f$, i.e.,
$\\#supp=\\{x\in\mathbb{F}_{2^{n}}\mid f(x)=1\\}$.
Let $\alpha$ be a primitive element of $\mathbb{F}_{2^{n}}$. Let $f$ be a
Boolean function of $\mathbb{F}_{2^{n}}$ such that $f(\alpha^{2^{m}+1}x)=f(x)$
and $f(0)=0$. This function $f$ is a hyper-bent function if and only if the
weight of the vector $(f(1),f(\alpha),f(\alpha^{2}),\cdots,f(\alpha^{2^{m}}))$
equals $2^{m-1}$.
Let $f:\mathbb{F}_{2^{m}}\rightarrow\mathbb{F}_{2}$ and $u\in U_{2^{m}+1}$,
the exponential sum $\Lambda(f)$ in $\mathbb{F}_{2^{n}}$ is defined as

$\displaystyle~{}\Lambda(f)=\sum\limits_{u\in U_{2^{m}+1}}(-1)^{f(u)},$ (1)

and $f$ is hyper-bent if and only if $\Lambda(f)=1$. It is a perfect tool to
study the hyper-bentness of Boolean functions $f$ of the polynomial forms.

###  2.2 Möbius transformations

A Möbius transformation is also called the fractional linear mapping that is a
rational function of the form

$~{}\frac{ax+b}{cx+d},~{}~{}a,b,c,d\in\mathbb{F}_{2^{m}}~{}and~{}ad+bc\neq 0,$

where variable $x\in\mathbb{F}_{2^{m}}\cup{\\{\infty\\}}$. Fix $u_{0}\in
U_{2^{m}+1}\setminus{\\{1\\}}$. By using Möbius transformation, we can present
the elements of $U_{2^{m}+1}$ as

$\displaystyle~{}U_{2^{m}+1}=\left\\{\frac{(u_{0}+1)x+1}{(u_{0}+1)x+u_{0}}:x\in\mathbb{F}_{2^{m}}\cup{\\{\infty\\}}\right\\}$
(2)

Specially, we select the element $u$ of $U_{2^{m}+1}$. If $x=\infty$, then
$u=1$. Assume that $\rho_{0}=\frac{u_{0}}{1+u^{2}_{0}}$, we get the results

$\displaystyle~{}u+\frac{1}{u}=\frac{1}{x^{2}+x+\rho_{0}}$ (3)

and

$\displaystyle~{}u_{0}u+\frac{1}{u_{0}u}=\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}.$
(4)

They play a vital role for us to research the hyper-bentness of Boolean
funtions in the extension fields $\mathbb{F}_{2^{2m}}$ of finite field
$\mathbb{F}_{2^{m}}$.

###  2.3 Dickson polynomial

Dickson polynomial of degree $r$ in $\mathbb{F}_{2}[X]$ is defined as

$D_{r}(X)=\sum\limits^{\lfloor{\frac{r}{2}}\rfloor}_{i=0}\frac{r}{r-i}C^{i}_{r-i}X^{r-2i},r=2,3,\cdots$

where $\lfloor{r/2}\rfloor$ denotes the largest integer less than or equal to
$r/2$. It has many applications and interesting properties [36]. Dickson
polynomials $D_{r}(X)$ satisfy the following recurrence relation

$D_{i+2}(X)=XD_{i+1}(X)+D_{i}(X),$

the initial value of Dickson polynomial $D_{0}(X)=0$, $D_{1}(X)=X$. For any
non-zero positive integer $r$ and $p$, there are the following properties
about Dickson polynomials

$D_{rp}(X)=D_{r}(D_{p}(X)).$

Dickson polynomials satisfy with $X=1/x$ and $x\neq 0$,

$D_{r}(x+x^{-1})=x^{r}+x^{-r}.$

##  3 The characterization of hyper-bent function in the extension field

Recently, Claude Carlet et al. [37] have researched the hyper-bentness of
Boolean functions in trace representation given by Youssef and Gong [23] in
the following form:

$\displaystyle~{}f_{a_{r}}(x)=\sum\limits_{r\in
R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)}),$ (5)

where $R$ is a set of representatives of the cyclotomic classes modulo
$2^{m}+1$, and the coefficient $a_{r}$ lives in this field
$\mathbb{F}_{2^{n}}$ rather than its subfield $\mathbb{F}_{2^{m}}$. In this
section, we use the technology of Claude Carlet et al. to treat the hyper-
bentness of Charpin-Gong like functions with an additional trace term over
$\mathbb{F}_{4}$:

$\displaystyle~{}f^{(r)}_{a_{r},b}(x)=\sum\limits_{r\in
R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})+\mathrm{Tr}^{2}_{1}(bx^{\frac{2^{n}-1}{3}}),$
(6)

where the coefficient $b$ is in $\mathbb{F}_{4}$, and the coefficient $a_{r}$
lies in this field $\mathbb{F}_{2^{n}}$.

Theorem 3.1 Let $u_{0}$ be a primitive $(2^{m}+1)$-th root of unity and
$\rho_{0}=\frac{u_{0}}{1+u^{2}_{0}}$. Let $f^{(r)}_{a_{r},b}(x)$ be a Boolean
function defined by (6). With the help of Boolean functions

$g_{a_{r},b}(x)=\sum_{r\in
R}\mathrm{Tr}^{m}_{1}\left(a^{\prime}_{r}D_{r}\left(\frac{1}{x^{2}+x+\rho_{0}}\right)+a^{\prime\prime}_{r}D_{r}\left(\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)\right)+\mathrm{Tr}^{2}_{1}(b),$

where
$a^{\prime}_{r}=\frac{a_{r}u^{-r}+a^{2^{m}}_{r}u^{r}_{0}}{u^{r}_{0}+u^{-r}_{0}}$
and $a^{\prime\prime}_{r}=\frac{a_{r}+a^{2^{m}}_{r}}{u^{r}_{0}+u^{-r}_{0}}$. A
Boolean function $f^{(r)}_{a_{r},b}(x)$ is said to be a hyper-bent function if
and only if

$\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{g_{a_{r},b}(x)}=1-(-1)^{f^{(r)}_{a_{r},b}(1)}.$

###### Proof.

Note that the facts

$f^{(r)}_{a_{r},b}(x)=\sum_{r\in
R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})+\mathrm{Tr}^{2}_{1}(bx^{\frac{2^{n}-1}{3}}),$

and (1). A direct verification confirms that
$a^{\prime}_{r},~{}a^{\prime\prime}_{r}\in\mathbb{F}_{2^{m}}$ and
$a_{r}=a^{\prime}_{r}+a^{\prime\prime}_{r}u^{r}_{0}$, where the choice of
$u_{0}$ is satisfied for the total functions $f^{(r)}_{a_{r},b}(x)$ with their
coefficients $a_{r}$ in the extension field $\mathbb{F}_{2^{n}}$. One has

$\displaystyle\Lambda(f^{(r)}_{a_{r},b})~{}$ $\displaystyle=\sum_{u\in
U_{2^{m}+1}}(-1)^{\sum\limits_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime}_{r}(u^{r}+u^{-r})+a^{\prime\prime}_{r}((u_{0}u)^{r}+(u_{0}u)^{-r}))+\mathrm{Tr}^{2}_{1}(b)}$

We depend on the facts

$u+\frac{1}{u}=\frac{1}{x^{2}+x+\rho_{0}},~{}u_{0}u+\frac{1}{u_{0}u}=\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})},$

and draw support from the properties of Dickson polynomials:

$u^{r}+u^{-r}=D_{r}(u+u^{-1}),~{}{(u_{0}u)}^{r}+{(u_{0}u)}^{-r}=D_{r}((u_{0}u)+{(u_{0}u)}^{-1}).$

Note that
$U_{2^{m}+1}\setminus{\\{1\\}}=\left\\{\frac{(u_{0}+1)x+1}{(u_{0}+1)x+u_{0}}:x\in\mathbb{F}_{2^{m}}\right\\}$.
Further, we have

$\displaystyle\Lambda(f^{(r)}_{a_{r},b})~{}$ $\displaystyle=\sum_{u\in
U_{2^{m}+1}}(-1)^{\sum\limits_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime}_{r}D_{r}(u+u^{-1})+a^{\prime\prime}_{r}D_{r}((u_{0}u)+(u_{0}u)^{-1}))+\mathrm{Tr}^{2}_{1}(b)}~{}$
$\displaystyle=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{\sum_{r\in
R}\mathrm{Tr}^{m}_{1}\left(a^{\prime}_{r}D_{r}\left(\frac{1}{x^{2}+x+\rho_{0}}\right)+a^{\prime\prime}_{r}D_{r}\left(\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)\right)+\mathrm{Tr}^{2}_{1}(b)}+(-1)^{f^{(r)}_{a_{r},b}(1)}.$

According to $f^{(r)}_{a_{r},b}(x)$ is hyper-bent function if and only if
$\Lambda(f^{(r)}_{a_{r},b})=1$. We say that this function
$f^{(r)}_{a_{r},b}(x)$ is hyper-bent function if and only if

$\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{\sum_{r\in
R}\mathrm{Tr}^{m}_{1}\left(a^{\prime}_{r}D_{r}\left(\frac{1}{x^{2}+x+\rho_{0}}\right)+a^{\prime\prime}_{r}D_{r}\left(\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)\right)+\mathrm{Tr}^{2}_{1}(b)}=1-(-1)^{f^{(r)}_{a_{r},b}(1)}.$

This is completed the proof. ∎

Theorem 3.2 Let $u_{0}$ be a primitive $(2^{m}+1)$-th root of unity. Let
$f_{a_{r}}(x)$ be the Boolean function defined on $\mathbb{F}_{2^{n}}$ by (5),
where $a_{r}\in\mathbb{F}_{2^{n}}$ and
$a_{r}=a^{\prime}_{r}+a^{\prime\prime}_{r}u^{r}_{0}$. Denote that
$g_{a^{\prime}_{r}}(x)=\sum_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime}_{r}D_{r}(x))$ and
$g_{a^{\prime\prime}_{r}}(x)=\sum_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime\prime}_{r}D_{r}(x))$. We have

$\displaystyle~{}\sum\limits_{u\in
U_{2^{m}+1}}\chi(f_{a_{r}}(u^{p}))=1+\sum\limits_{u\in
U_{2^{m}+1}\setminus\\{1\\}}\chi(g_{a^{\prime}_{r}}(D_{p}(u+u^{-1})))+g_{a^{\prime\prime}_{r}}(D_{p}(u^{\frac{1}{p}}_{0}u+(u^{\frac{1}{p}}_{0}u)^{-1}))).$
(7)

###### Proof.

Let $u$ be an element of cyclic group $U_{2^{m}+1}$ and the choice of $u$ have
no impact on the correctness of Theorem 3.2. The formula can be attained as
follows:

$\displaystyle f_{a_{r}}(u^{p})=\sum_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime}_{r}D_{rp}(u+u^{-1})+a^{\prime\prime}_{r}D_{rp}(u^{\frac{1}{p}}_{0}u+(u^{\frac{1}{p}}_{0}u)^{-1})).$

By straightforword calculations, one has

$\displaystyle\sum\limits_{u\in U_{2^{m}+1}}\chi(f_{a_{r}}(u^{p}))~{}$
$\displaystyle=\sum\limits_{u\in U_{2^{m}+1}}\chi(\sum_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime}_{r}D_{rp}(u+u^{-1})+a^{\prime\prime}_{r}D_{rp}(u^{\frac{1}{p}}_{0}u+(u^{\frac{1}{p}}_{0}u)^{-1})))~{}$
$\displaystyle=1+\sum\limits_{u\in U_{2^{m}+1}\setminus\\{1\\}}\chi(\sum_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime}_{r}D_{rp}(u+u^{-1})+a^{\prime\prime}_{r}D_{rp}(u^{\frac{1}{p}}_{0}u+(u^{\frac{1}{p}}_{0}u)^{-1})))~{}$
$\displaystyle=1+\sum\limits_{u\in
U_{2^{m}+1}\setminus\\{1\\}}\chi(g_{a^{\prime}_{r}}(D_{p}(u+u^{-1})))+g_{a^{\prime\prime}_{r}}(D_{p}(u^{\frac{1}{p}}_{0}u+(u^{\frac{1}{p}}_{0}u)^{-1}))).$

This is completed the proof. ∎

We denote the equation $T_{1}$ as

$\mathrm{T}_{1}(g_{a^{\prime}}(c_{1})+g_{a^{\prime\prime}_{r}}(c_{2}))=\sum\limits_{u\in
U_{2^{m}+1}\setminus\\{1\\}}\chi(g_{a^{\prime}_{r}}(u+u^{-1})+g_{a^{\prime\prime}_{r}}(u_{0}u+(u_{0}u)^{-1})),$

where $c_{1}=u+u^{-1}$ and $c_{2}=u_{0}u+(u_{0}u)^{-1}$. We employ the fact
that Boolean function $f$ is hyper-bent if and only if $\sum\limits_{u\in
U_{2^{m}+1}}(-1)^{f(u)}=1$, which is leading to the following the corollary.
Corollary 3.1 Let $f_{a_{r}}(x)=\sum_{r\in
R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})$ be a Boolean function, where
$a_{r}\in\mathbb{F}_{2^{n}}$. Then, $f_{a_{r}}(x)$ is hyper-bent if and only
if $\mathrm{T}_{1}(g_{a^{\prime}}(c_{1})+g_{a^{\prime\prime}_{r}}(c_{2}))=0$.

###### Proof.

A Boolean function $f_{a_{r}}(x)$ is hyper-bent with Dillion exponents defined
over $\mathbb{F}_{2^{m}}$ if and only if

$\sum\limits_{u\in U_{2^{m}+1}}(-1)^{f_{a_{r}}(u)}=1.$

By this equation
$\Lambda(f_{a_{r}}(u))=1+\mathrm{T}_{1}(g_{a^{\prime}}(c_{1})+g_{a^{\prime\prime}_{r}}(c_{2}))$,
we get the ultimate fact. ∎

Let $\alpha$ be a primitive element in $\mathbb{F}_{2^{n}}$. If
$U={\\{u\in\mathbb{F}^{\ast}_{2^{n}}\mid u^{2^{m}+1}=1\\}}$, then
$\xi=\alpha^{2^{m}-1}$ be a generator of the cyclic group $U$ and $U$ can be
decomposed as: $U=\bigcup^{2}_{i=0}\xi^{i}V$ with $V=\\{u^{3}:u\in U\\}$. Let
$S_{i}$ be the sums

$S_{i}=\sum\limits_{v\in V}\chi(f_{a_{r}}(\xi^{i}v)),\forall i\in\\{0,1,2\\},$

it is an obvious result that $\sum\limits^{2}_{i=0}S_{i}=\sum\limits_{u\in
U}\chi(f_{a_{r}}(u))$.
Definition 3.1 For $i\in\mathbb{Z}$, define $S_{i}$ and $\overline{S}_{i}$ to
the partial exponential sums

$S_{i}=\sum\limits_{v\in
V}\chi(f_{a_{r}}(\zeta^{i}v)),\overline{S}_{i}=\sum\limits_{v\in
V}\chi(\overline{f}_{a_{r}}(\zeta^{i}v)).$

Remark 3.1 It is straightforward to see that

$\sum\limits^{2}_{i=0}S_{i}=\Lambda(f_{a_{r}}(u))=1+\mathrm{T}_{1}(g_{a^{\prime}}(c_{1})+g_{a^{\prime\prime}_{r}}(c_{2})),$

thus, $f_{a_{r}}(x)$ is hyper-bent if and only if
$\sum\limits^{2}_{i=0}S_{i}=1$. Furthermore, we present an equivalent
characterization of hyper-bent function as

$\mathrm{T}_{1}(g_{a^{\prime}}(c_{1})+g_{a^{\prime\prime}_{r}}(c_{2}))=0.$

Proposition 3.1 Let $s$ be the order of subgroup of the field
$\mathbb{F}^{\ast}_{2^{n}}$ and $f^{(r)}_{a_{r},b}(x)$ be a Boolean function
defined on $\mathbb{F}_{2^{n}}$ by (6), then $f^{(r)}_{a_{r},b}(x)$ is hyper-
bent if and only if
$\sum\limits^{2}_{i=0}\chi(\mathrm{Tr}^{2}_{1}(b\xi^{i}))\overline{S}_{i}=1$.

###### Proof.

As a matter of fact, we can refer to this way of proving in [21] for the value
of $\Lambda(f^{(r)}_{a_{r},b})$. Since this function $f^{(r)}_{a_{r},b}(x)$ is
hyper-bent if and only if $\Lambda(f^{(r)}_{a_{r},b})=1$, we get the result
that $f^{(r)}_{a_{r},b}(x)$ is hyper-bent if and only if
$\sum\limits^{2}_{i=0}\chi(\mathrm{Tr}^{2}_{1}(b\xi^{i}))\overline{S}_{i}(a)=1$.
∎

Theorem 3.3 Let $f_{a_{r}}(x)=\sum_{r\in
R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})$ be a Boolean function with
$a_{r}\in\mathbb{F}_{2^{n}}$ and $\rho_{0}=\frac{u_{0}}{1+u^{2}_{0}}$ with
$u_{0}\in U_{2^{m}+1}\backslash\\{1\\}$, then $f_{a_{r}}(x)$ is said to be a
hyper-bent if and only if

$wt\left(g_{a^{\prime}_{r}}\left(\frac{1}{x^{2}+x+\rho_{0}}\right)+g_{a^{\prime\prime}_{r}}\left(\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)\right)=2^{m-1}.$

###### Proof.

It is worthy of attention that both $\mathrm{Tr}^{n}_{1}(x^{2}+x+\rho_{0})=1$
and $\mathrm{Tr}^{n}_{1}({(\rho_{0}/x)}^{2}+(\rho_{0}/x)+\rho_{0})=1$ can be
used to determine the hyper-bentness of Boolean function $f_{a_{r}}(x)$ with
$\mathrm{Tr}^{n}_{1}(\rho_{0})=1$. Hence, we get the results

$\displaystyle\sum\limits_{u\in U_{2^{m}+1}}\chi(f_{a_{r}}(u))~{}$
$\displaystyle=1+\sum\limits_{u\in
U_{2^{m}+1}\setminus\\{1\\}}\chi(g_{a^{\prime}_{r}}(u+u^{-1})+g_{a^{\prime\prime}_{r}}(u_{0}u+(u_{0}u)^{-1}))~{}$
$\displaystyle=1+\sum\limits_{x\in\mathbb{F}_{2^{m}}}\chi\left(g_{a^{\prime}_{r}}\left(\frac{1}{x^{2}+x+\rho_{0}}\right)+g_{a^{\prime\prime}_{r}}\left(\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)\right)~{}$
$\displaystyle=1+2\sum\limits_{x\in\mathbb{F}_{2^{m}},~{}\mathrm{Tr}^{m}_{1}(1/x)=1}\chi(g_{a^{\prime}_{r}}(x)+g_{a^{\prime\prime}_{r}}(x)).$

For a Boolean function $h$ defined on $\mathbb{F}_{2^{n}}$, it is satisfied
with the equation
$\sum\limits_{x\in\mathbb{F}_{2^{n}}}\chi(h(x))=2^{n}-2wt(h)$. Thus,

$\sum\limits_{x\in\mathbb{F}_{2^{m}}}\chi\left(g_{a^{\prime}_{r}}\left(\frac{1}{x^{2}+x+\rho_{0}}\right)+g_{a^{\prime\prime}_{r}}\left(\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)\right)=2^{m}-2wt\left(g_{a^{\prime}_{r}}\left(\frac{1}{x^{2}+x+\rho_{0}}\right)+g_{a^{\prime\prime}_{r}}\left(\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)\right).$

By Corollary 3.1, $f_{a_{r}}(x)$ is hyper-bent if and only if

$\sum\limits_{x\in\mathbb{F}_{2^{m}},~{}\mathrm{Tr}^{m}_{1}(1/x)=1}\chi(g_{a^{\prime}_{r}}(x)+g_{a^{\prime\prime}_{r}}(x))=0.$

This implies that

$wt\left(g_{a^{\prime}_{r}}\left(\frac{1}{x^{2}+x+\rho_{0}}\right)+g_{a^{\prime\prime}_{r}}\left(\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)\right)=2^{m-1}.$

∎

Remark 3.2 If $a_{r}\in\mathbb{F}_{2^{m}}$, then $a^{\prime}_{r}=a_{r}$,
$a^{\prime\prime}_{r}=0$ and $f^{(r)}_{a_{r},b}(1)=\mathrm{Tr}^{2}_{1}(b)$.
Under the conditions, $f^{(r)}_{a_{r},b}(x)$ is said to be a hyper-bent
function if and only if

$\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{\sum\limits_{r\in
R}\mathrm{Tr}^{m}_{1}(a_{r}D_{r}(\frac{1}{x^{2}+x+\rho_{0}}))}=1-(-1)^{\mathrm{Tr}^{2}_{1}(b)}.$

Theorem 3.4 Let $n=2m$ and $a_{r}\in\mathbb{F}_{2^{n}}$, $u_{0}$ be a
primitive $(2^{m}+1)$-th root of unity and
$\rho_{0}=\frac{u_{0}}{1+u^{2}_{0}}$. Let $g_{a_{r},1}(x)$ be a Boolean
function defined on $\mathbb{F}_{2^{m}}$ by

$g_{a_{r},1}(x)=\sum_{r\in
R}\mathrm{Tr}^{m}_{1}\left(a^{\prime}_{r}D_{r}\left(\frac{1}{x^{2}+x+\rho_{0}}\right)+a^{\prime\prime}_{r}D_{r}\left(\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)\right),$

where
$a^{\prime}_{r}=\frac{a_{r}u^{-r}_{0}+a^{2^{m}}_{r}u^{r}_{0}}{u^{r}_{0}+u^{-r}_{0}}$
and $a^{\prime\prime}_{r}=\frac{a_{r}+a^{2^{m}}_{r}}{u^{r}_{0}+u^{-r}_{0}}$.
Then, $f^{(r)}_{a_{r},1}(x)$ is hyper-bent function if and only if

$\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{g_{a_{r},1}(x)}=1-(-1)^{f^{(r)}_{a_{r},1}(1)}.$

###### Proof.

Let $\Lambda(f^{(r)}_{a_{r},1})=\sum_{u\in
U_{2^{m}+1}}(-1)^{f^{(r)}_{a_{r},1}(u)}$. Applying this relation
$a_{r}=a^{\prime}_{r}+a^{\prime\prime}_{r}u^{r}_{0}$, one has

$\displaystyle\Lambda(f^{(r)}_{a_{r},1})~{}$ $\displaystyle=\sum_{u\in
U_{2^{m}+1}}(-1)^{\sum\limits_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime}_{r}(u^{r}+u^{-r})+a^{\prime\prime}_{r}((u_{0}u)^{r}+(u_{0}u)^{-r}))}~{}$
$\displaystyle=\sum_{u\in U_{2^{m}+1}}(-1)^{\sum\limits_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime}_{r}D_{r}(u+u^{-1})+a^{\prime\prime}_{r}D_{r}((u_{0}u)+(u_{0}u)^{-1}))}~{}$
$\displaystyle=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{\sum\limits_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime}_{r}D_{r}(u+u^{-1})+a^{\prime\prime}_{r}D_{r}((u_{0}u)+(u_{0}u)^{-1}))}+(-1)^{f^{(r)}_{a_{r},1}(1)},$

where $a^{\prime}_{r},~{}a^{\prime\prime}_{r}\in\mathbb{F}_{2^{m}}$. Then,
Boolean function $f^{(r)}_{a_{r},1}(x)$ is hyper-bent function if and only if

$\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{\sum\limits_{r\in
R}\mathrm{Tr}^{m}_{1}(a^{\prime}_{r}D_{r}(u+u^{-1})+a^{\prime\prime}_{r}D_{r}((u_{0}u)+(u_{0}u)^{-1}))}=1-(-1)^{f^{(r)}_{a_{r},1}(1)}.$

This is completed the proof. ∎

Let us investigate this Boolean function $f_{a,b,c}(x)$ together. We define

$\displaystyle~{}f_{a,b,c}(x)=\mathrm{Tr}^{n}_{1}((a+cu_{0})x^{(2^{m}-1)}+bx^{\frac{2^{n}-1}{3}}),~{}~{}a,b,c\in\mathbb{F}_{2^{m}},$
(8)

where $m$ is an odd integer and
$u_{0}\in\mathbb{F}_{4}\setminus{\mathbb{F}_{2}}$. Then, we get that

$\displaystyle g(x)~{}$
$\displaystyle=\mathrm{Tr}^{m}_{1}\left(aD_{1}\left(\frac{1}{x^{2}+x+1}\right)+cD_{1}\left(\frac{x^{2}}{x^{2}+x+1}\right)\right)~{}$
$\displaystyle=\mathrm{Tr}^{m}_{1}\left(\frac{cx^{2}+a}{x^{2}+x+1}\right).$

Theorem 3.5 Let $f_{a,b,c}(x)$ be defined by (8) with $u_{0}\in
U_{2^{m}+1}\setminus{\\{1\\}}$, where $m$ is an odd integer. Then,
$f_{a,b,c}(x)$ is said to be a hyper-bent function if and only if

$\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}^{m}_{1}\left(\frac{cx^{2}+a}{x^{2}+x+1}\right)}=1-(-1)^{\mathrm{Tr}^{m}_{1}(c)}.$

###### Proof.

Since $u_{0}$ is a primitive third root of unity, this equation
$u_{0}+\frac{1}{u_{0}}=1$ can be obtained from
$\rho_{0}=\frac{u_{0}}{1+u^{2}_{0}}$. It is easy to get $\Lambda(f_{a,b,c})$
from the proof process of Theorem 3.1.

$\displaystyle\Lambda(f_{a,b,c})~{}$
$\displaystyle=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{g\left(x\right)}+\left(-1\right)^{f_{a,b,c}(x)}~{}$
$\displaystyle=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}^{m}_{1}\left(\frac{cx^{2}+a}{x^{2}+x+1}\right)}+(-1)^{\mathrm{Tr}^{m}_{1}(c)}.$

Therefore, by Theorem 3.1, this function $f_{a,b,c}(x)$ is a hyper-bent
function if and only if

$\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{g\left(x\right)}=1-(-1)^{\mathrm{Tr}^{m}_{1}(c)}.$

This is completed the proof. ∎

We have the following result that describes the hyper-bentness of this
function $f_{a,b,c}(x)$ with the hyperelliptic curves.
Theorem 3.6 A Boolean function $f_{a,b,c}(x)$ defined by (8). The curve $C$ is
written as

$\mathcal{C}:~{}~{}y^{2}+yD(x)=cx^{2}D(x)+aD(x),$

where $D(x)=x^{2}+x+1$. This function $f_{a,b,c}(x)$ is hyper-bent function if
and only if

$\\#\mathcal{C}(\mathbb{F}_{2^{m}})=2^{m}+2-(-1)^{\mathrm{Tr}^{m}_{1}(c)}.$

###### Proof.

Let the sum

$\displaystyle L~{}$ $\displaystyle=1-(-1)^{\mathrm{Tr}^{m}_{1}(c)}~{}$
$\displaystyle=\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{\mathrm{Tr}^{m}_{1}\left(\frac{cx^{2}+a}{x^{2}+x+1}\right)}.$

Denote $T$ is the number of $x\in\mathbb{F}_{2^{m}}$ such that
$\mathrm{Tr}^{m}_{1}\left(\frac{cx^{2}+a}{x^{2}+x+1}\right)=0$, then we have

$2T=\\#\mathcal{C}_{1}(\mathbb{F}_{2^{m}})-1.$

The curve $\mathcal{C}_{1}$ is denoted as

$\displaystyle~{}t^{2}+t=\frac{cx^{2}+a}{x^{2}+x+1}.$ (9)

The substitution $y=t(x^{2}+x+1)$ into (9) is a bijective transformation
between $\mathcal{C}_{1}(\mathbb{F}_{2^{m}})$ and
$\mathcal{C}(\mathbb{F}_{2^{m}})$. Furthermore, one has

$L=\\#\mathcal{C}(\mathbb{F}_{2^{m}})-1-2^{m}.$

We characterize the hyper-bentness of this function $f_{a,b,c}(x)$ with
hyperelliptic curves in terms of Theorem 3.5, i.e., this function
$f_{a,b,c}(x)$ is hyper-bent if and only if
$\\#\mathcal{C}(\mathbb{F}_{2^{m}})=2^{m}+2-(-1)^{\mathrm{Tr}^{m}_{1}(c)}$.
This proof is completed. ∎

Example 3.1 To illustrate our results, Let
$\rho_{0}=\frac{u_{0}}{1+u^{2}_{0}}$, $u_{0}\in U_{2^{m}+1}\setminus{\\{1\\}}$
and
$f^{(1)}_{a_{1},b}(x)=\mathrm{Tr}^{n}_{1}(a_{1}x^{2^{m}-1})+\mathrm{Tr}^{2}_{1}(bx^{\frac{2^{n}-1}{3}})$
be a Boolean function defined on $\mathbb{F}_{2^{n}}$ with
$a_{1}=a^{\prime}_{1}+a^{\prime\prime}_{1}u^{r}_{0}$, where
$a^{\prime}_{1},a^{\prime\prime}_{1}\in\mathbb{F}_{2^{m}}$. Let $g_{a_{1},b}$
be a Boolean function defined on $\mathbb{F}_{2^{m}}$ by

$g_{a_{1},b}(x)=\mathrm{Tr}^{m}_{1}\left(a^{\prime}_{1}\frac{1}{x^{2}+x+\rho_{0}}+a^{\prime\prime}_{1}\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)+\mathrm{Tr}^{2}_{1}(b).$

Then, by Theorem 3.1, $f^{(1)}_{a_{1},b}$ is hyper-bent if and only if

$\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{g_{a_{1},b}(x)}=1-(-1)^{f^{(r)}_{a_{1},b}(1)}.$

Example 3.2 Let us consider a particular Boolean function
$f^{(r)}_{a_{r},1}(x)$ defined on $\mathbb{F}_{2^{n}}$ as

$f^{(r)}_{a_{r},1}(x)=\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})+\mathrm{Tr}^{2}_{1}(x^{\frac{2^{n}-1}{3}}),~{}\forall
x\in\mathbb{F}_{2^{n}},$

where $a_{r}\in\mathbb{F}_{2^{n}}$. Let $u_{0}$ be a primitive $(2^{m}+1)$-th
root of unity and $\rho_{0}=\frac{u_{0}}{1+u^{2}_{0}}$, $g_{a_{r},1}$ be a
Boolean functions defined on $F_{2^{m}}$ by

$g_{a_{r},1}(x)=\mathrm{Tr}^{m}_{1}\left(a^{\prime}_{r}D_{r}\left(\frac{1}{x^{2}+x+\rho_{0}}\right)+a^{\prime\prime}_{r}D_{r}\left(\frac{x^{2}}{\rho_{0}(x^{2}+x+\rho_{0})}\right)\right),$

where
$a^{\prime}_{r}=\frac{a_{r}u^{-r}+a^{2^{m}}_{r}u^{r}_{0}}{u^{r}_{0}+u^{-r}_{0}}$
and $a^{\prime\prime}_{r}=\frac{a_{r}+a^{2^{m}}_{r}}{u^{r}_{0}+u^{-r}_{0}}$.
Then, by Theorem 3.4, $f^{(r)}_{a_{r},1}(x)$ is hyper-bent function if and
only if

$\sum_{x\in\mathbb{F}_{2^{m}}}(-1)^{g_{a_{r},1}(x)}=1-(-1)^{f^{(r)}_{a_{r},1}(1)}.$

##  4 Conclusion

We answer an open problem left in [34] that determines hyper-bentness of
Charpin-Gong like functions $f^{(r)}_{a_{r},b}(x)=\sum\limits_{r\in
R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})+\mathrm{Tr}^{2}_{1}(bx^{\frac{2^{n}-1}{3}})$,
where the coefficient $b$ is in $\mathbb{F}_{4}$, some of the coefficients
$a_{r}$ are in $\mathbb{F}_{2^{n}}$, but not in $\mathbb{F}_{2^{m}}$. In this
paper, we provide several characterizations of hyper-bentness for Charpin-Gong
like functions $f^{(r)}_{a_{r},b}(x)$ with the oefficient $a_{r}$ is in
$\mathbb{F}_{2^{n}}$ by applying the technology in [37]. A relative problem is
to character the hyper-bentness of this family of Boolean function
$f^{(r)}_{a_{r},b}(x)=\sum\limits_{r\in
R}\mathrm{Tr}^{n}_{1}(a_{r}x^{r(2^{m}-1)})+\mathrm{Tr}^{4}_{1}(bx^{\frac{2^{n}-1}{5}})$,
where the same restriction $a_{r}$ is in $\mathbb{F}_{2^{n}}$ rather than in
$\mathbb{F}_{2^{m}}$ and satisfies $m\equiv 2(mod4)$ and the coefficient $b$
is in $\mathbb{F}_{16}$. In additional, to generalize the results of this
paper to $p$-ary hyper-bent functions is also a fascinating problem.

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