A class of hyper-bent functions and Kloosterman sums

Abstract

This paper is devoted to the characterization of hyper-bent functions. Several classes of hyper-bent functions have been studied, such as Charpin and Gong’s family $\sum \limits _{r\in R}\text {Tr}_{1}^{n} (a_{r}x^{r(2^{m}-1)})$ and Mesnager’s family $\sum \limits _{r\in R}\text {Tr}_{1}^{n}(a_{r}x^{r(2^{m}-1)}) +\text {Tr}_{1}^{2}(bx^{\frac {2^{n}-1}{3}})$ . In this paper, we generalize these results by considering the following class of Boolean functions over $\mathbb {F}_{2^{n}}$:

$$\sum\limits_{r\in R}\sum\limits_{i=0}^{2}T{r^{n}_{1}}(a_{r,i} x^{r(2^{m}-1)+\frac{2^{n}-1}{3}i}) +T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), $$

where $n=2m$, m is odd, $b\in \mathbb {F}_{4}$, and $a_{r,i}\in \mathbb {F}_{2^{n}}$. With the restriction of $a_{r,i}\in \mathbb {F}_{2^{m}}$, we present a characterization of hyper-bentness of these functions in terms of crucial exponential sums. For some special cases, we provide explicit characterizations for some hyper-bent functions in terms of Kloosterman sums and cubic sums. Finally, we explain how our results on binomial, trinomial and quadrinomial hyper-bent functions can be generalized to the general case where the coefficients $a_{r,i}$ belong to the whole field $\mathbb {F}_{2^{n}}$.

1 Introduction

Bent functions are maximally nonlinear Boolean functions with even numbers of variables whose Hamming distance to the set of all affine functions equals $2^{n-1}\pm 2^{\frac {n}{2}-1}$. These functions introduced by Rothaus [29] as interesting combinatorial objects have been extensively studied for their applications not only in cryptography, but also in coding theory [3, 25, 31] and combinatorial design. A bent function can be considered as a Boolean function defined over $\mathbb {F}_{2}^{n}$, $\mathbb {F}_{2^{m}}\times \mathbb {F}_{2^{m}} (n=2m)$ or $\mathbb {F}_{2^{n}}$. Thanks to the different structures of the vector space $\mathbb {F}_{2}^{n}$ and the Galois field $\mathbb {F}_{2^{n}}$, bent functions can be well studied. Hyper-bent functions as a subclass of bent functions [15, 35] achieve the maximal minimum distance to all the coordinate functions of all bijective monomials (i.e., functions of the form $\text {Tr}_{1}^{n}(ax^{i})+ \epsilon$, $\gcd (i, 2^{n}-1) = 1$ ). It is still elusive to completely characterize bent and hyper-bent functions. Much research on bent and hyper-bent functions on $\mathbb {F}_{2^{n}}$ can be found in [1, 5, 7–10, 14, 17, 18, 22, 24–26, 36].

Charpin and Gong [5] studied the hyper-bent functions with multiple trace terms of the form

$$f(x) = \sum\limits_{r\in R}\text{Tr}_{1}^{n}(a_{r}x^{r(2^{m}-1)}), $$

where $n=2m$, R is a set of representations of the cyclotomic cosets modulo $2^{m}+1$ of maximal size n ¹ and $a_{r}\in \mathbb {F}_{2^{m}}$. The characterization of these hyper-bent functions was presented by the exponential sums on $\mathbb {F}_{2^{m}}$. Lisoněk [20] presented another characterization of Charpin and Gong’s hyper-bent functions in terms of the number of rational points on certain hyper-elliptic curves. He provided an algorithm for determining such hyper-bent functions with time complexity and space complexity $O(r_{max}^{a}m^{b})$, where $r_{max}$ is the biggest element in R, and $a, b$ are some positive constants irrelevant to $r_{max}$ and m. In particular, when $R=\{r\}$ and $(r,2^{m}+1) = 1$, these hyper-bent functions are monomial functions via Dillon-like exponent. Many authors have proved that the monomial function $T{r^{n}_{1}}(ax^{r(2^{m}-1)}) (a\in \mathbb {F}_{2^{m}})$ is hyper-bent if and only if $K_{m}(a) = 0$ (a proof can be found for instance in [17]).

In [22–25], Mesnager studied and characterized the hyper-bentness of the functions of the form (which are distinct from those of Charpin and Gong’s functions)

$$f(x) = \sum\limits_{r\in R}\text{Tr}_{1}^{n}(a_{r}x^{r(2^{m}-1)})+T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}). $$

She exhibited firstly binomial hyper-bent functions in [22, 23] and studied the case of multiple trace terms in [24, 25].

Afterward, Mesnager and Flori [27] considered a general class of Boolean functions and characterized the hyper-bentness of these Boolean functions of the form

$$f(x) = \sum\limits_{r\in R}\text{Tr}_{1}^{n}(a_{r}x^{r(2^{m}-1)})+T{r^{t}_{1}}(bx^{s(2^{m}-1)}), $$

where $s|(2^{m}+1)$, $t=o(s(2^{m}-1))$ (i.e. t is the size of the cyclotomic coset of s modulo $2^{m}+1$), $a_{r}\in \mathbb {F}_{2^{m}}$, and $b\in \mathbb {F}_{2^{t}}$. The so-called Mesnager’s functions correspond then to the case where $t=2$ and $s=\frac {2^{m}+1}{3}$. For the case: $t=4$ and $s=\frac {2^{m}+1}{5}$, explicit characterizations were given in [32–34]. When $r_{max}$ is small, Flori and Mesnager [11, 12] used the number of rational points on hyper-elliptic curves to determine those classes of hyper-bent functions.

Next, Li et al. [19] characterized a class of Boolean functions of the form

$$f(x) = {\sum}^{q-1}_{i=0}T{r^{n}_{1}}(a_{i}x^{i(q-1)})+T{r^{l}_{1}}(\epsilon x^{\frac{q^{2}-1}{e}}), $$

where $n=2m$, $q=p^{m}$ (p is a prime), $e|(q+1)$, $a_{i}\in \mathbb {F}_{q^{2}}$, $\epsilon \in \mathbb {F}_{p^{l}}$, and l is the smallest positive integer satisfying $l|n$ and $e|p^{l}-1$.

The coefficients $a_{r}$ in these above Boolean function are restricted to the subfield $\mathbb {F}_{2^{m}}$. This paper considers a class of Boolean functions of the form

$$\sum\limits_{r\in R}\sum\limits_{i=0}^{2}T{r^{n}_{1}}(a_{r,i}x^{r(2^{m}-1)+\frac{2^{n}-1}{3}i})+T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), $$

where $n=2m$, m is odd, $b\in \mathbb {F}_{4}$, and $a_{r,i}\in \mathbb {F}_{2^{n}}$. We first characterize the hyper-bentness of this class of Boolean functions in terms of exponential sums on $\mathbb {F}_{2^{m}}$. Using these results, we exhibit some hyper-bent functions via the so-called Dillon-like exponent in terms of the well-known cubic sums and Kloosterman sums. Further, our characterizations can also be applied in the general case for $a_{r,i}\in \mathbb {F}_{2^{n}}$.

The following paper is organized as follows: Section 2 introduces some notations and background. Section 3 considers a class of Boolean functions and presents the characterization of hyper-bentness of these functions with exponential sums on $\mathbb {F}_{2^{m}}$. Section 4 gives an explicit characterization of some special hyper-bent functions with multiple trace terms in terms of Kloosterman sums and cubic sums. Section 5 provides a conclusion.

2 Preliminaries

2.1 Boolean functions

Let n be a positive integer, $\mathbb {F}_{2^{n}}$ be a finite field with $2^{n}$ elements, $\mathbb {F}_{2^{n}}^{*}$ be the multiplicative group of $\mathbb {F}_{2^{n}}$, and $\mathbb {F}_{2^{k}}$ be a subfield of $\mathbb {F}_{2^{n}}$. The trace function from $\mathbb {F}_{2^{n}}$ to $\mathbb {F}_{2^{k}}$ is denoted by $\text {Tr}_{k}^{n} (x) = {\sum }_{i=0}^{n/k-1}x^{2^{ik}}$ . When k=1 , $\text {Tr}_{1}^{n}(\cdot )$ is called the absolute trace function.

A Boolean function over $\mathbb {F}_{2^{n}}$ can be represented by

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

where

• ${\Gamma }_{n}$ is the set of integers obtained by choosing one element in each cyclotomic class of 2 module $2^{n}-1$ (j is often chosen as the smallest element in its cyclotomic class, called the coset leader of the class);

• $o(j)$ is the size of the cyclotomic coset of 2 modulo $2^{n}-1$ containing j;

• $a_{j}\in \mathbb {F}_{2^{o(j)}}$;

• $\epsilon =wt(f)\pmod 2$, where $\text {wt}(f):=\#\{ x\in \mathbb {F}_{2^{n}}|f(x) = 1 \}$.

The “sign” function of a Boolean function f is defined by χ(f) := (−1)^f. The Walsh-Hadamard transform of f over $\mathbb {F}_{2^{n}}$ is defined by $\widehat {\chi }_{f}(w):= {\sum }_{x\in \mathbb {F}_{2^{n}}}(-1)^{f(x) +\text {Tr}_{1}^{n}(wx)},$ where $w\in \mathbb {F}_{2^{n}}$. Then we can define the bent functions.

Definition 1

A Boolean function $f:\mathbb {F}_{2^{n}}\rightarrow \mathbb {F}_{2}$ is called a bent function, if $\widehat {\chi }_{f}(w) =\pm 2^{\frac {n}{2}} (\forall w\in \mathbb {F}_{2^{n}})$.

If f is a bent function, n must be even. Further, $\text {deg}(f)\leq \frac {n}{2}$ [2]. Hyper-bent functions are an important subclass of bent functions. The definition of hyper-bent functions is given below.

Definition 2

A bent function $f:\mathbb {F}_{2^{n}}\rightarrow \mathbb {F}_{2}$ is called a hyper-bent function, if, for any i satisfying $(i,2^{n}-1) = 1$, $f(x^{i})$ is also a bent function.

It has been proved in [3] and [35] that if f is a hyper-bent function, then $\text {deg}(f) =\frac {n}{2}$. For Boolean functions over $\mathbb {F}_{2^{\frac {n}{2}}}\times \mathbb {F}_{2^{\frac {n}{2}}}$, we have a class of hyper-bent functions $\mathcal {P}\mathcal {S}_{ap}$ [3].

Definition 3

Let $n=2m$, the $\mathcal {P}\mathcal {S}_{ap}$ class is the set of all the Boolean functions of the form $f(x,y) = g(\frac {x}{y})$, where $x,y\in \mathbb {F}_{2^{m}}$, g is a balanced Boolean functions (i.e., $\text {wt}(f) = 2^{m-1}$) and $g(0) = 0$. When $y=0$, let $\frac {x}{y} =xy^{2^{n}-2}=0$.

Each Boolean function f in $\mathcal {P}\mathcal {S}_{ap}$ satisfies $f(\beta z) = f(z)$ and $f(0) = 0$, where $\beta \in \mathbb {F}_{2^{m}}^{*}$ and $z\in \mathbb {F}_{2^{m}}\times \mathbb {F}_{2^{m}}$. Youssef and Gong [35] studied these functions over $\mathbb {F}_{2^{n}}$ and gave the following property.

Proposition 1

Let $n=2m$, α be a primitive element in $\mathbb {F}_{2^{n}}$ , and f be a Boolean function over $\mathbb {F}_{2^{n}}$ such that $f(\alpha ^{2^{m}+1} x) =f(x) (\forall x\in \mathbb {F}_{2^{n}})$ and f(0) = 0 , then f is a hyper-bent function if and only if the weight of (f(1), f(α) , f(α ²), ⋯, $f(\alpha ^{2^{m}}))$ is 2^m−1.

Further, [3] proved the following result.

Proposition 2

Let f be a Boolean function defined in Proposition 1. If f (1) = 0, then f is in $\mathcal {P}\mathcal {S}_{ap}$ . If $f(1) = 1$, then there exists a Boolean function g in $\mathcal {P}\mathcal {S}_{ap}$ and $\delta \in \mathbb {F}_{2^{n}}^{*}$ satisfying $f(x) = g(\delta x)$.

Charpin and Gong [5] expressed Proposition 2 in a different version below.

Proposition 3

Let n = 2m, α be a primitive element of $\mathbb {F}_{2^{n}}$ and f be a Boolean function over $\mathbb {F}_{2^{n}}$ satisfying $f(\alpha ^{2^{m+1}}x) =f(x) (\forall x\in \mathbb {F}_{2^{n}})$ and $f(0) = 0$. Let ξ be a primitive $2^{m}+1$ -th root in $\mathbb {F}_{2^{n}}^{*}$. Then f is a hyper-bent function if and only if the cardinality of the set {i|f(ξ ⁱ) = 1, 0 ≤ i ≤ 2^m} is 2^m−1.

2.2 Dickson polynomials

For r > 0, Dickson polynomials are given by

$$D_{r}(x) = \sum\limits_{i=0}^{\lfloor \frac{r}{2}\rfloor} \frac{r}{r-i}{{r-i \choose i}}x^{r-2i}, r=2,3,{\cdots} . $$

Further, Dickson polynomials can be also defined by the following recurrence relation

$$D_{i+2}(x) = xD_{i+1}+D_{i}(x) $$

with initial values $D_{0}(x) = 0$ and $D_{1}(x) = x$.

Some properties of Dickson polynomials are given below.

• $\text {deg}(D_{r}(x)) = r$.

• $D_{rp}(x) = D_{r}(D_{p}(x))$.

• $D_{r}(x+x^{-1}) = x^{r}+x^{-r}$.

More results on Dickson polynomials over 𝔽₂ can be found in [28].

2.3 Kloosterman sums and cubic sums

In this subsection, we introduce some results on some special binary exponential sums.

Definition 4

The binary Kloosterman sums associated with a are

$$K_{m}(a) = \sum\limits_{x\in \mathbb{F}_{2^{m}}}\chi(T{r^{m}_{1}}(\frac{1}{x}+ax)),a\in \mathbb{F}_{2^{m}}. $$

When $x=0$, we set $\frac {1}{x}=0$.

Some properties of binary Kloosterman sums are given in the following propositions [16].

Proposition 4

Let $a\in \mathbb {F}_{2^{m}}$, then $K_{m}(a)\in [-2^{(m+2)/2}+1,2^{(m+2)/2}+1]$ and $K_{m}(a)\equiv 0 \mod 4$.

Another property of Kloosterman sums is stated in the following proposition [6].

Proposition 5

Let $m\geq 3$ be an odd integer and $a \in \mathbb {F}_{2^{m}}$. Then

$$K_{m}(a)\equiv 1 \mod 3 \text{if and only if} T{r^{m}_{1}}(a^{1/3}) = 0.$$

Definition 5

The binary cubic sums on $\mathbb {F}_{2^{m}}$ are

$$C_{m}(a,b) = \sum\limits_{x\in \mathbb{F}_{2^{m}}}\chi(T{r^{m}_{1}}(ax^{3}+bx)),a\in \mathbb{F}_{2^{m}}^{*},b\in \mathbb{F}_{2^{m}}. $$

Carlitz [4] computed the exact values of the cubic sums in the following proposition.

Proposition 6

Let m be a positive integer. Then

• (1) $C_{m}(1,1) = (-1)^{(m^{2}-1)/8}2^{(m+1)/2}$.

• (2) If $T{r^{m}_{1}}(c) = 0$ , then $C_{m}(1,c) = 0$.

• (3) If $T{r^{m}_{1}}(c) = 1$ and $c\neq 1$, then $C_{m}(1,c) = \chi (T{r^{m}_{1}}(\gamma ^{3}+\gamma )) (\frac {2}{m})2^{(m+1)/2}$, where $c=\gamma ^{4}+\gamma +1, (\frac {2}{m})$ is the Jacobi symbol , and $\gamma \in \mathbb {F}_{2^{m}}$.

From Proposition 5 and Proposition 6, we have the following corollary.

Corollary 1

Let m be odd and $a\in \mathbb {F}_{2^{m}}^{*}$, the following results are equivalent:

• (1) $K_{m}(a)\equiv 1\mod 3;$

• (2) $C_{m}(a,a) = 0;$

• (3) $T{r^{m}_{1}}(a^{\frac {1}{3}}) = 0$.

2.4 Exponential sums

Let m be an odd integer, $U=\{u:u^{2^{m}+1}=1,u\in \mathbb {F}_{2^{n}}\}$ and $V=U^{3}=\{u^{3}:u\in U\}$. Let ξ be a generator of U and $w=\xi ^{\frac {2^{m}+1}{3}}$. Let R be a set of representations of the cyclotomic cosets modulo 2^m + 1 and $a_{r}\in \mathbb {F}_{2^{m}}$. For simplicity, some notations on exponential sums are defined below.

$$\begin{array}{@{}rcl@{}} &&S_{i}((a_{r})_{r\in R}) = \sum\limits_{v\in V}\chi(T{r^{n}_{1}}(\sum\limits_{r\in R}a_{r}(\xi^{i}v)^{r(2^{m}-1)})) ; T_{i}((a_{r})_{r\in R}) = \sum\limits_{x\in \mathbb{F}_{2^{m}},T{r^{m}_{1}}(x^{-1}) = i}\chi(T{r^{m}_{1}}(\sum\limits_{r\in R}a_{r} D_{r}(x))). \\&&{T_{i}^{3}}((a_{r})_{r\in R}) = \sum\limits_{x\in \mathbb{F}_{2^{m}},T{r^{m}_{1}}(x^{-1}) = i}\chi(T{r^{m}_{1}}(\sum\limits_{r\in R}a_{r} D_{r}(D_{3}(x)))); {\Xi}((a_{r})_{r\in R}) = \!\!\sum\limits_{x\in \mathbb{F}_{2^{m}}}\chi(T{r^{m}_{1}}(\sum\limits_{r\in R}a_{r}D_{r}(x)));\\&&\overline{\Xi}((a_{r})_{r\in R}) = \sum\limits_{x\in \mathbb{F}_{2^{m}}}\chi(T{r^{m}_{1}}(\frac{1}{x}+\sum\limits_{r\in R}a_{r}D_{r}(x))); {\Xi}^{3}((a_{r})_{r\in R}) = \sum\limits_{x\in \mathbb{F}_{2^{m}}}\chi(T{r^{m}_{1}}(\sum\limits_{r\in R}a_{r}D_{r}(D_{3}(x)))). \\&&\overline{\Xi}^{3}((a_{r})_{r\in R}) = \sum\limits_{x\in \mathbb{F}_{2^{m}}}\chi(T{r^{m}_{1}}(\frac{1}{x}+\sum\limits_{r\in R}a_{r}D_{r}(D_{3}(x)))). \end{array} $$

In all evidence, if i ≡ j mod 2^m + 1, then $S_{i}((a_{r})_{r\in R}) = S_{j}((a_{r})_{r\in R})$. Some relationships of these exponential sums are given in the following proposition [27].

Proposition 7

• (1) $S_{0}((a_{r})_{r\in R}) = \frac {1}{3}(1+2{T_{1}^{3}}((a_{r})_{r\in R})) = \frac {1}{3}[1+2{\Xi }^{3}((a_{r})_{r\in R})-2T_{0}((a_{r})_{r\in R})]$; $S_{1}((a_{r})_{r\in R}) = S_{2}((a_{r})_{r\in R})$; $S_{0}((a_{r})_{r\in R})+S_{1}((a_{r})_{r\in R})+S_{2}((a_{r})_{r\in R}) = 1+2T_{1}((a_{r})_{r\in R})$.

• (2) ${T_{0}^{3}}((a_{r})_{r\in R}) = T_{0}((a_{r})_{r\in R})$; ${T_{1}^{3}}((a_{r})_{r\in R}) = {\Xi }^{3}((a_{r})_{r\in R})-T_{0}((a_{r})_{r\in R})$; $T_{i}((a_{r})_{r\in R}) = \frac {1}{2}[{\Xi }((a_{r})_{r\in R})+(-1)^{i}\overline {\Xi }((a_{r})_{r\in R})]$; ${T_{i}^{3}}((a_{r})_{r\in R}) = \frac {1}{2}[{\Xi }^{3}((a_{r})_{r\in R})+(-1)^{i}\overline {\Xi }^{3}((a_{r})_{r\in R})]$.

When #R = 1, we write (a) _R for (a) _r∈R and a for (a)_{1}. From the definitions and Proposition 7, we have the following lemma.

Lemma 1

Let $a\in \mathbb {F}_{2^{m}}$ and $gcd(r,\frac {2^{m}+1}{3}) = 1$, then

• (1) $S_{0}((a)_{\{r\}}) = S_{0}(a)$;

• (2) If 3 ∤ r, $S_{1}((a)_{\{r\}}) = S_{1}(a)$; If $3\mid r$, $S_{1}((a)_{\{r\}}) = S_{0}(a)$.

Actually, $S_{0}(a)$ and $S_{1}(a)$ can be expressed by cubic sums and Kloosterman sums [22].

Lemma 2

Let $a\in \mathbb {F}_{2^{m}}^{*}$, then

• (1) $S_{0}(a) = \frac {1}{3}[-K_{m}(a)+2C_{m}(a,a)+1]$;

• (2) $S_{1}(a) = \frac {1}{3}[-K_{m}(a)-C_{m}(a,a)+1]$.

Obviously, when $a\in \mathbb {F}_{2^{m}}$, we have $S_{i}(a) = S_{i}(a^{2})$. From Lemma 2, Corollary 1, Proposition 4, and Proposition 6, we have the following lemma.

Lemma 3

Let $m\geq 5$, $m\equiv 1 \mod 2$, and $a\in \mathbb {F}_{2^{m}}^{*}$ . The following four conditions are equivalent:

• (1) $K_{m}(a) = 4$;

• (2) $-S_{0}(a) = 1$;

• (3) $-2S_{0}(a)+S_{1}(a) = 1;$

• (4) $S_{0}(a)-2S_{1}(a) = 1$.

3 A class of hyper-bent functions with multiple trace terms

In this section, we will consider a class of Boolean functions of the form

$$ f_{a,b}(x) = \sum\limits_{r\in R}\sum\limits_{i=0}^{2}T{r^{n}_{1}}(a_{r,i}x^{r(2^{m}-1)+\frac{2^{n}-1}{3}i})+T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), $$

(1)

where $n=2m$, m is odd, $a_{r,i}\in \mathbb {F}_{2^{n}}$ and $b\in \mathbb {F}_{4}$. With notations given in the previous section, we present the characterization of hyper-bent functions in (1).

Theorem 1

Let $f_{a,b}$ be a Boolean function defined in (1) and $a_{r,i}^{\prime }$ be defined by

$$ \left(\begin{array}{c} a_{r,0}^{\prime}\\a_{r,1}^{\prime}\\a_{r,2}^{\prime} \end{array}\right) = \left[ \begin{array}{ccc} 1&1&1\\ 1&w&w^{2}\\ 1&w^{2}&w \end{array} \right] \left(\begin{array}{c}a_{r,0}\\a_{r,1}\\a_{r,2} \end{array}\right). $$

(2)

where w is a primitive 3rd root of unity. Then $f_{a,b}$ is hyper-bent if and only if

$${\Lambda}(f_{a,b}) = \sum\limits_{u\in U}\chi(f_{a,b}(u)) = 1. $$

Further, we have that

$${\Lambda}(f_{a,b}) = \chi(T{r^{2}_{1}}(b))S_{0}((a_{r,0}^{\prime})_{r\in R})+\chi(T{r^{2}_{1}}(bw))S_{1}((a_{r,1}^{\prime})_{r\in R})+\chi(T{r^{2}_{1}}(bw^{2}))S_{2}((a_{r,2}^{\prime})_{r\in R}). $$

If $a_{r,2}^{\prime }\in \mathbb {F}_{2^{m}}$ , then

$${\Lambda}(f_{a,b}) = \chi(T{r^{2}_{1}}(b))S_{0}((a_{r,0}^{\prime})_{r\in R})+\chi(T{r^{2}_{1}}(bw))S_{1}((a_{r,1}^{\prime})_{r\in R})+\chi(T{r^{2}_{1}}(bw^{2}))S_{1}((a_{r,2}^{\prime})_{r\in R}). $$

Proof

Let $\alpha$ be a primitive element in $\mathbb {F}_{2^{n}}$. From the definition of $f_{a,b}$, for $x\in \mathbb {F}_{2^{m}}$ we have that

$$f_{a,b}(\alpha^{2^{m}+1}x) = f_{a,b}(x), \quad f(0) = 0, $$

From Proposition 3, $f_{a,b}(x)$ is hyper-bent if and only if ${\Lambda }(f_{a,b}) = 1$ [24, 25]. Since $U=V\cup \xi V \cup \xi ^{2} V$,

$${\Lambda}(f_{a,b}) = \sum\limits_{u\in U}\chi(f_{a,b}(u)) = \sum\limits_{v\in V}\chi(f_{a,b}(v))+\sum\limits_{v\in V}\chi(f_{a,b}(\xi v))+\sum\limits_{v\in V}\chi(f_{a,b}(\xi^{2}v)). $$

For any $v\in V$, $v^{\frac {2^{n}-1}{3}}=1$. Hence

$$\begin{array}{@{}rcl@{}} {\Lambda}(f_{a,b})&=&\chi(T{r^{2}_{1}}(b))\sum\limits_{v\in V}\chi(\sum\limits_{r\in R}\sum\limits_{i=0}^{2}T{r^{n}_{1}}(a_{r,i}v^{r(2^{m}-1)}))\\ &+&\chi(T{r^{2}_{1}}(b\xi^{\frac{2^{n}-1}{3}}))\sum\limits_{v\in V}\chi(\sum\limits_{r\in R}\sum\limits_{i=0}^{2}T{r^{n}_{1}}(a_{r,i}\xi^{\frac{2^{n}-1}{3}i}(\xi v)^{r(2^{m}-1)}))\\ &+&\chi(T{r^{2}_{1}}(b\xi^{\frac{2^{n}-1}{3}\cdot 2}))\sum\limits_{v\in V}\chi(\sum\limits_{r\in R}\sum\limits_{i=0}^{2}T{r^{n}_{1}}(a_{r,i}\xi^{\frac{2^{n}-1}{3}\cdot2i}(\xi^{2} v)^{r(2^{m}-1)})). \end{array} $$

Note that $\xi ^{\frac {2^{n}-1}{3}}=(\xi ^{\frac {2^{m}+1}{3}})^{2^{m}-1}=w^{2^{m}+1-2}=w$. We have that

$$\begin{array}{@{}rcl@{}} {\Lambda}(f_{a,b})&=&\chi(T{r^{2}_{1}}(b))\sum\limits_{v\in V}\chi(\sum\limits_{r\in R}T{r^{n}_{1}}(({\sum}_{i=0}^{2}a_{r,i})v^{r(2^{m}-1)}))\\ &+&\chi(T{r^{2}_{1}}(bw))\sum\limits_{v\in V}\chi(\sum\limits_{r\in R}T{r^{n}_{1}}((\sum\limits_{i=0}^{2}a_{r,i}w^{i})(\xi v)^{r(2^{m}-1)}))\\ &+&\chi(T{r^{2}_{1}}(bw^{2}))\sum\limits_{v\in V}\chi(\sum\limits_{r\in R}T{r^{n}_{1}}((\sum\limits_{i=0}^{2}a_{r,i}w^{2i})(\xi^{2} v)^{r(2^{m}-1)})). \end{array} $$

From the definitions of $S_{i}((.)_{r\in R})$ and $a_{r,i}^{\prime }$,

$$\begin{array}{@{}rcl@{}} {\Lambda}(f_{a,b})&=& \chi(T{r^{2}_{1}}(b))S_{0}((a_{r,0}^{\prime})_{r\in R})+\chi(T{r^{2}_{1}}(bw))S_{1}((a_{r,1}^{\prime})_{r\in R})\\ &&+\chi(T{r^{2}_{1}}(bw^{2}))S_{2}((a_{r,2}^{\prime})_{r\in R}). \end{array} $$

Note that if $a_{r,2}^{\prime }\in \mathbb {F}_{2^{m}}$, $S_{2}((a_{r,2}^{\prime })_{r\in R}) = S_{1}((a_{r,2}^{\prime })_{r\in R})$. Then we have

$$\begin{array}{@{}rcl@{}} {\Lambda}(f_{a,b})&=&\chi(T{r^{2}_{1}}(b)) S_{0}((a_{r,0}^{\prime})_{r\in R})+\chi(T{r^{2}_{1}}(bw))S_{1}((a_{r,1}^{\prime})_{r\in R})\\ &&+\chi(T{r^{2}_{1}}(bw^{2}))S_{1}((a_{r,2}^{\prime})_{r\in R}). \end{array} $$

The result follows. □

The values of $S_{0}((a_{r,0}^{\prime })_{r\in R})$ and $S_{1}((a_{r,1}^{\prime })_{r\in R})$ can be computed by means of exponential sums on $\mathbb {F}_{2^{m}}$. From Proposition 7, the following result is obtained.

Lemma 4

Let $a_{r}\in \mathbb {F}_{2^{m}}$ , then (1) $S_{0}((a_{r})_{r\in R}) = \frac {1}{3}[2{\Xi }^{3}((a_{r})_{r\in R})-{\Xi }((a_{r})_{r\in R})-\overline {\Xi }((a_{r})_{r\in R})+1];$ (2) $S_{1}((a_{r})_{r\in R}) = \frac {1}{3}[-{\Xi }^{3}((a_{r})_{r\in R})+2{\Xi }((a_{r})_{r\in R})-\overline {\Xi }((a_{r})_{r\in R})+1]$.

For the function $f_{a,b}$ with coefficients $a_{r}\in \mathbb {F}_{2^{m}}$, we can compute ${\Lambda }(f_{a,b})$ with exponential sums on $\mathbb {F}_{2^{m}}$. Note that $\chi (T{r^{2}_{1}}(0)) = \chi (T{r^{2}_{1}}(1)) = 1$ and $\chi (T{r^{2}_{1}}(w)) = \chi (T{r^{2}_{1}}(w^{2})) = -1$. From Theorem 1 and Lemma 4, the value of ${\Lambda }(f_{a,b})$ can be computed by exponential sums ${\Xi }^{3}((a_{r,0}^{\prime })_{r\in R})$, ${\Xi }((a_{r,0}^{\prime })_{r\in R})$, $\overline {\Xi }((a_{r,0}^{\prime })_{r\in R})$, ${\Xi }^{3}((a_{r,1}^{\prime })_{r\in R})$, ${\Xi }((a_{r,1}^{\prime })_{r\in R})$, $\overline {\Xi }((a_{r,1}^{\prime })_{r\in R})$, ${\Xi }^{3}((a_{r,2}^{\prime })_{r\in R})$, $2{\Xi }((a_{r,2}^{\prime })_{r\in R})$, and $\overline {\Xi }((a_{r,2}^{\prime })_{r\in R})$.

4 Hyper-bent functions and Kloosterman sums

In this section, we charcterize the hyper-bentness of certain functions in some particular cases by means of Kloosterman sums and cubic sums. Further, we try to generalize the characterization of some special hyper-bent functions for general cases: $a_{r,i}\in \mathbb {F}_{2^{n}}$.

4.1 Explicit characterization of hyper-bent functions with coefficients in $\mathbb {F}_{2^{m}}$

Theorem 2

Let $gcd(r_{i},\frac {2^{m}+1}{3}) = 1,a,c,d\in \mathbb {F}_{2^{m}}$ and w be a primitive 3rd root of unity. Let $f(x)$ be defined by

$$\begin{array}{@{}rcl@{}} f(x)&=& T{r^{n}_{1}}(a(x^{r_{0}(2^{m}-1)}+x^{r_{0}(2^{m}-1)+\frac{2^{n}-1}{3}}+x^{r_{0}(2^{m}-1)+2\frac{2^{n}-1}{3}}))\\ &+&T{r^{n}_{1}}(c(x^{r_{1}(2^{m}-1)}+w^{2}x^{r_{1}(2^{m}-1)+\frac{2^{n}-1}{3}}+wx^{r_{1}(2^{m}-1)+2\frac{2^{n}-1}{3}}))\\ &+&T{r^{n}_{1}}(d(x^{r_{2}(2^{m}-1)}+wx^{r_{2}(2^{m}-1)+\frac{2^{n}-1}{3}}+w^{2}x^{r_{2}(2^{m}-1)+2\frac{2^{n}-1}{3}}))\\ &+&T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}). \end{array} $$

(3)

Then

• (1) If b = 0, f(x) is hyper-bent if and only if $S_{0}(a)+S_{1}((c)_{\{r_{1}\}})+S_{1}((d)_{\{r_{2}\}}) = 1$;

• (2) If b = 1, 0 f(x) is hyper-bent if and only if $S_{0}(a)-S_{1}((c)_{\{r_{1}\}})-S_{1}((d)_{\{r_{2}\}}) = 1$;

• (3) If b = w, f(x) is hypre-bent if and only if $-S_{0}(a)-S_{1}((c)_{\{r_{1}\}})+S_{1}((d)_{\{r_{2}\}}) = 1$;

• (4) If $b=w^{2}$, $f(x)$ is hyper-bent if and only if $-S_{0}(a)+S_{1}((c)_{\{r_{1}\}})-S_{1}((d)_{\{r_{2}\}}) = 1$.

Proof

Let $R=\{r_{0},r_{1},r_{2}\}$. Take $a_{r_{0},0}=a_{r_{0},1}=a_{r_{0},2}=a$, $a_{r_{1},0}=c$, $a_{r_{1},1}=cw^{2}$, $a_{r_{1},2}=cw$, $a_{r_{2},0}=d$, $a_{r_{2},1}=dw$, and $a_{r_{2},2}=dw^{2}$. Then

$$\begin{array}{@{}rcl@{}} a_{r_{0},0}^{\prime}&=&a, a_{r_{1},0}^{\prime}=0, a_{r_{2},0}^{\prime}=0, a_{r,0}^{\prime}=0 (r\not \in R),\\ a_{r_{0},1}^{\prime}&=&0, a_{r_{1},1}^{\prime}=c, a_{r_{2},1}^{\prime}=0, a_{r,1}^{\prime}=0 (r\not \in R),\\ a_{r_{0},2}^{\prime}&=&0, a_{r_{1},2}^{\prime}=0, a_{r_{2},2}^{\prime}=d, a_{r,2}^{\prime}=0 (r\not \in R), \end{array} $$

From Lemma 1, we have

$$\begin{array}{@{}rcl@{}} S_{0}((a_{r,0}^{\prime})_{r\in R})&=S_{0}((a)_{\{r_{0}\}}) = S_{0}(a); \; S_{1}((a_{r,1}^{\prime})_{r\in R}) =S_{1}((c)_{\{r_{1}\}});\; S_{2}((a_{r,2}^{\prime})_{r\in R}) =S_{2}((a)_{\{r_{2}\}}). \end{array} $$

From $T{r^{2}_{1}}(w) = T{r^{2}_{1}}(w^{2}) = 1,T{r^{2}_{1}}(1) = 0$, and Theorem 1, this theorem can be immediately obtained. □

Corollary 2

Let $gcd(r_{i},\frac {2^{m}+1}{3}) = 1(i=1,2)$, $a,c,d\in \mathbb {F}_{2^{m}}$, $T{r^{m}_{1}}(a^{\frac {1}{3}}) = T{r^{m}_{1}}(c^{\frac {1}{3}}) = T{r^{m}_{1}}(d^{\frac {1}{3}}) = 0$ , w be a primitive 3rd root of unity, and $f(x)$ be defined by (3). Then

• (1) If b = 0, f(x) is hyper-bent if and only if $K_{m}(a)+K_{m}(c)+K_{m}(d) = 0$;

• (2) If b = 1, f(x) is hyper-bent if and only if $-K_{m}(a)+K_{m}(c)+K_{m}(d) = 4$;

• (3) If b = w, f(x) is hyper-bent if and only if $K_{m}(a)+K_{m}(c)-K_{m}(d) = 4$;

• (4) If $b=w^{2}$, $f(x)$ is hyper-bent if and only if $K_{m}(a)-K_{m}(c)+K_{m}(d) = 4$.

Proof

From $T{r^{m}_{1}}(a^{\frac {1}{3}}) = 0$ and Corollary 1, we have $C_{m}(a,a) = 0$. From Lemma 2, we obtain $S_{0}(a) = S_{1}(a) = \frac {1}{3}(-K_{m}(a)+1)$. Similarly, $S_{0}(c) = S_{1}(c) = \frac {1}{3}(-K_{m}(c)+1)$ and $S_{0}(d) = S_{1}(d) = \frac {1}{3}(-K_{m}(d)+1)$. The corollary follows from Lemma 1 and Theorem 2. □

Example 1

Let m = 23, $\mathbb {F}_{2^{m}}\cong \mathbb {F}_{2}[x]/(x^{23} + x^{5} + 1)$. Consider positive integers $r_{i}$ such that $gcd(r_{i},\frac {2^{m}+1}{3}) = 1(i=1,2)$. Set

$$\begin{array}{@{}rcl@{}} a&=&{a_{0}^{3}}=x^{23} + x^{16} + x^{15} + x^{13} + x^{9} + x^{8} + x^{7} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1,\\ c&=&{c_{0}^{3}}=x^{23} + x^{21} + x^{20} + x^{17} + x^{14} + x^{12} + x^{11} + x^{10} + x^{7} + x^{5} + 1,\\ d&=&{d_{0}^{3}}=x^{23} + x^{21} + x^{19} + x^{17} + x^{12} + x^{11} + x^{10} + x^{8} + x^{6} + x^{4} + x^{2} + x + 1,\\ b&=&0; \end{array} $$

where $a_{0}=x^{23} + x^{21} + x^{19} + x^{16} + x^{14} + x^{12} + x^{11} + x^{10} + x^{8} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$, $c_{0}=x^{23} + x^{21} + x^{18} + x^{13} + x^{12} + x^{8} + x^{7} + x^{4} + x^{3} + x + 1$, $d_{0}=x^{23} + x^{21} + x^{19} + x^{18} + x^{17} + x^{13} + x^{12} + x^{10} + x^{7} + x^{6} + x^{3} + x^{2} + 1$. Then $T{r^{m}_{1}}(a_{0}) = 0,T{r^{m}_{1}}(c_{0}) = 0$, $T{r^{m}_{1}}(d_{0}) = 0$, and $K_{m}(a) = 1120,K_{m}(c) = 2920$, $K_{m}(d) = -4040$, $K_{m}(a)+K_{m}(c)+K_{m}(d) = 0$. Hence, $f(x)$ in (3) is a hyper-bent function with nine trace terms.

Many hyper-bent functions of the form (3) exist. From an exhaustive search, when m = 5, 7, the number of the hyper-bent functions in Result (1) in Corollary 2 are 1500 and 58653 respectively, and the number of the hyper-bent functions in Result (2) in Corollary 2 are 1500 and 57624 respectively.

For some special cases, from Lemma 1, Lemma 2, Lemma 3, Theorem 1, and Theorem 2, we can straightforwardly obtain the following theorem on the explicit characterization of hyper-bent functions $f_{a,b}$ in (3).

Theorem 3

Let $gcd(r_{i},\frac {2^{m}+1}{3}) = 1$, $a\in \mathbb {F}_{2^{m}}$, and w be a primitive 3rd root of unity. Let $f(x)$ be defined by (3) with $a=c=d$. Then

• (1) If b = 0 and $3|r_{i}(i=1,2)$, f(x) is not hyper-bent;

• (2) If $b=0$ and $3\nmid r_{i}(i=1,2)$, $f(x)$ is hyper-bent if and only if $K_{m}(a) = 0$;

• (3) If $b=0$ and $\#\{r_{i}:r_{i}\equiv 0 \mod 3,i=1,2\}=1$, $f(x)$ is hyper-bent if and only if $K_{m}(a) = C_{m}(a,a)$;

• (4) For the following cases: i) $b=1$, $\#\{r_{i}:r_{i}\equiv 0 \mod 3,i=1,2\}=1$ ; ii) $b=w$, $3\nmid r_{1}$, $3\mid r_{2}$ ; and iii) $b=w^{2}$, $3\mid r_{1}$, $3\nmid r_{2}$; $f(x)$ is hyper-bent if and only if $K_{m}(a) = -C_{m}(a,a)+4$;

• (5) For the rest cases, $f(x)$ is hyper-bent if and only if $K_{m}(a) = 4$.

Remark 1

The value a such that $K_{m}(a) = 0,4$ or $-C_{m}(a,a)+4$ can be used to construct monomial hyper-bent functions [5, 8, 13, 17] or binomial hyper-bent functions by Mesnager [22, 26]. From the above theorem, the value a such that $K_{m}(a) = C_{m}(a,a)$ can be used to construct hyper-bent functions. From Corollary 1, if $K_{m}(a) = C_{m}(a,a)$, then $T{r^{m}_{1}}(a^{1/3}) = 1$. Obviously, if a satisfies $K_{m}(a) = C_{m}(a,a)$, any Frobenius conjugate $a^{2^{i}}$ of a also satisfies $K_{m}(a^{2^{i}}) = C_{m}(a^{2^{i}},a^{2^{i}})$. Actually, If $m=5,7,9$, just one conjugacy class satisfies $K_{m}(a) = C_{m}(a,a)$, and if $m=11,13,15$, there are 3,8,9 conjugacy classes.

Take $R=\{r_{0},r_{1}\}$, $a_{r_{0},0}=a_{r_{0},1}=a_{r_{0},2}=a$, $a_{r_{1},0}=0$, and $a_{r_{1},1}=a_{r_{1},2}=c$. From Lemma 1 and Theorem 1, the theorem comes straightforwardly.

Theorem 4

Let $a,c\in \mathbb {F}_{2^{m}}$, $b\in \mathbb {F}_{4}$ , and $gcd(r_{i},\frac {2^{m}+1}{3}) = 1$ . Let $f(x)$ be defined by

$$\begin{array}{@{}rcl@{}} f(x)&=& T{r^{n}_{1}}(a(x^{r_{0}(2^{m}-1)}+ x^{r_{0}(2^{m}-1)+\frac{2^{n}-1}{3}}+ x^{r_{0}(2^{m}-1)+2\frac{2^{n}-1}{3}}))\\ &+&T{r^{n}_{1}}(c(x^{r_{1}(2^{m}-1)+\frac{2^{n}-1}{3}}+x^{r_{1}(2^{m}-1)+2\frac{2^{n}-1}{3}}))+T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), \end{array} $$

(4)

Then

• (1) If $b=0$, $f(x)$ is hyper-bent if and only if $S_{0}(a)+2S_{1}((c)_{\{r_{1}\}}) = 1$;

• (2) If $b=1$, $f(x)$ is hyper-bent if and only if $S_{0}(a)-2S_{1}((c)_{\{r_{1}\}}) = 1$;

• (3) If b is a primitive 3rd root of unity, $f(x)$ is hyper-bent if and only if $S_{0}(a) = -1$.

From Corollary 1, Lemma 2, Lemma 1, and Theorem 4, the following corollary can be obtained straightforwardly.

Corollary 3

Let $a,c\in \mathbb {F}_{2^{m}}$, $T{r^{m}_{1}}(a^{\frac {1}{3}}) = T{r^{m}_{1}}(c^{\frac {1}{3}}) = 0$, $b\in \mathbb {F}_{4}$, $gcd(r_{i},\frac {2^{m}+1}{3}) = 1$ , and $f(x)$ be defined by (4). Then

• (1) If $b=0$, $f(x)$ is hyper-bent if and only if $K_{m}(a)+2K_{m}(c) = 0$;

• (2) If $b=1$, $f(x)$ is hyper-bent if and only if $-K_{m}(a)+2K_{m}(c) = 4$;

• (3) If b is a primitive 3rd root of unity, $f(x)$ is hyper-bent if and only if $K_{m}(a) = 4$.

Example 2

Let $m=23$, $\mathbb {F}_{2^{m}}\cong \mathbb {F}_{2}[x]/(x^{23} + x^{5} + 1)$. Take positive integers $r_{i}$ such that $gcd(r_{i},\frac {2^{m}+1}{3}) = 1$ $(i=0,1)$. Take

$$\begin{array}{@{}rcl@{}} a&=&{a_{0}^{3}}=x^{23} + x^{21} + x^{14} + x^{13} + x^{11} + x^{10} + x^{9} + x^{4} + x^{2} + x + 1,\\ c&=&{c_{0}^{3}}=x^{23} + x^{22} + x^{20} + x^{18} + x^{17} + x^{16} + x^{15} + x^{13} + x^{10} + x^{8} + x^{7} + x^{5} + x^{3} + x + 1,\\ b&=&0; \end{array} $$

where $a_{0}=x^{23} + x^{21} + x^{19} + x^{18} + x^{17} + x^{15} + x^{13} + x^{12} + x^{11} + x^{10} + x^{9} + x^{7} + x^{4} + x^{3} + x^{2} + x + 1$, $c_{0}=x^{23} + x^{20} + x^{19} + x^{17} + x^{15} + x^{12} + x^{9} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + 1$. Then $T{r^{m}_{1}}(a_{0}) = 0$, $T{r^{m}_{1}}(c_{0}) = 0$, and $K_{m}(a) = 1768,K_{m}(c) = -884,K_{m}(a)+2K_{m}(c) = 0$. Hence, $f(x)$ in (4) is a hyper-bent function with 5 trace terms.

From an exhaustive search, when $m=5,7,9$, the number of hyper-bent functions in Result (1) in Corollary 3 are 50, 735 and 5346 respectively, and the number of hyper-bent functions in Result (2) in Corollary 3 are 100, 588 and 5103 respectively.

Take $R=\{r_{0},r_{1}\}$, $a_{r_{0},0}=0$, $a_{r_{0},1}=aw$, $a_{r_{0},2}=aw^{2}$, $a_{r_{1},0}=c$, $a_{r_{1},1}=cw$, $a_{r_{1},2}=cw^{2}$. From Lemma 1 and Theorem 1, we get straightforwardly the following result.

Theorem 5

Let $a,c\in \mathbb {F}_{2^{m}}$, $b\in \mathbb {F}_{4}$, $gcd(r_{i},\frac {2^{m}+1}{3}) = 1$ . Let $f(x)$ be defined by

$$\begin{array}{@{}rcl@{}} f(x)&=& T{r^{n}_{1}}(a(wx^{r_{0}(2^{m}-1)+\frac{2^{n}-1}{3}}+ w^{2}x^{r_{0}(2^{m}-1)+2\frac{2^{n}-1}{3}}))\\ &+&T{r^{n}_{1}}(c(x^{r_{1}(2^{m}-1)}+ wx^{r_{1}(2^{m}-1)+\frac{2^{n}-1}{3}}+ w^{2}x^{r_{1}(2^{m}-1)+2\frac{2^{n}-1}{3}})) +T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), \end{array} $$

(5)

Then

• (1) If $b=0$, $f(x)$ is hyper-bent if and only if $S_{0}(a)+S_{1}((a)_{\{r_{0}\}})+S_{1}((c)_{\{r_{1}\}}) = 1$;

• (2) If $b=1$, $f(x)$ is hyper-bent if and only if $S_{0}(a)-S_{1}((a)_{\{r_{0}\}})-S_{1}((c)_{\{r_{1}\}}) = 1$;

• (3) If $b=w$, $f(x)$ is hyper-bent if and only if $-S_{0}(a)-S_{1}((a)_{\{r_{0}\}})+S_{1}((c)_{\{r_{1}\}}) = 1$;

• (4) If $b=w^{2}$, $f(x)$ is hyper-bent if and only if $-S_{0}(a)+S_{1}((a)_{\{r_{0}\}})-S_{1}((c)_{\{r_{1}\}}) = 1$.

From Corollary 1, Lemma 2, Lemma 1, and Theorem 5, this corollary can be obtained straightforwardly.

Corollary 4

Let $a,c\in \mathbb {F}_{2^{m}}$, $T{r^{m}_{1}}(a^{\frac {1}{3}}) = T{r^{m}_{1}}(c^{\frac {1}{3}}) = 0$, $b\in \mathbb {F}_{4}$, $gcd(r_{i},\frac {2^{m}+1}{3}) = 1$, and $f(x)$ be defined by (5). Then

• (1) If $b=0$, $f(x)$ is hyper-bent if and only if $K_{m}(a)+2K_{m}(c) = 0$;

• (2) If $b\in \{1,w^{2}\}$, $f(x)$ is hyper-bent if and only if $K_{m}(c) = 4$;

• (3) If $b=w$, $f(x)$ is hyper-bent if and only if $2K_{m}(a)-K_{m}(c) = 4$.

Theorem 6

Let $a,c_{r}\in \mathbb {F}_{2^{m}}$, $b\in \mathbb {F}_{4}$ be a primitive 3rd root of unity, and $gcd(r,\frac {2^{m}+1}{3}) = 1$ . Let $f(x)$ be defined by

$$\begin{array}{@{}rcl@{}} f(x) = T{r^{n}_{1}}(ax^{r_{0}(2^{m}-1)})+\sum\limits_{r\in R}T{r^{n}_{1}}(c_{r} (x^{r(2^{m}-1)+\frac{2^{n}-1}{3}}+x^{r(2^{m}-1)+2\frac{2^{n}-1}{3}})) +T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}). \end{array} $$

Then $f(x)$ is hyper-bent if and only if $K_{m}(a) = 4$.

Proof

Let $r_{0} \in R$. From the definition of $a_{r,i}^{\prime }$, we have that

$$a_{r_{0},0}^{\prime}=a, a_{r,0}^{\prime}=0 (r\neq r_{0}),\; a_{r,1}^{\prime} =0, a_{r,2}^{\prime}=0 (r\in R). $$

From Lemma 1,

$$\begin{array}{@{}rcl@{}} S_{0}((a_{r,0}^{\prime})_{r\in R})&=&S_{0}((a)_{\{r_{0}\}}) = S_{0}(a),\\ S_{1}((a_{r,1}^{\prime})_{r\in R})&=&S_{1}(0) = \frac{2^{m}+1}{3},\\ S_{2}((a_{r,2}^{\prime})_{r\in R})&=&S_{2}(0) = \frac{2^{m}+1}{3}. \end{array} $$

Note that $T{r^{2}_{1}}(w) = T{r^{2}_{1}}(w^{2}) = 1,T{r^{2}_{1}}(1) = 0$. The result follows from Theorem 1. □

4.2 Explicit characterization of hyper-bent functions with coefficients in $\mathbb {F}_{2^{n}}$

From the previous characterization of hyper-bent functions, we recover some well-known results and generalize those results in the case where the coefficients $a_{r,i}$ belong to the whole field $\mathbb {F}_{2^{n}}$.

Some results on binomial, trinomial and quadrinomial hyper-bent functions for ($p=2,e=3$) are given by Li et al. [19]. Let $R=\{r\}$, $a_{r,0}= a+c+d$, $a_{r,1}=a+cw^{2}+dw$, and $a_{r,2}=a+cw+dw^{2}$. From Lemma 1 and Theorem 1, we obtain the following results similar to Theorem 2 in [19].

Theorem 7

Let $a,c,d\in \mathbb {F}_{2^{m}}$, $b\in \mathbb {F}_{4}$ , and $gcd(r,\frac {2^{m}+1}{3}) = 1$ . Let $f(x)$ be defined by

$$\begin{array}{@{}rcl@{}} f(x)&=& T{r^{n}_{1}}((a+c+d)x^{r(2^{m}-1)})+ T{r^{n}_{1}}((a+cw^{2}+dw)x^{r(2^{m}-1)+\frac{2^{n}-1}{3}}) \\ &+&T{r^{n}_{1}}((a+cw+dw^{2})x^{r(2^{m}-1)+2\frac{2^{n}-1}{3}})+T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), \end{array} $$

(6)

Then

• (1) If $b=0$, $f(x)$ is hyper-bent if and only if $S_{0}(a)+S_{1}((c)_{\{r\}})+S_{1}((d)_{\{r\}}) = 1$;

• (2) If $b=1$, $f(x)$ is hyper-bent if and only if $S_{0}(a)-S_{1}((c)_{\{r\}})-S_{1}((d)_{\{r\}}) = 1$;

• (3) If $b=w$, $f(x)$ is hyper-bent if and only if $-S_{0}(a)-S_{1}((c)_{\{r\}})+S_{1}((d)_{\{r\}}) = 1$;

• (4) If $b=w^{2}$, $f(x)$ is hyper-bent if and only if $-S_{0}(a)+S_{1}((c)_{\{r\}})-S_{1}((d)_{\{r\}}) = 1$.

From Corollary 1, Lemma 2, Lemma 1, and Theorem 7, this corollary can be obtained immediately.

Corollary 5

Let $a,c,d\in \mathbb {F}_{2^{m}}$, $T{r^{m}_{1}}(a^{\frac {1}{3}}) = T{r^{m}_{1}}(c^{\frac {1}{3}}) = T{r^{m}_{1}}(d^{\frac {1}{3}}) = 0$, $b\in \mathbb {F}_{4}$, $gcd(r,\frac {2^{m}+1}{3}) = 1$, and $f(x)$ be defined by (6). Then

• (1) If $b=0$, $f(x)$ is hyper-bent if and only if $K_{m}(a)+K_{m}(c)+K_{m}(d) = 0$;

• (2) If $b=1$, $f(x)$ is hyper-bent if and only if $-K_{m}(a)+K_{m}(c)+K_{m}(d) = 4$;

• (3) If $b=w$, $f(x)$ is hyper-bent if and only if $K_{m}(a)+K_{m}(c)-K_{m}(d) = 4$;

• (4) If $b=w^{2}$, $f(x)$ is hyper-bent if and only if $K_{m}(a)-K_{m}(c)+K_{m}(d) = 4$.

Parameters $a, c, d$ considered in the above results are restricted in the subfield $\mathbb {F}_{2^{m}}$. Actually, this restriction is not necessary. For convenience, we shall provide some explanations for this fact by considering Theorem 7 as an example.

Some notations are given first. Let $A\in \mathbb {F}_{2^{n}}^{*}$ with unique polar decomposition $A=\widetilde {A}\xi ^{I(A)}$, where $\widetilde {A}\in \mathbb {F}_{2^{m}}^{*}$, $0\leq I(A)\leq 2^{m}$, and ξ is a primitive $2^{m}+1$-th root of unity. Define I(0) = 0 and $\widetilde {0}=0$. If $a\in \mathbb {F}_{2^{m}}$, I(a) = 0 and $\widetilde {a}=a$. Then we have a general result of Lemma 1.

Lemma 5

Let $A\in \mathbb {F}_{2^{n}}$ and $gcd(r,\frac {2^{m}+1}{3}) = 1$ . Then $S_{i}((A)_{\{r\}}) = S_{ri+I(A)} (\widetilde {A}) = S_{(ri+I(A))\mod 3} (\widetilde {A}).$

Proof

We have

$$S_{i}((A)_{\{r\}}) = \sum\limits_{v\in V}\chi(T{r^{n}_{1}}(A(\xi^{i}v)^{r(2^{m}-1)})) = \sum\limits_{v\in V}\chi(T{r^{n}_{1}}(A(\xi^{ri}v^{r})^{2^{m}-1})). $$

Since $gcd(r,\frac {2^{m}+1}{3}) = 1$, $v\mapsto v^{r}$ is a transform for V. Then

$$S_{i}((A)_{\{r\}}) = \sum\limits_{v\in V}\chi(T{r^{n}_{1}}(A(\xi^{ri}v)^{2^{m}-1})) = \sum\limits_{v\in V}\chi(T{r^{n}_{1}}(\widetilde{A}\xi^{i(A)} (\xi^{ri}v)^{2^{m}-1})). $$

Let l be an integer satisfying $(2^{m}-1)l\equiv 1\mod 2^{m}+1$. Since $2^{m}-1\equiv 1\mod 3$, $l\equiv 1\mod 3$. Then

$$S_{i}((A)_{\{r\}}) = \sum\limits_{v\in V}\chi(T{r^{n}_{1}}(\widetilde{A} (\xi^{ri+lI(A)}v)^{2^{m}-1})) = \sum\limits_{v\in V}\chi(T{r^{n}_{1}}(\widetilde{A} (\xi^{ri+I(A)}v)^{2^{m}-1})) = S_{ri+I(A)\mod3}(\widetilde{A}). $$

Hence, this lemma follows. □

From Lemma 5 and Theorem 1, this theorem can be obtained immediately.

Theorem 8

[Theorem 2 in [19]] Let $a,c,d\in \mathbb {F}_{2^{n}}$, $b\in \mathbb {F}_{4}$, $gcd(r,\frac {2^{m}+1}{3}) = 1$ , and $f(x)$ be defined in (6). Then

• (1) If $b=0$, $f(x)$ is hyper-bent if and only if $S_{I(a)}(\widetilde {a})+ S_{r+I(c)\mod 3}(\widetilde {c})+ S_{2r+I(d)\mod 3}(\widetilde {d}) = 1$;

• (2) If $b=1$, $f(x)$ is hyper-bent if and only if $S_{I(a)}(\widetilde {a})- S_{r+I(c)\mod 3}(\widetilde {c})- S_{2r+I(d)\mod 3}(\widetilde {d}) = 1$;

• (3) If $b=w$, $f(x)$ is hyper-bent if and only if $-S_{I(a)}(\widetilde {a})- S_{r+I(c)\mod 3}(\widetilde {c})+ S_{2r+I(d)\mod 3}(\widetilde {d}) = 1$;

• (4) If $b=w^{2}$, $f(x)$ is hyper-bent if and only if $-S_{I(a)}(\widetilde {a})+ S_{r+I(c)\mod 3}(\widetilde {c})- S_{2r+I(d)\mod 3}(\widetilde {d}) = 1$.

The above theorem is equivalent to Theorem 2 in [19] for the case $p=2, e=3$.

Corollary 6

Let $a,c\in \mathbb {F}_{2^{m}}$, $b\in \mathbb {F}_{4}$, $gcd(r,\frac {2^{m}+1}{3}) = 1$, $T{r^{m}_{1}}((A+C)^{1/3}) = 0$ , and $T{r^{m}_{1}}((A^{2}+AC+C^{2})^{1/3}) = 0$ . Let $f(x)$ be defined by

$$ f(x) = T{r^{n}_{1}}(Ax^{r(2^{m}-1)})+T{r^{n}_{1}}(Cx^{r(2^{m}-1)+\frac{2^{n}-1}{3}})+ T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}). $$

(7)

Then

• (1) If $b=0$, $f(x)$ is hyper-bent if and only if $K_{m}(A+C)+2K_{m}(A^{2}+AC+C^{2}) = 0$;

• (2) If $b=1$, $f(x)$ is hyper-bent if and only if $-K_{m}(A+C)+2K_{m}(A^{2}+AC+C^{2}) = 4$;

• (3) If b is a primitive 3rd root of unity, $f(x)$ is hyper-bent if and only if $K_{m}(A+C) = 4$.

Proof

Take $a, c, d$ as $A+C$, $A+Cw$, $A+Cw^{2}$ respectively. Note that

$$\widetilde{c}^{2}=c^{2^{m}+1}=A^{2}+AC+C^{2}, \quad \widetilde{d}^{2} =d^{2^{m}+1}=A^{2}+AC+C^{2}. $$

Then $S_{I(a)}(\widetilde {a}) =S_{I(a)}(A+C)$, $S_{r+I(c)\mod 3}(\widetilde {c}) =S_{r+I(c) \mod 3}(\widetilde {c}^{2}) = S_{r+i(c)}(A^{2}+AC+C^{2})$ , and $S_{2r+I(d)\mod 3}(\widetilde {d}) = S_{2r+I(d)\mod 3}(\widetilde {d}^{2}) = S_{2r+I(d)\mod 3}(A^{2}+AC+C^{2}).$ Note that $T{r^{m}_{1}}((A+C)^{1/3}) = 0,T{r^{m}_{1}}((A^{2}+AC+C^{2})^{1/3}) = 0$. From Corollary 1 and Lemma 2, we have

$$\begin{array}{@{}rcl@{}} S_{I(a)}(A+C)&=\frac{-K_{m}(A+C)+1}{3},\\ S_{r+I(c)\mod3}(A^{2}+AC+C^{2})&= \frac{-K_{m}(A^{2}+AC+C^{2})+1}{3},\\ S_{2r+I(d)\mod3}(A^{2}+AC+C^{2})&=\frac{-K_{m}(A^{2}+AC+C^{2})+1}{3}. \end{array} $$

Then

$$\begin{array}{@{}rcl@{}} S_{I(a)}(\widetilde{a})&=&\frac{-K_{m}(A+C)+1}{3},\\ S_{r+I(c)\mod3}(\widetilde{c})&=&\frac{-K_{m}(A^{2}+AC+C^{2})+1}{3},\\ S_{2r+I(d)\mod3}(\widetilde{d})&=&\frac{-K_{m}(A^{2}+AC+C^{2})+1}{3}, \end{array} $$

The result follows from Theorem 8. □

Example 3

Let m = 23 and $\mathbb {F}_{2^{m}}\cong \mathbb {F}_{2}[x]/(x^{23} + x^{5} + 1)$. Take a positive integer r such that $gcd(r,\frac {2^{m}+1}{3}) = 1$. Take

$$\begin{array}{@{}rcl@{}} A&=&x^{23} + x^{20} + x^{17} + x^{16} + x^{15} + x^{14} + x^{12} + x^{11} + x^{10} + x^{9} + x^{6} + x^{4} + x^{2} + x + 1, \\ C&=&x^{23} + x^{14} + x^{13} + x^{7} + x^{6} + x^{5} + x^{3} + x^{2} + 1, \; b =0; \end{array} $$

Then $A+C=x^{23} + x^{21} + x^{20} + x^{14} + x^{12} + x^{11} + x^{10} + x^{8} + x^{7} + x^{5} + x^{4} + x + 1$, $A^{2}+AC+C^{2}=x^{23} + x^{22} + x^{20} + x^{19} + x^{18} + x^{16} + x^{15} + x^{8} + x^{6} + x^{3} + x^{2} + x + 1$and $T{r^{m}_{1}}((A+C)^{1/3}) = 0$, $T{r^{m}_{1}}((A^{2}+AC+C^{2})^{1/3}) = 0$, $K_{m}(A+C) = -4280$, $K_{m}(A^{2}+AC+C^{2}) = 2140$, $K_{m}(A+C)+2K_{m}(A^{2}+AC+C^{2}) = 0$. From Result (1) in Corollary 6, $f(x)$ in (7) is a binomial hyper-bent function.

From an exhaustive search, when $m=5, 7, 9$, the number of hyper-bent functions in Result (1) in Corollary 6 are 40, 728 and 5346 respectively. And the number of hyper-bent functions in Result (2) in Corollary 6 are 100, 546 and 5112 respectively.

From Theorem 8 and Lemma 2, we can completely characterize all the quadrinomial hyper-bent functions in Theorem 8 by Kloosterman sums and cubic sums. The following lemma explains that exponents of x in the trace functions would be very big.

Lemma 6

Let $r_{m,1}$ be the smallest integer in the cyclotomic coset of 2 modulo $2^{m}+1$ containing $1+\frac {2^{m}+1}{3}$ , and let $r_{m,2}$ be the smallest integer in the cyclotomic coset of 2 modulo $2^{m}+1$ containing $1+2\frac {2^{m}+1}{3}$ , i.e., $r_{m,1}=min_{0\leq i\leq 2m-1} (1+\frac {2^{m}+1}{3})\cdot 2^{i}\mod 2^{m}+1$, $r_{m,2}=min_{0\leq i\leq \ 2m-1} (1+2\frac {2^{m}+1}{3})\cdot 2^{i}\mod 2^{m}+1$ . Then $r_{m,1}=\frac {2^{m-2}+1}{3}$ and $r_{m,2}=\frac {2^{m-1}-1}{3}$.

Proof

We first prove $r_{m,1}=\frac {2^{m-2}+1}{3}$.

• (i) If $j=0,2,\cdots ,m-1$, $3|(2^{j}-1)$. We have

  $$(1+\frac{2^{m}+1}{3})\cdot2^{j}=\frac{2^{j}-1}{3}(2^{m}+1)+\frac{2^{m}+1}{3}+2^{j}, $$

  where $0<\frac {2^{m}+1}{3}+2^{j}<2^{m}+1$. Then

  $$(1+\frac{2^{m}+1}{3})\cdot2^{j}\mod (2^{m}+1) = \frac{2^{m}+1}{3}+2^{j} \geq \frac{2^{m}+1}{3}+1. $$

  When $j=0$, the equality on $r_{m,1}$ holds.

• (ii) If $j=1,3,\cdots ,m-2$, $3|(2^{j}-2)$. We have

  $$(1+\frac{2^{m}+1}{3})\cdot2^{j}=\frac{2^{j}-2}{3}(2^{m}+1)+\frac{2(2^{m}+1)}{3}+2^{j}, $$

  where $0<\frac {2(2^{m}+1)}{3}+2^{j}<2^{m}+1$. Then

  $$(1+\frac{2^{m}+1}{3})\cdot2^{j}\mod (2^{m}+1) = \frac{2(2^{m}+1)}{3}+2^{j} \geq \frac{2(2^{m}+1)}{3}+2 $$

  When $j=1$, the equality on $r_{m,1}$ holds.

• (iii) If $j=m+1,m+3,\cdots ,2m-2$, $3|(2^{j}-1)$. We have

  $$(1+\frac{2^{m}+1}{3})\cdot2^{j}=(\frac{2^{j}-1}{3}+2^{j-m})(2^{m}+1)+\frac{2^{m}+1}{3}-2^{j-m}, $$

  where $0<\frac {2^{m}+1}{3}-2^{j-m}<2^{m}+1$. Then

  $$(1+\frac{2^{m}+1}{3})\cdot2^{j}\mod (2^{m}+1) = \frac{2^{m}+1}{3}-2^{j-m} \geq \frac{2^{m}+1}{3}-2^{m-2}=\frac{2^{m-2}+1}{3}. $$

  When $j=2m-2$, the equality on $r_{m,1}$ holds.

• (iv) If $j=m,m+2,\cdots ,2m-1$, $3|(2^{j}-2)$. We have

  $$(1+\frac{2^{m}+1}{3})\cdot2^{j}=(\frac{2^{j}-2}{3}+2^{j-m})(2^{m}+1)+\frac{2(2^{m}+1)}{3}-2^{j-m}, $$

  where $0<\frac {2(2^{m}+1)}{3}-2^{j-m}<2^{m}+1$. Then

  $$(1+\frac{2^{m}+1}{3})\cdot2^{j}\mod (2^{m}+1) = \frac{2(2^{m}+1)}{3}-2^{j-m} \geq \frac{2(2^{m}+1)}{3}-2^{m-1}=\frac{2^{m-1}+2}{3}. $$

  When $j=2m-1$, the equality on $r_{m,1}$ holds.

Hence, $r_{m,1}=\frac {2^{m-2}+2}{3}$. From the similar discussion, $r_{m,2}=\frac {2^{m-1}-1}{3}$. □

Further, we analyze a subclass of functions in Theorem 8 of the form

$$f(x) = T{r^{n}_{1}}(a_{0}x^{2^{m}-1})+T{r^{n}_{1}}(a_{1}x^{(2^{m}-1)+\frac{2^{n}-1}{3}}) +T{r^{n}_{1}}(a_{2}x^{(2^{m}-1)+2\frac{2^{n}-1}{3}})+T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), $$

where $a_{i}\in \mathbb {F}_{2^{m}} (i=0,1,2)$ and $b\in \mathbb {F}_{4}$. From Lemma 6, $f(x)$ can be transformed into another function

$$ f^{\prime}(x) = T{r^{n}_{1}}(a_{0}x^{2^{m}-1})+T{r^{n}_{1}}(a_{1}^{\prime}x^{r_{m,1}(2^{m}-1)})+T{r^{n}_{1}}(a_{2}^{\prime}x^{r_{m,2}(2^{m}-1)})+T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), $$

(8)

where $r_{m,1}=\frac {2^{m-2}+1}{3}$ and $r_{m,2}=\frac {2^{m-1}-1}{3}$. If $b=0$, $f(x)$ in (8) belongs to cases studied by Charpin and Gong [5]. If $b\neq 0$, $f(x)$ is studied by Mesnager [25]. From results in [5] and [25], some exponential sums on $\mathbb {F}_{2^{m}}$ should be computed to determine the hyper-bent functions in (8). From Lisoněk [20] and Flori, Mesnager [11,12], to determine the hyper-bentness of $f(x)$ in (8) is equivalent to counting the number of rational points on hyper-elliptic curves of genus $g\in \{\frac {3r_{m,2}+1}{2}$, $\frac {3r_{m,2}-1}{2}$, $\frac {r_{m,2}+1}{2},\frac {r_{m,2}-1}{2}\}$ over $\mathbb {F}_{2^{m}}$, where $r_{m,2}=\frac {2^{m-1}-1}{3}$. The genus grows exponentially with m, hence algorithms for counting rational points cannot be applied to determine hyper-bentness of $f(x)$ for big m. From Theorem 8, to determine hyper-bentness of $f(x)$, we just need to compute $S_{I(a)}(a)$, $S_{1+I(c)\mod 3}(\widetilde {c})$ and $S_{1+I(d)\mod 3}(\widetilde {d})$, where $a=a_{0}+a_{1}+a_{2}$, $c=a_{0}+a_{1}w+a_{2}w^{2}$, $d=a_{0}+a_{1}w^{2}+a_{2}w$. Values of $C_{m}(a,a)$, $C_{m}(\widetilde {c},\widetilde {c})$ and $C_{m}(\widetilde {d},\widetilde {d})$ can be computed by Proposition 6. From Lemma 2, we have just to compute Kloosterman sums $K_{m}(a)$, $K_{m}(\widetilde {c})$ and $K_{m}(\widetilde {d})$ on $\mathbb {F}_{2^{m}}$. They can be computed by counting algorithms on elliptic curves or hyper-elliptic curves [21], such as Schoof algorithm [30]. Hence, it explains that our techniques can efficiently characterize the hyper-bentness property of some special functions studied by Charpin, Gong [5] and Mesnager [25].

5 Conclusion

This paper generalizes classes of hyper-bent functions proposed by Charpin, Gong [5] and Mesnager [25], and presents a characterization of a more general class of hyper-bent functions. These hyper-bent functions with coefficients in $\mathbb {F}_{2^{m}}$ are characterized by exponential sums. For many special cases, we give an explicit characterization of the obtained hyper-bent functions by means of Kloosterman sums and cubic sums. For some special cases, we present some attempts to generalize our characterization in the case where the coefficients belong to the whole ambient space $\mathbb {F}_{2^{n}}$.