A Generic Construction of Plateaued Functions via Support Modification of Quadratic Functions

Abstract

Plateaued Boolean functions play an important role in sequence design and cryptography due to their favorable properties such as high nonlinearity, resilience and low additive autocorrelation. In this paper, by modifying the support of certain quadratic functions, we propose a generic construction of plateaued Boolean functions. Additionally, we analyze the algebraic normal form of the newly constructed functions. As applications of the proposed construction, we obtain infinite families of near-bent and semi-bent functions.

1. Introduction

Plateaued Boolean functions are Boolean functions whose squared Walsh transform takes either zero or a fixed nonzero value. Proposed by Zheng and Zhang in 1999, such functions are are regarded as good candidates for designing cryptographic functions [1]. Due to their desirable characteristics, such as high nonlinearity, resilience, and low additive autocorrelation [2,3], these functions are of great significance in sequence design and cryptography [1]. Plateaued Boolean functions bring together multiple nonlinear properties and include three significant classes of Boolean functions, bent functions, near-bent functions and semi-bent functions, all of which have been actively studied owing to their interesting cryptographic, algebraic, and combinatorial properties (see surveys [2,3,4,5]). Furthermore, the concept of plateaued Boolean functions have been extended to p-ary plateaued functions (see e.g., [6]) as well as vectorial p-ary plateaued functions (see, e.g., [7]).

The construction of plateaued functions remains a challenging problem. There are two categories of constructions of plateaued functions. Primary constructions are also referred to as direct constructions. Constructions that build functions from scratch are called primary, whereas those that employ some initial functions to produce new functions are called secondary [4,5]. Typical examples of secondary constructions include modifying the supports or adding indicator functions to existing Boolean functions.

It is widely acknowledged in the literature that general design methods for the explicit and direct constructions of plateaued functions are highly desirable [8]. However, apart from methods inspired by the well-known Maiorana-McFarland constructions or those derived from adaptations of the secondary constructions of bent functions, progress in direct constructions of plateaued functions has been relatively limited [8]. In [8], Li et al. proposed a framework of new generic constructions of bent and plateaued functions by studying Boolean functions of the form $h\left(x\right)=f\left(x\right)+F(f_1\left(x\right),\dots ,f_r\left(x\right))$, where $f_i\left(x\right)=f\left(x\right)+f(x+{\mu }_i)$ for each $1\le i\le r$, obtaining several plateaued functions of high algebraic degrees. In [9], Carlet gave a survey of construction techniques for Boolean plateaued functions and provided characterizations for vectorial plateaued functions, extending the early characterization efforts reported in [10]. In [1,9,11], plateaued functions were derived through the generalized Maiorana-McFarland constructions by concatenation of quadratic functions with special forms. In [12], efficient algorithms were developed to specify optimal plateaued functions.

Regarding secondary constructions, several explicit design methods of bent and plateaued functions were proposed by the tool of the compositional representation of Boolean functions [13]. In [14], the authors proposed a secondary construction of plateaued functions using three suitable bent functions. Recently, some secondary constructions of plateaued Boolean functions were proposed via the addition of indicator functions [15]. Surveys on constructions of plateaued Boolean functions can be found in [2,3,4,5,9].

The purpose of this paper is to present a secondary construction of plateaued Boolean functions. For any subset $S\subset {\mathbb{F}}_2^m$ with $\left|S\right|\ge 2$, we propose a generic construction of r-plateaued Boolean functions on ${\mathbb{F}}_2^n$ related to S, by modifying the support of a family of quadratic functions, where $n=2m+r$ and $m,r$ are arbitrary positive integers. Notably, the construction in [16] is a special case of our framework when $r=0$, and is generalized here to r-plateaued functions for $r>0$ by introducing the additional vector $x^{‴}\in {\mathbb{F}}_2^r$. This structural extension allows us to generate r-plateaued functions, yielding near-bent functions when $r=1$ and semi-bent functions when $r=2$. Moreover, we derive the algebraic normal form of the constructed functions, and provide concrete examples to demonstrate that our construction readily yields numerous infinite families of near-bent and semi-bent functions.

The rest of this paper is organized as follows. Section 2 introduces the required notations and necessary background. Section 3 presents our construction of plateaued Boolean functions, along with concrete examples of near-bent and semi-bent functions. Section 4 analyzes the algebraic normal form of the functions obtained from our construction. Finally, Section 5 summarizes the work.

2. Preliminaries

Let ${\mathbb{F}}_2=\{0,1\}$ be the binary field, n a positive integer, and ${\mathbb{F}}_2^n$ the n-dimensional vector space over ${\mathbb{F}}_2$. Given a subset $\Omega \subseteq {\mathbb{F}}_2^n$, its cardinality $|\Omega |$ denotes the number of elements in $\Omega$. The characteristic function ${\chi }_{\Omega }$ associated with $\Omega$ satisfies ${\chi }_{\Omega }\left(x\right)=1$ for $x\in \Omega$ and ${\chi }_{\Omega }\left(x\right)=0$ otherwise. For any vector $x=(x_1,\dots ,x_n)\in {\mathbb{F}}_2^n$, its support is given by $\mathrm{supp}\left(x\right)=\{1\le i\le n\mid x_i=1\}$, and the set of zero coordinates of x is defined as $\mathrm{Z}\left(x\right)=\{1\le i\le n\mid x_i=0\}$, which will be used as a shorthand in subsequent formulas. The symbols ${\mathbf{1}}_n$ and ${\mathbf{0}}_n$ refer to the all-one and all-zero vectors of length n, respectively.

For clarity, we distinguish two addition operations: + denotes addition over $\mathbb{Z}$, while ⊕ stands for addition over ${\mathbb{F}}_2$. For two vectors $a=(a_1,\dots ,a_n)$ and $b=(b_1,\dots ,b_n)\in {\mathbb{F}}_2^n$, their sum over ${\mathbb{F}}_2$ is defined by $a\oplus b=(a_1\oplus b_1,\dots ,a_n\oplus b_n)$. Their inner product over ${\mathbb{F}}_2$ is defined as $a·b=a_1{b}_1\oplus \cdots \oplus a_n{b}_n$. Note that this is different from the ordinary Euclidean inner product, as it uses the modulo-2 addition ⊕. Furthermore, we say a covers b if $a_i\ge b_i$ for all $1\le i\le n$, and write $a⪰b$ for this relation.

An n-variable Boolean function f is a mapping from ${\mathbb{F}}_2^n$ to ${\mathbb{F}}_2$. Its support is defined as $\mathrm{supp}\left(f\right)=\{x\in {\mathbb{F}}_2^n\mid f\left(x\right)=1\}$. Boolean functions can be represented in multiple ways. Two well-known representations are their truth table and the algebraic normal form. The truth table representation of a Boolean function f on ${\mathbb{F}}_2^n$ is a binary string of length $2^n$, given by

$$\left(f(0,\cdots ,0,0),f(0,\cdots ,0,1),\cdots ,f(1,\cdots ,1,1)\right).$$

The algebraic normal form (ANF) provides a unique polynomial representation of f on ${\mathbb{F}}_2^n$:

$$f\left(x\right)={\displaystyle \underset{\beta \in {\mathbb{F}}_2^n}{⨁}}{c}_{\beta }{x}^{\beta },c_{\beta }\in {\mathbb{F}}_2,$$

where $c_{\beta }$ is the coefficient of the term $x^{\beta }=x_1^{{\beta }_1}\cdots x_n^{{\beta }_n}$ for $x=(x_1,\dots ,x_n)$, $\beta =({\beta }_1,\dots ,{\beta }_n)\in {\mathbb{F}}_2^n$, and

$$c_{\beta }={\displaystyle \underset{x\in {\mathbb{F}}_2^n,\beta ⪰x}{⨁}}f\left(x\right).$$

The Walsh transform of f at $\alpha \in {\mathbb{F}}_2^n$ is defined as

$$W_f\left(\alpha \right)=\sum _{x\in {\mathbb{F}}_2^n}{(-1)}^{f\left(x\right)\oplus \alpha ·x},$$

where $\alpha ·x={\displaystyle \underset{i=1}{\overset{n}{⨁}}}{\alpha }_i{x}_i$ denotes the inner product of $\alpha =({\alpha }_1,\dots ,{\alpha }_n)$ and $x=(x_1,\dots ,x_n)$.

Definition 1

([5]). A Boolean function f: ${\mathbb{F}}_2^n\to {\mathbb{F}}_2$ is said to be bent if $W_f\left(\alpha \right)=\pm 2^{{\displaystyle \frac{n}{2}}}$ for all $\alpha \in {\mathbb{F}}_2^n$.

Definition 2

([5]). Let r be an integer with $0\le r\le n$. A Boolean function f on ${\mathbb{F}}_2^n$ is called r-plateaued if $W_f\left(\alpha \right)\in \left\{0,\pm 2^{{\displaystyle \frac{n+r}{2}}}\right\}$ for all $\alpha \in {\mathbb{F}}_2^n$. The value $2^{{\displaystyle \frac{n+r}{2}}}$ is called the amplitude of f.

In fact, for an r-plateaued Boolean function f on ${\mathbb{F}}_2^n$, the integers n and r must share the same parity [5].

Plateaued Boolean functions include three important subclasses: bent functions, near-bent functions and semi-bent functions. Specifically, bent functions correspond to 0-plateaued functions, while near-bent functions (referred to as semi-bent functions over odd-dimensional spaces) are exactly 1-plateaued functions, and semi-bent functions coincide with 2-plateaued functions. 0-plateaued functions and 2-plateaued functions on ${\mathbb{F}}_2^n$ are only available for even n, while 1-plateaued functions on ${\mathbb{F}}_2^n$ exist when n is odd [5]. For $r\in \{0,1,2\}$, r-plateaued functions have been widely investigated and have attracted much attention because of their cryptographic, algebraic, and combinatorial properties [5].

We next present two exponential sum formulas that will be employed to prove our main theorem.

Lemma 1.

Let a be a vector in ${\mathbb{F}}_2^n$. Then, we have

$$\sum _{x\in {\mathbb{F}}_2^n}{(-1)}^{a·x}=\left\{\begin{array}{cc}{2}^n,\hfill & \mathrm{if}a={\mathbf{0}}_n;\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Lemma 2.

Let $n>0$, $a\in {\mathbb{F}}_2^n\setminus \left\{{\mathbf{0}}_n\right\}$ and $H=\{x\in {\mathbb{F}}_2^n\mid a·x=\epsilon \}$, where $\epsilon =0$ or 1. Then

$$\sum _{x\in H}{(-1)}^{b·x}=\left\{\begin{array}{cc}{2}^{n-1},\hfill & \mathit{if}b={\mathbf{0}}_n;\hfill \\ {(-1)}^{\epsilon }·2^{n-1},\hfill & \mathit{if}b=a;\hfill \\ 0,\hfill & \mathit{if}b\in {\mathbb{F}}_2^n\setminus \{{\mathbf{0}}_n,a\}.\hfill \end{array}\right.$$

Proof. 

Note that $\left|H\right|=2^{n-1}$. If $b={\mathbf{0}}_n$, then ${\displaystyle \sum _{x\in H}}{(-1)}^{x·b}=\left|H\right|=2^{n-1}$. If $b=a$, then

$$\sum _{x\in H}{(-1)}^{x·b}=\left\{\begin{array}{cc}{2}^{n-1},\hfill & \mathrm{if}\epsilon =0;\hfill \\ -2^{n-1},\hfill & \mathrm{if}\epsilon =1.\hfill \end{array}\right.$$

If $b\in {\mathbb{F}}_2^n\setminus \{{\mathbf{0}}_n,a\}$, then

$$\left|\{x\in H\mid b·x=0\}\right|=|\{x\in {\mathbb{F}}_2^n\mid a·x=0\mathrm{and}b·x=0\}|=2^{n-2},$$

and then $\left|\{x\in H\mid b·x=1\}\right|=2^{n-2}$. So $\sum _{x\in H}{(-1)}^{x·b}=0$. This finishes the proof. □

3. A Construction of Plateaued Boolean Functions

This section presents a construction of plateaued Boolean functions. In our previous work [16], we proposed a construction of bent functions by modifying a class of quadratic functions associated with a suitably chosen set $S\subset {\mathbb{F}}_2^m$. In what follows, we extend this construction to plateaued functions.

Throughout this section, let $m\ge 1$ and $r\ge 0$ be integers, and $n=2m+r$. Given a vector $x=(x_1,x_2,\dots ,x_n)\in {\mathbb{F}}_2^n$, we always denote $x^{\prime }=(x_1,x_2,\dots ,x_m)$, $x^{″}=(x_{m+1},x_{m+2},\dots ,x_{2m})$, and $x^{‴}=(x_{2m+1},\dots ,x_{2m+r})$. Thus, any vector $x\in {\mathbb{F}}_2^n$ can be uniquely written as $x=(x^{\prime },x^{″},x^{‴})$, where $x^{\prime }\in {\mathbb{F}}_2^m$, $x^{″}\in {\mathbb{F}}_2^m$, and $x^{‴}\in {\mathbb{F}}_2^r$.

3.1. The Method of Constructing Plateaued Boolean Functions

We introduce the n-variable Boolean function

$$f_0\left(x\right)=x^{\prime }·x^{″}={\displaystyle \underset{i=1}{\overset{m}{⨁}}}{x}_i{x}_{m+i},$$

(1)

where $x^{\prime }$ and $x^{″}$ are defined as above. Using $f_0$, we construct a new class of n-variable r-plateaued functions by modifying the support of $f_0$.

Let S be a nonempty subset of ${\mathbb{F}}_2^m$ with $\left|S\right|\ge 2$. Note that there exists a family of nonzero vectors ${\left\{a_s\right\}}_{s\in S}$ such that

$$S=\{s\oplus a_s\mid s\in S\}.$$

(2)

Such a family of nonzero vectors ${\left\{a_s\right\}}_{s\in S}$ always exists whenever $\left|S\right|\ge 2$. To see this, take any fixed-point-free permutation $\pi$ on S and set $a_s=s\oplus \pi \left(s\right)$. Then $s\oplus a_s=\pi \left(s\right)\in S$, which gives (2). For each $s\in S$, the vector $a_s$ is referred to as the associated vector of s. For every $s\in S$, we define

• $H_s=\{x\in {\mathbb{F}}_2^m\mid a_s·x={\epsilon }_s\},\mathrm{where}{\epsilon }_s=0\mathrm{or}1$;
• ${\Omega }_s=\{x=(x^{\prime },x^{″},x^{‴})\in {\mathbb{F}}_2^m\times {\mathbb{F}}_2^m\times {\mathbb{F}}_2^r\mid x^{\prime }\in H_s,x^{″}=x^{\prime }\oplus s\}$;
• $\Omega ={\bigcup }_{s\in S}{\Omega }_s$.

The set $\Omega$ is fully determined by the sets S together with the selection of parameters $\{(a_s,{\epsilon }_s)\mid s\in S\}$. Based on $\Omega$, we introduce the Boolean function $f:{\mathbb{F}}_2^n\to {\mathbb{F}}_2$ as below

$$f\left(x\right)=\left\{\begin{array}{cc}{f}_0\left(x\right)\oplus 1,\hfill & \mathrm{if}x\in \Omega ,\hfill \\ f_0\left(x\right),\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

(3)

where $f_0\left(x\right)$ is given in (1). In the sequel, the plateauedness of $f\left(x\right)$ is analyzed. We first establish a useful lemma.

Lemma 3.

${\Omega }_{s_1}\cap {\Omega }_{s_2}=\varnothing$ for any distinct $s_1,s_2\in S$.

Proof. 

Assume there exists some element $x=(x^{\prime },x^{″},x^{‴})\in {\Omega }_{s_1}\cap {\Omega }_{s_2}$. By the definitions of ${\Omega }_{s_1}$ and ${\Omega }_{s_2}$, we have $x^{\prime }\in H_{s_1}\cap H_{s_2}$ and $x^{″}=x^{\prime }\oplus s_1=x^{\prime }\oplus s_2$. This immediately gives $s_1=s_2$, leading to a contradiction. Thus, ${\Omega }_{s_1}\cap {\Omega }_{s_2}=\varnothing$. □

Theorem 1.

The function $f\left(x\right)$ defined by (3) is r-plateaued.

Proof. 

For any $\alpha \in {\mathbb{F}}_2^n$, the Walsh transform of f at $\alpha$ is

$$\begin{array}{ccc}\hfill W_f\left(\alpha \right)& =& {\displaystyle \sum _{x\in {\mathbb{F}}_2^n}}{(-1)}^{f\left(x\right)\oplus \alpha ·x}\hfill \\ & =& {\displaystyle \sum _{x\in \Omega }}{(-1)}^{f_0\left(x\right)\oplus 1\oplus \alpha ·x}+{\displaystyle \sum _{x\in {\mathbb{F}}_2^n\setminus \Omega }}{(-1)}^{f_0\left(x\right)\oplus \alpha ·x}\hfill \\ & =& {\displaystyle \sum _{x\in {\mathbb{F}}_2^n}}{(-1)}^{f_0\left(x\right)\oplus \alpha ·x}-2{\displaystyle \sum _{x\in \Omega }}{(-1)}^{f_0\left(x\right)\oplus \alpha ·x}.\hfill \end{array}$$

We have

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{x\in {\mathbb{F}}_2^n}}{(-1)}^{f_0\left(x\right)\oplus \alpha ·x}& =& {\displaystyle \sum _{x\in {\mathbb{F}}_2^n}}{(-1)}^{x^{\prime }·x^{″}\oplus {\alpha }^{\prime }·x^{\prime }\oplus {\alpha }^{″}·x^{″}\oplus {\alpha }^{‴}·x^{‴}}\hfill \\ & =& {\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{{\alpha }^{‴}·x^{‴}}{\displaystyle \sum _{x^{″}\in {\mathbb{F}}_2^m}}{(-1)}^{{\alpha }^{″}·x^{″}}{\displaystyle \sum _{x^{\prime }\in {\mathbb{F}}_2^m}}{(-1)}^{x^{\prime }·(x^{″}\oplus {\alpha }^{\prime })}\hfill \\ & =& {(-1)}^{{\alpha }^{″}·{\alpha }^{\prime }}·2^m{\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{{\alpha }^{‴}·x^{‴}},\hfill \end{array}$$

where the last equality holds by Lemma 1. Applying Lemma 1 again, we obtain

$${\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{{\alpha }^{‴}·x^{‴}}=\left\{\begin{array}{cc}{2}^r,\hfill & \mathrm{if}{\alpha }^{‴}=0;\hfill \\ 0,\hfill & \mathrm{if}{\alpha }^{‴}\ne 0.\hfill \end{array}\right.$$

By Lemma 3, we get

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{x\in \Omega }}{(-1)}^{f_0\left(x\right)\oplus \alpha ·x}& =& {\displaystyle \sum _{s\in S}}{\displaystyle \sum _{x\in {\Omega }_s}}{(-1)}^{f_0\left(x\right)\oplus \alpha ·x}\hfill \\ & =& {\displaystyle \sum _{s\in S}}{\displaystyle \sum _{\begin{array}{c}{x}^{\prime }\in H_s,\\ x^{″}=x^{\prime }\oplus s\end{array}}}{\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{x^{\prime }·x^{″}\oplus {\alpha }^{\prime }·x^{\prime }\oplus {\alpha }^{″}·x^{″}\oplus {\alpha }^{‴}·x^{‴}}\hfill \\ & =& {\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{{\alpha }^{‴}·x^{‴}}{\displaystyle \sum _{s\in S}}{\displaystyle \sum _{x^{\prime }\in H_s}}{(-1)}^{x^{\prime }·(x^{\prime }\oplus s)\oplus {\alpha }^{\prime }·x^{\prime }\oplus {\alpha }^{″}·(x^{\prime }\oplus s)}\hfill \\ & =& {\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{{\alpha }^{‴}·x^{‴}}{\displaystyle \sum _{s\in S}}{\displaystyle \sum _{x^{\prime }\in H_s}}{(-1)}^{x^{\prime }·({\mathbf{1}}_m\oplus s\oplus {\alpha }^{\prime }\oplus {\alpha }^{″})\oplus {\alpha }^{″}·s}\hfill \\ & =& {\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{{\alpha }^{‴}·x^{‴}}{\displaystyle \sum _{s\in S}}{(-1)}^{{\alpha }^{″}·s}{\displaystyle \sum _{x^{\prime }\in H_s}}{(-1)}^{x^{\prime }·({\mathbf{1}}_m\oplus s\oplus {\alpha }^{\prime }\oplus {\alpha }^{″})},\hfill \end{array}$$

where the fourth equality holds since $x^{\prime }·x^{\prime }=x^{\prime }·{\mathbf{1}}_m$ over ${\mathbb{F}}_2$.

(1) If ${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus {\mathbf{1}}_m\notin S$, then ${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus {\mathbf{1}}_m\ne s$ for all $s\in S$, i.e.,

$${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s\oplus {\mathbf{1}}_m\ne {\mathbf{0}}_m$$

(4)

for all $s\in S$. It follows that

$${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s\oplus {\mathbf{1}}_m\ne a_s$$

(5)

for all $s\in S$. Indeed, if ${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s\oplus {\mathbf{1}}_m=a_s$ for some $s\in S$, then setting $s^{\prime }=s\oplus a_s$ would give $s^{\prime }\in S$ and ${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s^{\prime }\oplus {\mathbf{1}}_m={\mathbf{0}}_m$, which contradicts (4).

Therefore, by (4), (5) and Lemma 2, we obtain

$$\sum _{x^{\prime }\in H_s}{(-1)}^{x^{\prime }·({\mathbf{1}}_m\oplus s\oplus {\alpha }^{\prime }\oplus {\alpha }^{″})}=0$$

for any $s\in S$, which implies that

$$\sum _{x\in \Omega }{(-1)}^{f_0\left(x\right)\oplus \alpha ·x}=0.$$

Finally, by Lemma 1, we derive

$$\begin{array}{ccc}\hfill W_f\left(\alpha \right)& =& {(-1)}^{{\alpha }^{″}·{\alpha }^{\prime }}·2^m{\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{{\alpha }^{‴}·x^{‴}}\hfill \\ & =& \left\{\begin{array}{cc}{(-1)}^{{\alpha }^{″}·{\alpha }^{\prime }}·2^{m+r},\hfill & \mathrm{if}{\alpha }^{‴}={\mathbf{0}}_r,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

(2) If ${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus {\mathbf{1}}_m\in S$, suppose ${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus {\mathbf{1}}_m=s_1$ for some $s_1\in S$. Recall from Equation (2) that every element of S can be represented as $s\oplus a_s$ for some $s\in S$. Thus there exists $s_2\in S$ together with its associated vector $a_{s_2}$ satisfying $s_1=s_2\oplus a_{s_2}$. This directly gives

$${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s_1\oplus {\mathbf{1}}_m={\mathbf{0}}_m$$

(6)

and

$${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s_2\oplus {\mathbf{1}}_m=a_{s_2}.$$

(7)

Then by Lemma 2, we obtain

$$\sum _{x\in H_{s_1}}{(-1)}^{x·({\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s_1\oplus 1_m)}=\sum _{x\in H_{s_1}}{(-1)}^{x·{\mathbf{0}}_m}=2^{m-1}$$

and

$$\sum _{x\in H_{s_2}}{(-1)}^{x·({\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s_2\oplus {\mathbf{1}}_m)}=\sum _{x\in H_{s_2}}{(-1)}^{x·a_{s_2}}={(-1)}^{{\epsilon }_{s_2}}·2^{m-1}.$$

For each $s\in S\setminus \{s_1,s_2\}$, from (6) and (7) one gets

$${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s\oplus {\mathbf{1}}_m\ne {\mathbf{0}}_m$$

and

$${\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s\oplus {\mathbf{1}}_m\ne a_s$$

respectively. Thus, by Lemma 2, for any $s\in S\setminus \{s_1,s_2\}$, we have

$$\sum _{x\in H_s}{(-1)}^{x·({\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s\oplus 1_m)}=0.$$

Consequently,

$$\begin{array}{ccc}& & \sum _{x\in \Omega }{(-1)}^{f_0\left(x\right)\oplus \alpha ·x}\hfill \\ & =& \sum _{x^{‴}\in {\mathbb{F}}_2^r}{(-1)}^{{\alpha }^{‴}·x^{‴}}({(-1)}^{{\alpha }^{″}·s_1}\sum _{x\in H_{s_1}}{(-1)}^{x·({\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s_1\oplus 1_m)}\hfill \\ & & +{(-1)}^{{\alpha }^{″}·s_2}\sum _{x\in H_{s_2}}{(-1)}^{x·({\alpha }^{\prime }\oplus {\alpha }^{″}\oplus s_2\oplus 1_m)}+\sum _{s\in S\setminus \{s_1,s_2\}}{(-1)}^{{\alpha }^{″}·s}\sum _{x^{\prime }\in H_s}{(-1)}^{x^{\prime }·({\mathbf{1}}_m\oplus s\oplus {\alpha }^{\prime }\oplus {\alpha }^{″})})\hfill \\ & =& \sum _{x^{‴}\in {\mathbb{F}}_2^r}{(-1)}^{{\alpha }^{‴}·x^{‴}}\left({(-1)}^{{\alpha }^{″}·s_1}·2^{m-1}+{(-1)}^{{\alpha }^{″}·s_2}·{(-1)}^{{\epsilon }_{s_2}}·2^{m-1}\right)\hfill \\ & =& \sum _{x^{‴}\in {\mathbb{F}}_2^r}{(-1)}^{{\alpha }^{‴}·x^{‴}}\left({(-1)}^{{\alpha }^{″}·({\alpha }^{\prime }\oplus {\alpha }^{″}\oplus {\mathbf{1}}_m)}·2^{m-1}+{(-1)}^{{\alpha }^{″}·({\alpha }^{\prime }\oplus {\alpha }^{″}\oplus {\mathbf{1}}_m\oplus a_{s_2})}·{(-1)}^{{\epsilon }_{s_2}}·2^{m-1}\right)\hfill \\ & =& \sum _{x^{‴}\in {\mathbb{F}}_2^r}{(-1)}^{{\alpha }^{‴}·x^{‴}}\left({(-1)}^{{\alpha }^{″}·{\alpha }^{\prime }}·2^{m-1}+{(-1)}^{{\alpha }^{″}·({\alpha }^{\prime }\oplus a_{s_2})}·{(-1)}^{{\epsilon }_{s_2}}·2^{m-1}\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill W_f\left(\alpha \right)& =& {(-1)}^{{\alpha }^{″}·{\alpha }^{\prime }}·2^m{\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{{\alpha }^{‴}·x^{‴}}-2{\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{{\alpha }^{‴}·x^{‴}}({(-1)}^{{\alpha }^{″}·{\alpha }^{\prime }}·2^{m-1}\hfill \\ & & +{(-1)}^{{\alpha }^{″}·({\alpha }^{\prime }\oplus a_{s_2})}·{(-1)}^{{\epsilon }_{s_2}}·2^{m-1})\hfill \\ & =& {(-1)}^{{\alpha }^{″}·{\alpha }^{\prime }\oplus {\alpha }^{″}·a_{s_2}\oplus {\epsilon }_{s_2}\oplus 1}·2^m{\displaystyle \sum _{x^{‴}\in {\mathbb{F}}_2^r}}{(-1)}^{{\alpha }^{‴}·x^{‴}}.\hfill \end{array}$$

Therefore, by Lemma 1,

$$W_f\left(\alpha \right)=\left\{\begin{array}{cc}{(-1)}^{{\alpha }^{″}·{\alpha }^{\prime }\oplus {\alpha }^{″}·a_{s_2}\oplus {\epsilon }_{s_2}\oplus 1}·2^{m+r},\hfill & \mathrm{if}{\alpha }^{‴}={\mathbf{0}}_r,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

According to Cases (1) and (2), we obtain $W_f\left(\alpha \right)\in \{0,\pm 2^{m+r}\}$, i.e., $W_f\left(\alpha \right)\in \{0,\pm 2^{{\displaystyle \frac{n+r}{2}}}\}$, which implies that the n-variable function $f\left(x\right)$ defined by (3) is r-plateaued. This completes the proof. □

Remark 1.

In our construction, given a subset $S\subseteq {\mathbb{F}}_2^m$ with $\left|S\right|\ge 2$, we assign a family of nonzero vectors ${\left\{a_s\right\}}_{s\in S}$ to S so that (2) holds. As shown at the beginning of Section 3.1, such a family always exists. However, the associated vectors $a_s$ are not uniquely determined and may admit multiple valid selections (see Example 1 below). Moreover, distinct vectors $s\in S$ may share an identical associated vector $a_s$ (see Example 2 below).

Example 1.

Let $m=2$ and $S=\{s_1,s_2,s_3\}$, where $s_1=(0,0)$, $s_2=(0,1)$, $s_3=(1,0)$. Below we give two different valid selections.

(1) 

Choose $a_{s_1}=(0,1)$, $a_{s_2}=(1,1)$, $a_{s_3}=(1,0)\in {\mathbb{F}}_2^2$. Then we get

$$S=\{s_1\oplus a_{s_1},s_2\oplus a_{s_2},s_3\oplus a_{s_3}\}.$$

(8)

(2) 

Alternatively, take $a_{s_1}=(1,0)$, $a_{s_2}=(0,1)$, $a_{s_3}=(1,1)\in {\mathbb{F}}_2^2$. Then (8) also holds.

Example 2. (1) Let $m=3$, $s_1=(1,1,1)$, $s_2=(0,1,1)$, and $S=\{s_1,s_2\}$. Assign $a=a_{s_1}=a_{s_2}=(1,0,0)\in {\mathbb{F}}_2^3$. Then

$$a\oplus S=\left\{\right(1,0,0)\oplus (1,1,1),(1,0,0)\oplus (0,1,1\left)\right\}=\left\{\right(0,1,1),(1,1,1\left)\right\}=S.$$

(2) Consider a larger set: $S=\{s_1,s_2,s_3,s_4\}$ with $s_1=(1,1,1)$, $s_2=(0,1,1)$, $s_3=(0,0,0)$, $s_4=(1,0,0)$. Assign $a_{s_i}=(1,0,0)$ for all $1\le i\le 4$. Then,

$$\begin{array}{ccc}\hfill a\oplus S& =& \left\{\right(1,0,0)\oplus (1,1,1),(1,0,0)\oplus (0,1,1),(1,0,0)\oplus (0,0,0),(1,0,0)\oplus (1,0,0\left)\right\}\hfill \\ & =& \left\{\right(0,1,1),(1,1,1),(1,0,0),(0,0,0\left)\right\}\hfill \\ & =& S.\hfill \end{array}$$

Both cases show that distinct elements of S may share an identical associated vector.

Remark 2.

It is worth noting the fundamental differences between our approach and the recent results in [8,15].

For the underlying quadratic functions, Li, Y. et al. [8] introduces a generic direct construction framework by studying Boolean functions of the form $h\left(x\right)=f\left(x\right)+F(f_1\left(x\right),\dots ,f_r\left(x\right))$, whereas we start from a specific quadratic function $f_0=x^{\prime }·x^{″}$.

Regarding the role and structure of S and Ω, the secondary construction in [15] modifies plateaued functions by adding indicators on simple linear subspaces (e.g., $1_{E_1}\left(x\right)1_{E_1}\left(y\right)$). In contrast, our set Ω is uniquely and strictly determined by the subset S and its associated vectors $a_s$ via the hyperplanes $H_s=\{x\in {\mathbb{F}}_2^m:a_s·x={\epsilon }_s\}$ and the mapping $x^{″}=x^{\prime }\oplus s$. This intrinsic algebraic structure of Ω fundamentally distinguishes our approach from [15].

Overall, the classes of Boolean functions generated by our method are controlled by arbitrary sets S and parameters $a_s,{\epsilon }_s$, leading to new infinite families not covered by [8,15].

Remark 3.

When $r=0$, we have $n=2m$, and the constructed set Ω degenerates to the set defined in [16]. Thus, the resulting Boolean function f coincides with the bent functions constructed in [16]. However, when $r>0$, the introduction of the variable $x^{‴}$ fundamentally changes the structure of the set Ω, allowing the resulting function to be r-plateaued. Therefore, the function families obtained for $r>0$ extend the results of [16].

Corollary 1.

If $r=1$, then the function $f\left(x\right)$ in (3) is near-bent; if $r=2$, then it is semi-bent.

3.2. Applications

As the applications of the construction of plateaued Boolean functions in the previous section, in this section we give some examples of near-bent and semi-bent functions generated by this construction.

3.2.1. Near-Bent Functions

Taking $r=1$ in our construction, the function $f\left(x\right)$ in (3) becomes near-bent. We now illustrate this with a concrete example. We compute the plateauedness of the Boolean function $f\left(x\right)$ defined by (3) with the set S given in Example 1 (1), and prove it is a near-bent function.

Let $m=2$, $r=1$, and $S=\{s_1,s_2,s_3\}$, where $s_1=(0,0)$, $s_2=(0,1)$, $s_3=(1,0)$. Choose $a_{s_1}=(0,1)$, $a_{s_2}=(1,1)$, $a_{s_3}=(1,0)\in {\mathbb{F}}_2^2$, and set ${\epsilon }_{s_1}=0$ and ${\epsilon }_{s_2}={\epsilon }_{s_3}=1$. The corresponding sets $H_{s_i}$ and ${\Omega }_{s_i}$ for $1\le i\le 3$ in our construction are listed in the following Table 1.

Table 1. The sets of $H_{s_i}$ and ${\Omega }_{s_i}$ for $1\le i\le 3$.

i	${\mathit{H}}_{{\mathit{s}}_{\mathit{i}}}$	${\mathbf{\Omega }}_{{\mathit{s}}_{\mathit{i}}}$
1	$\left\{\right(0,0),(1,0\left)\right\}$	$\left\{\right(0,0,0,0,0),(0,0,0,0,1),(1,0,1,0,0),(1,0,1,0,1\left)\right\}$
2	$\left\{\right(1,0),(0,1\left)\right\}$	$\left\{\right(1,0,1,1,0),(1,0,1,1,1),(0,1,0,0,0),(0,1,0,0,1\left)\right\}$
3	$\left\{\right(1,0),(1,1\left)\right\}$	$\left\{\right(1,0,0,0,0),(1,0,0,0,1),(1,1,0,1,0),(1,1,0,1,1\left)\right\}$

Let $\Omega ={\displaystyle \bigcup _{i=1}^3}{\Omega }_{s_i}$ with ${\Omega }_{s_i}$ given in Table 1. Then the Boolean function $f\left(x\right)$ on ${\mathbb{F}}_2^5$ defined by (3) is

$$f(x_1,x_2,x_3,x_4,x_5)=x_1{x}_3\oplus x_2{x}_4\oplus {\chi }_{\Omega }(x_1,x_2,x_3,x_4,x_5).$$

The following Table 2 lists its truth table.

Table 2. The truth table of the Boolean function $f\left(x\right)$.

x	$\mathit{f}\left(\mathit{x}\right)$	x	$\mathit{f}\left(\mathit{x}\right)$	x	$\mathit{f}\left(\mathit{x}\right)$	x	$\mathit{f}\left(\mathit{x}\right)$
$(0,0,0,0,0)$	1	$(0,1,0,0,0)$	1	$(1,0,0,0,0)$	1	$(1,1,0,0,0)$	0
$(0,0,0,0,1)$	1	$(0,1,0,0,1)$	1	$(1,0,0,0,1)$	1	$(1,1,0,0,1)$	0
$(0,0,0,1,0)$	0	$(0,1,0,1,0)$	1	$(1,0,0,1,0)$	0	$(1,1,0,1,0)$	0
$(0,0,0,1,1)$	0	$(0,1,0,1,1)$	1	$(1,0,0,1,1)$	0	$(1,1,0,1,1)$	0
$(0,0,1,0,0)$	0	$(0,1,1,0,0)$	0	$(1,0,1,0,0)$	0	$(1,1,1,0,0)$	1
$(0,0,1,0,1)$	0	$(0,1,1,0,1)$	0	$(1,0,1,0,1)$	0	$(1,1,1,0,1)$	1
$(0,0,1,1,0)$	0	$(0,1,1,1,0)$	1	$(1,0,1,1,0)$	0	$(1,1,1,1,0)$	0
$(0,0,1,1,1)$	0	$(0,1,1,1,1)$	1	$(1,0,1,1,1)$	0	$(1,1,1,1,1)$	0

From the truth table of $f\left(x\right)$, we now calculate the values of the Walsh transform.

As can be seen from Table 3, $W_f\left(\alpha \right)\in \{0,\pm 8\},$ which means the Boolean function $f\left(x\right)$ is 1-plateaued, i.e., near-bent. That is to say, the result in this example is consistent with Theorem 1.

Table 3. The Values of the Walsh transform of the Boolean function $f\left(x\right)$.

$\mathit{\alpha }$	${\mathit{W}}_{\mathit{f}}\left(\mathit{\alpha }\right)$	$\mathit{\alpha }$	${\mathit{W}}_{\mathit{f}}\left(\mathit{\alpha }\right)$	$\mathit{\alpha }$	${\mathit{W}}_{\mathit{f}}\left(\mathit{\alpha }\right)$	$\mathit{\alpha }$	${\mathit{W}}_{\mathit{f}}\left(\mathit{\alpha }\right)$
$(0,0,0,0,0)$	8	$(0,1,0,0,0)$	8	$(1,0,0,0,0)$	$-8$	$(1,1,0,0,0)$	8
$(0,0,0,0,1)$	0	$(0,1,0,0,1)$	0	$(1,0,0,0,1)$	0	$(1,1,0,0,1)$	0
$(0,0,0,1,0)$	$-8$	$(0,1,0,1,0)$	$-8$	$(1,0,0,1,0)$	8	$(1,1,0,1,0)$	$-8$
$(0,0,0,1,1)$	0	$(0,1,0,1,1)$	0	$(1,0,0,1,1)$	0	$(1,1,0,1,1)$	0
$(0,0,1,0,0)$	−8	$(0,1,1,0,0)$	−8	$(1,0,1,0,0)$	−8	$(1,1,1,0,0)$	8
$(0,0,1,0,1)$	0	$(0,1,1,0,1)$	0	$(1,0,1,0,1)$	0	$(1,1,1,0,1)$	0
$(0,0,1,1,0)$	−8	$(0,1,1,1,0)$	−8	$(1,0,1,1,0)$	−8	$(1,1,1,1,0)$	8
$(0,0,1,1,1)$	0	$(0,1,1,1,1)$	0	$(1,0,1,1,1)$	0	$(1,1,1,1,1)$	0

3.2.2. Semi-Bent Functions

When $r=2$ is adopted in our proposed construction, the derived function $f\left(x\right)$ in (3) is verified to be semi-bent. In this section, we calculate the plateauedness of the Boolean function $f\left(x\right)$ defined by (3) with the set S provided in Example 2 (1), and prove its semi-bent property.

Let $m=3$, $r=2$, and $S=\{s_1,s_2\}$, where $s_1=(1,1,1),s_2=(0,1,1)$. Set $a_{s_1}=a_{s_2}=(1,0,0)$, and define ${\epsilon }_{s_1}=0$ and ${\epsilon }_{s_2}=1$. Then the sets of $H_{s_i}$ and ${\Omega }_{s_i}$ for $1\le i\le 2$ in our construction are listed in the following Table 4.

Table 4. The sets of $H_{s_i}$ and ${\Omega }_{s_i}$ for $1\le i\le 2$.

i	Elements of ${\mathit{H}}_{{\mathit{s}}_{\mathit{i}}}$	Elements of ${\mathbf{\Omega }}_{{\mathit{s}}_{\mathit{i}}}$
1	$(0,0,0)$	$(0,0,0,1,1,1,0,0),(0,0,0,1,1,1,0,1),(0,0,0,1,1,1,1,0),(0,0,0,1,1,1,1,1)$
	$(0,0,1)$	$(0,0,1,1,1,0,0,0),(0,0,1,1,1,0,0,1),(0,0,1,1,1,0,1,0),(0,0,1,1,1,1,1,1)$
	$(0,1,0)$	$(0,1,0,1,0,1,0,0),(0,1,0,1,0,1,0,1),(0,1,0,1,0,1,1,0),(0,1,0,1,0,1,1,1)$
	$(0,1,1)$	$(0,1,1,1,0,0,0,0),(0,1,1,1,0,0,0,1),(0,1,1,1,0,0,1,0),(0,1,1,1,0,0,1,1)$
2	(1,0,0)	$(1,0,0,1,1,1,0,0),(1,0,0,1,1,1,0,1),(1,0,0,1,1,1,1,0),(1,0,0,1,1,1,1,1)$
	$(1,0,1)$	$(1,0,1,1,1,0,0,0),(1,0,1,1,1,0,0,1),(1,0,1,1,1,0,1,0),(1,0,1,1,1,0,1,1)$
	$(1,1,0)$	$(1,1,0,1,0,1,0,0),(1,1,0,1,0,1,0,1),(1,1,0,1,0,1,1,0),(1,1,0,1,0,1,1,1)$
	$(1,1,1)$	$(1,1,1,1,0,0,0,0),(1,1,1,1,0,0,0,1),(1,1,1,1,0,0,1,0),(1,1,1,1,0,0,1,1)$

Let $\Omega ={\displaystyle \bigcup _{i=1}^2}{\Omega }_{s_i}$ with ${\Omega }_{s_i}$ given in Table 4. Then the Boolean function $f\left(x\right)$ on ${\mathbb{F}}_2^8$ defined by (3) is

$$f\left(x\right)=x_1{x}_4\oplus x_2{x}_5\oplus x_3{x}_6\oplus {\chi }_{\Omega }\left(x\right),x\in {\mathbb{F}}_2^8.$$

By a similar calculation as in Section 3.2.1, we get $W_f\left(\alpha \right)\in \{0,\pm 32\},$ which implies the Boolean function $f\left(x\right)$ is 2-plateaued, i.e., semi-bent. That is to say, the result in this example is consistent with Theorem 1.

4. The Algebraic Normal Form of the Plateaued Boolean Functions in (3)

For a vector $x=(x_1,x_2,\dots ,x_n)\in {\mathbb{F}}_2^n$ with $x^{\prime }=(x_1,x_2,\dots ,x_m)$, $x^{″}=(x_{m+1},x_{m+2},$$\dots ,x_{2m})$, and $x^{‴}=(x_{2m+1},\dots ,x_{2m+r})$, whenever $x^{\prime }+x^{″}$ is used in this section, where $x^{\prime }+x^{″}=(x_1+x_{m+1},x_2+x_{m+2},\dots ,x_m+x_{2m})$, we adopt the convention that the corresponding components of $x^{\prime }$ and $x^{″}$ are not both 1, i.e., $x_i{x}_{m+i}=0$ for all $1\le i\le m$. Under this restriction, the symbol + denotes componentwise addition over the integers, which coincides with the operation ⊕ over ${\mathbb{F}}_2$.

We now proceed to compute the algebraic normal form of the plateaued Boolean function defined by (3). Recall that the function f in (3) can be rewritten as

$$f\left(x\right)=f_0\left(x\right)\oplus {\chi }_{\Omega }\left(x\right),$$

where ${\chi }_{\Omega }$ denotes the characteristic function of $\Omega$. Therefore, it suffices to determine the algebraic normal form of ${\chi }_{\Omega }$.

Theorem 2.

Let S, ${\Omega }_s$ and Ω be the sets in Theorem 1. Then we obtain ${\chi }_{\Omega }$ of Ω is

$${\chi }_{\Omega }\left(x\right)={\displaystyle \underset{s\in S}{⨁}}{\chi }_{{\Omega }_s}\left(x\right),$$

where

$${\chi }_{{\Omega }_s}\left(x\right)=\left(a_s·x^{\prime }\oplus {\epsilon }_s\oplus 1\right){\displaystyle \underset{{\beta }^{\prime }+{\beta }^{″}⪰s}{⨁}}{x}^{\prime {\beta }^{\prime }}{x}^{″{\beta }^{″}}.$$

Proof. 

By Lemma 3, it follows that

$${\chi }_{\Omega }\left(x\right)={\displaystyle \underset{s\in S}{⨁}}{\chi }_{{\Omega }_s}\left(x\right),$$

where ${\chi }_{{\Omega }_s}$ denotes the characteristic function of the set ${\Omega }_s$.

From the definitions of $H_s$ and ${\Omega }_s$, one may directly check that

$${\chi }_{{\Omega }_s}\left(x\right)=\left(a_s·x^{\prime }\oplus {\epsilon }_s\oplus 1\right){\displaystyle \prod _{i=1}^m}(x_i\oplus x_{m+i}\oplus s_i\oplus 1),$$

where $s=(s_1,s_2,\dots ,s_m)\in S$ and $a_s$ is the associated vector of s. Recall that $\mathrm{supp}\left(s\right)=\{1\le i\le m\mid s_i=1\}$, and $\mathrm{Z}\left(s\right)=\{1\le i\le m\mid s_i=0\}$. By calculation, we obtain

$$\begin{array}{ccc}& & {\displaystyle \prod _{i=1}^m}(x_i\oplus x_{m+i}\oplus s_i\oplus 1)\hfill \\ & =& \prod _{i\in \mathrm{supp}\left(s\right)}(x_i\oplus x_{m+i})·\prod _{i\in \mathrm{Z}\left(s\right)}(x_i\oplus x_{m+i}\oplus 1)\hfill \\ & =& \prod _{i\in \mathrm{supp}\left(s\right)}\left(\underset{\begin{array}{c}{\beta }_i,{\beta }_{m+i}\in {\mathbb{F}}_2,\\ {\beta }_i+{\beta }_{m+i}=1\end{array}}{⨁}{x}_i^{{\beta }_i}{x}_{m+i}^{{\beta }_{m+i}}\right)·\prod _{i\in \mathrm{Z}\left(s\right)}\left(\underset{\begin{array}{c}{\beta }_i,{\beta }_{m+i}\in {\mathbb{F}}_2,\\ {\beta }_i+{\beta }_{m+i}\in \{0,1\}\end{array}}{⨁}{x}_i^{{\beta }_i}{x}_{m+i}^{{\beta }_{m+i}}\right)\hfill \\ & =& \prod _{i\in \mathrm{supp}\left(s\right)}\left(\underset{\begin{array}{c}{\beta }_i,{\beta }_{m+i}\in {\mathbb{F}}_2,\\ {\beta }_i+{\beta }_{m+i}\ge s_i\end{array}}{⨁}{x}_i^{{\beta }_i}{x}_{m+i}^{{\beta }_{m+i}}\right)·\prod _{i\in \mathrm{Z}\left(s\right)}\left(\underset{\begin{array}{c}{\beta }_i,{\beta }_{m+i}\in {\mathbb{F}}_2,\\ {\beta }_i+{\beta }_{m+i}\ge s_i\end{array}}{⨁}{x}_i^{{\beta }_i}{x}_{m+i}^{{\beta }_{m+i}}\right)\hfill \\ & =& {\displaystyle \prod _{i=1}^m}\left(\underset{\begin{array}{c}{\beta }_i,{\beta }_{m+i}\in {\mathbb{F}}_2,\\ {\beta }_i+{\beta }_{m+i}\ge s_i\end{array}}{⨁}{x}_i^{{\beta }_i}{x}_{m+i}^{{\beta }_{m+i}}\right)\hfill \\ & =& \underset{\begin{array}{c}1\le i\le m,\\ {\beta }_i,{\beta }_{m+i}\in {\mathbb{F}}_2,\\ {\beta }_i+{\beta }_{m+i}\ge s_i\end{array}}{⨁}{\displaystyle \prod _{i=1}^m}{x}_i^{{\beta }_i}{x}_{m+i}^{{\beta }_{m+i}}.\hfill \end{array}$$

Denote ${\beta }^{\prime }=({\beta }_1,{\beta }_2,\dots ,{\beta }_m)$ and ${\beta }^{″}=({\beta }_{m+1},{\beta }_{m+2},\dots ,{\beta }_{2m})$. Then ${\displaystyle \prod _{i=1}^m}{x}_i^{{\beta }_i}{x}_{m+i}^{{\beta }_{m+i}}=x^{\prime {\beta }^{\prime }}{x}^{″{\beta }^{″}}$. Observe that the component-wise condition ${\beta }_i+{\beta }_{m+i}\ge s_i$ for all $1\le i\le m$ is exactly the relation ${\beta }^{\prime }+{\beta }^{″}⪰s$. So

$$\underset{\begin{array}{c}1\le i\le m,\\ {\beta }_i,{\beta }_{m+i}\in {\mathbb{F}}_2,\\ {\beta }_i+{\beta }_{m+i}\ge s_i\end{array}}{⨁}{\displaystyle \prod _{i=1}^m}{x}_i^{{\beta }_i}{x}_{m+i}^{{\beta }_{m+i}}=\underset{{\beta }^{\prime }+{\beta }^{″}⪰s}{⨁}{x}^{\prime {\beta }^{\prime }}{x}^{″{\beta }^{″}}.$$

Combining the above equalities, we arrive at

$${\displaystyle \prod _{i=1}^m}(x_i\oplus x_{m+i}\oplus s_i\oplus 1)=\underset{{\beta }^{\prime }+{\beta }^{″}⪰s}{⨁}{x}^{\prime {\beta }^{\prime }}{x}^{″{\beta }^{″}}.$$

Thus,

$${\chi }_{{\Omega }_s}\left(x\right)=\left(a_s·x^{\prime }\oplus {\epsilon }_s\oplus 1\right)\underset{{\beta }^{\prime }+{\beta }^{″}⪰s}{⨁}{x}^{\prime {\beta }^{\prime }}{x}^{″{\beta }^{″}}.$$

The proof is finished. □

Example 3.

We compute the algebraic normal form for ${\displaystyle \underset{{\beta }^{\prime }+{\beta }^{″}⪰s}{⨁}}{x}^{\prime {\beta }^{\prime }}{x}^{″{\beta }^{″}}$ with $s=(0,1)$. Under the condition ${\beta }^{\prime }+{\beta }^{″}⪰s$, the possible values for ${\beta }^{\prime }+{\beta }^{″}$ are $(0,1)$ and $(1,1)$. The detailed enumeration and ANF computation are presented in the following Table 5.

Table 5. The ANF of ${\displaystyle \underset{{\beta }^{\prime }+{\beta }^{″}⪰s}{⨁}}{x}^{\prime {\beta }^{\prime }}{x}^{″{\beta }^{″}}$ for $s=(0,1)$.

$\mathit{s}=(0,1)$	${\mathit{\beta }}^{\prime }+{\mathit{\beta }}^{″}=(0,1)$		${\mathit{\beta }}^{\prime }+{\mathit{\beta }}^{″}=(1,1)$
${\beta }^{\prime }$	$(0,0)$	$(0,1)$	$(0,0)$	$(0,1)$	$(1,0)$	$(1,1)$
${\beta }^{″}$	$(0,1)$	$(0,0)$	$(1,1)$	$(1,0)$	$(0,1)$	$(0,0)$
$x^{\prime {\beta }^{\prime }}{x}^{″{\beta }^{″}}$	$x_4$	$x_2$	$x_3{x}_4$	$x_2{x}_3$	$x_1{x}_4$	$x_1{x}_2$
${\displaystyle \underset{{\beta }^{\prime }+{\beta }^{″}⪰s}{⨁}}{x}^{\prime {\beta }^{\prime }}{x}^{″{\beta }^{″}}$	$x_4\oplus x_2\oplus x_3{x}_4\oplus x_2{x}_3\oplus x_1{x}_4\oplus x_1{x}_2$

Example 4.

Let S and Ω be the sets defined in Section 3.2.1. Then by Theorem 2 we derive the characteristic function of the sets ${\Omega }_{s_1}$ is

$$\begin{array}{ccc}\hfill {\chi }_{{\Omega }_{s_1}}\left(x\right)& =& \left(a_{s_1}·x^{\prime }\oplus {\epsilon }_{s_1}\oplus 1\right)\underset{{\beta }^{\prime }+{\beta }^{″}⪰s_1}{⨁}{x}^{\prime {\beta }^{\prime }}{x}^{″{\beta }^{″}}\hfill \\ & =& (x_2\oplus 1)(x_1{x}_2\oplus x_1{x}_4\oplus x_2{x}_3\oplus x_3{x}_4\oplus x_1\oplus x_2\oplus x_3\oplus x_4\oplus 1).\hfill \end{array}$$

Similarly, we derive the characteristic functions for ${\Omega }_{s_2}$ and ${\Omega }_{s_3}$,

$$\begin{array}{ccc}& & {\chi }_{{\Omega }_{s_2}}\left(x\right)=(x_1\oplus x_2)(x_1{x}_2\oplus x_1{x}_4\oplus x_2{x}_3\oplus x_3{x}_4\oplus x_2\oplus x_4),\hfill \\ & & {\chi }_{{\Omega }_{s_3}}\left(x\right)=x_1(x_1{x}_2\oplus x_1{x}_4\oplus x_2{x}_3\oplus x_3{x}_4\oplus x_1\oplus x_3).\hfill \end{array}$$

Furthermore, the characteristic function for the whole set Ω is

$${\chi }_{\Omega }\left(x\right)={\displaystyle \underset{i=1}{\overset{3}{⨁}}}{\chi }_{{\Omega }_{s_i}}\left(x\right)=x_1{x}_2\oplus x_1{x}_3\oplus x_3{x}_4\oplus x_3\oplus x_4\oplus 1.$$

5. Conclusions

This paper is focused on the construction of plateaued Boolean functions, which are of significant interest in cryptography and sequence design. A generic construction of $2m+r$-variables r-plateaued functions is proposed by modifying the support of a class of quadratic functions, associated with a nonempty subset $S\subseteq {\mathbb{F}}_2^m$. Furthermore, we analyzed the algebraic normal form of the resulting functions. As applications of the proposed construction, we obtained near-bent and semi-bent functions. Notably, by introducing the additional vector $x^{‴}\in {\mathbb{F}}_2^r$, our construction generalizes the bent functions in [16] (when $r=0$) to the general class of r-plateaued functions for $r>0$. Future work will focus on extending this technique to design functions with improved nonlinearity and algebraic degree.