On $q$-nearly bent Boolean functions

TL;DR

This paper introduces the concept of $q$-nearly bent Boolean functions, proving their existence for certain classes and providing conditions to identify non-$q$-nearly bent functions, advancing understanding of Boolean function transformations.

Contribution

It establishes the existence of $q$-nearly bent functions for non-affine $q$ with weight one, answering an open question and providing criteria for non-$q$-nearly bent functions.

Findings

Balanced Boolean functions are $q$-nearly bent if $q$ has weight one.

Proved existence of non-affine $q$-nearly bent functions.

Provided a necessary condition to determine when a function isn't $q$-nearly bent.

Abstract

For each non-constant Boolean function $q$, Klapper introduced the notion of $q$-transforms of Boolean functions. The {\em $q$-transform} of a Boolean function $f$ is related to the Hamming distances from $f$ to the functions obtainable from $q$ by nonsingular linear change of basis. In this work we discuss the existence of $q$-nearly bent functions, a new family of Boolean functions characterized by the $q$-transform. Let $q$ be a non-affine Boolean function. We prove that any balanced Boolean functions (linear or non-linear) are $q$-nearly bent if $q$ has weight one, which gives a positive answer to an open question (whether there exist non-affine $q$-nearly bent functions) proposed by Klapper. We also prove a necessary condition for checking when a function isn't $q$-nearly bent.

Tables

Table 1. Table 1: Examples of q 𝑞 q with small w ​ t ​ ( q ) 𝑤 𝑡 𝑞 wt(q)

$n$	$wt\left(q\right)$	${\rho }_q$	Do $q$-nearly bent functions exist?
$n>2$	$4$	${\rho }_q=4$	No (Proved in Thm.6)
$n>3$	$5$	${\rho }_q\in \{4,5\}$	No (by Thm.3)
$n=4$	$6$	${\rho }_q=4$	No (Proved in Thm.7)
$n>4$	$6$	${\rho }_q=5$	No (by Thm.3)
$n=4$	$7$	${\rho }_q=2$	No (Proved in Thm.7)
$n=5$	$7$	${\rho }_q=5$	No (by Thm.3)
$n>5$	$7$	${\rho }_q=6$	?

Equations

$$W(f,g)=\sum _{a\in V_n}(-1{)}^{f(a)+g(a)}\in \mathbb{Z}.$$

$$W(f,g)=2^n-2d(f,g),$$

$$\sum _{v\in V_n}W(f)(v{)}^2=2^{2n}.$$

$$W_q(f)(0)=\sum _{a\in V_n}(-1{)}^{f(a)}\triangleq I_f,$$

$$\omega (A)=\frac{1}{N+N/(2^n-1)}=\frac{2^n-1}{2^{n}N}$$

$$\omega (0)=\frac{N/(2^n-1)}{N+N/(2^n-1)}=\frac{1}{2^n},$$

$$E^{\prime }[W_q(f)(A{)}^2]=(2^{2n}-I_f^2)/(2^n-1),$$

$$E[W_q(f)(A{)}^2]=2^n.$$

$$|W_q(f)(A)|\le \lceil {\left(\frac{2^{2n}-I_q^2}{2^n-1}\right)}^{1/2}\rceil$$

$${\rho }_q=\lceil {\left(\frac{2^{2n}-I_q^2}{2^n-1}\right)}^{1/2}\rceil .$$

$$|W_q(f)(A)|\in \{0,4,8,\dots ,2w-4,2w\}$$

$$|W_q(f)(A)|\in \{2,6,10,\dots ,2w-4,2w\}.$$

$$wt(f+q_A)\in \{2^{n-1}+w,2^{n-1}+w-2,2^{n-1}+w-4,\dots ,2^{n-1}-w\}.$$

$$({\rho }_q-1{)}^2<\frac{2^{2n}-I_q^2}{2^n-1}\le {\rho }_q^2.$$

$${\left(\frac{2^{2n}-I_q^2}{2^n-1}\right)}^{1/2}\le {\rho }_q\text{and}{\left(\frac{2^{2n}-I_q^2}{2^n-1}\right)}^{1/2}+1>{\rho }_q.$$

$$|W_q(f)(A)|=2={\left(\frac{2^{2n}-(2^n-2{)}^2}{2^n-1}\right)}^{1/2}={\rho }_q.$$

$$\rho_{q}=\left\{\begin{array}[]{ll}3,&\mathrm{if}\,\ wt(q)=2\,\ \mathrm{and}\,\ n\geq 3,\\ 3,&\mathrm{if}\,\ wt(q)=3\,\ \mathrm{and}\,\ n=3,\\ 4,&\mathrm{if}\,\ wt(q)=3\,\ \mathrm{and}\,\ n>3.\end{array}\right.$$

$$\left[\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right],\left[\begin{array}[]{ccc}0&1&0\\ 1&0&0\\ 0&0&1\end{array}\right],\left[\begin{array}[]{ccc}1&1&1\\ 0&1&0\\ 0&0&1\end{array}\right],$$

$$\left[\begin{array}[]{ccc}1&1&1\\ 1&0&0\\ 0&0&1\end{array}\right],\left[\begin{array}[]{ccc}0&1&0\\ 1&1&1\\ 0&0&1\end{array}\right],\text{ and }\left[\begin{array}[]{ccc}1&0&0\\ 1&1&1\\ 0&0&1\end{array}\right].$$

$$q({\beta }_{i}M)=q({\alpha }_i)=1\text{and}f({\beta }_i)=1.$$

$$|W_q(f)(M)|=\left|\sum _{a\in V_n}(-1{)}^{f(a)+q(aM)}\right|=8.$$

$$q({\gamma }_{i}M)=q({\alpha }_i)=1\text{and}f({\gamma }_i)=1$$

$${\eta }_1={\gamma }_1+{\gamma }_2,{\eta }_2={\gamma }_1+{\gamma }_3,{\eta }_4={\gamma }_2+{\gamma }_3.$$

$$q({\eta }_{i}M)=q({\alpha }_i)=1\text{but}f({\eta }_i)=0$$

$$2^{n-1}-2^{n/2-1}\le wt(q)\le 2^{n-1},$$

$$\begin{array}[]{ll}\displaystyle 2^{n}+1<\frac{2^{2n}-I^{2}_{q}}{2^{n}-1}<2^{n}+2,&\text{if}~{}~{}~{}I^{2}_{q}=0,\\[12.0pt] \displaystyle 2^{n}<\frac{2^{2n}-I^{2}_{q}}{2^{n}-1}<2^{n}+1,&\text{if}~{}~{}~{}0<I^{2}_{q}<2^{n},\text{and}\\[12.0pt] \displaystyle\frac{2^{2n}-I^{2}_{q}}{2^{n}-1}=2^{n},&\text{if}~{}~{}~{}I^{2}_{q}=2^{n}.\end{array}$$

$$2^{n/2}<{\rho }_q\le \lceil \sqrt{2^n+2}\rceil \le 2^{n/2}+1,$$

$$|W_q(f)(A)|\le 2^{n/2}.$$

Full text

In this work we discuss the existence of $q$-nearly bent functions, a new family of Boolean functions characterized by the $q$-transform. Let $q$ be a non-affine Boolean function. We prove that any balanced Boolean functions (linear or non-linear) are $q$-nearly bent if $q$ has weight one, which gives a positive answer to an open question (whether there exist non-affine $q$-nearly bent functions) proposed by Klapper. We also prove a necessary condition for checking when a function isn’t $q$-nearly bent.

Keywords. Boolean function, Walsh-Hadamard transform, $q$-transform, bent function, plateaued functions, $q$-nearly bent function

1 Introduction

Boolean functions play a central role in cryptography. For example, they appear as feedback functions in the state changes of nonlinear feedback shift registers, as combining functions for nonlinear combiners [8], and as components in S-boxes in block ciphers.

Linear or affine functions are cryptographically vulnerable and in general should not be used. A critical tool in the analysis of symmetric key cryptosystems is the Walsh-Hadamard transform [1, 3, 10]. It has been used, for example, to measure how closely a given Boolean function can be approximated by an affine function. For security of cryptographic schemes, we generally need to use functions that are ‘far’ from affine functions, or even from simple functions (e.g., of low degree, depending on few variables, or of low rank). This reasoning has led to a new transform (see Definition 1) that is related to the Hamming distance from a fixed (not necessarily affine) function [6]. We (partially with other coauthors) have explored some issues concerning the q-transform [2, 4, 6, 7].

The topic of this work is related to the concept of bent functions, which we will review. It is well known that for even dimension $n$, every bent function achieves the upper bound $2^{n-1}-2^{n/2-1}$ on the distance from the set of affine functions. We introduce below some basic knowledge on Boolean functions.

Let $n$ be a positive integer, let $V_{n}=\{0,1\}^{n}$, treated as row vectors, and let ${\cal B}_{n}=\{f:V_{n}\to\{0,1\}\}$, the set of Boolean functions of dimension $n$. We denote $(0,0,\ldots,0)\in V_{n}$ by $0^{n}$. We refer the reader to Carlet’s book chapter [1] and Cusick and Stănică’s monograph [3] for background on Boolean functions.

For $f\in{\cal B}_{n}$, let $wt(f)=|\{a\in V_{n}:f(a)=1\}|$ be the Hamming weight of $f$. It is the cardinality of the support of $f$.

For $f,g\in{\cal B}_{n}$, let

[TABLE]

Also let $d(f,g)=|\{a\in V_{n}:f(a)\neq g(a)\}|=wt(f+g)$ be the Hamming distance from $f$ to $g$. Then

[TABLE]

which means, in particular, that $W(f,g)$ is an even number.

Let $g_{v}(a)=v\cdot a\in{\cal B}_{n}$ be linear, where $v\in V_{n}$. The Walsh-Hadamard transform coefficient of $f\in{\cal B}_{n}$ at $v$ is $W(f)(v)=W(f,g_{v})$. We recall that Parseval’s identity says that

[TABLE]

A function $f\in{\cal B}_{n}$ with $|W(f)(v)|=2^{n/2}$ for all $v\in V_{n}$ is called a bent function. For bent functions on $n$ bits to exist it is necessary that $n$ be even. It is well known that bent functions exist for all even $n$. For example, rank $n$ quadratic functions are bent.

In recent years, Klapper generalized the Walsh-Hadamard transform to a new transform called the $q$-transform, which is associated with a fixed non-constant function $q\in{\cal B}_{n}$ [6]. Let $GL_{n}=GL_{n}(\mathbb{F}_{2})$ be the group of $n\times n$ invertible matrices over $\mathbb{F}_{2}$. Let $N=|GL_{n}|=(2^{n}-1)(2^{n}-2)(2^{n}-4)\ldots(2^{n}-2^{n-1})$. Let $q_{A}$ denote the function $q_{A}(a)=q(aA)$ for $q\in{\cal B}_{n}$ and $A\in GL_{n}$. Let $\mathbf{0}$ denote the $n\times n$ zero matrix.

Definition 1**.**

Let $q\in{\cal B}_{n}$. If $f\in{\cal B}_{n}$ and $A\in GL_{n}$, then the $q$-transform coefficient of $f$ at $A$ is $W_{q}(f)(A)=W(f,q_{A})$. Also let

[TABLE]

where $I_{f}$ is referred to as the imbalance of $f$.

We note that $W_{q}(f)(A)$ is even for all $A\in GL_{n}\cup\{\mathbf{0}\}$ and $I_{f}=0$ if and only if $f$ is balanced.

Klapper considered the statistical behavior of the $q$-transform with respect to two probability distributions [6]. For a random variable $X$ on $GL_{n}\cup\{\mathbf{0}\}$, let $E^{\prime}[X]$ denote the expected value of $X$ with respect to the uniform distribution on $GL_{n}$. Let $E[X]$ denote the expected value of $X$ on $GL_{n}\cup\{\mathbf{0}\}$ with respect to $\omega$, the probability distribution on $GL_{n}\cup\{\mathbf{0}\}$ defined by

[TABLE]

for $A\in GL_{n}$ and

[TABLE]

where $N=|GL_{n}|$. Then, for balanced $q\in{\cal B}_{n}$, by some computations involving $\omega$ that we shall omit [6, p. 2801], we get

[TABLE]

and so

[TABLE]

This is a generalization of Parseval’s equation. Equation (2) leads to the notion of $q$-bent functions.

Definition 2**.**

([6]) Let $q\in{\cal B}_{n}$ be balanced. A function $f\in{\cal B}_{n}$ is $q$-bent if $|W_{q}(f)(A)|=2^{n/2}$ for all $A\in GL_{n}\cup\{\mathbf{0}\}$.

As with bent functions, $q$-bent functions do not exist if $n$ is odd. If $q$ is linear and non-zero, then $f$ is $q$-bent if and only if it is bent. However, for balanced non-affine $q$, we have proved the following.

Theorem 1**.**

([2]) Let $n\geq 4$ be even. No $q$-bent functions exist for balanced non-affine $q\in{\cal B}_{n}$.

Theorem 1 gave a positive answer to a conjecture in Klapper’s paper [6, Conj. 3, p. 2802]. So we consider generalizations of $q$-bent functions. Equation (1) leads to the notion of $q$-nearly bent functions [6], where $W_{q}(f)(\mathbf{0})=I_{f}$ is not considered. Due to the fact that balanced Boolean functions are especially important in cryptography, we consider the partial expected value and second moment of the $q$-transform of such a function. From [6, Thms.1(1) and 4], the $q$-nearly bent functions are introduced in the following definition.

Definition 3**.**

([6, Def.3]) Let $q\in{\cal B}_{n}$. A balanced function $f\in{\cal B}_{n}$ is * $q$-nearly bent if*

[TABLE]

for all $A\in GL_{n}$, where $I_{q}=\sum_{a\in V_{n}}(-1)^{q(a)}$ as before.

Note that Definition 3 is valid for odd $n$ as well as even. It follows that if $q\in{\cal B}_{n}$ is a quadratic function with rank $n$, then any non-zero linear function $f$ is $q$-nearly bent. Klapper left an open question as to whether there exist non-linear (balanced) $f$ so that $f$ is $q$-nearly bent for a given non-linear $q$ [6, p. 2803].

In this work, our main contribution is to prove that there do exist non-linear (balanced) $q$-nearly bent. We organize the work as follows. In Section 2 we present some necessary conditions for $q$-nearly bent functions. In Section 3 we give a family of $q$-nearly bent functions. In Section 4 we prove some partial results of impossibility of being $q$-nearly bent.

Throughout this work, let

[TABLE]

Since $I_{1+q}=-I_{q}$ for any $q\in{\cal B}_{n}$, we see that a balanced $f$ is $(1+q)$-nearly bent if $f$ is $q$-nearly bent. Thus hereafter we assume that $wt(q)\leq 2^{n-1}$.

2 Necessary conditions for $q$-nearly bent functions

In the section we show certain properties of the parameter $\rho_{q}$ for any $q$-nearly bent functions. We first give two lemmas.

Lemma 1**.**

Let $n>2$ be any integer (odd or even) and $q\in{\cal B}_{n}$. For any balanced $f\in{\cal B}_{n}$, if $wt(q)=w\leq 2^{n-1}$, we have for all $A\in GL_{n}$

[TABLE]

if $w$ is even, and otherwise

[TABLE]

Proof. Since $wt(f)=2^{n-1}$ and $wt(q_{A})=wt(q)=w$ for all $A\in GL_{n}$, we derive

[TABLE]

The lemma follows from the fact that $W_{q}(f)(A)=2^{n}-2wt(f+q_{A})$, which was shown in our earlier paper [7]. ∎

Lemma 2**.**

For any $q\in{\cal B}_{n}$, we have

[TABLE]

Proof. These bounds can be proven using the the following

[TABLE]

∎

Now we study the parameter $\rho_{q}$.

Theorem 2**.**

Let $n>2$ be any integer (odd or even) and $q\in{\cal B}_{n}$. If $f\in{\cal B}_{n}$ is (balanced) $q$-nearly bent, then there exists a matrix $A_{0}\in GL_{n}$ such that $|W_{q}(f)(A_{0})|=\rho_{q}$.

Proof. Suppose $|W_{q}(f)(A)|<\rho_{q}$ for all $A\in GL_{n}$. This implies that $|W_{q}(f)(A)|^{2}\leq(\rho_{q}-1)^{2}$ for all $A\in GL_{n}$, and hence that $E^{\prime}[W_{q}(f)(A)^{2}]\leq(\rho_{q}-1)^{2}$. However, from Equation (1)111Indeed here we use $W_{q}(f)(A)=W_{f}(q)(A^{-1})$ by Thm. 1(1) in [6]. and Lemma 2, we see that $(\rho_{q}-1)^{2}<E^{\prime}[W_{q}(f)(A)^{2}]\leq\rho_{q}^{2}$, a contradiction. ∎

Theorem 3**.**

Let $n>2$ be any integer (odd or even) and $q\in{\cal B}_{n}$. If there exists a (balanced) $q$-nearly bent function in ${\cal B}_{n}$, then, either $\rho_{q}\equiv 0\pmod{4}$ when $wt(q)$ is even or $\rho_{q}\equiv 2\pmod{4}$ when $wt(q)$ is odd.

Proof. Suppose that $f$ is $q$-nearly bent. Then there exists a matrix $A_{0}\in GL_{n}$ such that $|W_{q}(f)(A_{0})|=\rho_{q}$ by Theorem 2. The statements can be proved by Lemma 1. ∎

Theorem 3 can be used to check whether there exist $q$-nearly bent functions for certain $q$. We prove the non-existence of $q$-nearly bent functions for certain $q$ in Section 4. But in the coming section, we present a family of $q$-nearly bent functions.

3 Existence of $q$-nearly bent functions

Since linear functions are $q$-nearly bent for any bent function $q$, Klapper asked whether there exist non-linear $q$ and non-linear balanced $f$ so that $f$ is $q$-nearly bent [6, p. 2803]. Theorem 4 below answers this question.

Theorem 4**.**

Let $n>2$ be any integer (odd or even) and $q\in{\cal B}_{n}$. If $wt(q)=1$, then any balanced $f$ (linear or non-linear) is $q$-nearly bent.

Proof. By Lemma 1 we see that $W_{q}(f)(A)\in\{-2,2\}$ for all $A\in GL_{n}$. Now using the fact that $I_{q}=2^{n}-2wt(q)=2^{n}-2$, we can prove

[TABLE]

So by Definition 3, $f$ is $q$-nearly bent. ∎

It is natural to consider the cases of other values of $wt(q)$. In the following section, we find different phenomena.

4 Impossibility of being $q$-nearly bent

In this section we derive conditions under which no $q$-nearly bent functions can exist. These conditions are expressed in terms of $wt(q)$ and $\rho_{q}$. They apply when the weight of $q$ is low.

Theorem 5**.**

Let $n>2$ be an integer (odd or even) and $q\in{\cal B}_{n}$. If $wt(q)\in\{2,3\}$, then there are no non-linear $q$-nearly bent functions.

Proof. Using $I_{q}=2^{n}-2wt(q)$, we compute

[TABLE]

Then the proof is finished using Theorem 3. ∎

Table 1 below lists some experimental results. Some of these results can be checked using Theorem 3, but others cannot.

We see that for $wt(q)=4$, Theorem 3 is of no use to us. So we need further analysis.

Theorem 6**.**

Let $n>2$ be any integer (odd or even) and $q\in{\cal B}_{n}$. If $wt(q)=4$, then no non-linear $q$-nearly bent function exists.

Proof. We first prove the case when $n=3$. Let $q\in{\cal B}_{3}$ have $wt(q)=4$. In this case $q$ is balanced. We only need to deal with the balanced functions with value [math] at $0^{3}=(0,0,0)$. There are only $\binom{7}{4}=35$ balanced functions (with value [math] at $0^{3}$), which include 7 of degree 1 and 28 of degree 2. Note that there are no balanced functions of degree 3 222There are no balanced functions with degree $n$ in ${\cal B}_{n}$..

On the other hand, the number of $3\times 3$ invertible matrices over $\mathbb{F}_{2}$ is $|GL_{3}|=(8-1)(8-2)(8-4)=168$. Consider the balanced function $q(x)=q(x_{1},x_{2},x_{3})=x_{1}x_{2}+x_{3}\in{\cal B}_{3}$. There are exactly six $A\in{\cal B}_{3}$’s such that $q(xA)=q(x)$. They are

[TABLE]

[TABLE]

These six matrices form $q$’s stabilizer subgroup under the action of $GL_{3}$ on ${\cal B}_{3}$. Hence for all $B\in GL_{3}$, there are $168\div 6=28$ distinct $q_{B}(x)$’s. This implies that all balanced functions of degree 2 are of the form $q_{B}(x)$ for some $B\in GL_{3}$.

Thus for any balanced $f$ of degree 2, we have $f(x)=q(xA_{0})$ for some $A_{0}\in GL_{3}$. Hence $|W_{q}(f)(A_{0})|=8>\rho_{q}=4$. That is, no non-linear $q$-nearly bent function exists for balanced $q\in{\cal B}_{3}$ of degree 2.

Now we turn to the case when $n\geq 4$. Since $wt(q)=4$ we may assume that $q(0^{n})=0$ and select $\alpha_{i}\in V_{n}$ for $1\leq i\leq 4$ such that $q(\alpha_{i})=1$, and $q(\beta)=0$ for $\beta\not\in\{\alpha_{i}:i=1,\cdots,4\}$. We will show that for any non-linear balanced $f$ there exists an $M\in GL_{n}$ such that $|W_{q}(f)(M)|=8$. This implies that $|W_{q}(f)(M)|>\rho_{q}=4$ and hence $f$ isn’t $q$-nearly bent. If $f(\alpha_{i})=0$ for all $i\in\{1,2,3,4\}$ or $f(\alpha_{i})=1$ for all $i\in\{1,2,3,4\}$, we get immediately that $|W_{q}(f)(E)|=8$, where $E$ is the identity matrix. Otherwise, we give a proof in two cases. We assume below that $f$ is non-linear and balanced, and that $f(0^{n})=0$.

Suppose $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ are linearly independent. The support of $f$ has cardinality $2^{n-1}$ and is not contained in a proper subspace of $V_{n}$ (otherwise the support would equal the subspace, which would imply that $f$ is affine). We select four row vectors $\beta_{1},\beta_{2},\beta_{3},\beta_{4}\in\text{supp}(f)$ that are linearly independent. Hence there exists an $M\in GL_{n}$ such that $\alpha_{i}=\beta_{i}M$ for $1\leq i\leq 4$. Then for $1\leq i\leq 4$ we have

[TABLE]

So we get

[TABLE]

Now suppose $\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}$ are linearly dependent. Without loss of generality, we suppose that $\alpha_{1},\alpha_{2},\alpha_{3}$ are linearly independent and $\alpha_{4}$ is of the form $\alpha_{4}=\alpha_{1}+\alpha_{2}+\alpha_{3}$ or $\alpha_{4}=\alpha_{1}+\alpha_{2}$.

First, if $\alpha_{4}=\alpha_{1}+\alpha_{2}+\alpha_{3}$, then there are $2^{n-1}(2^{n-1}-1)$ vectors of the form $\gamma_{i}+\gamma_{j}$ with $\gamma_{i}\neq\gamma_{j}\in\text{supp}(f)$. We can select four pairwise distinct vectors $\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\in\text{supp}(f)$ such that $\gamma_{1}+\gamma_{2}=\gamma_{3}+\gamma_{4}$. Hence there exists an $M\in GL_{n}$ such that $\alpha_{i}=\gamma_{i}M$ for $1\leq i\leq 3$, from which we get $\alpha_{4}=\gamma_{4}M$ and

[TABLE]

for $1\leq i\leq 4$. So we get $|W_{q}(f)(M)|=8$.

Second, we suppose $\alpha_{4}=\alpha_{1}+\alpha_{2}$. If there exist pairwise distinct $\gamma_{1},\gamma_{2},\gamma_{3}\in\text{supp}(f)$ such that $\gamma_{3}\neq\gamma_{1}+\gamma_{2}$, then $\gamma_{1},\gamma_{2},\gamma_{3}$ are linearly independent. We write $\gamma_{4}\triangleq\gamma_{1}+\gamma_{2}$. If $\gamma_{4}\in\text{supp}(f)$, and then we can construct $M\in GL_{n}$ such that $\alpha_{i}=\gamma_{i}M$ for $1\leq i\leq 3$ and calculate $|W_{q}(f)(M)|=8$. If $\gamma_{4}\not\in\text{supp}(f)$, that is, if any vector $\gamma_{i}+\gamma_{j}\in V_{n}\setminus\text{supp}(f)$ for distinct $\gamma_{i},\gamma_{j}\in\text{supp}(f)$, then we choose pairwise distinct $\gamma_{1},\gamma_{2},\gamma_{3}\in\text{supp}(f)$ and set

[TABLE]

Then $\eta_{1},\eta_{2},\eta_{4}\in V_{n}\setminus\text{supp}(f)$. We further select $0^{n}\neq\eta_{3}\in V_{n}\setminus\text{supp}(f)$ such that $\eta_{3}\neq\eta_{4}=\eta_{1}+\eta_{2}$. Hence we can construct $M\in GL_{n}$ such that $\alpha_{i}=\eta_{i}M$ for $1\leq i\leq 3$, from which we get $\alpha_{4}=\eta_{4}M$ and

[TABLE]

for $1\leq i\leq 4$. So we get $|W_{q}(f)(M)|=8$. This completes the proof. ∎

We hope that the method used in Theorem 6 can be useful for checking $q$ with other $wt(q)$. For example, if $4<wt(q)\leq n$ and all vectors in $\mathrm{supp}(q)$ are linearly independent, then any non-linear balanced function isn’t $q$-nearly bent. However, if the vectors in $\mathrm{supp}(q)$ are linearly dependent, the proof seems to be more complicated.

We remark that the $wt(q)$ considered above is small, so we prove a result for large $wt(q)$.

Theorem 7**.**

Let $n>2$ be even and $q\in{\cal B}_{n}$. If

[TABLE]

then no non-linear $q$-nearly bent function exists.

Proof. Suppose that non-linear, balanced $f$ is $q$-nearly bent. We have $|\{x:q(x)=1\}|=wt(q)$, and $|\{x:q(x)=0\}|=2^{n}-wt(q)$. Thus $I_{q}=|\{x:q(x)=0\}|-|\{x:q(x)=1\}|=2^{n}-2wt(q)$. We have

[TABLE]

Note that $\sqrt{2^{n}+2}<2^{n/2}+1$. Thus for $0\leq I^{2}_{q}<2^{n}$ we have

[TABLE]

and hence $\rho_{q}=2^{n/2}+1$ (an odd number), which is impossible by Theorem 3.

Now for $I^{2}_{q}=2^{n}$ and $A\in GL_{n}$ we have by Definition 3

[TABLE]

Then we see that $|W_{f}(q)(A)|=|W_{q}(f)(A^{-1})|\leq 2^{n/2}$ for all $A\in GL_{n}$ and $|W_{f}(q)(\mathbf{0})|=I_{q}=2^{n/2}$. Equation (2) implies that $|W_{f}(q)(A)|=2^{n/2}$ for all $A\in GL_{n}\cup\{\mathbf{0}\}$ and hence $q$ is $f$-bent, which contradicts Theorem 1. This completes the proof. ∎

We have immediately the following corollary, which says Proposition 5 in [6] is vacuous for even $n$.

Corollary 1**.**

Let $n>2$ be even and $q\in{\cal B}_{n}$. If $q$ is balanced (so $wt(q)=2^{n-1}$), then no non-linear (balanced) $q$-nearly bent function exists.

Unfortunately, we cannot prove a similar result based on Theorem 7 for odd $n\geq 3$. For $n=3$, the first part of the proof of Theorem 6 says that any non-linear balanced $f\in{\cal B}_{3}$ of degree 2 is not $q$-nearly bent for non-linear balanced $q\in{\cal B}_{3}$. For odd $n>3$, we haven’t any results, although sometimes it can be checked by Theorem 3. We list blow some experimental data. We cannot prove the cases when $n=7$ or $n=9$.

[TABLE]

Table 2. Examples of $q$ with $wt(q)=2^{n-1}$

For $q(x)=q(x_{1},x_{2},\ldots,x_{n})=x_{1}x_{2}\in{\cal B}_{n}$, we see that $wt(q)=2^{n-2}(<2^{n-1}-2^{n/2-1})$ and check that there are no $q$-nearly bent functions when $n=4$ by Theorem 6 and when $5\leq n\leq 8$ by Theorem 3 (by checking $\rho_{q}$). We do not know whether $q$-nearly bent functions exist if $n=9$, in which case $\rho_{q}=20$.

The results and examples seem to indicate that there are no non-linear $q$-nearly bent functions when $1<wt(q)\leq 2^{n-1}$, so we make this a conjecture.

Conjecture 1**.**

Let $n>2$ be any integer (odd or even) and $q\in{\cal B}_{n}$. If $1<wt(q)\leq 2^{n-1}$, then no non-linear (balanced) $q$-nearly bent function exists.

5 Conclusions

In this work, we have studied the existence and non-existence of $q$-nearly bent functions with respect to $q$-transform for a non-affine Boolean function $q$. We have shown that all balanced functions are $q$-nearly bent if $q$ has weight one, which confirms the existence of $q$-nearly bent functions. We have also pointed out a necessary condition for $q$-nearly bent functions to exist. That is, if a $q$-nearly bent function exists then either $\rho_{q}\not\equiv 0\pmod{4}$ when $wt(q)$ is even or $\rho_{q}\not\equiv 2\pmod{4}$ when $wt(q)$ is odd. The statement can be used to determine the non-existence of $q$-nearly bent functions. Results and examples lead us to the conjecture that there are no non-linear $q$-nearly bent functions when $1<wt(q)\leq 2^{n-1}$.

To weaken the notion of q-bentness, it is natural to generalize plateaued functions [5, 9], we would say $f\in{\cal{B}}_{n}$ is $q$-plateaued if all its $q$-transform coefficients are in $\{0,\pm\lambda\}$ for some positive integer $\lambda$. When $n=3$ and $q(x_{1},x_{2},x_{3})=x_{1}x_{2}+x_{3}$, we can check that any $f\in{\cal{B}}_{3}$ with $wt(f)=2$ is $q$-plateaued. In this case, $\lambda=4$. Then we ask, for $n>3$ and a given non-affine $q$, do there exist non-affine $q$-plateaued functions?

Acknowledgments

Z. Chen was partially supported by the National Natural Science Foundation of China under grant No. 61772292, by the Provincial Natural Science Foundation of Fujian under grant No. 2018J01425 and by the Program for Innovative Research Team in Science and Technology in Fujian Province University under grant No. 2018-49.

A. Klapper was partially supported by the National Science Foundation under Grant No. CNS-1420227. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.