Root-Hadamard transforms and complementary sequences

TL;DR

This paper introduces a new root-Hadamard transform for generalized Boolean functions, unifying various existing transforms, and explores its properties and the concept of complementarity akin to Golay sequences.

Contribution

It defines the root-Hadamard transform for generalized Boolean functions and establishes a new notion of complementarity based on this transform.

Findings

The root-Hadamard transform generalizes multiple existing transforms.

Behavior of the transform is characterized in terms of binary components.

A new concept of complementarity for Boolean functions is introduced.

Abstract

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, $2^k$-Hadamard, consta-Hadamard and all $HN$-transforms. We describe the behavior of what we call the root- Hadamard transform for a generalized Boolean function $f$ in terms of the binary components of $f$. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set.

Tables

Table 1. Table 1 : Set of Boolean functions S F subscript 𝑆 𝐹 S_{F} .

$x_1{x}_2+x_3{x}_2+x_1{x}_4+x_3{x}_4+x_4$

$x_1{x}_2+x_3+x_1{x}_4+x_4$

$x_1{x}_2+x_3{x}_2+x_3+x_1{x}_4+x_3{x}_4+x_4$

$x_1+x_2{x}_3+x_3{x}_4+x_4$

$x_2{x}_1+x_4{x}_1+x_1+x_2{x}_3+x_3{x}_4+x_4$

$x_2{x}_1+x_4{x}_1+x_1+x_3+x_4$

$x_2{x}_1+x_4{x}_1+x_1+x_2{x}_3+x_3+x_3{x}_4+x_4$

$x_1+x_2+x_2{x}_3+x_3{x}_4$

$x_2{x}_1+x_4{x}_1+x_1+x_2+x_2{x}_3+x_3{x}_4$

$x_1+x_2+x_2{x}_3+x_3+x_3{x}_4$

$x_2{x}_1+x_4{x}_1+x_1+x_2+x_3$

$x_2{x}_1+x_4{x}_1+x_1+x_2+x_2{x}_3+x_3+x_3{x}_4$

Table 2. Table 2 : Set of Boolean functions S G subscript 𝑆 𝐺 S_{G} .

$x_1{x}_2+x_1{x}_3{x}_2+x_3{x}_2+x_1{x}_3+x_1{x}_4+x_1{x}_3{x}_4$

$x_1{x}_2+x_1{x}_3{x}_2+x_4{x}_2+x_1{x}_3+x_1{x}_4+x_1{x}_3{x}_4+x_3{x}_4+x_4$

$x_1{x}_2+x_1{x}_3{x}_2+x_3{x}_2+x_1{x}_3+x_3+x_1{x}_4+x_1{x}_3{x}_4$

$x_1{x}_2+x_1{x}_3{x}_2+x_4{x}_2+x_1{x}_3+x_3+x_1{x}_4+x_1{x}_3{x}_4+x_3{x}_4+x_4$

$x_2{x}_1+x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_4{x}_1+x_1+x_2{x}_3$

$x_2{x}_1+x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_4{x}_1+x_1+x_2{x}_4+x_3{x}_4+x_4$

$x_2{x}_1+x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_4{x}_1+x_1+x_2{x}_3+x_3$

$x_2{x}_1+x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_4{x}_1+x_1+x_3+x_2{x}_4+x_3{x}_4+x_4$

$x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_1+x_2+x_2{x}_4+x_3{x}_4$

$x_2{x}_1+x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_4{x}_1+x_1+x_2+x_2{x}_4+x_3{x}_4$

$x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_1+x_2+x_2{x}_3+x_4$

$x_2{x}_1+x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_4{x}_1+x_1+x_2+x_2{x}_3+x_4$

$x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_1+x_2+x_3+x_2{x}_4+x_3{x}_4$

$x_2{x}_1+x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_4{x}_1+x_1+x_2+x_3+x_2{x}_4+x_3{x}_4$

$x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_1+x_2+x_2{x}_3+x_3+x_4$

$x_2{x}_1+x_2{x}_3{x}_1+x_3{x}_1+x_3{x}_4{x}_1+x_4{x}_1+x_1+x_2+x_2{x}_3+x_3+x_4$

Equations

$$f(x_1,\dots ,x_n)=a_0+\sum _{1\le i\le n}{a}_i{x}_i+\sum _{1\le i<j\le n}{a}_{ij}{x}_i{x}_j+\cdots +a_{12\dots n}{x}_1{x}_2\dots x_n,$$

$$f(\mathbf{x})=a_0(\mathbf{x})+2a_1(\mathbf{x})+\cdots +2^{k-1}{a}_{k-1}(\mathbf{x}),\text{mbox}forall\mathbf{x}\in {\mathbb{F}}_2^n.$$

$${\mathcal{H}}_f^{(q)}(\mathbf{u})=2^{-n/2}\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\zeta }_q^{f(\mathbf{x})}(-1{)}^{⟨\mathbf{u},\mathbf{x}⟩},$$

$${\mathcal{W}}_f(\mathbf{u})=2^{-n/2}\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}(-1{)}^{f(\mathbf{x})+⟨\mathbf{u},\mathbf{x}⟩}.$$

$${\mathcal{C}}_{f,g}(\mathbf{z})=\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\zeta }^{f(\mathbf{x}+\mathbf{z})-g(\mathbf{x})}$$

$$D_{\mathbf{a}}f(\mathbf{x})=f(\mathbf{x}+\mathbf{a})-f(\mathbf{x}),\text{mbox}forall\mathbf{x}\in {\mathbb{F}}_2^n.$$

$${\mathcal{N}}_f^{(q)}(\mathbf{u})=2^{-\frac{n}{2}}\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\zeta }^{f(\mathbf{x})}(-1{)}^{\mathbf{u}\cdot \mathbf{x}}{ı}^{wt(\mathbf{x})}.$$

$${\mathcal{C}}_{f,g}^n(\mathbf{z})=\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\zeta }^{f(\mathbf{x}+\mathbf{z})-g(\mathbf{x})}(-1{)}^{\mathbf{x}\cdot \mathbf{z}}$$

$${\mathcal{C}}_f^n(\mathbf{u})=\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\zeta }^{f(\mathbf{x}+\mathbf{z})-f(\mathbf{x})}(-1{)}^{\mathbf{x}\cdot \mathbf{z}}.$$

$${\mathcal{U}}_{L,A,f}^{(2^k)}(\mathbf{u})=2^{-\frac{n}{2}}\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\zeta }^{f(\mathbf{x})}(-1{)}^{\mathbf{u}\cdot \mathbf{x}}{\lambda }_L(\mathbf{x}).$$

$${\mathcal{C}}_{L,A,f,g}^p(\mathbf{z})=\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\zeta }^{f(\mathbf{x}+\mathbf{z})-g(\mathbf{x})}\prod _{s\in K}{\mu }_s^{{\mathbf{x}}_{R_s}\odot {\mathbf{z}}_{R_s}}$$

$${\mathcal{C}}_{L,A,f}^p(\mathbf{z})=\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\zeta }^{f(\mathbf{x}+\mathbf{z})-f(\mathbf{x})}\prod _{s\in K}{\mu }_s^{{\mathbf{x}}_{R_s}\odot {\mathbf{z}}_{R_s}}.$$

$$f(\mathbf{x})=x_1{x}_2+x_2{x}_3+x_2{x}_4+x_1{x}_4+x_3{x}_4+2\left(x_1{x}_2+x_1{x}_3+x_3{x}_4\right)$$

$${\zeta }^{f(\mathbf{y})}=\frac{1}{2^{\frac{n}{2}}{\lambda }_L(\mathbf{y})}\sum _{\mathbf{w}\in {\mathbb{F}}_2^n}{\mathcal{U}}_{L,A,f}(\mathbf{w})(-1{)}^{\mathbf{y}\cdot \mathbf{w}}.$$

$$2^{-\frac{n}{2}}\sum _{\mathbf{w}\in {\mathbb{F}}_2^n}{\mathcal{U}}_{L,A,f}(\mathbf{w})(-1{)}^{\mathbf{y}\cdot \mathbf{w}}$$

$$=2^{-n}\sum _{\mathbf{w}\in {\mathbb{F}}_2^n}\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\zeta }^{f(\mathbf{x})}(-1{)}^{\mathbf{w}\cdot \mathbf{x}}\prod _{s\in K}{\alpha }_s^{wt({\mathbf{x}}_{R_s})}(-1{)}^{\mathbf{y}\cdot \mathbf{w}}$$

$$=2^{-n}\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\zeta }^{f(\mathbf{x})}\prod _{s\in K}{\alpha }_s^{wt({\mathbf{x}}_{R_s})}\sum _{\mathbf{w}\in {\mathbb{F}}_2^n}(-1{)}^{(\mathbf{x}+\mathbf{y})\cdot \mathbf{w}}$$

$$={\zeta }^{f(\mathbf{y})}\prod _{s\in K}{\alpha }_s^{wt({\mathbf{y}}_{R_s})},$$

$$\mathrm{wt}(\mathbf{x}+\mathbf{y})=\mathrm{wt}(\mathbf{x})+\mathrm{wt}(\mathbf{y})-2\mathrm{wt}(\mathbf{x}\star \mathbf{y}).$$

$$\begin{array}{r}\sum _{\mathbf{u}\in {\mathbb{F}}_2^n}{\mathcal{C}}_{f,g}(\mathbf{u})(-1{)}^{⟨\mathbf{u},\mathbf{x}⟩}={\mathcal{H}}_f(\mathbf{x})\overline{{\mathcal{H}}_g(\mathbf{x})},\\ {\mathcal{C}}_{f,g}(\mathbf{u})=2^{-n}\sum _{\mathbf{x}\in {\mathbb{F}}_2^n}{\mathcal{H}}_f(\mathbf{x})\overline{{\mathcal{H}}_g(\mathbf{x})}(-1{)}^{⟨\mathbf{u},\mathbf{x}⟩}.\end{array}$$

$${\mathcal{C}}_{f,g}^n(\mathbf{z})={ı}^{wt(\mathbf{z})}\sum _{\mathbf{u}\in {\mathbb{F}}_2^n}{\mathcal{N}}_f(\mathbf{u})\overline{{\mathcal{N}}_g(\mathbf{u})}(-1{)}^{\mathbf{u}\cdot \mathbf{z}}.$$

$${\mathcal{C}}_{L,f,g}^p(\mathbf{z})$$

$$\sum _{\mathbf{u}\in {\mathbb{F}}_2^n}|{\mathcal{U}}_{L,A,f}(\mathbf{u}){|}^2=2^n.$$

$${\lambda }_L(\mathbf{z})\sum _{\mathbf{u}\in {\mathbb{F}}_2^n}{\mathcal{U}}_{L,A,f}(\mathbf{u})\overline{{\mathcal{U}}_{L,A,g}(\mathbf{u})}(-1{)}^{\mathbf{u}\cdot \mathbf{z}}$$

$$=$$

$${\mathcal{C}}_{L,A,f}^p(\mathbf{z})$$

$$={\lambda }_L(\mathbf{z})\sum _{\mathbf{u}\in {\mathbb{F}}_2^n}{\mathcal{U}}_{L,A,f}(\mathbf{u})\overline{{\mathcal{U}}_{L,A,f}(\mathbf{u})}(-1{)}^{\mathbf{u}\cdot \mathbf{z}},$$

$${\mathcal{A}}_{\mathbf{a}}(k)+{\mathcal{A}}_{\mathbf{b}}(k)=0,\text{ for }k\ne 0.$$

$${\mathcal{C}}_{\mathbf{a}}(k)+{\mathcal{C}}_{\mathbf{b}}(k)=0,\text{ for }k\ne 0,$$

$${\mathcal{C}}_{\mathbf{a}}^n(k)+{\mathcal{C}}_{\mathbf{b}}^n(k)=0,\text{ for }k\ne 0,$$

$$A(x)A(x^{-1})+B(x)B(x^{-1})=2N.$$

Full text

Keywords: Golay pairs, Boolean functions, correlations, Generalized transforms. MSC 2000: 06E30, 11B83, 94A55, 94C10

1 (Generalized) Boolean functions

Let $\mathbb{F}_{2}^{n}$ be an $n$-dimensional vector space over the binary field ${\mathbb{F}}_{2}$ (here, we consider both ${\mathbb{F}}_{2}^{n}$ and the finite field ${\mathbb{F}}_{2^{n}}$) and, for an integer $q$, let ${\mathbb{Z}}_{q}$ be the ring of integers modulo $q$. By ‘$+$’ and ‘$-$’ we respectively denote addition and subtraction in the appropriate environment (it will be obvious from the context).

A function $F:\mathbb{F}_{2}^{n}\rightarrow{\mathbb{F}}_{2}$, $n>0$, is called a Boolean function in $n$ variables, whose set will be denoted by $\mathcal{B}_{n}$. A Boolean function can be regarded as a multivariate polynomial over $\mathbb{F}_{2}$, called the algebraic normal form (ANF)

[TABLE]

where the coefficients $a_{0},a_{ij},\ldots,a_{12\ldots n}\in\mathbb{F}_{2}$. The maximum number of variables in a monomial is called the (algebraic) degree. The (Hamming) weight of ${\mathbf{x}}=(x_{1},\ldots,x_{n})\in\mathbb{F}_{2}^{n}$ is denoted by $wt({\mathbf{x}})$ and equals $\sum_{i=1}^{n}x_{i}$ (the Hamming weight of a function is the weight of its truth table, that is, the weight of its output vector). The cardinality of a set $S$ is denoted by $|S|$.

We order $\mathbb{F}_{2}^{n}$ lexicographically, and denote ${\bf v}_{0}=(0,\ldots,0,0)$, ${\bf v}_{1}=(0,\ldots,0,1)$, ${\bf v}_{2^{n}-1}=(1,\ldots,1,1)$. The truth table of a Boolean function $f\in{\mathcal{B}}_{n}$ is the binary string of length $2^{n}$, $[f({\bf v}_{0}),$ $f({\bf v}_{1}),\ldots,f({\bf v}_{2^{n}-1})]$ (we will often disregard the commas).

If ${\mathbf{x}}=(x_{1},\ldots,x_{n})$ and ${\mathbf{y}}=(y_{1},\ldots,y_{n})$ are two vectors in $\mathbb{F}_{2}^{n}$ we define the scalar $($or inner$)$ product, by ${\mathbf{x}}\cdot{\mathbf{y}}=x_{1}y_{1}+x_{2}y_{2}+\cdots+x_{n}y_{n}.$ The scalar/inner product ${\mathbf{x}}\odot{\mathbf{y}}$ in $\mathbb{C}\times\mathbb{C}$ is similar, with the sum over $\mathbb{C}$. The intersection of two vectors ${\mathbf{x}},{\mathbf{y}}$ in some vector space under discussion is ${\mathbf{x}}\star{\mathbf{y}}=(x_{1}y_{1},x_{2}y_{2},\ldots,x_{n}y_{n}).$ We write $a=\Re(z),b=\Im(z)$ for the real part, respectively, imaginary part of the complex number $z=a+b\,i\in\mathbb{C}$, where $i^{2}=-1$, and $a,b\in\mathbb{R}$. Further, $|z|=\sqrt{a^{2}+b^{2}}$ is the absolute value of $z$, and $\overline{z}=a-b\,i$ denotes the complex conjugate of $z$.

We call a function from $\mathbb{F}_{2}^{n}$ to ${\mathbb{Z}}_{q}$ ($q\geq 2$) a generalized Boolean function on $n$ variables, and denote the set of all generalized Boolean functions by $\mathcal{GB}_{n}^{q}$ and, when $q=2$, by ${\mathcal{B}}_{n}$, as previously mentioned. If $q=2^{k}$ for some $k\geq 1$, we can associate to any $f\in\mathcal{GB}_{n}^{q}$ a unique sequence of Boolean functions $a_{i}\in{\mathcal{B}}_{n}$ ($i=0,1,\ldots,k-1$) such that

[TABLE]

For a Boolean function $f\in{\mathcal{B}}_{n}$, we define its sign function $\hat{f}$ by $\hat{f}({\mathbf{x}})=(-1)^{f({\mathbf{x}})}$. In general, the sign function of $f\in\mathcal{GB}_{n}^{q}$ is $\hat{f}({\mathbf{x}})=\zeta_{q}^{f({\mathbf{x}})}$, where $\zeta_{q}=e^{\frac{2\pi i}{q}}$, the $q$-complex root of $1$.

For a generalized Boolean function $f:\mathbb{F}_{2}^{n}\to{\mathbb{Z}}_{q}$ we define the (normalized) generalized Walsh-Hadamard transform to be the complex valued function

[TABLE]

where $\zeta_{q}=e^{\frac{2\pi i}{q}}$ (for easy writing, we sometimes use $\zeta$, ${\mathcal{H}}_{f}$, instead of $\zeta_{q}$, respectively, ${\mathcal{H}}_{f}^{(q)}$, when $q$ is fixed) and $\langle{\bf u},{\mathbf{x}}\rangle$ is a nondegenerate inner product: for ${\mathbb{F}}_{2}^{n}$, the vector space of the $n$-tuples over ${\mathbb{F}}_{2}$ we use the conventional dot product ${\bf u}\cdot{\mathbf{x}}$ for $\langle{\bf u},{\mathbf{x}}\rangle$, and for ${\mathbb{F}}_{2^{n}}$, the standard inner product of $u,x\in{\mathbb{F}}_{2^{n}}$ is ${\rm Tr_{n}}(ux)$, where ${\rm Tr_{n}}(z)$ denotes the absolute trace of $z\in{\mathbb{F}}_{2^{n}}$. For $q=2$, we obtain the usual Walsh-Hadamard transform

[TABLE]

The sum

[TABLE]

is the crosscorrelation of $f$ and $g$ at ${\mathbf{z}}\in\mathbb{F}_{2}^{n}$. The autocorrelation of $f\in\mathcal{GB}_{n}^{q}$ at ${\bf u}\in\mathbb{F}_{2}^{n}$ is ${\mathcal{C}}_{f,f}({\bf u})$ above, which we denote by ${\mathcal{C}}_{f}({\bf u})$.

Given a generalized Boolean function $f$, the derivative $D_{{\bf a}}f$ of $f$ with respect to a vector ${\bf a}\in\mathbb{F}_{2}^{n}$, is the generalized Boolean function defined by

[TABLE]

A function $f:\mathbb{F}_{2}^{n}\rightarrow{\mathbb{Z}}_{q}$ is called generalized bent (gbent) if $|\mathcal{H}_{f}({\bf u})|=2^{n/2}$ for all ${\bf u}\in\mathbb{F}_{2}^{n}$. For descriptions of generalized bents, the reader can consult [10, 18, 20].

The nega–Hadamard transform of $f\in\mathcal{GB}_{n}^{q}$ at any vector ${\bf u}\in\mathbb{F}_{2}^{n}$ is the complex valued function:

[TABLE]

As customary, we may drop (some of) the superscripts in all of our notations, if convenient. In [13], some of us considered generalized bent criteria for Boolean functions by analyzing Boolean functions which have flat spectrum with respect to one or more transforms chosen from a set of unitary transforms. The transforms chosen are $n$-fold tensor products of the identity mapping \left(\begin{array}[]{cc}1&0\\ 0&1\end{array}\right), the Walsh–Hadamard transformation \frac{1}{\sqrt{2}}\left(\begin{array}[]{lr}1&1\\ 1&-1\end{array}\right), and the nega–Hadamard transform \frac{1}{\sqrt{2}}\left(\begin{array}[]{lr}1&\imath\\ 1&-\imath\end{array}\right), where $\imath^{2}=-1$. That choice is motivated by local unitary transforms that play an important role in the structural analysis of pure $n$-qubit stabilizer quantum states.

A function $f:\mathbb{F}_{2}^{n}\rightarrow{\mathbb{Z}}_{q}$ is said to be (generalized) negabent if the nega–Hadamard transform is flat in absolute value, namely $|{\mathcal{N}}_{f}^{(q)}({\bf u})|=1$ for all ${\bf u}\in\mathbb{F}_{2}^{n}$. The sum

[TABLE]

is the nega–crosscorrelation of $f$ and $g$ at $z$. We define the nega–autocorrelation of $f$ at ${\bf u}\in\mathbb{F}_{2}^{n}$ by

[TABLE]

For more on (generalized) Boolean functions, the reader can consult [2, 3, 10, 14, 18, 20] and the references therein.

2 The root-Hadamard transform

Both Walsh-Hadamard and nega-Hadamard transforms can be generalized into what we will call the root-Hadamard transform:

Definition 1**.**

Let $f\in\mathcal{GB}_{n}^{2^{k}}$, $\zeta_{{2^{k}}}$ a $2^{k}$-complex root of $1$ $($when convenient we drop the index$)$, $A=\{\alpha_{1},\ldots,\alpha_{r}\}$ a set of roots of unity $\alpha_{j}=e^{\frac{2\pi i}{k_{j}}}$, $K=\{k_{j}\}_{1\leq j\leq r}$ and $L=\{R_{s}\}_{s\in K}$ be a partition of the index set $\{0,\ldots,n-1\}=\bigsqcup_{s\in K}R_{s}$, $|L|=r$ $($for convenience, we index the partition by the elements of $K)$. For $L=\{R_{s}\}_{s\in K}$, we let ${\mathbf{x}}_{R_{s}}=(x_{j})_{j\in R_{s}}$ and $\lambda_{L}({\mathbf{x}})=\prod_{s\in K}\alpha_{s}^{wt({\mathbf{x}}_{R_{s}})}$, where $\displaystyle wt({\mathbf{x}}_{R_{s}})=\sum_{j\in R_{s}}x_{j}\in{\mathbb{N}}$ $($observe that $\lambda_{L}({\bf 0})=1$, for any partition $L$). We define the root-Hadamard transform of $f$ at any vector ${\bf u}\in\mathbb{F}_{2}^{n}$ as the complex valued function:

[TABLE]

If $f$ is a Boolean function, we let ${\mathcal{T}}_{L,A,f}:={\mathcal{U}}_{L,A,f}^{(2)}$. A function $f$ on $\mathcal{GB}_{n}^{2^{k}}$ is said to be root-bent if the root-Hadamard transform is flat in absolute value, namely $|{\mathcal{U}}_{L,A,f}({\bf u})|=1$ for all ${\bf u}\in\mathbb{F}_{2}^{n}$. The sum

[TABLE]

is the root–crosscorrelation of $f$ and $g$ at $z$ (recall that $\mu_{s}=\alpha_{s}^{2}$). We define the root–autocorrelation of $f$ at ${\bf u}\in\mathbb{F}_{2}^{n}$ by

[TABLE]

When the sets $L,A$ are understood from the context, to simplify the notation, we may write ${\mathcal{U}}_{f},T_{f},{\mathcal{C}}_{f,g}^{p},{\mathcal{C}}_{f}^{p}$ in lieu of ${\mathcal{U}}_{L,A,f}^{(2^{k})}$, ${\mathcal{T}}_{L,A,f}^{(2^{k})}$, ${\mathcal{C}}_{L,A,f,g}^{p}$, ${\mathcal{C}}_{L,A,f}^{p}$.

Example 2**.**

Consider $A=\left\{e^{\frac{2\pi i}{4}},e^{\frac{2\pi i}{8}}\right\}$ and $L=\{(0,2),(1,3)\}$. The generalized Boolean function

[TABLE]

is root-bent, that is, $|{\mathcal{U}}_{L,A,f}({\bf u})|=1$ for all ${\bf u}\in\mathbb{F}_{2}^{n}$.

This transform is related to (but more general than) the concept of consta-Hadamard transform (see [11, 17]). First, we show that it is a proper kernel transform, so we show its invertibility.

Proposition 3**.**

Let $f\in\mathcal{GB}_{n}$, $A$ be a set of complex roots of $1$ and $\{R_{k}\}_{k\in K}$ be a partition of $\{0,1,\ldots,n-1\}$, indexed by the elements in $A$. Then, for any ${\mathbf{y}}\in\mathbb{F}_{2}^{n}$, we have that

[TABLE]

Proof.

We perform the following computation:

[TABLE]

and the claim follows. ∎

If $\alpha$ is a complex root of 1, we let $\mu=\alpha^{2}$ (recall the scalar product ${\mathbf{x}}\odot{\mathbf{z}}$ is computed over $\mathbb{C}$). We will make use throughout of the well-known identity on binary vectors (see [9])

[TABLE]

The following result is a collection of facts from [18, 19].

Proposition 4**.**

We have:

• $(1)$

If $f,g\in\mathcal{GB}_{n}^{q}$, then

[TABLE]

In particular, if $f=g$, then $\displaystyle{\mathcal{C}}_{f}({\bf u})=2^{-n}\sum_{{\mathbf{x}}\in\mathbb{F}_{2}^{n}}|{\mathcal{H}}_{f}({\mathbf{x}})|^{2}(-1)^{\langle{\bf u},{\mathbf{x}}\rangle}.$

• $(2)$

If $f,g\in{\mathcal{B}}_{n}$, then the nega–crosscorrelation equals

[TABLE]

We now prove a result similar to Proposition 4 for this newly defined transform.

Theorem 5**.**

Let $f,g\in{\mathcal{B}}_{n}^{2^{k}}$, $A$ be a set of complex roots of $1$ and $\{R_{k}\}_{k\in K}$ be a partition of $\{0,1,\ldots,n-1\}$ as before. The root-crosscorrelation of $f,g$ is

[TABLE]

Furthermore, the root-Parseval identity holds

[TABLE]

Moreover, $f$ is root-bent if and only if ${\mathcal{C}}_{L,A,f}^{p}({\bf u})=0$, for all ${\bf u}\neq{\bf 0}$.

Proof.

Using [3, Lemma 2.9] and identity (2), we write

[TABLE]

If $f=g$, then we get

[TABLE]

and by replacing ${\mathbf{z}}={\bf 0}$, then we get the root-Parseval identity. The last claim is also implied by the previous identity. ∎

Example 6**.**

The generalized Boolean function $f({\bf x})$ in Example 2 satisfies ${\mathcal{C}}^{p}_{L,A,f}({\bf u})=0$ for all ${\bf u}\neq{\bf 0}$ $($as predicted by Theorem 5$)$.

3 Complementary sequences

We next give a brief overview of Golay complementary pairs (CP) (see, for example, [4, 5, 8, 16] or the reader’s preferred reference on CP). Let ${\bf a}=\{a_{i}\}_{i=0}^{N-1}$ be a sequence of $\pm 1$ (bipolar) and let the aperiodic autocorrelation of ${\bf a}$ at $k$ be defined by ${\mathcal{A}}_{\bf a}(k)=\sum_{i=0}^{N-k-1}a_{i}a_{i+k}$, $0\leq k\leq N-1$. The periodic autocorrelation of ${\bf a}$ at $0\leq k\leq N-1$ is defined by ${\mathcal{C}}_{\bf a}(k)=\sum_{i=0}^{N-1}a_{i}a_{i+k}$, $0\leq k\leq N-1$, where we take indices modulo $N$. The negaperiodic autocorrelation is ${\mathcal{C}}^{n}_{f}({\bf u})=\sum_{i=0}^{2^{n}-1}a_{i}a_{i+k}(-1)^{\lfloor{(k+i)/2^{n}}\rfloor}$.

Two bipolar sequences ${\bf a},{\bf b}$ form a (Golay) complementary pair if

[TABLE]

We call them a $P$-complementary pair if

[TABLE]

and $N$-complementary pair if

[TABLE]

We associate a polynomial $A$ to the sequence ${\bf a}$ by $A(x)=a_{0}+a_{1}x+\cdots+a_{N-1}x^{N-1}$. It is rather straightforward to show that two sequences ${\bf a},{\bf b}$ (with corresponding polynomials $A,B$) form a Golay complementary pair if and only if

[TABLE]

Similarly, they form a $P$-complementary pair if

[TABLE]

and a $N$-complementary pair if

[TABLE]

Let $U,V$ be the following $N\times N$ matrices, defined by

[TABLE]

We can quickly see that the periodic and negaperiodic autocorrelations satisfy ${\mathcal{C}}_{\bf a}(k)={\bf a}\cdot{\bf a}U^{k}$ and ${\mathcal{C}}^{n}_{\bf a}(k)={\bf a}\cdot{\bf a}V^{k}$. It is interesting to note that a pair of sequences is complementary if and only if it is $P$-complementary and $N$-complementary, which easily follows from the identities

[TABLE]

All of the above concepts can be extended to a set of sequences $S=\{{\bf a}_{i}\}_{1\leq i\leq M}$ by imposing the sum of autocorrelations to be zero at nonzero shift, and we shall such, a complementary set) (with respect to some fixed autocorrelation). For example, the set $S$ is $P$-complementary (respectively, $N$-complementary) if $\displaystyle\sum_{i=1}^{M}{\mathcal{A}}_{{\bf a}_{i}}(k)=0$ (respectively, $\displaystyle\sum_{i=1}^{M}{\mathcal{C}}_{{\bf a}_{i}}(k)=0$), for $k\neq 0$. We shall also define the notion of pairwise complementary set $S$, by assuming that any pair within the set is complementary with respect to some autocorrelation.

The previous concepts take different forms for Boolean functions, since when we add vectors in the input of a function, we lose the circular permutation property (though, it can regarded as negacyclic permutation property. For two functions $f,g\in\mathcal{GB}_{n}^{q}$, we let the periodic/negaperiodic correlation of $f,g$ to be

[TABLE]

We say that two Boolean functions are complementary if and only if they are both $P$-complementary and $N$-complementary.

We will now define our new concept of (periodic and aperiodic) complementarity.

Definition 7**.**

We say that two pairs of functions $(a_{1},a_{2}),(b_{1},b_{2})$ are:

• $(i)$

$A$-crosscomplementary* if*

[TABLE]

• $(ii)$

$P$-crosscomplementary* if*

[TABLE]

• $(iii)$

$N$-crosscomplementary* if*

[TABLE]

4 Complementary pairs and components of generalized Boolean functions

The following lemma from [10, 18, 20], provides a relationship between the generalized Walsh-Hadamard transform and the classical transform. Recall the “canonical bijection” ${{\iota}\,:\,{{\mathbb{F}}_{2}^{k-1}}\,\rightarrow\,{{\mathbb{Z}}_{2^{k-1}}}}$, which is defined by $\iota({\bf c})=\sum_{j=0}^{k-2}c_{j}2^{j}$ where ${\bf c}=(c_{0},c_{1},\dots,c_{k-2})$.

Lemma 8**.**

For a generalized Boolean $f\in\mathcal{GB}_{n}^{2^{k}}$, $f({\mathbf{x}})=a_{0}({\mathbf{x}})+2a_{1}({\mathbf{x}})+\cdots+2^{k-1}a_{k-1}({\mathbf{x}})$, $a_{i}\in{\mathcal{B}}_{n}$, we have

[TABLE]

where $f_{\bf c}({\mathbf{x}})=c_{0}a_{0}({\mathbf{x}})\oplus c_{1}a_{1}({\mathbf{x}})\oplus\cdots\oplus c_{k-2}a_{k-2}({\mathbf{x}})\oplus a_{k-1}({\mathbf{x}})$ are the component functions of $f$.

The next lemma is known and easy to show.

Lemma 9**.**

If $b,c$ are bits and $z$ is a complex number, then

[TABLE]

In the next theorem, we generalize Lemma 8 with respect to our root-Hadamard transforms. The proof is interestingly enough, similar to the proof of Lemma 8, with some appropriate changes.

Theorem 10**.**

For a generalized Boolean $f\in\mathcal{GB}_{n}^{2^{k}}$, $A$ and $L$ as in Definition 1, and $f({\mathbf{x}})=a_{0}({\mathbf{x}})+2a_{1}({\mathbf{x}})+\cdots+2^{k-1}a_{k-1}({\mathbf{x}})$, $a_{i}\in{\mathcal{B}}_{n}$, we have

[TABLE]

where $f_{\bf c}({\mathbf{x}})=c_{0}a_{0}({\mathbf{x}})\oplus c_{1}a_{1}({\mathbf{x}})\oplus\cdots\oplus c_{k-2}a_{k-2}({\mathbf{x}})\oplus a_{k-1}({\mathbf{x}})$ are the component functions of $f$.

Proof.

Denoting $\displaystyle\gamma_{\bf c}=\prod_{i=0}^{k-2}\left(1+(-1)^{c_{i}}\zeta_{2^{k-i}}\right)$, where ${\bf c}=(c_{0},c_{1},\ldots,c_{k-2})\in{\mathbb{F}}_{2}^{k-1}$, we compute

[TABLE]

The theorem follows after easily expressing $\gamma_{\bf c}$ as a sum of powers of the complex roots of $1$ (or simply using [20, Lemma 5]). ∎

The next lemma will be used later.

Lemma 11** (Inversion Lemma).**

We let $F_{\bf u}:{\mathbb{F}}_{2}^{n}\to{\mathbb{C}}$ be a class of complex-valued functions indexed by ${\bf u}\in{\mathbb{F}}_{2}^{k-1}$. Then, for every ${\bf c}\in{\mathbb{F}}_{2}^{k-1}$, we have

[TABLE]

Proof.

Observe that

[TABLE]

by [3, Lemma 2.9], and our result follows. ∎

In the spirit of the Hadamard and nega-Hadamard complementary notions, we define a root-transform complementarity notion next. For $L=\{R_{j}\}_{j\in K}$, a partition of $\{0,1,\ldots,n-1\}$ and $A=\{\alpha_{j}\}_{j\in K}$, a set of complex roots of $1$, we say that a set $S=\{g_{i}\}_{i=1}^{M}$ of functions in $\mathcal{GB}_{n}^{q}$ ($q$ may be $2$) is $LA$-complementary if

[TABLE]

Moreover, two tuples $S_{1}=(f_{i})_{i=1}^{M},S_{2}=(g_{i})_{i=1}^{M}$ of (generalized or not) Boolean functions are $LA$-crosscomplementary if

[TABLE]

We give such an example below.

Example 12**.**

Let $A=\left\{e^{\frac{2\pi i}{4}},e^{\frac{2\pi i}{8}}\right\}$ and $L=\{\{0,2\},\{1,3\}\}$. Consider the generalized Boolean function $f\in\mathcal{GB}_{4}^{4}$, defined by $f({\bf x})=x_{1}+x_{2}+x_{3}+x_{4}+2f_{1}({\bf x})$ where $f_{1}$ is any Boolean function in the set $S_{F}$ $($see Table 1$)$. Then,

[TABLE]

Furthermore, let $g\in\mathcal{GB}_{4}^{4}$, defined by $g({\bf x})=x_{1}x_{2}+x_{3}+x_{4}+x_{1}x_{4}+2g_{1}({\bf x})$, where $g_{1}$ is a Boolean function in the set $S_{G}$ $($see Table 2$)$. Then,

[TABLE]

Clearly, ${\mathcal{C}}^{p}_{L,A,f}({\bf u})+{\mathcal{C}}^{p}_{L,A,g}({\bf u})=0$. In other words, the generalized Boolean functions $f,g$ are $LA$-complementary.

Theorem 13**.**

Let $S=\{f_{i}\}_{i\in I}$, $f_{i}\in\mathcal{GB}_{n}^{2^{k}}$, be a set of generalized Boolean functions and $A,L$ as in Definition 1. Then $S$ forms an $LA$-complementary set if and only if for all ${\bf a},{\bf c}\in{\mathbb{F}}_{2}^{k-1}$, $S_{\bf a}=\{f_{\bf a}\}_{f\in S},S_{\bf c}=\{f_{\bf c}\}_{f\in S}$ form a binary $LA$-crosscomplementary set.

Proof.

We first assume that $S$ forms an LA-complementary set. Then $\displaystyle\sum_{f\in S}{\mathcal{C}}_{L,A,f}^{p}({\bf v})=0$, for ${\bf v}\neq{\bf 0}$. From Theorem 5, we know that ${\mathcal{C}}_{L,A,f}^{p}({\bf v})=\lambda_{L}({\bf v})\sum_{{\bf u}\in{\mathbb{F}}_{2}^{n}}|{\mathcal{U}}_{L,A,f}({\bf u})|^{2}(-1)^{{\bf v}\cdot{\bf u}}$, and using our assumption along with Theorem 10 we obtain (we divide throughout by $\lambda_{L}({\bf v})$)

[TABLE]

and since $\{\zeta_{{2^{k}}}^{\iota({\bf d})}\}_{{\bf d}\in{\mathbb{F}}_{2}^{k-1}}$ is a basis of $\mathbb{Q}(\zeta_{{2^{k}}})$, then, for all ${\bf d}\in{\mathbb{F}}_{2}^{k-1}$,

[TABLE]

which, by the same reason as above, renders, for all ${\bf b},{\bf d}\in{\mathbb{F}}_{2}^{k-1}$,

[TABLE]

Inverting the previous equation using Lemma 11, we obtain $\displaystyle\sum_{f\in S}{\mathcal{C}}^{p}_{L,A,f_{\bf a},f_{\bf c}}({\bf v})=0$. The reciprocal follows easily from the identity

[TABLE]

The theorem is shown. ∎

Corollary 14**.**

Let $\{f,g\}$ be two generalized Boolean functions in $\mathcal{GB}_{n}^{2^{k}}$. Then $\{f,g\}$ forms an $P$-complementary $($respectively, $N$-complementary$)$ pair if and only if for all ${\bf a},{\bf c}\in{\mathbb{F}}_{2}^{k-1}$, $\{f_{\bf a},f_{\bf c}\}$ and $\{g_{\bf a},g_{\bf c}\}$ form a $($binary$)$ $P$-crosscomplementary $($respectively, $N$-crosscomplementary$)$ pair.

5 Transforms and complementary constructions

It is well known [12] that a Boolean function $f$ has flat spectrum with respect to the nega-Hadamard spectrum if $f+s_{2}$ (where $s_{2}$ is the quadratic elementary symmetric polynomial) has flat spectrum with respect to the Walsh-Hadamard transform. Thus, it is natural to ask whether some connection exists between the Walsh-Hadamard and the nega-Hadamard transforms defined on generalized Boolean functions. To that effect, we show the first claim of our next result (see [1] for the particular case of $k=1$). We can also relate the root-Hadamard transform and the generalized Hadamard transform.

We let $A=\{\alpha_{1},\ldots,\alpha_{r}\}$ be a set of roots of unity $\alpha_{j}=e^{\frac{2\pi i}{k_{j}}}$ such that $k_{j}=2^{m_{j}},\,m_{j}\leq k$, $K=\{k_{j}\}_{1\leq j\leq r}$ and $L=\{R_{s}\}_{s\in K}$ be a partition of the index set $\{0,\ldots,n-1\}=\bigsqcup_{s\in K}R_{s}$, $|L|=|K|=r$. For $J\subseteq K$, we let $A_{J}$ be $A$ with $\alpha_{s}$ replaced by $1$ for all $s\in J$. Let $s_{1}({\mathbf{x}})=\displaystyle\bigoplus_{j=1}^{n}x_{j}$, $s_{2}({\mathbf{x}})=\displaystyle\bigoplus_{1\leq j<k\leq n}x_{j}x_{k}$, and in general $s_{t}({\mathbf{x}})=\displaystyle\bigoplus_{1\leq j_{1}<\ldots<j_{t}\leq n}x_{j_{1}}\cdots x_{j_{1}}$ be the symmetric polynomials of degree 1, 2, $t$, respectively, all reduced modulo 2. We will use here Lemma 5 from [17]:

Lemma 15** ([17]).**

Let ${\mathbf{x}}\in{\mathbb{V}}_{n}$. Then,

[TABLE]

Theorem 16**.**

We have:

• $(i)$

*Let $n\geq 1$, $k\geq 1$, $f\in\mathcal{GB}_{n}^{2^{k}}$ and $g\in\mathcal{GB}_{n}^{2^{k+1}}$ defined by $g({\mathbf{x}})=2f({\mathbf{x}})+2^{k-1}s_{1}({\mathbf{x}})+2^{k}s_{2}({\mathbf{x}})$ *$($the sum is taken modulo $2^{k+1})$.Then, ${\mathcal{N}}_{f}^{(2^{k})}({\bf u})={\mathcal{H}}_{g}^{(2^{k+1})}({\bf u})$, for all ${\bf u}\in{\mathbb{F}}_{2}^{n}$.

• $(ii)$

*Let $n\geq 1$, $k\geq 1$, $f,\,h_{K},h_{J}\in\mathcal{GB}_{n}^{2^{k}}$ defined by $h_{K}({\mathbf{x}})=f({\mathbf{x}})-\sum_{s\in K}2^{k-m_{s}}\sum_{j=0}^{s-1}s_{2^{j}}({\mathbf{x}}_{R_{s}})$, $h_{J}({\mathbf{x}})=f({\mathbf{x}})-\sum_{s\in J}2^{k-m_{s}}\sum_{j=0}^{s-1}s_{2^{j}}({\mathbf{x}}_{R_{s}})$ *$($the sum is taken modulo $2^{k})$, where ${\mathbf{x}}_{R_{s}}$ is the restriction of ${\mathbf{x}}$ to the indices in $R_{s}$. Then,

[TABLE]

Proof.

We compute

[TABLE]

The two claims of $(ii)$ are similar so we will show only the first part. Note that, by Lemma 15, $wt({\mathbf{x}})\pmod{2^{s}}=\sum_{j=0}^{s-1}s_{2^{j}}({\mathbf{x}})$. We compute

[TABLE]

and the theorem is shown. ∎

Two of us introduced the next concept in [14]. We call a function $f\in\mathcal{GB}_{n}^{q}$ a landscape function if there exist $t\geq 1$, $m_{i}\in{\mathbb{N}}_{0},\ell_{i}\in 2{\mathbb{N}}_{0}+1$, $1\leq i\leq t$, such that

[TABLE]

We call the set of pairs $\{(m_{1},\ell_{1}),(m_{2},\ell_{2}),\ldots\}$, the levels of $f$, and $t+1$ (if [math] belongs to the Walsh-Hadamard spectrum), or $t$ (if [math] is not in the spectrum) the length of $f$.

We can deduce this corollary (the second claim of the next result was also shown in [1]).

Corollary 17**.**

Let $f,h\in{\mathcal{B}}_{n}$, where $h({\mathbf{x}})=f({\mathbf{x}})+s_{2}({\mathbf{x}})$. If $n$ is even, then $f$ is negabent if and only if $h$ is bent. Furthermore, $f$ is negaplateaued if and only if $h$ is plateaued. In general, $f$ is negalandscape $($defined as above, via the nega-Hadamard transform$)$ if and only if $h$ is landscape.

Proof.

By Theorem 16 $(i)$, for $k=1$, we have that if $f\in{\mathcal{B}}_{n}$ and $g\in\mathcal{GB}_{n}^{4}$ defined by $g({\mathbf{x}})=2f({\mathbf{x}})+s_{1}({\mathbf{x}})+2s_{2}({\mathbf{x}})$, then, ${\mathcal{N}}_{f}^{(2)}({\bf u})={\mathcal{H}}_{g}^{(4)}({\bf u})$, for all ${\bf u}\in{\mathbb{F}}_{2}^{n}$. Since the decomposition of $g$ is $g({\mathbf{x}})=a_{0}({\mathbf{x}})+2a_{1}({\mathbf{x}})$, where $a_{0}({\mathbf{x}})=s_{1}({\mathbf{x}})$, and $a_{1}({\mathbf{x}})=f({\mathbf{x}})+s_{2}({\mathbf{x}})$, this implies, by [10], that, when $n$ is even, $g$ is gbent if and only if $a_{1}$ and $a_{0}+a_{1}$ are bent Boolean functions. This implies that $g$ is gbent if and only if $f+s_{2}$ and $s_{1}+f+s_{2}$ are bent Boolean functions. Since $s_{1}$ is a linear Boolean function, this implies that $f$ is negabent if and only if $h=f+s_{2}$ is bent, giving yet another proof to this known result [12].

Further, by Corollary 1 of [14], we see that, if $g:{\mathbb{F}}_{2}^{n}\rightarrow{\mathbb{Z}}_{2^{k}}$ is a function given by $g({\mathbf{x}})=a_{0}({\mathbf{x}})+2a_{1}({\mathbf{x}})+\cdots+2^{k-1}a_{k-1}$, and $s\geq 0$ is an integer, then, $g$ is $s$-gplateaued if and only if, for each ${\bf c}\in{\mathbb{F}}_{2}^{k-1}$, the Boolean function $g_{\bf c}$ defined as $g_{\bf c}({\mathbf{x}})={\bf c}\cdot(a_{0}({\mathbf{x}}),\ldots,a_{k-2}({\mathbf{x}}))+a_{k-1}({\mathbf{x}})$ is an $s$-plateaued (if $n+s$ is even), respectively, an $(s+1)$-plateaued function (if $n+s$ is odd), with some extra conditions on the Walsh-Hadamard coefficients. In particular, taking $k=1$, this implies that $f$ is negaplateaued if and only if $f+s_{1}+s_{2}$ is plateaued (no extra conditions on the Walsh-Hadamard coefficients), which again implies that $f+s_{2}$ is plateaued.

Using Theorem 3.2 of [14], this argument can be also extended to landscape functions, in a similar way as in the plateaued case. ∎

6 Further comments

In this paper we defined a class of transforms which generalize many others, like generalizes the Walsh-Hadamard, nega-Hadamard, $2^{k}$-Hadamard [17], consta-Hadamard [11] and $HN$-transforms. For generalized Boolean functions, we describe its behavior on the binary components. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set. Some concrete examples are provided.

There are many questions one can ask on the new transforms. For example, it would be interesting to provide more constructions of root-bent and more generally, root-plateaued functions (surely, Theorem 16 may help). Certainly, finding connections between these transforms, their values, and (relative) difference sets would be quite interesting, as well.