On the Normality of Boolean quartics

Valérie Gillot , Philippe Langevin Imath, university of Toulon {valerie.gillot,philippe.langevin}@univ-tln.fr and Alexandr Polujan Otto-von-Guericke-Universität [email protected]

(Date: April 30th 2024)

Abstract.

In the BFA’2023 conference paper, A. Polujan, L. Mariot and S. Picek exhibited the first example of a non-normal but weakly normal bent function in dimension 8. In this note, we present numerical approaches based on the classification of Boolean spaces to explore in detail the normality of bent functions of 8 variables and we complete S. Dubuc’s results for dimensions less or equal to 7. Based on our investigations, we show that all bent functions in 8 variables are normal or weakly normal. Finally, we conjecture that more generally all Boolean functions of degree at most 4 in 8 variables are normal or weakly normal.

1. Introduction

In the context of random generation of non-normal bent functions in 8 variables, a non-normal but weakly normal bent quartic was presented by A. Polujan, L. Mariot and S. Picek at BFA’2023; see [9]. Furthermore, a non-normal function of degree 6 in 8 variables was presented by S. Dubuc in her PhD thesis, and we verify that this function is in fact weakly normal. For that dimension, the main question concerns the existence of Boolean functions that are neither normal nor weakly normal, we will say abnormal. For that purpose, we propose a numerical approach based on the classification of Boolean spaces under the action of the affine general linear group to study the relative degree of Boolean functions. We namely use most of the old [8] and recent [5] classification results obtained in the last decade to analyse the maximal relative degree that a Boolean function can have. Our main objective is to update the knowledge of Boolean functions in small dimensions, thus completing the results in the PhD work of S. Dubuc [4], and answering definitively on the (weak)-normality of bent function and related questions in 8 variables.

Let ${\mathbb{F}}_{2}$ be the finite field of order $2$ . Let $m$ be a positive integer. We denote $B(m)$ the set of Boolean functions $f\colon{\mathbb{F}}^{m}_{2}\rightarrow{\mathbb{F}}_{2}$ . Every Boolean function has a unique algebraic reduced representation:

(1)

f(x_{1},x_{2},\ldots,x_{m})=f(x)=\sum_{S\subseteq\{1,2,\ldots,m\}}a_{S}X_{S},\quad a_{S}\in{\mathbb{F}}_{2},\ {\color[rgb]{0,0,0}X_{S}=\prod_{s\in S}x_{s}}.

The degree of $f$ is the maximal cardinality of $S$ with $a_{S}=1$ in the algebraic form. In this paper, we conventionally fix the degree of the null function to zero. We denote by $B(s,t,m)$ the space of Boolean functions of valuation greater than or equal to $s$ and of degree less than or equal to $t$ . Note that $B(s,t,m)=\{0\}$ whenever $s>t$ . The affine general linear group of ${\mathbb{F}}^{m}_{2}$ , denoted by ${\textsc{agl}}(m,2)$ , acts naturally over all these spaces. A system of representatives of $B(s,t,m)$ is denoted $\widetilde{B}(s,t,m)$ . In next sections, we are using the classifications available in the project [7], more precisely those which interest us here are listed in Table 1.

Table 1. Useful spaces and the corresponding number of classes

$B(s,t,m)$	$B(1,5,5)$	$B(1,6,6)$	$B(1,3,7)$	$B(4,7,7)$	$B(2,3,8)$	$B(4,4,8)$
$\sharp\widetilde{B}(s,t,m)$	206	$7\,888\,299$	$1\,890$	$68\,443$	$20\,748$	$999$

2. Normality and Relative degree

Let $V$ be a linear subspace of ${\mathbb{F}}^{m}_{2}$ , we denote $\mathcal{B}(V)$ the space of Boolean functions from $V$ to ${\mathbb{F}}_{2}$ . For $a\in{\mathbb{F}}^{m}_{2}$ , we denote $f_{a,V}$ the Boolean function of $\mathcal{B}(V)$ defined by $v\in V\mapsto f(a+v)$ .

The notion of normality was introduced by Dobbertin [3]. In this paper, we use the following terminologies of P. Charpin [2]. A Boolean function $f\in B(m)$ is said to be normal if there exists an affine subspace $V+a$ of ${\mathbb{F}}^{m}_{2}$ with dimension $\lceil m/2\rceil$ where $f_{a,V}$ is constant i.e. $\deg(f_{a,V})=0$ . The function $f$ is weakly normal when $f_{a,V}$ is affine, and not constant i.e. $\deg(f_{a,V})=1$ , on an affine space $V+a$ of dimension $\lceil m/2\rceil$ .

Example 1.

For a positive integer $t$ , the quadric $x_{1}x_{t+1}+x_{2}x_{t+2}+\cdots+x_{t}x_{2t}+x_{2t+1}$ is not normal, but weakly normal in $B(2t+1)$ .

The normality of Boolean functions is an invariant under the action of ${\textsc{agl}}(m,2)$ .

Now, we introduce the relative degree that generalizes normality notions. The relative degree of $f$ with respect the affine space $a+V$ is the degree of $f_{a,V}$ , denoted by: $\deg_{a+V}(f)=\deg(f_{a,V})$ . Note that if $f_{a,V}=0$ , then $\deg_{a+V}(f)=0$ . We define the $r$ -degree of $f\in B(m)$ as the minimal relative degree of $f$ for all affine spaces $a+V$ , where $\dim(V)=r$ :

0\leq\deg_{r}(f)=\min\{\deg_{a+V}(f)\mid\dim V=r\quad\text{and}\quad a\in{\mathbb{F}}^{m}_{2}\}.

Notions of normality are translated into

\deg_{\lceil m/2\rceil}(f)=\begin{cases}0,&\text{$f$ is normal};\\ 1,&\text{$f$ is non-normal and weakly normal};\\ \geq 2,&\text{ $f$ is abnormal}.\\ \end{cases}

For integers $r\leq m$ and $k\leq m$ , we propose to study the following combinatorial parameter

D_{r}(k,m)=\max\{\deg_{r}(f)\mid\deg(f)\leq k\}.

As we will see, it may be interesting to look at more closely what is happening by degree, for that the denote by $D_{r}^{\dagger}(k,m)$ the maximum over the Boolean functions of degree equal to $k$ . In general, the determination of $D_{r}(k,m)$ seems to be a hard problem. Since the affine general linear group acts on the set of affine space of given dimension preserving the degree of functions, whence the relative degrees are affine invariants of $B(m)$ . So, we can use the result of classification of Boolean functions to determine some of these parameters. Each space of dimension $r$ ( $r$ -space) of ${\mathbb{F}}^{m}_{2}$ has $2^{m-r}$ cosets and the number of $r$ -spaces is given by Gauss binomial coefficient $\genfrac{[}{]}{0.0pt}{}{m}{r}$ . Taking into account the cost of the computation of the degree of a Boolean function, the work factor to determine the $r$ -relative degree of function in $B(s,t,m)$ by brute force:

W(r,s,t,m)=\sharp\widetilde{B}(s,t,m)\times 2^{m-r}\genfrac{[}{]}{0.0pt}{}{m}{r}\times r2^{r},\quad\genfrac{[}{]}{0.0pt}{}{m}{r}=\prod_{i=0}^{r-1}\frac{2^{m}-2^{i}}{2^{r}-2^{i}}.

For $m=6$ , $r=4$ and $\widetilde{B}(1,6,6)=7\,888\,299\approx 2^{23}$ , the work factor is about $2^{40}$ . So, we complete the tables $D^{\dagger}_{r}(k,6)$ and $D^{\dagger}_{r}(k,5)$ in a matter of minutes, using a 80-core computer. But, since $\widetilde{B}(2,4,7)=118\,140\,881\,980\approx 2^{37}$ , we can not reasonably use brute force in dimension $7$ .

Table 2. The values of

D_{r}^{\dagger}(k,m)

, for a fixed

m

$D^{\dagger}_{r}(k,5)$

$r\backslash k$	2	3	4	5
4	2	2	3	2
3	1	1	1	1
2	0	0	0	0

$D^{\dagger}_{r}(k,6)$

$r\backslash k$	2	3	4	5	6
5	2	3	4	3	4
4	2	2	2	2	2
3	0	0	0	0	0
2	0	0	0	0	0

$D^{\dagger}_{r}(k,7)$

$r\backslash k$	2	3	4	5	6	7
6	2	3	$\geq 4$	$\geq 4$	5	$\geq 4$
5	2	3	$\geq 3$	$\geq 3$	$\geq 3$	$\geq 3$
4	1	1	$\geq 1$	$\geq 1$	$\geq 1$	$\geq 1$
3	0	0	0	0	0	0
2	0	0	0	0	0	0

Numerical fact 1.

Using the classification of $B(1,6,6)$ , we complete the table $D^{\dagger}_{r}(k,6)$ . As proved in [4], all the functions in $B(6)$ are normal. We point this implies that $D_{3}(k,m)=0$ , for all $m\geq 6$ .

Among the 12 members of $\widetilde{B}(5,7,7)$ , all have 6-relative degree 0 except

h(x)=x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}+x_{2}x_{3}x_{4}x_{5}x_{7}+x_{1}x_{3}x_{4}x_{6}x_{7}+x_{1}x_{2}x_{5}x_{6}x_{7},

for which $\deg_{6}(h)=5$ fixing the entry (6,6) of the table $D^{\dagger}_{r}(k,7)$ .

Numerical fact 2.

Using the classification of $B(1,3,7)$ , we fill the left part of the table $D^{\dagger}_{r}(k,7)$ . All cubics in $B(7)$ are normal or weakly normal.

The minoration in the right part of the table $D^{\dagger}_{r}(k,7)$ were obtained at random, by adding a random cubic to the members of $\widetilde{B}(4,7,7)$ . The computation suggests the non-existence of abnormal quartic :

Conjecture 1.

All the quartics of $B(7)$ are normal or weakly normal.

Remark 1.

Verifying the previous conjecture, which consists of listing all the functions of the form form $q+h$ with $q\in B(2,2,7)$ and $h\in\widetilde{B}(3,4,7)$ i.e. $2^{21}\times 2^{16}$ functions, does not seem numerically out of reach.

3. Normality in dimension 8

In her PhD thesis, S. Dubuc gave and presented the first example of a non-normal function. It has degree 6 comprising 140 monomials, but we checked that this function is weakly normal and equivalent to the following one:

\begin{split}f(x)=&x_{2}x_{3}x_{4}x_{5}x_{7}x_{8}+x_{2}x_{3}x_{4}x_{5}x_{8}+x_{2}x_{3}x_{4}x_{7}x_{8}+x_{2}x_{3}x_{4}x_{6}+x_{2}x_{3}x_{5}x_{6}\\ +&x_{2}x_{3}x_{4}x_{8}+x_{2}x_{4}x_{6}x_{8}+x_{2}x_{5}x_{7}x_{8}+x_{1}x_{2}x_{3}+x_{3}x_{4}x_{5}+x_{2}x_{5}x_{6}+x_{2}x_{4}x_{7}\\ +&x_{3}x_{4}x_{7}+x_{4}x_{5}x_{8}+x_{3}x_{6}x_{8}+x_{3}x_{7}x_{8}+x_{2}x_{3}+x_{2}x_{5}+x_{3}x_{5}+x_{4}x_{6}+x_{2}x_{7}\\ +&x_{3}x_{8}+x_{7}x_{8}+x_{1}+x_{2}.\end{split}

She also proved that all the bent cubics in 8 variables are normal. A fact that we can improve strongly in two ways, by proving that all 8-bit cubics are normal or weakly normal and by proving that all 8-bit bent functions are normal or weakly normal.

Problem 1.

Does there exist a Boolean function $B(8)$ with 4-relative degree 2?

Proposition 1.

All the cubics in $B(8)$ are normal or weakly normal.

Proof.

The distribution by relative degrees of the $20\,748$ representatives of $B(2,3,8)$ is given in Table 3. For example, the entry (6,1) which is 21 means that there are 21 cubics of 6-relative degrees equal to 1. The distribution for $r=4$ gives the result. ∎

Table 3. Distribution of relative degrees of cubics in

\widetilde{B}(2,3,8)

$r\backslash\deg_{r}$	0	1	2	3
8	$1$	$0$	$0$	$20\,747$
7	$10$	$0$	$53$	$20\,712$
6	$130$	$21$	$1\,910$	$18\,714$
5	$5\,504$	$5\,227$	$10\,044$	$0$
4	$20\,748$	$0$	$0$	$0$

Remark 2.

An alternative proof of the previous proposition is to use the numerical fact 2. Indeed, for $f\in RM(3,8)$ and for any hyperplane $H\subset{\mathbb{F}}_{2}^{8}$ , the restriction of $f$ to $H$ is a cubic in $B(H)\sim B(7)$ .

We say that a homogeneous quartic $h$ in $B(m)$ is bentable, if there exists a Boolean cubic $c$ in $B(m)$ such that $h+c$ is bent. In that case, we say that $h+c$ has type $S_{N}$ , where $N$ is the order of the stabilizer of $h$ in $B(4,4,8)$ . It is known [8] that 536 of the 999 classes of $B(4,4,8)$ , are bentable and they can be arranged in 418 classes using the notion of the dual bent function. At BFA’2023, A. Polujan, L. Mariot and S. Picek [9] presented the first example of a non-normal bent quartic in 8 variables. This example belongs to the partial spread class and has type $S_{1}$ .

Numerical fact 3.

It is known that up duality, there are 25 bentable homogeneous quartics of type $S_{1}$ , providing $26\,532\,848\approx 2^{24.7}$ classes of bent functions, in which we identify 361 classes of non-normal but weakly normal bent functions.

To decide on the normality of a large set of 8-bit bent functions, we use a rather approach initialized by the pre-computation of the $3\,108\,960$ affine 3-spaces as a list of $97\,155$ 3-spaces and their 32 cosets. A Boolean function is weakly normal is there exists a pair of cosets of the same 3-space on which $f$ is constant. If $f$ takes the same value on these cosets then it is normal. Using a bit programming implementation in C, we were able to show the following result.

Numerical fact 4.

All the bent functions in $B(8)$ are normal or weakly normal.

We have adapted the counting method presented in [8] to enumerate a cover-set of 8-bit bent functions of degree 4. This cover set contains $355\,073\,617\approx 2^{28.52}$ bent functions and data are available in [6]. Checking the normality or weak normality of these functions, we were able to otain fact 4. Since 8-bit bent functions have a degree of at most 4, the previous result leads us to consider a more general phenomenon :

Conjecture 2.

All the Boolean quartics in $B(8)$ are normal or weakly normal.

In the space of homogeneous quadratic forms of 8 variables, a space of dimension 28, the degree of Boolean indicator of bent functions has degree 4 in B(8) . This last conjecture is an attempt to precise the structure of the set of bent functions within the space of components of a 8-bit APN function.

Acknowledgments

Philippe Langevin is partially supported by the French Agence Nationale de la Recherche through the SWAP project under Contract ANR-21-CE39-0012.

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