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\title[A quadratic part of a bent function can be any]{A quadratic part of a bent function can be any}

\author{{N.N. TOKAREVA}}%
\address{Natalia Nikolaevna Tokareva
	\newline\hphantom{iii} Sobolev Institute of Mathematics,
	\newline\hphantom{iii} pr. Koptyuga, 4,
	\newline\hphantom{iii} 630090, Novosibirsk, Russia}%
\email{tokareva@math.nsc.ru}%

\thanks{\sc Tokareva, N.N.,
A quadratic part of a bent function can be any}
\thanks{\copyright \ 2022 Tokareva, N.N.}
\thanks{\rm The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation and Laboratory of Cryptography JetBrains Research.
}
\thanks{\it Received  March, 1, 2022, published  March, 1,  2022.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. } Boolean functions in $n$ variables that are on the maximal possible 
Hamming distance from all affine Boolean functions in $n$ variables 
are called bent functions ($n$ is even). They are intensively studied 
since sixties of XX century in relation to applications in cryptography and discrete mathematics. Often, 
bent functions are represented in their algebraic normal form (ANF). 
It is well known that the linear part of ANF of a bent function can be 
arbitrary. In this note we prove that a quadratic part of a bent function 
can be arbitrary too. \medskip

\noindent{\bf Keywords:} Boolean function, bent function, linear function,
quadratic function, homogeneous function.
 \end{quote}
}

\section{Introduction}

Recall that Boolean functions in even number of variables that are
on the maximal possible Hamming distance from the set of all affine
Boolean functions are called bent functions \cite{Rothaus-1976}. Bent functions play an
important role in constructions of symmetric ciphers since they help
to defend ciphers against linear cryptanalysis\cite{Matsui} and have many applications in discrete mathematics and communications, see \cite{Tokareva-book-2015}. It is well known
that every Boolean function can be in the unique way represented in
its Algebraic Normal Form (ANF). This representation is used very
often for property description and realization of a Boolean
function. It is known that bent functions are too far from
classification. No conditions on ANF of a Boolean function are
known in order to say that the function is bent.

In this paper a new problem in bent functions is stated and studied: is
it true that an arbitrary homogeneous Boolean function of degree $k$
in $n$ variables ($n$ is even) is a $k$-degree part in ANF of some
bent function in $n$ variables? For small $k$ it can be formulated
like this. Is it true that linear (quadratic, cubic, etc.) part of
ANF of a bent function can be arbitrary? For sure, this question is
interesting nor only for bent functions.


It is well known that a linear part in ANF of a bent function can be
arbitrary. Moreover, any linear function can be added to a bent
function without changing its property to be bent. In this paper we
prove that a quadratic part of a bent function can also be
arbitrary. Namely, we prove that an arbitrary quadratic homogeneous
Boolean function in $n$ variables is a quadratic part of some bent
function in $n$ variables, where $n$ is even, $n\geqslant 6$. For cubic parts the question remains open.




%Maximally nonlinear Boolean functions in $n$ variables, where $n$ is
%even, are called bent functions. There are several ways to represent
%Boolean functions. One of the most useful is via algebraic normal
%form (ANF). What can we say about ANF of a bent function? We try to
%collect all known and new facts related to ANF of a bent function. A
%new problem in bent functions is stated and studied: is it true that
%a linear, quadratic, cubic, etc. part of ANF of a bent function can
%be arbitrary? The case of linear part is well studied before. In
%this paper we prove that a quadratic part of a bent function can be
%arbitrary too. %They are intensively studied since sixties of XX century 




\section{Preliminaries}

We use the following standard notation:

\medskip $\mathbb{F}_2^n$ --- the vector space over $\mathbb{F}_2$;

$x=(x_1,\ldots,x_n)$ --- a binary vector;

$f, g: \mathbb{F}_2^n \to \mathbb{F}_2$ --- Boolean functions;

$dist(f,g)$ --- {\em Hamming distance} between $f$ and $g$, i.~e.
the number of coordinates in which their vectors of values differ;





%$\langle x,y \rangle = x_1 y_1 \oplus \ldots \oplus x_n y_n$ --- the
%standard {\em inner product} modulo $2$;

%$W_f(y) = \sum_{x\in \mathbb{F}_2^n} (-1)^{\langle x,y \rangle
%\oplus f(x)}$ --- the {\em Walsh~---~Hadamard transform} of a
%Boolean function $f$;

$a_1 x_1 \oplus \ldots \oplus a_n x_n \oplus b$ --- an {\em affine function} in
variables $x_1,\ldots,x_n$, where $a\in \mathbb{F}_2^n$ and $b\in \mathbb{F}_2$, sign $\oplus$ stands for addition modulo $2$ (XOR);

{\em bent function} --- a Boolean function in $n$ variables ($n$ is
even) that is on the maximal possible Hamming distance from the set of all
affine functions. It is known \cite{Rothaus-1976} that this distance is equal to $2^{n-1}-2^{(n/2)-1}$;

$\mathcal{A}_n$ --- the set of all affine functions in $n$
variables;

$\mathcal{B}_n$ --- the set of all bent functions in $n$ variables.


\noindent \medskip Recall that any Boolean function can be uniquely represented
in its {\em algebraic normal form} (ANF):
$$f(x_1,\ldots,x_n) = \left( \bigoplus_{k=1}^{n} \bigoplus_{i_1,\ldots,i_k}
a_{i_1,\ldots,i_k} \; x_{i_1}\cdot \ldots \cdot x_{i_k} \right)
\oplus a_0,$$ where for each $k$ indices $i_1,\ldots, i_k$ are
pairwise distinct and sets $\{i_1,\ldots,i_k\}$ are exactly all
different nonempty subsets of the set $\{1,\ldots, n\}$;
coefficients $a_{i_1,\ldots,i_k}$, $a_0$ take values from
$\mathbb{F}_2$. For a Boolean function $f$ the number of variables in the longest
item of its ANF is called the {\em algebraic degree} of a function
(or briefly {\em degree}) and is denoted by $deg(f)$. A Boolean
function is {\em affine}, {\em quadratic}, {\em cubic} and so on if
its degree is not more than $1$, or equal to $2$, $3$, etc.


%\section{Properties of the ANF of a bent function}

In what follows let $n$ be an even number. 

According to O.Rothaus
(1966, 1976) \cite{Rothaus-1976} and V.~A.~Eliseev,
O.~P.~Stepchenkov (1962) \cite{Tokareva-book-2015}, degree $deg(f)$ of a bent function $f$ in
$n\geqslant 4$ variables is not more than $n/2$. If $n=2$ a bent
function is quadratic. For any possible degree from $2$ to $n/2$ it is not difficult
to construct a bent function of such degree.

%It is not difficult to prove that all variables are presented 
%in ANF of an arbitrary bent functions.

Several restrictions on ANF of bent functions  can be naturally considered. 
A~bent function is called {\em homogeneous} if all monomials of its
ANF are of the same degree. C.~Qu, J.~Seberri and J.~Pieprzyk
proved \cite{QuSeberryPieprzyk2000} that there are $30$ homogeneous
bent functions of degree $3$ in $6$ variables. Partial results
on classification of cubic homogeneous bent functions in $8$
variables were obtained by C.Charnes et al. in 
\cite{Charnes-Dempwolff-Pieprzyk-2008}. C.~Charnes, M.~Rotteler and T.~Beth \cite{CharnesRottelerBeth2002}
have proved the following fact that we will use further.

\begin{proposition}\label{theorem-exist-cubic-homogeneous}
 There exist cubic homogeneous bent
functions in each even number of variables $n$ for $n\geqslant 6$.
\end{proposition}


For the homogeneous bent functions of higher degrees it is known only a little. 



\section{On the quadratic part of ANF of a bent function}

It is well known that the class of bent functions is closed under
addition of affine functions and under affine transformations of
variables, see \cite{Cusick-Stanica}. In other words it holds

\begin{proposition}\label{th-affine-transform}
For any bent function $g$ in $n$ variables ($n$ is even, $n\geqslant
2$) the function
$g'(x) = g(Ax\oplus b) \oplus c_1 x_1 \oplus \ldots \oplus c_n x_n \oplus d$
is also bent, where $A$ is a nonsingular matrix, $b$, $c$ are
arbitrary binary vectors of length $n$, $d$ is a constant from $\mathbb{F}_2$.
\end{proposition}

Functions $g$ and $g'$ are called {\em EA-equivalent}.

Note that we can add an arbitrary affine function to a bent function without changing its property to be bent.
Recall that it is not possible to find a non affine Boolean function that does the same, since for any non affine Boolean function $f$
there exists a bent function $g$ such that $f\oplus g$ is not bent, 
see \cite{Tokareva-auto-2010}, \cite{Tokareva-duality-2012}. For instance, it is not possible even to add a quadratic function to all bent functions in order to save their property to be bent. But we want to prove that it is possible to find a bent function with an arbitrary quadratic part of ANF!

In this section we show that an arbitrary quadratic homogeneous 
Boolean function in $n$ variables is a quadratic part of some 
bent function in $n$ variables, where $n$ is even, $n\geqslant 6$.

To prove this fact, we need the following statements.

In \cite{on-graphs-in-6-vertices} one can find

\begin{proposition}\label{graphs-6} There exist exactly $156$ nonisomorphic graphs with $6$
vertices.
\end{proposition}

In \cite{on-graphs-in-6-vertices} all these graphs can be found. Let us
prove first the following result.

\begin{proposition}\label{prop-6} An arbitrary quadratic homogeneous Boolean function in $6$ variables is a
quadratic part of some bent function in $6$ variables.
\end{proposition}

\begin{proof} Let us put into the correspondence to an arbitrary
quadratic homogeneous Boolean function $f$ in $6$ variables a graph
$G_f$ on $6$ vertices by the following rule: vertices correspond to
variables; there is an edge between two vertices if and only if the
product of corresponding variables belongs to ANF of $f$.

Consider only those quadratic homogeneous Boolean functions that
correspond to nonisomorphic graphs. It is clear that if a quadratic
homogeneous function $f$ is a quadratic part of some bent function
then any quadratic homogeneous function $f'$ with graph $G_{f'}$
isomorphic to $G_f$ is also a quadratic part of some bent function.
It holds since any permutation on vertices produce an affine
transformation of variables and hence by Proposition
\ref{th-affine-transform} does not change a property of a function
to be bent.

According to Proposition \ref{graphs-6} there are exactly $156$
nonisomorphic graphs with $6$ variables. We prove the statement by
listing in the table in Appendix 1 all $156$ corresponding (to graphs)
homogeneous quadratic Boolean functions and cubic parts that can be
added to them in order to get a bent function in every case. So, the
function equal to the sum of the quadratic function from the second
column and cubic function from the third column of the table is
always bent. Note that we list quadratic parts in the lexicographical order. 
For every quadratic part we have found a cubic part of the minimal 
possible length. Sometimes it is of length $0$ and we put sign "$-$"\, in the table: it means that a 
quadratic part is already a bent function.
Symbol $|$ in both columns should be replaced by $\oplus$, and items like $12$ and $123$ by $x_1x_2$ and $x_1x_2x_3$ respectively. We use such short notation in the table for a compactness. Thus, we prove the statement. 
\end{proof}


The following iterative construction was proposed by O.~Rothaus
(1966, 1976) and J.~Dillon (1974), see \cite{Tokareva-book-2015}.

\medskip
\begin{proposition}\label{theo-Rothaus} Let $f'$, $f''$, $f'''$ be bent functions
in $n$ variables such that $f'\oplus f''\oplus f'''$ is a bent
function too. Then
$$g(x,x_{n+1},x_{n+2}) = f'(x)f''(x) \oplus  f'(x)f'''(x) \oplus f''(x)f'''(x)\oplus $$
$$x_{n+1} f'(x) \oplus x_{n+1} f''(x) \oplus x_{n+2} f'(x) \oplus x_{n+2} f'''(x) \oplus
x_{n+1} x_{n+2}$$ is a bent function in $n+2$ variables.
\end{proposition}

Now let us prove the main result.

\begin{theorem}\label{th-main} An arbitrary quadratic homogeneous Boolean function in $n$ variables is a
quadratic part of some bent function in $n$ variables, where $n$ is
even, $n\geqslant 6$.
\end{theorem}

\begin{proof}
Let us prove it by induction. For $n=6$ the result follows from Proposition \ref{prop-6}. Suppose that it is proven for some $n$. Consider the case of $n+2$
variables. Let $x$ be a vector of variables $(x_1,\ldots,x_n)$.
Assume that $q(x,x_{n+1},x_{n+2})$ is an arbitrary homogeneous
quadratic Boolean function in $n+2$ variables. If $q$ is identically
zero, then by Proposition \ref{theorem-exist-cubic-homogeneous} there
exists a cubic homogeneous bent function in every number of
variables: it will be a bent function with an empty quadratic part.

Let us consider a nonzero $q$. Since it is nonzero, there exists at
least one item in its ANF. W.l.o.g. suppose that ANF of $q$ contains
item $x_{n+1}x_{n+2}$. Otherwise by renumbering of variables we turn
to this case. So, $q(x,x_{n+1},x_{n+2})$ is of the form:
$q(x,x_{n+1},x_{n+2}) = h(x) \oplus  a(x) x_{n+1} \oplus  b(x) x_{n+2} \oplus  x_{n+1} x_{n+2},$
where $h$ is a homogeneous quadratic Boolean function in $n$
variables, $a$, $b$ are some linear functions in $n$ variables.

Consider the quadratic homogeneous Boolean function $h(x) \oplus
a(x)b(x)$ in $n$ variables. By induction, there exists a cubic
homogeneous Boolean function $c(x)$ such that $f'(x) = c(x) \oplus
h(x) \oplus  a(x)b(x)$ is a bent function in $n$ variables. Let
$f''(x) = f'(x) \oplus  a(x)$ and $f'''(x) = f'(x) \oplus  b(x)$.
According to Proposition \ref{th-affine-transform} functions $f''$,
$f'''$ are bent too. Note that $f'\oplus f''\oplus f'''$ is also
bent by the same reason.

Then, by Proposition \ref{theo-Rothaus} a Boolean function
$$g(x,x_{n+1},x_{n+2}) = f'(x)f''(x) \oplus   f'(x)f'''(x) \oplus
f''(x)f'''(x)$$
$$\oplus  x_{n+1} f'(x) \oplus  x_{n+1} f''(x) \oplus  x_{n+2} f'(x) \oplus  x_{n+2} f'''(x) \oplus
x_{n+1} x_{n+2}$$ is a bent function in $n+2$ variables. We see that
$$g(x,x_{n+1},x_{n+2}) = f'(x)(f'(x)\oplus a(x)) \oplus   f'(x)(f'(x)\oplus b(x)) \oplus
(f'(x)\oplus a(x))(f'(x)\oplus b(x))$$
$$\oplus  x_{n+1} f'(x) \oplus  x_{n+1} (f'(x)\oplus a(x)) \oplus  x_{n+2} f'(x) \oplus  x_{n+2} (f'(x)\oplus b(x)) \oplus
x_{n+1} x_{n+2} = $$
$$f'(x) \oplus  a(x)b(x) \oplus  a(x) x_{n+1} \oplus  b(x) x_{n+2} \oplus  x_{n+1} x_{n+2}.$$
Hence, we get a bent function $$g(x,x_{n+1},x_{n+2}) = c(x) \oplus
h(x) \oplus  a(x) x_{n+1} \oplus  b(x) x_{n+2} \oplus  x_{n+1}
x_{n+2} = c(x) \oplus  q(x,x_{n+1},x_{n+2})$$ in $n+2$ variables
with prescribed quadratic part $q(x,x_{n+1},x_{n+2})$.
\end{proof}

%{\bf Remark.} Note that a cubic part for the fixed quadratic
%function $q(x,x_{n+1},x_{n+2})$ was found easy enough. Moreover it
%was ```reused'' from the previous step of induction. May be it can
%be explained by ``a high degree of freedom'' in construction of bent
%function's ANF? We think so.


\section{Future remarks}

Can a $k$-degree part of ANF of a bent function be any?

In particular, is it true that the cubic part of a bent function can be arbitrary?

$\bullet\;$ In case $n=6$ the answer is {\bf no}, since there exists
only three classes of nonequivalent cubic bent functions: $123 \oplus 14
\oplus 25 \oplus 36$, $123 \oplus 245 \oplus 12 \oplus 14 \oplus 26 \oplus 35 \oplus 45$ and $123 \oplus 245 \oplus
346 \oplus 14 \oplus 26 \oplus 34 \oplus 35 \oplus 36 \oplus 45 \oplus 46$, but there are five classes
of nonequivalent homogeneous cubic Boolean functions in $6$
variables. So, we need to have items of the next degree in order to
have a possibility to ``put'' all variants of the cubic part in a
bent function. Here in notation we again use $123$ for $x_1x_2x_3$ and so on.

$\bullet\;$ Case $n=8$ is still open. The problem is that the
existing classification of quartic bent functions in $8$ variables
(obtained by P.~Langevin and G.~Leander in 2011, see
\cite{LangevinLeander--2011}) does not include the list of
representatives of EA-classes.

\medskip
We think it is a very interesting open problem to study in the
general case.


%\section{Bent decomposition problem in terms of ANF}

In 2011 we have formulated the following hypothesis, see
\cite{Tokareva-amc-2011}.

\medskip \noindent {\bf Hypothesis 1}. \emph{Any Boolean
function in $n$ variables of degree not more than $n/2$ can be
represented as the sum of two bent functions in $n$ variables ($n$
is even, $n\geqslant 2$).}

\medskip
The problem to prove or disprove this hypothesis is known now as the
{\em Bent sum decomposition problem}. 
It is closely connected to the problem of asymptotic of the number
of all bent functions.

\medskip
For now the following is known in relation to this
hypothesis.

$\bullet\;$ Hypothesis is confirmed for $n=2, 4, 6$ (see
\cite{Tokareva-amc-2011} and \cite{QuLi}).

$\bullet\;$ Hypothesis was proved for quadratic Boolean functions,
Maiorana---McFarland bent functions, partial spread functions, see
\cite{QuLi}.

$\bullet\;$ A weakened variant of the hypothesis was proved: any Boolean function of
degree not more than $n/2$ can be represented as the sum of the fixed number of
bent functions in $n$ variables, that depends on $n$, see \cite{Tokareva-semr-2014}.

Hypothesis 1 can be reformulated like this: \emph{an arbitrary ANF
of degree not more than $n/2$ can be ``divided'' into two parts ---
every part gives the ANF of a bent function.}

\medskip
Here we just give an idea that follows from Hypothesis 1 (assuming
it holds): {\em $k$-degree part of the ANF of a bent function
``tends'' to be arbitrary}. It is necessary that at least
$\sqrt{2^{\binom{n}{k}}}$ different variants of $k$-degree part of ANF
should be realized in a bent function. Recall that the total number
of all such variants is $2^{\binom{n}{k}}.$


\section{Conclusion}

It is very interesting to study if it is possible to
define a bent function through the conditions on ANF. Of course,
these questions are interesting in respect to an arbitrary class of
cryptographic Boolean functions, not only to bent functions. The author is very grateful to E.\,Ponomareva for valuable
contribution in proving of Theorem \ref{th-main}, to V.\,Idrisova for kind help and remarks,
and to the reviewer of this paper for the careful reading of the work and comments.



\bigskip

\begin{thebibliography}{9}



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{\it The eight variable homogeneous degree three bent
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\bibitem{CharnesRottelerBeth2002} C. Charnes, M. Rotteler, T. Beth, 
{\it Homogeneous bent functions, invariants, and designs}, Designs, Codes and Cryptography, \textbf{26}:1--3 (2002), 139--154.


\bibitem{Cusick-Stanica} T. Cusick, P. Stanica, 
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\bibitem{Matsui} M. Matsui, 
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\bibitem{LangevinLeander--2011} P. Langevin, G. Leander, 
{\it Counting all bent functions in dimension eight 99270589265934370305785861242880}, Designs, Codes and Cryptography, \textbf{59} (2011), 193--205.


\bibitem{on-graphs-in-6-vertices} 
{\it List of all $156$ nonisomorphic graphs
on $6$ vertices}, \textbf{https://users.cecs.anu.edu.au/~bdm/data/graphs.html}.

%\bibitem{online-encyclopedia} 
%{\it The On-Line Encyclopedia of Integer
%Sequences}, Edited by N.J.A.Sloane, \textbf{https://oeis.org/}.



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\bibitem{Tokareva-book-2015} N. Tokareva, 
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\end{thebibliography}



\newpage
\section*{Appendix 1n}

{\footnotesize \begin{center} 
\begin{tabular}{|l|l|l|}
\hline
       N & homogeneous quadratic function & homogeneous cubic function\\
	\hline

     1	& $-$  &  $123|125|126|134|136|145|146|156|234|235|245|246|256|345|346|356$ \\
     2	& $12$  &  $134|136|145|156|235|236|245|246|345|346|356|456$ \\
     3	& $12|13$  &  $126|136|146|156|234|235|236|245|256|345|346|456$ \\
     4	& $12|13|14$  &  $234|235|236|246|256|345|356|456$ \\
     5	& $12|13|14|15$  &  $123|124|126|146|234|236|245|246|256|345|346|356$ \\
     6	& $12|13|14|15|16$  &  $123|125|134|145|234|235|245|246|256|345|346|356$ \\
     7	& $12|13|14|15|16|23$  &  $146|156|245|256|345|346|456$ \\
     8	& $12|13|14|15|16|23|24$  &  $125|134|135|345|356|456$ \\
     9	& $12|13|14|15|16|23|24|25$  &  $124|125|134|145|345|346|356$ \\
    10	& $12|13|14|15|16|23|24|25|26$  &  $123|125|135|136|146|234|245|345|346|356$ \\
    11	& $12|13|14|15|16|23|24|25|26|34$  &  $125|135|156|256|356$ \\
    12	& $12|13|14|15|16|23|24|25|26|34|35$  &  $124|134|146|246|346$ \\
    13	& $12|13|14|15|16|23|24|25|26|34|35|36$  &  $124|125|135|145|234|245|345$ \\
    14	& $12|13|14|15|16|23|24|25|26|34|35|36|45$  &  $-$ \\
    15	& $12|13|14|15|16|23|24|25|26|34|35|36|45|46$  &  $145|156|235|256|345$ \\
    16	& $12|13|14|15|16|23|24|25|26|34|35|36|45|46|56$  &  $-$ \\
    17	& $12|13|14|15|16|23|24|25|26|34|35|45$  &  $123|125|134|136|146$ \\
    18	& $12|13|14|15|16|23|24|25|26|34|36|45$  &  $-$ \\
    19	& $12|13|14|15|16|23|24|25|26|35|36|45|46$  &  $134|145|156|234|235|256|345$ \\
    20	& $12|13|14|15|16|23|24|25|26|35|46$  &  $-$ \\
    21	& $12|13|14|15|16|23|24|25|34$  &  $-$ \\
    22	& $12|13|14|15|16|23|24|25|34|35$  &  $124|145|234|246|346$ \\
    23	& $12|13|14|15|16|23|24|25|34|35|45$  &  $-$ \\
    24	& $12|13|14|15|16|23|24|25|34|35|46|56$  &  $124|134|135|145|245$ \\
    25	& $12|13|14|15|16|23|24|25|34|36$  &  $-$ \\
    26	& $12|13|14|15|16|23|24|25|35|46$  &  $123|126|136$ \\
    27	& $12|13|14|15|16|23|24|26|34|35|45$  &  $-$ \\
    28	& $12|13|14|15|16|23|24|34$  &  $125|145|235|256|356$ \\
    29	& $12|13|14|15|16|23|24|34|56$  &  $-$ \\
    30	& $12|13|14|15|16|23|25|26|34$  &  $125|135|235$ \\
    31	& $12|13|14|15|16|23|25|26|34|35|46$  &  $123|124|134$ \\
    32	& $12|13|14|15|16|23|25|34$  &  $-$ \\
    33	& $12|13|14|15|16|24|25|26|34|35$  &  $124|145|245$ \\
    34	& $12|13|14|15|16|24|25|26|34|35|36$  &  $123|125|134|145|235|246|345$ \\
    35	& $12|13|14|15|16|24|25|34|35$  &  $123|124|145|245|246|256$ \\
    36	& $12|13|14|15|16|24|25|34|36$  &  $-$ \\
    37	& $12|13|14|15|16|24|25|35|36|46$  &  $-$ \\
    38	& $12|13|14|15|16|24|25|36$  &  $124|134|234$ \\
    39	& $12|13|14|15|16|24|35$  &  $-$ \\
    40	& $12|13|14|15|23$  &  $126|136|245|256|345|346|456$ \\
    41	& $12|13|14|15|23|24$  &  $126|134|136|346|356|456$ \\
    42	& $12|13|14|15|23|24|25$  &  $124|125|126|135|136|145|146|246|345|346|356$ \\
    43	& $12|13|14|15|23|24|25|34$  &  $126|136|156|256|356$ \\
    44	& $12|13|14|15|23|24|25|34|35$  &  $124|126|136|146|234|246|346$ \\
    45	& $12|13|14|15|23|24|25|34|35|45$  &  $146|156|236|256|346$ \\
    46	& $12|13|14|15|23|24|25|34|36|46|56$  &  $-$ \\
    47	& $12|13|14|15|23|24|25|35|46$  &  $-$ \\
    48	& $12|13|14|15|23|24|25|36$  &  $123|145|156|234|456$ \\
    49	& $12|13|14|15|23|24|25|36|46$  &  $123|135|136|235|356$ \\
    50	& $12|13|14|15|23|24|25|36|46|56$  &  $134|145|234|235|345$ \\
    51	& $12|13|14|15|23|24|26$  &  $123|345|346|356$ \\
    52	& $12|13|14|15|23|24|26|34$  &  $-$ \\
    53	& $12|13|14|15|23|24|26|34|35$  &  $123|145|146|156$ \\
    54	& $12|13|14|15|23|24|26|34|35|45$  &  $-$ \\
    55	& $12|13|14|15|23|24|26|34|36|45$  &  $123|125|126|136|156$ \\
    56	& $12|13|14|15|23|24|26|34|36|46|56$  &  $124|125|126|136|145$ \\
    57	& $12|13|14|15|23|24|26|34|56$  &  $123|125|135$ \\
    58	& $12|13|14|15|23|24|26|35|36$  &  $124|134|234$ \\
    59	& $12|13|14|15|23|24|26|35|36|46$  &  $123|124|234$ \\
    60	& $12|13|14|15|23|24|26|36$  &  $-$ \\

    
    
    

        \hline
\end{tabular}\end{center}}

{\footnotesize \begin{center} %\rotatebox{90}
{
\begin{tabular}{|l|l|l|}
\hline
       N & homogeneous quadratic function & homogeneous cubic function\\
\hline

    61	& $12|13|14|15|23|24|26|36|45$  &  $-$ \\
    62	& $12|13|14|15|23|24|26|36|46|56$  &  $123|126|135|136|235|356$ \\
    63	& $12|13|14|15|23|24|26|56$  &  $123|135|235$ \\
    64	& $12|13|14|15|23|24|34$  &  $126|146|236|256|356$ \\
    65	& $12|13|14|15|23|24|34|56$  &  $-$ \\
    66	& $12|13|14|15|23|24|36$  &  $-$ \\
    67	& $12|13|14|15|23|24|36|46$  &  $134|235|236|356$ \\
    68	& $12|13|14|15|23|24|36|46|56$  &  $134|135|345$ \\
    69	& $12|13|14|15|23|24|56$  &  $123|136|236$ \\
    70	& $12|13|14|15|23|25|26|34|35|46$  &  $-$ \\
    71	& $12|13|14|15|23|25|26|34|36|46|56$  &  $123|126|136$ \\
    72	& $12|13|14|15|23|25|34$  &  $126|136|236$ \\
    73	& $12|13|14|15|23|25|36|46$  &  $-$ \\
    74	& $12|13|14|15|23|26$  &  $124|345|346|456$ \\
    75	& $12|13|14|15|23|26|36$  &  $126|134|245|246|456$ \\
    76	& $12|13|14|15|23|26|36|45$  &  $124|126|246$ \\
    77	& $12|13|14|15|23|26|36|46$  &  $-$ \\
    78	& $12|13|14|15|23|26|36|46|56$  &  $126|135|136|145|246|345$ \\
    79	& $12|13|14|15|23|26|46$  &  $-$ \\
    80	& $12|13|14|15|23|26|46|56$  &  $124|145|245$ \\
    81	& $12|13|14|15|24|25|26|34|35$  &  $124|145|146|156|245$ \\
    82	& $12|13|14|15|24|25|26|34|35|36|46|56$  &  $124|126|135|136|145|235|236|245|246|345$ \\
    83	& $12|13|14|15|24|25|26|34|36$  &  $-$ \\
    84	& $12|13|14|15|24|25|26|35|36|46$  &  $-$ \\
    85	& $12|13|14|15|24|25|34|35$  &  $123|124|136|146|234|256|346$ \\
    86	& $12|13|14|15|24|25|35|36|46$  &  $123|124|134$ \\
    87	& $12|13|14|15|24|26|34|35$  &  $145|146|156$ \\
    88	& $12|13|14|15|24|26|35$  &  $-$ \\
    89	& $12|13|14|15|24|26|35|36$  &  $145|146|156$ \\
    90	& $12|13|14|15|24|26|35|36|46$  &  $-$ \\
    91	& $12|13|14|15|24|26|35|36|46|56$  &  $123|126|136$ \\
    92	& $12|13|14|15|24|35$  &  $126|136|236$ \\
    93	& $12|13|14|15|24|36$  &  $-$ \\
    94	& $12|13|14|15|25|36|46$  &  $123|136|236$ \\
    95	& $12|13|14|15|26$  &  $124|125|234|245|345|346|356$ \\
    96	& $12|13|14|15|26|36$  &  $123|124|126|136|146|234|245|345$ \\
    97	& $12|13|14|15|26|36|46$  &  $123|134|234|235|245$ \\
    98	& $12|13|14|15|26|36|46|56$  &  $123|126|135|145|146|234|235|236|246|345$ \\
    99	& $12|13|14|23$  &  $136|235|246|256|345|356|456$ \\
   100	& $12|13|14|23|24$  &  $134|135|136|146|156|345|356|456$ \\
   101	& $12|13|14|23|24|34$  &  $125|126|136|156|235|256|356$ \\
   102	& $12|13|14|23|24|34|56$  &  $-$ \\
   103	& $12|13|14|23|24|35|46$  &  $-$ \\
   104	& $12|13|14|23|24|56$  &  $134|135|136|146|345$ \\
   105	& $12|13|14|23|25$  &  $126|346|356|456$ \\
   106	& $12|13|14|23|25|34|56$  &  $124|126|146$ \\
   107	& $12|13|14|23|25|35$  &  $125|136|246|256|456$ \\
   108	& $12|13|14|23|25|35|45$  &  $123|124|136|146|256|346$ \\
   109	& $12|13|14|23|25|35|46|56$  &  $-$ \\
   110	& $12|13|14|23|25|36$  &  $-$ \\
   111	& $12|13|14|23|25|36|45$  &  $124|126|146$ \\
   112	& $12|13|14|23|25|36|45|56$  &  $-$ \\
   113	& $12|13|14|23|25|45|56$  &  $-$ \\
   114	& $12|13|14|23|25|56$  &  $-$ \\
   115	& $12|13|14|23|26|45|56$  &  $134|136|146$ \\
   116	& $12|13|14|23|56$  &  $-$ \\
   117	& $12|13|14|24|25|35$  &  $126|136|236$ \\
   118	& $12|13|14|24|25|35|36|45$  &  $125|126|156$ \\
   119	& $12|13|14|24|26|35|36|45|56$  &  $125|126|156$ \\
   120	& $12|13|14|24|35$  &  $126|156|256$ \\

   
   \hline
\end{tabular}
}\end{center}}


{\footnotesize \begin{center} %\rotatebox{90}
{
\begin{tabular}{|l|l|l|}
\hline
       N & homogeneous quadratic function & homogeneous cubic function\\
	\hline

   121	& $12|13|14|24|35|56$  &  $-$ \\
   122	& $12|13|14|25$  &  $126|234|236|346|356|456$ \\
   123	& $12|13|14|25|26$  &  $123|135|156|345|346|356$ \\
   124	& $12|13|14|25|26|34|35|56$  &  $124|125|145$ \\
   125	& $12|13|14|25|26|34|56$  &  $-$ \\
   126	& $12|13|14|25|26|35$  &  $-$ \\
   127	& $12|13|14|25|26|35|36|45$  &  $123|124|126|136|146|234$ \\
   128	& $12|13|14|25|26|35|36|45|46$  &  $124|135|146|236|256|345$ \\
   129	& $12|13|14|25|26|35|45$  &  $123|125|134|136|146|234$ \\
   130	& $12|13|14|25|26|35|46$  &  $-$ \\
   131	& $12|13|14|25|26|56$  &  $234|235|345$ \\
   132	& $12|13|14|25|35$  &  $234|236|246|256|356|456$ \\
   133	& $12|13|14|25|35|45$  &  $123|126|134|136|146|234|236|246$ \\
   134	& $12|13|14|25|35|46$  &  $125|126|256$ \\
   135	& $12|13|14|25|35|46|56$  &  $123|126|236$ \\
   136	& $12|13|14|25|35|56$  &  $234|236|246$ \\
   137	& $12|13|14|25|36$  &  $-$ \\
   138	& $12|13|14|25|56$  &  $234|236|346$ \\
   139	& $12|13|14|26|35|56$  &  $-$ \\
   140	& $12|13|14|56$  &  $234|235|236|246|345$ \\
   141	& $12|13|23$  &  $125|126|134|146|156|234|245|256|345|346|456$ \\
   142	& $12|13|23|45$  &  $126|134|146|156|234|256|346$ \\
   143	& $12|13|23|45|46$  &  $125|134|145|156|234|256|345$ \\
   144	& $12|13|23|45|46|56$  &  $124|126|134|135|145|146|156$ \\
   145	& $12|13|24$  &  $136|145|146|156|235|236|345|356|456$ \\
   146	& $12|13|24|34$  &  $123|124|126|134|135|145|146|156|235|236|256|356$ \\
   147	& $12|13|24|34|56$  &  $124|135|136|145|156|245|246$ \\
   148	& $12|13|24|45$  &  $236|256|356$ \\
   149	& $12|13|25|34|45$  &  $126|146|246$ \\
   150	& $12|13|25|36|45|46$  &  $123|124|134$ \\
   151	& $12|13|25|45|46$  &  $-$ \\
   152	& $12|13|25|46$  &  $-$ \\
   153	& $12|13|45$  &  $125|134|156|235|236|246|256|346$ \\
   154	& $12|13|45|46$  &  $124|125|145|146|156|235|236|256$ \\
   155	& $12|34$  &  $125|126|135|146|156|236|245|256|356|456$ \\
   156	& $12|34|56$  &  $-$ \\
        \hline
\end{tabular}
}\end{center}}


\end{document}
