Leopardi-Bent-functions-Taipa-2018-talk.tex

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\title[Yet another database of strongly regular graphs]{Yet another database of
\\
strongly regular graphs:
\\
Classifying bent functions by
\\
their Cayley graphs}
\author{Paul Leopardi}

\date{For Tight Frames and Approximation
\\
Taipa, February 2018}

\institute{University of Melbourne
\\
Australian Government - Bureau of Meteorology}
\titlegraphic{
%\includegraphics[angle=0,width=10mm]{../../common/beamer-anu-colourlogo.png}
%\includegraphics[angle=0,width=20mm]{../../common/carma_logo.jpg}
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\begin{document}

\frame{\titlepage}
\begin{frame}
\frametitle{Acknowledgements}
\begin{center}
Robin Bowen,
An Braeken,
Nathan Clisby,
Robert Craigen,
Joanne Hall,
David Joyner,
Philippe Langevin,
Matthew Leingang,
William Martin,
Padraig {\'O} Cath{\'a}in,
Judy-anne Osborn,
Dima Pasechnik,
William Stein,
Natalia Tokareva, and
Sanming Zhou.

~

Australian National University. University of Newcastle, Australia. University of Melbourne.

Australian Government - Bureau of Meteorology.

~

National Computational Infrastructure.

~

SageMath, CoCalc, Bliss, Nauty, MPI4py, SQLite3, DB Browser for SQLite, PostgreSQL, Psycopg2.
\end{center}
\end{frame}

\begin{frame}
\frametitle{Serious Question}
\begin{center}
\vspace{+10mm}
\large{}
What would \Emph{you} do with a ``large'' database of strongly regular graphs?
\normalsize{}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Overview}
%\begin{center}
\begin{itemize}
\item
Classifing bent functions by their Cayley graphs.
\begin{itemize}
\item
Preliminaries.
\item
Cayley graphs and linear codes.
\item
Equivalence of bent functions.
\item
Computational results for quadratic forms.
\item
Block designs.
\item
Computational results for low dimensions.
\item
Questions.
\end{itemize}

~

\item
Databases of strongly regular graphs
\begin{itemize}
\item
Precedents: databases of strongly regular graphs.

\item
Preliminary database designs.

\item
Prototype databases.

\item
Prospects.

\item
Source code and documentation.
\end{itemize}

\end{itemize}

%\end{center}
\end{frame}

\section{Classifying bent functions}

\subsection{Preliminaries}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{Motivation}

In a construction for Hada\-mard matrices, I encountered
two sequences of \Emph{bent} Boolean functions,
\begin{align*}
\sigma_m &: \F_2^{2m} \To \F_2, \quad \tau_m : \F_2^{2m} \To \F_2,
\end{align*}
whose Cayley graphs are \Emph{strongly regular} with parameters
\begin{align*}
(v,k,\lambda,\mu) &= (4^m, 2^{2 m - 1} - 2^{m-1}, 2^{2 m - 2} - 2^{m-1}, 2^{2 m - 2} - 2^{m-1}),
\end{align*}
but the graphs for $\sigma_m$ and $\tau_m$ are isomorphic only when

$m=1,2$ or $3.$

~

Question: \Emph{Which strongly regular graphs arise as Cayley graphs of bent Boolean functions?}

\slidecite{L 2014, 2015, 2017}
\end{frame}
\end{colortheme}

\begin{colortheme}{seagull}

\begin{frame}
\frametitle{Bent functions}
% Bent functions can be defined in a number of equivalent ways.
% The definition used here involves the Walsh Hadamard Transform.
\begin{Def}
\label{def-Walsh-Hadamard-transform}
The Walsh Hadamard transform of
a Boolean function $f : \F_2^{2m} \To \F_2$ is
\begin{align*}
W_f(x)
&:=
\sum_{y \in \F_2^{2m}} (-1)^{f(y) + \langle x, y \rangle}
\end{align*}
\end{Def}

\begin{Def}
\label{def-Bent-function}
A Boolean function $f : \F_2^{2m} \To \F_2$ is \Emph{bent}
if and only if its Walsh Hada\-mard transform has constant absolute value $2^{m}$.
% \cite[p. 74]{Dil74}
% \cite[p. 300]{Rot76}.
\end{Def}
\slidecite{Dillon 1974; Rothaus 1976}
\end{frame}
\begin{frame}
\frametitle{Dual bent functions}

\begin{Def}
\label{def-dual-Bent-function}
For a bent function  $f : \F_2^{2m} \To \F_2$, the function $\dual{f}$, defined by
\begin{align*}
(-1)^{\dual{f}(x)} &:= 2^{-m} W_f(x)
\end{align*}
is called the \Emph{dual} of $f$.
\end{Def}

~

The function $\dual{f}$ is also a bent function on $\F_2^{2m}$.

~

~

\slidecite{Dillon 1974; Rothaus 1976}
\end{frame}

\begin{frame}
\frametitle{Example}

The function  $f : \F_2^2 \To \F_2$  defined by $f(x) := x_0 x_1$
is bent, since
\begin{align*}
W_f(x)
=&
\sum_{y \in \F_2^2} (-1)^{f(y) + \langle x, y \rangle}
\\
=&\ (-1)^{f(0,0) + \langle x, (0,0) \rangle}
 + (-1)^{f(1,0) + \langle x, (1,0) \rangle} +
\\
\phantom{=}&\ (-1)^{f(0,1) + \langle x, (0,1) \rangle}
 + (-1)^{f(1,1) + \langle x, (1,1) \rangle}
\\
=&\ (-1)^{0 + 0} + (-1)^{0 + x_0} + (-1)^{0 + x_1} + (-1)^{1 + x_0 + x_1}
\\
=&\ 1 + (-1)^{x_0} + (-1)^{x_1} - (-1)^{x_0 + x_1}
\\
=&\ 2 \times (-1)^{f(x)},
\end{align*}
so $\dual{f} = f$, and $f$ is \Emph{self-dual}.
%
\end{frame}

\begin{frame}
\frametitle{Bent functions and affine functions}
Bent functions are at maximum Hamming distance from affine functions.
For $f : \F_2^2 \To \F_2,$ this distance is 1 \slidecite{Meier and Staffelbach 1989}.
\scriptsize{}
\begin{align*}
\begin{array}{|cccc|}
\hline
(0,0)& (1,0)& (0,1)& (1,1)
\\
\hline
0 & 0 & 0 & 0
\\
\Red{0}& \Red{0} & \Red{0} & \Red{1}
\\
\Red{0}& \Red{0} & \Red{1} & \Red{0}
\\
0 & 0 & 1 & 1
\\
\Red{0}& \Red{1} & \Red{0} & \Red{0}
\\
0 & 1 & 0 & 1
\\
0 & 1 & 1 & 0
\\
\Red{0}& \Red{1} & \Red{1} & \Red{1}
\\
\Red{1}& \Red{0} & \Red{0} & \Red{0}
\\
1 & 0 & 0 & 1
\\
1 & 0 & 1 & 0
\\
\Red{1}& \Red{0} & \Red{1} & \Red{1}
\\
1 & 1 & 0 & 0
\\
\Red{1}& \Red{1} & \Red{0} & \Red{1}
\\
\Red{1}& \Red{1} & \Red{1} & \Red{0}
\\
1 & 1 & 1 & 1
\\
\hline
\end{array}
\end{align*}
\normalsize{}
\end{frame}

\end{colortheme}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{Weights and weight classes}
\begin{Def}
The \Emph{weight} of a binary function is the cardinality of its \Emph{support}.
For $f$ on $\F_2^{2m}$
\begin{align*}
\support{f} &:= \{x \in \F_2^{2m} \mid f(x)=1 \}.
\end{align*}

A bent function $f$ on $\F_2^{2m}$ has weight
\begin{align*}
\weight{f} &= 2^{2 m - 1} - 2^{m-1} \quad (\text{weight class~} \weightclass{f}=0), \text{~or}
\\
\weight{f} &= 2^{2 m - 1} + 2^{m-1} \quad (\text{weight class~} \weightclass{f}=1).
\end{align*}
% If $f(0)=0$ then $\weightclass{\Cay{f}} := \weightclass{f}$.
\end{Def}
\end{frame}

\end{colortheme}

\subsection{Cayley graphs and linear codes}

\begin{colortheme}{seagull}

\begin{frame}
\frametitle{The Cayley graph of a Boolean function}
%\begin{center}
The \Emph{Cayley graph} $\Cay{f}$ of a Boolean function

~

\begin{align*}
%
f : \F_2^n \To \F_2 \quad \text{where} \quad f(0) = 0
%
\end{align*}

~

is
an undirected graph with

\begin{align*}
V(\Cay{f}) &:= \F_2^n, \quad (x,y) \in E(\Cay{f}) \Leftrightarrow f(x+y) = 1.
\end{align*}

~

\slidecite{Bernasconi and Codenotti 1999}
\end{frame}

\begin{frame}
\frametitle{Strongly regular graphs}
%\begin{center}
A simple graph $\Gamma$ of order $v$ is \Emph{strongly regular} with parameters
$(v,k,\lambda,\mu)$ if

~

\begin{itemize}
 \item
each vertex has degree $k,$

~
 \item
each adjacent pair of vertices has $\lambda$ common neighbours, and

~
\item
each nonadjacent pair of vertices has $\mu$ common neighbours.
\end{itemize}

~

~

\slidecite{Brouwer, Cohen and Neumaier 1989}

%\end{center}
\end{frame}

\begin{frame}
\frametitle{Bent functions and strongly regular graphs}

\begin{Proposition}
\smallcite{Bernasconi and Codenotti 1999}

The Cayley graph $\Cay{f}$ of a bent function $f$ on $\F_2^{2m}$

with $f(0)=0$ is a strongly regular graph with $\lambda = \mu.$
\end{Proposition}

The parameters of $\Cay{f}$ are
\begin{align*}
(v,k,\lambda) = &(4^m, 2^{2 m - 1} - 2^{m-1}, 2^{2 m - 2} - 2^{m-1})
\\
  \text{or} \quad &(4^m, 2^{2 m - 1} + 2^{m-1}, 2^{2 m - 2} + 2^{m-1}).
\end{align*}

~

\slidecite{Menon 1962; Dillon 1974; Bernasconi and Codenotti 1999}
%\end{center}
\end{frame}

\begin{frame}
\frametitle{Projective two-weight binary codes}

\begin{Def}
A \Emph{two-weight binary code} with parameters $[n,k,d]$ is a $k$ dimensional subspace of $\F_2^n$
with
minimum Hamming distance $d$, such that the set of Hamming weights of the non-zero vectors has size
2.

~

``A \Emph{generator matrix} $G$ of a linear code $[n, k]$ code $C$ is any matrix
of rank $k$ (over $\F_2$) with rows from $C.$''

~

``A linear $[n, k]$ code is called \Emph{projective} if no two columns of a generator matrix
$G$ are linearly dependent, i.e., if the columns of $G$ are pairwise different points in a
projective $(k-1)$-dimensional space.''
In the case of $\F_2$, no two columns are equal.

~

\end{Def}

\slidecite{Tonchev 1996; Bouyukliev, Fack, Willems and Winne 2006}

\end{frame}

\begin{frame}
\frametitle{From bent function to linear code (1)}
\begin{Def}

\smallcite{Carlet 2007; Ding and Ding 2015, Corollary 10}

For a bent function $f : \F_2^{2m} \To \F_2$,
define the linear code $C(f)$ by the generator matrix
\begin{align*}
M C(f) &\in \F_2^{4^m \times \weight{f}},
\\
M C(f)_{x,y} &:= \langle x, \support{f}(y) \rangle,
\end{align*}
with $x$ in lexicographic order of $\F_2^{2m}$
and $\support{f}(y)$ in lexicographic order of $\support{f}$.

The $4^m$ words of the code $C(f)$ are the rows of the generator matrix $M C(f)$.
\end{Def}

\slidecite{Carlet 2007; Ding and Ding 2015, Corollary 10}

\end{frame}
\begin{frame}
\frametitle{From bent function to linear code (2)}
\begin{Proposition}
\smallcite{Carlet 2007, Prop. 20; Ding and Ding 2015, Corollary 10}

For a bent function $f : \F_2^{2m} \To \F_2$, the linear code $C(f)$
is a projective two-weight code.

~

The possible weights of non-zero code words are:
\begin{align*}
\begin{cases}
2^{2m-2}, 2^{2m-2} - 2^{m-1} & \text{if~} \weightclass{f}=0.
\\
2^{2m-2}, 2^{2m-2} + 2^{m-1} & \text{if~} \weightclass{f}=1.
\end{cases}
\end{align*}

\end{Proposition}

\slidecite{Carlet 2007, Prop. 20; Ding and Ding 2015, Corollary 10}

\end{frame}

\begin{frame}
\frametitle{From linear code to strongly regular graph}
\begin{Def}
\label{R-f-def}
Given $f : \F_2^{2m} \To \F_2$, form the linear code $C(f)$.

The graph $R(f)$ is defined as:

Vertices of $R(f)$ are code words of $C(f)$.

For $v,w \in C(f)$, edge $(u,v) \in R(f)$ if and only if
\begin{align*}
\begin{cases}
\weight{u+v} = 2^{2m-2} - 2^{m-1} & (\text{if~}\weightclass{f}=0).
\\
\weight{u+v} = 2^{2m-2} + 2^{m-1} & (\text{if~}\weightclass{f}=1).
\end{cases}
\end{align*}

\end{Def}
Since $C(f)$ is a projective two-weight code,
$R(f)$ is a strongly regular graph.

\slidecite{Delsarte 1972, Theorem 2}
\end{frame}

\end{colortheme}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{The strongly regular graph $R(f)$ is \\ the Cayley graph of the dual}

\begin{Theorem}
For $f : \F_2^{2m} \To \F_2$, with $f(0)=0$,
\begin{align*}
R(f) &\equiv \Cay{\dual{f} + \dual{f}(0)} = \Cay{\dual{f} + \weightclass{f}}.
\end{align*}
\end{Theorem}

\end{frame}

\end{colortheme}

\subsection{Equivalence of bent functions}

\begin{colortheme}{seagull}

\begin{frame}
\frametitle{Extended affine equivalence}

\begin{Def}
For bent functions $f,g : \F_2^{2m} \To \F_2$,

$f$ is \Emph{extended affine equivalent} to $g$ if and only if
\begin{align*}
g(x) &= f(A x + b) + \langle c, x \rangle + \delta
\end{align*}
for some $A \in GL(2m,2)$, $b, c \in \F_2^{2m}$, $\delta \in \F_2$.
\end{Def}

~

~

\slidecite{Budaghyan, Carlet and Pott 2006; Carlet and Mesnager 2011}
\end{frame}

\end{colortheme}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{General linear equivalence}

\begin{Def}
For bent functions $f,g : \F_2^{2m} \To \F_2$,
$f$ is \Emph{general linear equivalent} to $g$ if and only if
\begin{align*}
g(x) &= f(A x)
\end{align*}
for some $A \in GL(2m,2)$.
\end{Def}
\end{frame}
\begin{frame}
\frametitle{Extended translation equivalence}

\begin{Def}
For bent functions $f,g : \F_2^{2m} \To \F_2$,

$f$ is \Emph{extended translation equivalent} to $g$ if and only if
\begin{align*}
g(x) &= f(x + b) + \langle c, x \rangle + \delta
\end{align*}
for $b, c \in \F_2^{2m}$, $\delta \in \F_2$.
\end{Def}
\end{frame}

\begin{frame}
\frametitle{Cayley equivalence}
\begin{Def}
%
For $f, g : \F_2^{2m} \To \F_2$, with both $f$ and $g$ bent,

we call $f$ and $g$ \Emph{Cayley equivalent},
and write $f \equiv g$,

if and only if $f(0)=g(0)=0$ and $\Cay{f} \equiv \Cay{g}$ as graphs.

~

Equivalently, $f \equiv g$ if and only if $f(0)=g(0)=0$ and

there exists a bijection $\pi : \F_2^{2m} \To \F_2^{2m}$ such that
\begin{align*}
g(x+y) &= f \big(\pi(x)+\pi(y)\big) \quad \text{for all~} x,y \in \F_2^{2m}.
\end{align*}
\end{Def}
\end{frame}
\begin{frame}
\frametitle{Extended Cayley equivalence}
\begin{Def}
For $f, g : \F_2^{2m} \To \F_2$, with both $f$ and $g$ bent,

if there exist $\delta, \epsilon \in \{0,1\}$ such that $f + \delta \equiv g + \epsilon$,

we call $f$ and $g$ \Emph{extended Cayley (EC) equivalent} and write $f \cong g$.
\end{Def}
Extended Cayley equivalence is an equivalence relation on the set of all bent functions on $\F_2^{2m}$.
\end{frame}

\begin{frame}
\frametitle{General linear equivalence \\ implies Cayley equivalence}

\begin{Theorem}
If $f$ is bent with $f(0)=0$ and $g(x) := f(A x)$ where $A \in GL(2m,2)$,
then $g$ is bent with $g(0)=0$ and $f \equiv g$.
\end{Theorem}
\begin{proof}
\begin{align*}
g(x+y) &= f\big(A(x+y)\big) = f(A x + A y)\quad \text{for all~} x,y \in \F_2^{2m}.
\end{align*}
\end{proof}

\end{frame}

\begin{frame}
\frametitle{Extended affine, extended translation, and extended Cayley equivalence (1)}

\begin{Theorem}
For $A \in GL(2m,2)$, $b, c \in \F_2^{2m}$, $\delta \in \F_2$,
$f : \F_2^{2m} \To \F_2$,

the function
\begin{align*}
h(x) &:= f(A x + b) + \langle c, x \rangle + \delta
\intertext{can be expressed as $h(x) = g(A x)$ where}
g(x) &:= f(x+b) + \langle (A^{-1})^T c, x \rangle + \delta,
\end{align*}
and therefore if $f$ is bent then $h \cong g$.
\end{Theorem}
\end{frame}

\begin{frame}
\frametitle{Extended affine, extended translation, and extended Cayley equivalence (2)}

Therefore, to determine the extended Cayley equivalence classes within the extended affine equivalence class of
a bent function $f : \F_2^{2m} \To \F_2$, for which $f(0)=0$, we need only examine
the extended translation equivalent functions of the form
\begin{align*}
f(x+b) + \langle c, x \rangle + f(b),
\end{align*}
for each $b, c \in \F_2^{2m}$.
\end{frame}

\begin{frame}
\frametitle{Quadratic bent functions have two \\ extended Cayley classes}
\begin{Theorem}
For each $m>0$, the extended affine equivalence class of quadratic bent functions
$q : \F_2^{2m} \To \F_2$ contains exactly two extended Cayley equivalence classes,
corresponding to the two possible weight classes of $x \mapsto q(x+b) + \langle c, x \rangle + q(b)$.
\end{Theorem}

\end{frame}

\end{colortheme}

\subsection{Computational results for quadratic bent functions in low dimensions}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{For 2 dimensions: quadratic case: $[f_{2,1}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c2_1_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{2,1}]$: 2 extended Cayley classes}
  \label{fig:q2_1_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{For 4 dimensions: quadratic case: $[f_{4,1}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c4_1_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{4,1}]$: 2 extended Cayley classes}
  \label{fig:q4_1_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{For 6 dimensions: quadratic case: $[f_{6,1}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c6_1_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{6,1}]$: 2 extended Cayley classes}
  \label{fig:q6_1_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{For 8 dimensions: quadratic case: $[f_{8,1}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_1_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,1}]$: 2 extended Cayley classes}
  \label{fig:q8_1_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
~
\end{frame}

\end{colortheme}

\subsection{Block designs}

\begin{colortheme}{seagull}

\begin{frame}
\frametitle{The two block designs of a bent function}

The first block design of a bent function $f$ is obtained by interpreting
the adjacency matrix of $\Cay{f}$ as the incidence matrix of a block design.
In this case we do not need $f(0)=0$.

~
\begin{Def}
The second block design of a bent function $f$ is defined by the incidence matrix
$D(f)$ where
\begin{align*}
D(f)_{c,x} &:= f(x) + \langle c, x \rangle + \dual{f}(c).
\end{align*}
This is a symmetric block design with the \Emph{symmetric difference property},
called the \Emph{SDP design} of $f$.
\end{Def}

~

\slidecite{Kantor 1975; Dillon and Schatz 1987; Neumann 2006}
\end{frame}

\end{colortheme}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{The weight class matrix is \\ the SDP design matrix}
\begin{Theorem}
For every bent function $f$, the \Emph{weight class matrix} of the ET class of $f$
equals the incidence matrix of the SDP design of $f$.

~

Specifically,
\begin{align*}
\weightclass{x \mapsto f(x+b) + \langle c, x \rangle + f(b)}
&=
f(b) + \langle c, b \rangle + \dual{f}(c)
\\
&=
D(f)_{c,b}.
\end{align*}

\end{Theorem}

\end{frame}

\end{colortheme}

\subsection{Computational results for low dimensions}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{For 2 dimensions: classes}

One extended affine class, containing the extended translation class $[f_{2,1}]$,
where $f_{2,1}(x) := x_0 x_1$ is self dual.

~

Two extended Cayley classes:
\begin{align*}
\begin{array}{|cccl|}
\hline
\text{Class} &
\text{Parameters} &
\text{2-rank} &
\text{Clique polynomial}
\\
\hline
1 &
(4, 1, 0, 0) & 4 &
2t^{2} + 4t + 1
\\
2 &
K_4 & 4 &
t^{4} + 4t^{3} + 6t^{2} + 4t + 1
\\
\hline
\end{array}
\end{align*}

\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{2,1}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c2_1_weight_class_matrix.png}
  \captionof{figure}{$[f_{2,1}]$: weight classes}
  \label{fig:c2_1_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c2_1_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{2,1}]$: extended Cayley classes}
  \label{fig:c2_1_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{For 4 dimensions: classes}

One extended affine class, containing the extended translation class $[f_{4,1}]$, where
$f_{4,1}(x) := x_0 x_1 + x_2 x_3$ is self dual.

~

Two extended Cayley classes:
\begin{align*}
\begin{array}{|cccl|}
\hline
\text{Class} &
\text{Parameters} &
\text{2-rank} &
\text{Clique polynomial}
\\
\hline
1 &
(16, 6, 2, 2) &
6 &
8t^{4} + 32t^{3} + 48t^{2} + 16t + 1
\\
2 &
(16, 10, 6, 6) &
6 &
\begin{array}{l}
16t^{5} + 120t^{4} + 160t^{3} +
\\
80t^{2} + 16t + 1
\end{array}
\\
\hline
\end{array}
\end{align*}
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{4,1}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c4_1_weight_class_matrix.png}
  \captionof{figure}{$[f_{4,1}]$: weight classes}
  \label{fig:c4_1_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c4_1_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{4,1}]$: extended Cayley classes}
  \label{fig:c4_1_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}

\end{colortheme}

\begin{colortheme}{seagull}

\begin{frame}
\frametitle{For 6 dimensions: ET classes}

Four extended affine classes, containing the following extended translation classes:

\begin{align*}
\def\arraystretch{1.2}
\begin{array}{|cl|}
\hline
\text{Class} &
\text{Representative}
\\
\hline
\,[f_{6,1}] & f_{6,1} :=
\begin{array}{l}
x_{0} x_{1} + x_{2} x_{3} + x_{4} x_{5}
\end{array}
\\
\,[f_{6,2}] & f_{6,2} :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{3} + x_{1} x_{4} + x_{2} x_{5}
\end{array}
\\
\,[f_{6,3}] & f_{6,3} :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{1} + x_{0} x_{3} + x_{1} x_{3} x_{4} + x_{1} x_{5}
\\
 +\, x_{2} x_{4} + x_{3} x_{4}
\end{array}
\\
\,[f_{6,4}] & f_{6,4} :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{3} + x_{1} x_{3} x_{4} + x_{1} x_{5} + x_{2} x_{3} x_{5}
\\
 +\, x_{2} x_{3} + x_{2} x_{4} + x_{2} x_{5} + x_{3} x_{4} + x_{3} x_{5}
\end{array}
\\
\hline
\end{array}
\end{align*}
\slidecite{Rothaus 1976; Tokareva 2015}
\end{frame}

\end{colortheme}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{For ET class $[f_{6,1}]$: EC classes}

Bent function
$f_{6,1}(x) = x_0 x_1 + x_2 x_3 + x_4 x_5$ is self dual.

 ~

Two extended Cayley classes corresponding to Tonchev's projective two-weight codes:
\begin{align*}
\def\arraystretch{1.2}
\begin{array}{|ccl|}
\hline
\text{Class} &
\text{Parameters} & \text{Reference}
\\
\hline
0 & [35,6,16] & \text{Table 1.156 1, 2 (complement)}
\\
1 & [27,6,12] & \text{Table 1.155 1 }
\\
\hline
\end{array}
\end{align*}

Graph for class 0 is also isomorphic to the complement of Royle's $(64,35,18,20)$ strongly regular
graph $X$.

\slidecite{Tonchev 1996, 2006; Royle 2008}
 \end{frame}
\begin{frame}
\frametitle{For ET class $[f_{6,1}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c6_1_weight_class_matrix.png}
  \captionof{figure}{$[f_{6,1}]$: weight classes}
  \label{fig:6_1_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c6_1_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{6,1}]$: 2 extended Cayley classes}
  \label{fig:6_1_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{6,2}]$: EC classes}

Bent function
$f_{6,2}(x) = x_{0} x_{1} x_{2} + x_{0} x_{3} + x_{1} x_{4} + x_{2} x_{5}$.

~

Three extended Cayley classes corresponding to Tonchev's projective two-weight codes:
\begin{align*}
\def\arraystretch{1.2}
\begin{array}{|ccl|}
\hline
\text{Class} &
\text{Parameters} & \text{Reference}
\\
\hline
0 & [35,6,16] & \text{Table 1.156 1, 2 (complement)}
\\
1 & [35,6,16] & \text{Table 1.156 3 (complement)}
\\
2 & [27,6,12] & \text{Table 1.155 2 }
\\
\hline
\end{array}
\end{align*}

Graph for class 0 is also isomorphic to that of $[f_{6,1}]$ class 0.

\slidecite{Tonchev 1996, 2006}
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{6,2}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c6_2_weight_class_matrix.png}
  \captionof{figure}{$[f_{6,2}]$: weight classes}
  \label{fig:6_2_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c6_2_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{6,2}]$: 3 extended Cayley classes}
  \label{fig:6_2_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{6,3}]$: EC classes}

Bent function
\begin{align*}
f_{6,3}(x) &= x_{0} x_{1} x_{2} + x_{0} x_{1} + x_{0} x_{3} + x_{1} x_{3} x_{4}
\\
           &+ x_{1} x_{5} + x_{2} x_{4} + x_{3} x_{4}.
\end{align*}

Four extended Cayley classes corresponding to Tonchev's projective two-weight codes:
\begin{align*}
\def\arraystretch{1.2}
\begin{array}{|ccl|}
\hline
\text{Class} &
\text{Parameters} & \text{Reference}
\\
\hline
0 & [35,6,16] & \text{Table 1.156 4 (complement)}
\\
1 & [27,6,12] & \text{Table 1.155 3 }
\\
2 & [35,6,16] & \text{Table 1.156 5 (complement)}
\\
3 & [27,6,12] & \text{Table 1.155 4 }
\\
\hline
\end{array}
\end{align*}

\slidecite{Tonchev 1996, 2006}
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{6,3}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c6_3_weight_class_matrix.png}
  \captionof{figure}{$[f_{6,3}]$: weight classes}
  \label{fig:6_3_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c6_3_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{6,3}]$: 4 extended Cayley classes}
  \label{fig:6_3_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{6,4}]$: EC classes}

Bent function
\begin{align*}
f_{6,4}(x) &= x_{0} x_{1} x_{2} + x_{0} x_{3} + x_{1} x_{3} x_{4} + x_{1} x_{5} + x_{2} x_{3} x_{5}
\\
           &+ x_{2} x_{3} + x_{2} x_{4} + x_{2} x_{5} + x_{3} x_{4} + x_{3} x_{5}.
\end{align*}

Three extended Cayley classes corresponding to Tonchev's projective two-weight codes:
\begin{align*}
\def\arraystretch{1.2}
\begin{array}{|ccl|}
\hline
\text{Class} &
\text{Parameters} & \text{Reference}
\\
\hline
0 & [35,6,16] & \text{Table 1.156 7 (complement)}
\\
1 & [35,6,16] & \text{Table 1.156 6 (complement)}
\\
2 & [27,6,12] & \text{Table 1.155 5 }
\\
\hline
\end{array}
\end{align*}

\slidecite{Tonchev 1996, 2006}
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{6,4}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c6_4_weight_class_matrix.png}
  \captionof{figure}{$[f_{6,4}]$: weight classes}
  \label{fig:6_4_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c6_4_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{6,4}]$: 3 extended Cayley classes}
  \label{fig:6_4_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}

\end{colortheme}

\begin{colortheme}{seagull}

\begin{frame}
\frametitle{For 8 dimensions: \\ number of bent functions and EA classes}

According to Langevin and Leander (2011)
there are $99\,270\,589\,265\,934\,370\,305\,785\,861\,242\,880 \approx 2^{106}$ bent functions in dimension 8.

~

The number of EA classes has not yet been published, let alone a list of representatives.

~

~

~

~

\slidecite{Langevin and Leander 2011}
\end{frame}

\begin{frame}
\frametitle{For 8 dimensions, up to degree 3: \\ extended translation classes}

Ten extended affine classes are listed in Braeken's thesis (2006),

containing the following extended translation classes:

\tiny{}
\begin{align*}
\def\arraystretch{1.2}
\begin{array}{|cl|}
\hline
\text{Class} &
\text{Representative}
\\
\hline
\,[f_{ 8 , 1 }] & f_{ 8 , 1 } :=
\begin{array}{l}
x_{0} x_{1} + x_{2} x_{3} + x_{4} x_{5} + x_{6} x_{7}
\end{array}
\\
\,[f_{ 8 , 2 }] & f_{ 8 , 2 } :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{3} + x_{1} x_{4} + x_{2} x_{5} + x_{6} x_{7}
\end{array}
\\
\,[f_{ 8 , 3 }] & f_{ 8 , 3 } :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{6} + x_{1} x_{3} x_{4} + x_{1} x_{5} + x_{2} x_{3} + x_{4} x_{7}
\end{array}
\\
\,[f_{ 8 , 4 }] & f_{ 8 , 4 } :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{2} + x_{0} x_{4} + x_{1} x_{3} x_{4} + x_{1} x_{5} + x_{2} x_{3} + x_{6} x_{7}
\end{array}
\\
\,[f_{ 8 , 5 }] & f_{ 8 , 5 } :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{6} + x_{1} x_{3} x_{4} + x_{1} x_{4} + x_{1} x_{5} + x_{2} x_{3} x_{5} + x_{2} x_{4} + x_{3} x_{7}
\end{array}
\\
\,[f_{ 8 , 6 }] & f_{ 8 , 6 } :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{2} + x_{0} x_{3} + x_{1} x_{3} x_{4} + x_{1} x_{6} + x_{2} x_{3} x_{5} + x_{2} x_{4} + x_{5} x_{7}
\end{array}
\\
\,[f_{ 8 , 7 }] & f_{ 8 , 7 } :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{1} + x_{0} x_{2} + x_{0} x_{3} + x_{1} x_{3} x_{4} + x_{1} x_{4} + x_{1} x_{5} + x_{2} x_{3} x_{5}
\\
+\,  x_{2} x_{4} + x_{6} x_{7}
\end{array}
\\
\,[f_{ 8 , 8 }] & f_{ 8 , 8 } :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{5} + x_{1} x_{3} x_{4} + x_{1} x_{6} + x_{2} x_{3} x_{5} + x_{2} x_{4} + x_{3} x_{7}
\end{array}
\\
\,[f_{ 8 , 9 }] & f_{ 8 , 9 } :=
\begin{array}{l}
x_{0} x_{1} x_{6} + x_{0} x_{3} + x_{1} x_{4} + x_{2} x_{3} x_{6} + x_{2} x_{5} + x_{3} x_{4} + x_{4} x_{5} x_{6} + x_{6} x_{7}
\end{array}
\\
\,[f_{ 8 , 10 }] & f_{ 8 , 10 } :=
\begin{array}{l}
x_{0} x_{1} x_{2} + x_{0} x_{3} x_{6} + x_{0} x_{4} + x_{0} x_{5} + x_{1} x_{3} x_{4} + x_{1} x_{6} + x_{2} x_{3} x_{5} + x_{2} x_{4}
\\
+\,  x_{3} x_{7}
\end{array}
\\
\hline
\end{array}
\end{align*}
\slidecite{Braeken 2006; Tokareva 2015}
\normalsize{}
\end{frame}

\end{colortheme}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{For ET class $[f_{8,1}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_1_weight_class_matrix.png}
  \captionof{figure}{$[f_{8,1}]$: weight classes}
  \label{fig:8_1_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_1_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,1}]$: 2 extended Cayley classes}
  \label{fig:8_1_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
~
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{8,2}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_2_weight_class_matrix.png}
  \captionof{figure}{$[f_{8,2}]$: weight classes}
  \label{fig:8_2_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_2_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,2}]$: 4 extended Cayley classes}
  \label{fig:8_2_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
Graph for class 0 is isomorphic to graph for class 0 of $[f_{8,1}]$.
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{8,3}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_3_weight_class_matrix.png}
  \captionof{figure}{$[f_{8,3}]$: weight classes}
  \label{fig:8_3_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_3_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,3}]$: 6 extended Cayley classes}
  \label{fig:8_3_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
~
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{8,4}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_4_weight_class_matrix.png}
  \captionof{figure}{$[f_{8,4}]$: weight classes}
  \label{fig:8_4_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_4_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,4}]$: 5 extended Cayley classes}
  \label{fig:8_4_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
~
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{8,5}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_5_weight_class_matrix.png}
  \captionof{figure}{$[f_{8,5}]$: weight classes}
  \label{fig:8_5_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_5_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,5}]$: 9 extended Cayley classes}
  \label{fig:8_5_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
~
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{8,6}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_6_weight_class_matrix.png}
  \captionof{figure}{$[f_{8,6}]$: weight classes}
  \label{fig:8_6_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_6_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,6}]$: 9 extended Cayley classes}
  \label{fig:8_6_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}

\begin{frame}
\frametitle{ET classes  $[f_{8,5}]$ and $[f_{8,6}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/re8_5_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,5}]$: 9 extended Cayley classes}
  \label{fig:re8_5_bent_cayley_graph_index_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/re8_6_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,6}]$: 9 extended Cayley classes}
  \label{fig:re8_6_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
The same 9 EC classes, with the same frequencies!

\slidecite{colormap: gist\_stern, colours matched to extended Cayley classes}
\end{frame}

\begin{frame}[fragile]
\frametitle{Functions $f_{8,5}$ and $f_{8,6}$ are linearly equivalent}

\Emph{Proof}

~

Apply the permutation $\pi := (x_0\ x_5\ x_4)(x_1\ x_2\ x_3)(x_6\ x_7)$ to
\footnotesize{
\begin{align*}
f_{8,5}
&=
x_{0} x_{1} x_{2} + x_{0} x_{6} + x_{1} x_{3} x_{4} + x_{1} x_{4} + x_{1} x_{5} + x_{2} x_{3} x_{5} + x_{2} x_{4} + x_{3} x_{7}
%\end{align*}
%}\normalsize{}
\intertext{\normalsize{to obtain}}
%\small{
%\begin{align*}
\pi(f_{8,5})
&=
x_{5} x_{2} x_{3} + x_{5} x_{7} + x_{2} x_{1} x_{0} + x_{2} x_{0} + x_{2} x_{4} + x_{3} x_{1} x_{4} + x_{3} x_{0} + x_{1} x_{6}
\\
&=
x_{0} x_{1} x_{2} + x_{0} x_{2} + x_{0} x_{3} + x_{1} x_{3} x_{4} + x_{1} x_{6} + x_{2} x_{3} x_{5} + x_{2} x_{4} + x_{5} x_{7}
\\
&= f_{8,6}.
\\
&\ \Box
\end{align*}
}\normalsize{}
\end{frame}

\begin{frame}
\frametitle{For ET class $[f_{8,7}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_7_weight_class_matrix.png}
  \captionof{figure}{$[f_{8,7}]$: weight classes}
  \label{fig:8_7_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_7_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,7}]$: 5 extended Cayley classes}
  \label{fig:8_7_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
~
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{8,8}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_8_weight_class_matrix.png}
  \captionof{figure}{$[f_{8,8}]$: weight classes}
  \label{fig:8_8_weight_class_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_8_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,8}]$: 6 extended Cayley classes}
  \label{fig:8_8_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
~
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{8,9}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_9_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,9}]$: 8 extended Cayley classes ~~ ~~~~ ~~~~ ~~~~~~~~~}
  \label{fig:c8_9_bent_cayley_graph_index_matrix}
\end{minipage}
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_9_dual_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,9}]$: 8 extended Cayley classes of dual bent functions}
  \label{fig:c8_9_dual_cayley_graph_index_matrix}
\end{minipage}%
\end{figure}
4 of the 8 classes form 2 dual pairs of classes.
\end{frame}
\begin{frame}
\frametitle{For ET class $[f_{8,10}]$: matrices}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_10_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,10}]$: 10 extended Cayley classes ~~ ~~~~ ~~~~ ~~~~~~~~~}
  \label{fig:c8_10_bent_cayley_graph_index_matrix}
\end{minipage}
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/c8_10_dual_cayley_graph_index_matrix.png}
  \captionof{figure}{$[f_{8,10}]$: 10 extended Cayley classes of dual bent functions}
  \label{fig:c8_10_dual_cayley_graph_index_matrix}
\end{minipage}%
\end{figure}
6 of the 10 classes form 3 dual pairs of classes.
\end{frame}

\end{colortheme}

\begin{colortheme}{seagull}

\begin{frame}[fragile]
\frametitle{For 8 dimensions: Bent functions from CAST-128 S-boxes}

The CAST-128 encryption algorithm is used in PGP and elsewhere.

CAST-128, including the S-boxes, is specified by IETF RFC 2144:
\begin{verbatim}
https://www.ietf.org/rfc/rfc2144.txt
\end{verbatim}

The algorithm uses 8 S-boxes,
each of which consists of 32 binary bent functions of degree 4 in 8 dimensions,
giving a total of 256 bent functions.

~

~

~

\slidecite{Adams 1997}
\end{frame}

\end{colortheme}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{CAST-128 ET class $[cast128_{1,0}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
\includegraphics[width=.9\linewidth]{../matrix_plot/cast128_1_0_weight_class_matrix.png}
  \captionof{figure}{$[cast128_{1,0}]$: weight classes ~~~~~~ ~~~~~~~~\\~~~~~}
  \label{fig:cast128_1_0_weight_class_matrix}
\end{minipage}
\begin{minipage}{.48\textwidth}
  \centering
\includegraphics[width=.9\linewidth]{../matrix_plot/cast128_1_0_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[cast128_{1,0}]$: $65\,536$ extended Cayley classes.\\Total including duals is $131\,072$.}
  \label{fig:cast128_1_0_bent_cayley_graph_index_matrix}
\end{minipage}%
\end{figure}
\slidecite{LHS colormap: gist\_stern; RHS colormap: jet}
\end{frame}

\begin{frame}
\frametitle{CAST-128 ET classes $[cast128_{2,1}]$ and $[cast128_{2,16}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
\includegraphics[width=.9\linewidth]{../matrix_plot/cast128_2_1_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[cast128_{2,1}]$: $8\,256$ extended Cayley classes.\\Total including duals is $8\,256$.}
  \label{fig:cast128_2_1_bent_cayley_graph_index_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
\includegraphics[width=.9\linewidth]{../matrix_plot/cast128_2_16_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[cast128_{2,16}]$: $32\,768$ extended Cayley classes.\\Total including duals is $65\,536$.}
  \label{fig:cast128_2_16_bent_cayley_graph_index_matrix}
\end{minipage}%
\end{figure}
\slidecite{colormap: jet}
\end{frame}

\begin{frame}
\frametitle{CAST-128 ET classes $[cast128_{4,27}]$ and $[cast128_{5,16}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
\includegraphics[width=.9\linewidth]{../matrix_plot/cast128_4_27_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[cast128_{4,27}]$: $65\,536$ extended Cayley classes.\\Total including duals is $65\,536$.}
  \label{fig:cast128_4_27_bent_cayley_graph_index_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
\includegraphics[width=.9\linewidth]{../matrix_plot/cast128_5_16_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[cast128_{5,16}]$: $33\,280$ extended Cayley classes.\\Total including duals is $66\,560$.}
  \label{fig:cast128_5_16_bent_cayley_graph_index_matrix}
\end{minipage}%
\end{figure}
\slidecite{colormap: jet}
\end{frame}
\begin{frame}
\frametitle{CAST-128 ET classes $[cast128_{5,27}]$ and $[cast128_{6,17}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
\includegraphics[width=.9\linewidth]{../matrix_plot/cast128_5_27_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[cast128_{5,27}]$: $6\,144$ extended Cayley classes.\\Total including duals is $6\,144$.}
  \label{fig:cast128_5_27_bent_cayley_graph_index_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
\includegraphics[width=.9\linewidth]{../matrix_plot/cast128_6_17_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[cast128_{6,17}]$: $65\,536$ extended Cayley classes.\\Total including duals is $65\,536$.}
  \label{fig:cast128_6_17_bent_cayley_graph_index_matrix}
\end{minipage}%
\end{figure}
\slidecite{colormap: jet}
\end{frame}

\begin{frame}
\frametitle{CAST-128 ET classes $[cast128_{7,15}]$ and $[cast128_{7,21}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
\includegraphics[width=.9\linewidth]{../matrix_plot/cast128_7_15_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[cast128_{7,15}]$: $32\,768$ extended Cayley classes.\\Total including duals is $65\,536$.}
  \label{fig:cast128_7_15_bent_cayley_graph_index_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
\includegraphics[width=.9\linewidth]{../matrix_plot/cast128_7_21_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[cast128_{7,21}]$: $32\,768$ extended Cayley classes.\\Total including duals is $65\,536$.}
  \label{fig:cast128_7_21_bent_cayley_graph_index_matrix}
\end{minipage}%
\end{figure}
\slidecite{colormap: jet}
\end{frame}
\end{colortheme}

\begin{colortheme}{seagull}

\begin{frame}[fragile]
\frametitle{For 8 dimensions: number of partial spread \\ bent functions and EA classes}

According to Langevin and Hou (2011)
there are $70\,576\,747\,237\,594\,114\,392\,064 \approx 2^{75.9}$ \Emph{partial spread} bent functions in dimension 8,
contained in $14\,758$ EA classes, of which $14\,756$ have degree 4.

~

The EA class representatives are listed at Langevin's web site

\begin{verbatim}
http://langevin.univ-tln.fr/project/spread/psp.html
\end{verbatim}

~

\slidecite{Langevin and Hou 2011}
\end{frame}

\end{colortheme}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{Example partial spread ET class $[psf_{9,5439}]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/psf_9_5439_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[psf_{9,5439}]$: 16 extended Cayley classes ~~ ~~~~ ~~~~ ~~~~~~~~~}
  \label{fig:psf_9_5439_bent_cayley_graph_index_matrix}
\end{minipage}
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/psf_9_5439_dual_cayley_graph_index_matrix.png}
  \captionof{figure}{$[psf_{9,5439}]$: 16 extended Cayley classes of dual bent functions}
  \label{fig:psf_9_5439_dual_cayley_graph_index_matrix}
\end{minipage}%
\end{figure}
6 of the 16 classes form 3 dual pairs of classes.

\slidecite{colormap: gist\_stern}
\end{frame}

\end{colortheme}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{For 4 dimensions: $[\sigma_2]$ and $[\tau_2]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/sigma_2_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[\sigma_2]$: 2 extended Cayley classes}
  \label{fig:sigma_2_bent_cayley_graph_index_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/tau_2_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[\tau_2]$: 2 extended Cayley classes}
  \label{fig:tau_2_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}

\begin{frame}
\frametitle{For 6 dimensions: $[\sigma_3]$ and $[\tau_3]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/sigma_3_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[\sigma_3]$: 2 extended Cayley classes}
  \label{fig:sigma_3_bent_cayley_graph_index_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/tau_3_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[\tau_3]$: 3 extended Cayley classes}
  \label{fig:tau_3_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}

\begin{frame}
\frametitle{For 8 dimensions: $[\sigma_4]$ and $[\tau_4]$}
\begin{figure}
\centering
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/sigma_4_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[\sigma_4]$: 2 extended Cayley classes}
  \label{fig:sigma_4_bent_cayley_graph_index_matrix}
\end{minipage}%
\begin{minipage}{.48\textwidth}
  \centering
  \includegraphics[width=.9\linewidth]{../matrix_plot/tau_4_bent_cayley_graph_index_matrix.png}
  \captionof{figure}{$[\tau_4]$: 5 extended Cayley classes}
  \label{fig:tau_4_bent_cayley_graph_index_matrix}
\end{minipage}
\end{figure}
\end{frame}
\end{colortheme}

\subsection{Questions}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{Open problems (1)}
Settled only for dimensions up to 6:
\begin{enumerate}
\item
How many EC classes are there for each dimension?
Are there ``Exponential numbers'' of classes?
\item
In $n$ dimensions,
which ET classes contain the maximum number, $4^n$, of different EC classes?
\item
Which EC classes overlap more than one ET class?
\item
Which bent functions are Cayley equivalent to their dual?
\item
Which bent functions are EA equivalent to their dual?
\end{enumerate}

\slidecite{Kantor 1983; Jungnickel and Tonchev 1991; Langevin, Leander and McGuire 2008}
\end{frame}

\begin{frame}
\frametitle{Open problems (2)}
Also:

~

\begin{enumerate}
\item
What are the remaining EA and EC classes of binary bent functions of dimension 8 and degree 4?

~

\item
How do extended Cayley classes of bent functions generalize to bent functions over $\F_p$, $p \neq 2$?
\end{enumerate}

\slidecite{Langevin and Leander 2011; Chee, Tan and Zhang 2011}
\end{frame}

\end{colortheme}

\section{Databases of strongly regular graphs}

\subsection{Precedents}

\begin{colortheme}{seagull}
\begin{frame}
\frametitle{Precedents: databases of strongly regular graphs}
\begin{itemize}
 \item
Spence's lists of strongly regular graphs on at most 64 vertices.
 \begin{itemize}
  \item
Exhaustive lists of non-isomorphic graphs for some tuples of feasible parameters.
  \item
Flat text files.
 \end{itemize}

~

 \item
Brouwer's database of parameters of strongly regular graphs.

~

 \item
Cohen and Pasechnik's Sage implementation of Brouwer's database, with constructions.
 \begin{itemize}
  \item
Includes an example for each tuple of feasible parameters, if known.
  \item
Sage interface.
 \end{itemize}
\end{itemize}
\slidecite{Spence 1995-2006; Brouwer 2008-2017; Cohen and Pasechnik 2016-2017}
\end{frame}

\begin{frame}
\frametitle{Other recent mathematical databases}
\begin{itemize}
 \item
ISGCI - Information System on Graph Classes and their Inclusions
 \begin{itemize}
  \item
``\ldots an encyclopaedia of graph classes with an accompanying Java application that helps you to research what's known about particular graph classes.''
  \item
Web-based and Java interfaces.
 \end{itemize}

~

 \item
LMFDB - The L-functions and modular forms database
 \begin{itemize}
  \item
``The LMFDB is an extensive database of mathematical objects arising in Number Theory.''
  \item
Based on MongoDB and Python.
  \item
Web-based and Sage interfaces.
 \end{itemize}
\end{itemize}
\slidecite{H.N. de Ridder et al. 2001-2016; The LMFDB Collaboration 2007-2017}
\end{frame}

\end{colortheme}

\subsection{Preliminary designs}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{Database tables}
\begin{figure}
\centering
\begin{minipage}{\textwidth}
  \centering
\includegraphics[width=.75\linewidth]{../graphics/Classification-schema-SQLite.png}
  \captionof{figure}{Database schema for SQLite version of the classification database.}
  \label{fig:Classification_schema_SQLite}
\end{minipage}%
\end{figure}
\end{frame}

\begin{frame}[fragile]
\frametitle{Sage interface: insert function}

\begin{verbatim}
insert_classification(conn, bfcgc, name=None):
\end{verbatim}

\begin{itemize}
 \item \texttt{conn}: Database connection.

~

 \item \texttt{bfcgc}: Cayley graph classification of the ET class of \\a bent function.

~

 \item \texttt{name}: Name of the ET class (optional).
\end{itemize}

\end{frame}

\begin{frame}[fragile]
\frametitle{Sage interface: select functions}

\begin{verbatim}
select_classification_where_bent_function(
    conn, bentf):
\end{verbatim}
\begin{itemize}
 \item \texttt{conn}: Database connection.
 \item \texttt{bentf}: Bent function representing an ET class.
\end{itemize}

~

\begin{verbatim}
select_classification_where_name(conn, name):
\end{verbatim}
\begin{itemize}
 \item \texttt{conn}: Database connection.
 \item \texttt{name}: Name of the ET class.
\end{itemize}

\end{frame}

\end{colortheme}

\subsection{Prototype databases}

\begin{colortheme}{jubata}

\begin{frame}
\frametitle{Prototype databases}
Three prototype relational databases, using SQLite:

\begin{itemize}
 \item\normalsize{}
Bent functions in 6 dimensions.
 \begin{itemize}
  \item\footnotesize{}
Database of 11 strongly regular graphs from 4 ET classes.
  \item\footnotesize{}
About 20 CPU minutes to calculate (2.93 GHz Intel Core i7, serial).
  \item\footnotesize{}
Database size 780 KB.
 \end{itemize}

 \item\normalsize{}
Bent functions in 8 dimensions, up to degree 3.
 \begin{itemize}
  \item\footnotesize{}
Database of 55 strongly regular graphs from 9 ET classes.
  \item\footnotesize{}
About 250 CPU hours to calculate (2.93 GHz Intel Core i7, serial).
  \item\footnotesize{}
Database size 65 MB.
 \end{itemize}

 \item\normalsize{}
Bent functions from the 8 S-boxes of CAST-128.
 \begin{itemize}
  \item\footnotesize{}
Database of more than \Emph{32 million} ($32\,914\,496$) strongly regular graphs from 256 ET classes and their duals.
  \item\footnotesize{}
About \Emph{9000 CPU hours} to calculate
(4500 CPU hours on\\NCI Raijin, MPI, 16 ranks, for 128 ET classes and their duals).
  \item\footnotesize{}
Database size \Emph{195 GB}.
 \end{itemize}
\end{itemize}

\normalsize{}
\end{frame}

\begin{frame}
\frametitle{Time to create SQLite CAST-128 database}
\begin{figure}
\centering
\begin{minipage}{.49\textwidth}
  \centering
  \includegraphics[width=1.0\linewidth]{../graphics/CAST128-database-insert-times-by-existing.png}
  \label{fig:CAST128_database_insert_times_by_existing}
\end{minipage}
\begin{minipage}{.49\textwidth}
  \centering
  \includegraphics[width=1.0\linewidth]{../graphics/CAST128-database-insert-times-by-time-of-day.png}
  \label{fig:CAST128_database_insert_times-By_time_of_day}
\end{minipage}%
\end{figure}
It took almost 3 days to insert 256 classifications \Emph{sequentially} into the SQLite version of the CAST-128 database.
\end{frame}

\begin{frame}
\frametitle{Using DB Browser with a prototype database}
\begin{figure}
\centering
\begin{minipage}{\textwidth}
  \centering
\includegraphics[width=1.00\linewidth]{../graphics/Browser-session-SQLite.png}
%  \captionof{figure}{DB Browser session with a prototype classification database.}
  \label{fig:Browser_session_SQLite}
\end{minipage}%
\end{figure}
\end{frame}

\end{colortheme}

\subsection{Prospects}

\begin{colortheme}{jubata}

\begin{frame}[fragile]
\frametitle{Possible next steps}

\begin{itemize}
 \item
Submit ``Classifying bent functions by their Cayley graphs.''

~

 \item
Submit code to Sage project for review.

~

 \item
Gauge demand for the database. Find collaborators.

~

 \item
Arrange for hosting. Research Data Services?

~

 \item
Scale up to the partial spread bent functions?

 \begin{itemize}
  \item
This could be a database of about \Emph{2 billion SRGs},
\\
from 14\,758 ET classes, taking about \Emph{500\,000 CPU hours}
\\
to calculate and about \Emph{10 TB} to store.
 \end{itemize}
\end{itemize}
\end{frame}

\end{colortheme}

\subsection{Source code}

\begin{colortheme}{jubata}

\begin{frame}[fragile]
\frametitle{Source code and documentation}
~

CoCalc: Public worksheets, Sage and Python source code

\begin{verbatim}
http://tinyurl.com/Boolean-Cayley-graphs
\end{verbatim}

~

GitHub: Sage and Python source code

\begin{verbatim}
https://github.com/penguian/Boolean-Cayley-graphs
\end{verbatim}

~

SourceForge: Documentation

\begin{verbatim}
https://boolean-cayley-graphs.sourceforge.io/
\end{verbatim}
\end{frame}

\end{colortheme}

\end{document}