\usepackage[francais]{babel}
\usepackage{rotating}
\usepackage{algorithm2e}
+\usepackage{stmaryrd}
\graphicspath{{Figures/}}
\newcommand{\hauteur}[2]{\raisebox{0pt}[#1][-#1]{#2}}
\def\oeuvre{\oe uvre }
\def\oeuvrepv{\oe uvre}
+\newcommand{\bleu}[1]{\color{blue}{#1}}
%\newenvironment{myitemize}[1]{
%% \setlength{\topsep}{#1mm}
\item For cryptography: cryptographically secure
\item Successful pass on PRNG batteries of tests:
NIST\footnote{E.~Barker and A.~Roginsky.
-\newblock Draft {N}{I}{S}{T} special publication 800-131 recommendation for the
+ Draft {N}{I}{S}{T} special publication 800-131 recommendation for the
transitioning of cryptographic algorithms and key sizes, 2010.},
DieHARD\footnote{G.~Marsaglia.
-\newblock DieHARD: a battery of tests of randomness.
-\newblock {\em http://stat.fsu.edu/~geo/diehard.html}, 1996}
+ DieHARD: a battery of tests of randomness.
+ {\em http://stat.fsu.edu/~geo/diehard.html}, 1996}
\item Should have chaos properties
\end{itemize}
\end{itemize}
%\end{scriptsize}
\end{algorithm}
\end{block}
+$$
+F_f: \Bool^{{\mathsf{N}}} \times \llbracket1;{\mathsf{N}} \rrbracket \to \Bool^{\mathsf{N}},
+F_f(x,i)=(x_1,\dots,x_{i-1},f_i(x),x_{i+1},\dots,x_{\mathsf{N}}).
+$$
\end{frame}
}
(x_2 \oplus x_3, \overline{x_1}\overline{x_3} + x_1\overline{x_2},
\overline{x_1}\overline{x_3} + x_1x_2)$$
\item Iteration graph $\Gamma(f^*)$ of this function:
-\includegraphics[width=0.45\textwidth]{iter_f0c}
+\includegraphics[width=0.45\textwidth]{images/iter_f0c}
\end{itemize}
\end{block}
\end{frame}
\begin{exampleblock}{Previous work}
To provide a PRNG with the properties of Devaney's chaos and of succeeding NIST test: a (non-chaotic) PRNG + iterating a Boolean maps~\footnote{J. Bahi, J.-F. Couchot, C. Guyeux, and A. Richard.
-\newblock On the link between strongly connected iteration graphs and chaotic
+ On the link between strongly connected iteration graphs and chaotic
Boolean discrete-time dynamical systems, {\em
- Fundamentals of Computation Theory}, volume 6914 of {\em Lecture Notes in
- Computer Science}, pages 126--137. Springer Berlin Heidelberg, 2011.}:
+ Fundamentals of Computation Theory}, volume 6914 of {\em LNCS}, pages 126--137. Springer, 2011.}:
\begin{itemize}
\item with strongly connected iteration graph $\Gamma(f)$
\item with doubly stochastic Markov probability matrix
\begin{itemize}
\item Focus on the generation of Hamiltonian cycles in the
$n$-cube
- \item To find cyclic Gray codes.
+ \item Find cyclic Gray codes.
\end{itemize}
\end{block}
\footnote{Couchot, J., Héam, P., Guyeux, C., Wang, Q., Bahi, J. M. [2014]
\end{frame}
}
+\frameselect{true}{
+ \begin{frame}
+ \frametitle{Exemple sur le 3-cube}
+
+ \begin{center}
+ \vspace{-.75em}
+ \includegraphics[width=.3\textwidth]{3-cube.pdf}
+ \vspace{-.75em}
+ \end{center}
+
+ \begin{block}{}
+ \small
+ \vspace{-1.5em}
+ \begin{center}
+ \begin{equation*}
+ \begin{array}[h]{c|cccccccccccc}
+ \text{arêtes} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
+ \hline
+ \text{init} & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
+ \hline
+ \text{ajout a} & d_1 & d_1 & 2 & 2 & d_1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\
+ \hline
+ \text{ajout b} & \alert{1} & \bleu{g_1} & 2 & \bleu{g_2} & \bleu{g_1} & 2 & 2 & 2 & 2 & 2 & \bleu{g_2} & 2 \\
+ & \alert{0} & \bleu{1} & 2 & \bleu{1} & \bleu{1} & 2 & 2 & 2 & 2 & 2 & \bleu{1} & 2 \\
+ \hline
+ \text{ajout c} & 1 & \alert{1} & \bleu{g_3} & g_2 & \bleu{0} & 2 & \bleu{g_3} & 2 & 2 & 2 & g_2 & 2 \\
+ & 1 & \alert{0} & \bleu{1} & g_2 & \bleu{1} & 2 & \bleu{1} & 2 & 2 & 2 & g_2 & 2 \\
+ & 0 & 1 & \bleu{g_3} & 1 & 1 & 2 & \bleu{g_3} & 2 & 2 & 2 & 1 & 2 \\
+ \end{array}
+ \end{equation*}
+ \end{center}
+ \end{block}
+ \end{frame}
+}
+
\frameselect{true}{
\begin{frame}
\frametitle{Adaptation au contexte de N-cube}
}
+ \begin{frame}
+ \frametitle{Perspectives}
+\begin{itemize}
+\item Qu'est-ce qu'un $\mathsf{N}$-cube?
+\item Justifier les trois arrêtes sortantes du n{\oe}ud $010$ de la figure~1.
+\item Illustrer à l'aide d'un graphe le fait que la fonction $f^*$, définie au milieu de la~page~3 est un 3-cube privé d'un cycle hamiltonien.
+\item Pourquoi l'extension de Robinson-Cohn (page 11) n'est-elle pas un algorithme? A quoi sert la démonstration de la section 5.2?
+\item Quel est l'objectif de la section~6?
+Ordonner les lemmes et théorèmes de cette section pour dégager le résultat
+final.
+\item Expliquer toutes les informations que l'on peut trouver dans la seconde
+ligne du tableau de la page 18 (ligne portant le libéllé \og function \textcircled{a}\fg{}.
+\end{itemize}
+\end{frame}
+
+
+
\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
