From: couchot Date: Fri, 6 Jan 2017 16:27:44 +0000 (+0100) Subject: ajout du dossier talk X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hdrcouchot.git/commitdiff_plain/d31be79de69d9af9371a3b1fcb3df81246c5928a ajout du dossier talk --- diff --git a/talk/#assync.tex# b/talk/#assync.tex# new file mode 100644 index 0000000..b5a9dd5 --- /dev/null +++ b/talk/#assync.tex# @@ -0,0 +1,31 @@ +\begin{itemize} +\item Calculs effectués sur chaque n{\oe}ud: réécriture de +$$ +w_l^{(k+1)} = w_l^{(k+1)} + \theta^{(k)}. \left( \sum_{i \in N} a_{il}.q_i^{(k)} \right) +$$ +\item Conditions pour la convergence asynchrone du calcul\footnote{Nedić, A., Bertsekas, D. P., \& Borkar, V. S. (2001). Distributed asynchronous incremental subgradient methods. Studies in Computational Mathematics, 8, 381-407.} +\begin{itemize} +\item Majoration des sous-gradients par une constante $C$ +\item $\theta$: sous la forme $\omega / t^{q}$ avec $3/4 < q \leq 1$ + \end{itemize} +\item Premières expérimentations: +\begin{itemize} +\item Pour chaque variable $X$ à modifier: + $X^{k+1} = f(X^{k}) if \textit{random()} + + Linear congruential generator visualisation + Visualisation of generation of pseudo-random numbers in [0, 8] using a linear congruential generator by CMG Lee. The top two rows show a generator with m = 0, a = 2 and c = 0 outputting numbers from left to right. A seed of 1 gives a cycle length of 6 but a seed of 3 gives a cycle length of only 2. Using a = 4 and c = 1 (bottom row) gives a full cycle length of 9 with any seed. + + + + + + + + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + + + + + + + seed = 1 + + + + + + output 2 + + + 1 + + + + + 2 + + + + + + + + output 4 + + + 1 + + + + + 2 + + + + + + + + output 8 + + + 1 + + + + + 2 + + + + + + + + output 7 + + + 1 + + + + + 2 + + + + + + + + output 5 + + + 1 + + + + + 2 + + + + + + + + output 1 + + + 1 + + + + + 2 + + + + + + + m= 9 + a= 2 + c= 0 + + + + + + seed = 3 + + + + + + output 6 + + + 1 + + + + + 2 + + + + + + + + output 3 + + + 1 + + + + + 2 + + + + + + + m= 9 + a= 2 + c= 0 + + + + + + seed = 0 + + + + + + output 1 + + + + + + + + + + + + + + + + + + + c + + + + + + + + output 5 + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + c + + + + + + + + output 3 + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + c + + + + + + + + output 4 + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + c + + + + + + + + output 8 + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + c + + + + + + + + output 6 + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + c + + + + + + + + output 7 + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + c + + + + + + + + output 2 + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + c + + + + + + + + output 0 + + + 1 + + + + + 2 + + + + + 3 + + + + + 4 + + + + + c + + + + + + + m= 9 + a= 4 + c= 1 + + + + + + + diff --git a/talk/asynchrone.tex b/talk/asynchrone.tex new file mode 100644 index 0000000..3a83f7e --- /dev/null +++ b/talk/asynchrone.tex @@ -0,0 +1,26 @@ +\begin{itemize} + \item Deux modes: +\begin{itemize} + \item + \emph{synchrone}: chaque élément attend la valeur des élements dont il dépend; + \item + \emph{asynchrone}: chaque élément met à jour sa valeur sans attendre. + \end{itemize} + +\item $(D^{t})^{t \in \Nats}$: suite de matrices de taille $\mathsf{N} \times \mathsf{N}$ t.q. + $$ + D_{ij}^{t} =\textrm{ date où $x_j$ est disponible au composant $i$} +$$ + + +\item $x^{t+1}_i= \left\{ + \begin{array}{l} + f_i( x_1^{D_{i1}^t},\ldots, x_{\mathsf{N}}^{D_{i{\mathsf{N}}}^t}) + \textrm{ si } i \in s^t\\ + x^{t}_i \textrm{ sinon } + \end{array} + \right.$ + + +%\item Délai : $\delta_{ij}^t = t - D_{ij}^{t} \le \delta_0\, \forall i, j$ +\end{itemize} diff --git a/talk/attracteurs.tex b/talk/attracteurs.tex new file mode 100644 index 0000000..edc0088 --- /dev/null +++ b/talk/attracteurs.tex @@ -0,0 +1,46 @@ +\begin{itemize} + \item $x$ \emph{point fixe} de $f$ si $x = f (x)$ + \item $A$ \emph{attracteurs} du graphe si +\begin{itemize} + \item pour tout arc $x \rightarrow y$ , si $x \in A$, alors + $y \in A$ et +\item $A$: le plus petit au sens de l'inclusion +\end{itemize} +\end{itemize} + +\vspace{-1em} +\begin{exampleblock}{Attracteurs de +$(x_1, x_2, x_3) \mapsto +((\overline{x_1} + \overline{x_2}).x_3, +x_1.x_3, +x_1 + x_2 + x_3)$.} +\vspace{-2em} +\begin{figure}[h] + \begin{center} + \subfloat[$A_1=\{000\}$ et $A_2=\{011, 101, 111\}$]{ + \begin{minipage}{0.33\textwidth} + \begin{center} + \includegraphics[scale=0.25]{fsig3} + \end{center} + \end{minipage} + \label{fig:fsigb} + } + \subfloat[$A_1=\{000\}$ et $A_2=\{001, 101,111, 011 \}$]{ + \begin{minipage}{0.33\textwidth} + \begin{center} + \includegraphics[scale=0.25]{faig3} + \end{center} + \end{minipage} + \label{fig:faigb} + } + \subfloat[$A_1=\{000\}$ et $A_2=\{001, 101,111, 011 \}$]{ + \begin{minipage}{0.33\textwidth} + \begin{center} + \includegraphics[scale=0.25]{fgig3} + \end{center} + \end{minipage} + \label{fig:fgigb} + } + \end{center} +\end{figure} +\end{exampleblock} diff --git a/talk/beamerthemefemto.sty b/talk/beamerthemefemto.sty new file mode 100644 index 0000000..0ab88ce --- /dev/null +++ b/talk/beamerthemefemto.sty @@ -0,0 +1,249 @@ +%------------------------------------------------------------------------------- +% LaTeX Beamer theme "femto" from the femto-st evaluation model (2011) JM Nicod and PC Heam (LIFC) +% +% author: Christophe Varnier et passage au ".sty" par G. Laurent +% Joel Abadie (12/06/2012) rajout de fonctionnalites tikzpicture +% Jean-Marc Nicod (logo + reglage pied de page) +%------------------------------------------------------------------------------- +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{beamerthemefemto}[2012/06/05 beamer theme femto package] +\RequirePackage{times} +\RequirePackage{tikz} +%------------------------------------------------------------------------------- + +% _ _ _ _ +% ___| |_ _ _| | ___ ___| |_ ___ ___ _ _| | ___ _ _ _ __ ___ +% / __| __| | | | |/ _ \ / _ \ __| / __/ _ \| | | | |/ _ \ | | | '__/ __| +% \__ \ |_| |_| | | __/ | __/ |_ | (_| (_) | |_| | | __/ |_| | | \__ \ +% |___/\__|\__, |_|\___| \___|\__| \___\___/ \__,_|_|\___|\__,_|_| |___/ +% |___/ +% + +% commande permettant de dessiner une page de fond +% appel depuis le document par : \setbeamertemplate{background}{\titrefemto} +\newcommand{\titrefemto}{ +\begin{tikzpicture} + %definition du node centre au dimension de la page + \node[minimum 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fond +% appel depuis le document par : \setbeamertemplate{background}{\pagefemto} +\newcommand{\pagefemto}{ +\begin{tikzpicture} + % definition du node centre au dimension de la page + \node[minimum height=\paperheight, minimum width=\paperwidth](centre) {}; + % placement des logos en bas à gauche +% \node[above right=3mm] at (centre.south west) {\includegraphics[width=1.2cm]{Femto-logo_couleur-2012.png}}; + \node[above right=0.75mm] at ([shift={(0.25,-0.1)}]centre.south west) {\includegraphics[width=1cm]{logo-FEMTO.jpg}}; + %placement des logos en haut à droite + \node[below left, inner sep=0](hexa) at (centre.north east) {\includegraphics[width=3.5cm]{Hexagones.jpg}}; + %trace des traits + \coordinate (A) at ([shift={(0.5,-1.4)}]centre.north west); + \coordinate (B) at (A-| hexa.west); + \draw[BleuFemtoTitre] (A) -- ([xshift=-3mm]B); + %trait inferieur +% \coordinate (A) at ([shift={(0.5,1.1)}]centre.south west); + \coordinate (A) at ([shift={(0.5,.6)}]centre.south west); +% 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+\setbeamercolor{structure}{fg=DarkGray} + +% _ _ _ _ _ _ +%| |_ ___ _ __ ___ _ __ | | __ _| |_ ___ | |_(_) |_ _ __ ___ +%| __/ _ \ '_ ` _ \| '_ \| |/ _` | __/ _ \ | __| | __| '__/ _ \ +%| || __/ | | | | | |_) | | (_| | || __/ | |_| | |_| | | __/ +% \__\___|_| |_| |_| .__/|_|\__,_|\__\___| \__|_|\__|_| \___| +% |_| + +%\pgfdeclareimage[height=\paperheight,width=\paperwidth]{titrefemto}{TitreFemto.png} + +\setbeamertemplate{title page} +{ +\vskip 5mm + +\usebeamerfont{title}\usebeamercolor[fg]{title}\Large\textcolor{BleuFemto}\inserttitle + +%\vspace{5mm} +\footnotesize +\begin{center} +\insertauthor\\ +\vspace{5mm} +\insertdate +\end{center} + +\begin{footnotesize} +Rapporteurs~: +\begin{tabular}{p{4cm}l} +Olivier {\sc Bournez}& Professeur à l'Ecole Polytechnique. \\ +Jean-Paul {\sc Comet}& Professeur à l'Université de Nice Sophia Antipolis\\ +Juan-Pablo {\sc Ortega} & Professeur à l'Université de St. Gallen--Suisse \\ +\end{tabular} + + +\vspace{5mm} +Examinateurs~: +\begin{tabular}{p{4cm}l} +Sylvain {\sc Contassot-Vivier} & {Professeur à l'Université de Lorraine} \\ +Raphaël {\sc Couturier}& {Professeur à l'Univ. Bourgogne Franche-Comté} \\ +Christophe {\sc Guyeux}& {Professeur à l'Univ. Bourgogne Franche-Comté} +\end{tabular} +\end{footnotesize} + +% Examinateurs: + +% \begin{footnotesize} +% \begin{tabular}{rll} +% +% +% +% \end{tabular} +% \end{footnotesize} + +} + +% _ _ _ +%| |_ ___ _ __ ___ _ __ | | __ _| |_ ___ _ __ __ _ __ _ ___ +%| __/ _ \ '_ ` _ \| '_ \| |/ _` | __/ _ \ | '_ \ / _` |/ _` |/ _ \ +%| || __/ | | | | | |_) | | (_| | || __/ | |_) | (_| | (_| | __/ +% \__\___|_| |_| |_| .__/|_|\__,_|\__\___| | .__/ \__,_|\__, |\___| +% |_| |_| |___/ + +%\pgfdeclareimage[height=\paperheight,width=\paperwidth]{pagefemto}{PageFemto.png} + +\setbeamertemplate{frametitle}{\vspace{6mm}\hspace{-5mm}\Large\bf\textcolor{BleuFemto} \insertframetitle} + +% pour les block +\definecolor{fondFemtotableau}{rgb}{1,0.71,0.754} +\setbeamertemplate{blocks}[rounded] +\setbeamercolor{block title}{fg=BleuClairFemto!60,bg=BleuFemto} +\setbeamercolor{block body}{bg=GrisClairFemto!40,fg=GrisFemto!50!black} +\setbeamerfont{block title}{series=\bfseries} +\setbeamercolor{title in head/foot}{fg=gray} +\newenvironment{femtoBlock}[1]{% + \textcolor{BleuFemto}{\normalsize\textbf{#1}}\normalsize} + +\setbeamertemplate{footline} +{% + \leavevmode% + \vspace{-2mm}% + \hbox{\begin{beamercolorbox}[wd=.85\paperwidth,ht=2.5ex,dp=-2.5ex,leftskip=.3cm plus1fill,rightskip=.3cm]{author in head/foot}% + \usebeamerfont{author in head/foot} \hfill \textcolor{GrisFemto}{\insertshortinstitute} + \end{beamercolorbox}% + \begin{beamercolorbox}[wd=.15\paperwidth,right,ht=2.5ex,dp=1.8ex,leftskip=.3cm,rightskip=.3cm plus1fil]{title in head/foot}% + \usebeamerfont{title in head/foot} \hfill \textcolor{BleuFemto}{\insertframenumber\ / \inserttotalframenumber}\hspace{0.25cm} + \end{beamercolorbox}}% + \vskip 10pt% +} + +\setbeamertemplate{itemize item}[circle] +\setbeamertemplate{itemize subitem}[circle] + +\partopsep=0ex % espace apres les blocks et avant le contenu +\topsep=5ex % debut du texte du slide +\parsep=0ex +\itemsep=0ex + +%------------------------------------------------------------------------------- + diff --git a/talk/conclusion.tex b/talk/conclusion.tex new file mode 100644 index 0000000..960b079 --- /dev/null +++ b/talk/conclusion.tex @@ -0,0 +1,25 @@ +\begin{itemize} +\item Peut-on prédire le comportement de réseaux bouléens (Sec.~\ref{sec:sdd}) +\begin{itemize} +\item Théorique: \alert{conditions théoriques} + nécessaires/suffisantes de convergence/divergence? +\item Pratique: vérification par simulation (\alert{exhaustive?}) +\end{itemize} +\item Itérations divergentes \alert{$ \leftrightarrow$ comportement chaotique} (Sec.~\ref{sec:sddchaos})? +\begin{itemize} +\item Caractérisation des réseaux bouléens chaotiques. +\item Génération et prédiction. +\end{itemize} +\item Générateurs de nombres pseudo-aléatoires (Sec.~\ref{sec:prng}): +\begin{itemize} +\item Caractérisation d'un PRNG chaotique. +\item Générations et qualité. +\end{itemize} +\item Du Chaos au masquage d'information (Sec.\ref{sec:ih}). +\begin{itemize} +\item D'un point de vue chaotique. +\item D'un point de vue analyse vectorielle discrète. +\end{itemize} +\end{itemize} + + diff --git a/talk/couchot.pdf b/talk/couchot.pdf new file mode 100644 index 0000000..be3635f Binary files /dev/null and b/talk/couchot.pdf differ diff --git a/talk/ctx.tex b/talk/ctx.tex new file mode 100644 index 0000000..a003116 --- /dev/null +++ b/talk/ctx.tex @@ -0,0 +1,19 @@ +\begin{exampleblock}{Utilisation des nombres aléatoires} +\begin{itemize} +\item Vie courante: localisation GPS, cryptage de données (RSA); +\item Science: statistiques, en probabilité, en simulation numérique. +\end{itemize} +\end{exampleblock} + +\begin{exampleblock}{Génération} +\begin{itemize} +\item Réellement aléatoire: processus physique $\leadsto$ non reproductible. +\item Pseudo-aléatoire (PRNG): algorithme $\leadsto$ reproductible, quid de la sécurité? +\end{itemize} +\end{exampleblock} + +\begin{alertblock}{Problématique} +Les systèmes dynamiques sont-ils un bon outil mathématique pour +générer des nombres pseudo-aléatoires? +Quelles sont les qualités minimums d'un PRNG? +\end{alertblock} diff --git a/talk/faig2.pdf b/talk/faig2.pdf new file mode 100644 index 0000000..1d0c10f Binary files /dev/null and b/talk/faig2.pdf differ diff --git a/talk/faig3.pdf b/talk/faig3.pdf new file mode 100644 index 0000000..14b4408 Binary files /dev/null and b/talk/faig3.pdf differ diff --git a/talk/fgig2.pdf b/talk/fgig2.pdf new file mode 100644 index 0000000..6fb1bde Binary files /dev/null and b/talk/fgig2.pdf differ diff --git a/talk/fgig3.pdf b/talk/fgig3.pdf new file mode 100644 index 0000000..91698fc Binary files /dev/null and b/talk/fgig3.pdf differ diff --git a/talk/figreg-eps-converted-to.pdf b/talk/figreg-eps-converted-to.pdf new file mode 100644 index 0000000..a7fbf49 Binary files /dev/null and b/talk/figreg-eps-converted-to.pdf differ diff --git a/talk/figreg.eps b/talk/figreg.eps new file mode 100644 index 0000000..336e714 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36 534 225 +end +restore +%%EOF diff --git a/talk/figreg.tex b/talk/figreg.tex new file mode 100644 index 0000000..17f137a --- /dev/null +++ b/talk/figreg.tex @@ -0,0 +1,11 @@ +\psfrag{fW}[cc][cc]{$a_k$} +\psfrag{wdjW}[cc][cc]{$a_{k+1}$} +\psfrag{OFFY}[cc][cc]{$a_{k+2}$} +\psfrag{uH8PA}[cc][cc]{$a_{k+r-1}$} +\psfrag{x_oF}[cc][cc]{$a_{k+r}$} +\psfrag{al}[cc][cc]{$q_0$} +\psfrag{Lt}[cc][cc]{$q_1$} +\psfrag{kG}[cc][cc]{$q_2$} +\psfrag{QMeO}[cc][cc]{$q_{r-1}$} +\psfrag{MG}[cc][cc]{$\sum$} +\psfrag{Hvb}[cc][cc]{$\mod p$} diff --git a/talk/fsig2.pdf b/talk/fsig2.pdf new file mode 100644 index 0000000..5a3f91d Binary files /dev/null and b/talk/fsig2.pdf differ diff --git a/talk/fsig3.pdf b/talk/fsig3.pdf new file mode 100644 index 0000000..d375541 Binary files /dev/null and b/talk/fsig3.pdf differ diff --git a/talk/gcldef.tex b/talk/gcldef.tex new file mode 100644 index 0000000..5e1078d --- /dev/null +++ b/talk/gcldef.tex @@ -0,0 +1,27 @@ +\begin{definition}[]%[Générateur congruentiel linéaire] +Pour $m$, $a$, $c \in \Nats$ t.q. $m \ge a,c$. +La suite d'entiers + +\begin{equation} +\tag{1} +\left\{ +\begin{array}{l} +\textrm{$x_0 \in \Nats$ t.q. $0 \le x_0 \le m-1$}\\ +x_i = a \times x_{i-1} +c \mod m. +\end{array} +\right. +\label{eq:glc} +\end{equation} +est un \emph{générateur congruentiel linéaire} +de \emph{graine} $x_0$. +\end{definition} + +\vspace{-2em} +\begin{center} +\includegraphics[scale=0.2]{lcgexemple.png} +\end{center} + +\vspace{-2em} +\begin{itemize} +\item \alert<1> {Bien choisir $m$, $a$, $c$ et $x_0$: primordial!} +\end{itemize} diff --git a/talk/gcldef.tex.bak b/talk/gcldef.tex.bak new file mode 100644 index 0000000..b000cd6 --- /dev/null +++ b/talk/gcldef.tex.bak @@ -0,0 +1,27 @@ +\begin{definition}[]%[Générateur congruentiel linéaire] +Pour $m$, $a$, $c \in \Nats$ t.q. $m \ge a,c$. +La suite d'entiers + +\begin{equation} +\tag{1} +\left\{ +\begin{array}{l} +\textrm{$x_0 \in \Nats$ t.q. $0 \le x_0 \le m-1$}\\ +x_i = a \times x_{i-1} +c \mod m. +\end{array} +\right. +\label{eq:glc} +\end{equation} +est un \emph{générateur congruentiel linéaire} +de \emph{graine} $x_0$. +\end{definition} + +\vspace{-2em} +\begin{center} +\includegraphics[scale=0.2]{lcgexemple.png} +\end{center} + +\vspace{-2em} +\begin{itemize} +\item \alert<1> {Bien choisir $m$, $a$, $c$ et $x_0$: primoridial!} +\end{itemize} diff --git a/talk/gclth1.tex b/talk/gclth1.tex new file mode 100644 index 0000000..4652f6e --- /dev/null +++ b/talk/gclth1.tex @@ -0,0 +1,21 @@ +\begin{theorem}[Théorème 1: Hull \& Dobell 1962]\label{theo:hull:dobell} +La période de la suite produite par le générateur congruentiel linéaire est $m$ si %et seulement si +\begin{enumerate} +\item\label{item:gcl:c1} $c$ et $m$: premiers entre eux, +\item\label{item:gcl:c2} $a \equiv 1 \mod p$ pour chaque facteur premier $p$ de $m$ et +\item\label{item:gcl:c3} $a \equiv 1 \mod 4$ si $4$ divise $m$. +\end{enumerate} +\end{theorem} + +\begin{exampleblock}{Analyse de l'exemple de la Figure~1} +On a $m=180$, $a=61$ et $c=7$. Or $m=180 = 2^2 \times 3^2 \times 5$. +\begin{enumerate} +\item $c$ et $m$: premiers entre eux, +\item $61 \equiv 1 \mod 2$, $61 \equiv 1 \mod 3$, $61 \equiv 1 \mod 5$ et +\item $61 \equiv 1 \mod 4$. +\end{enumerate} +La période d'un tel générateur est donc maximale. +\end{exampleblock} + + + diff --git a/talk/gclth1demo.tex b/talk/gclth1demo.tex new file mode 100644 index 0000000..09d708d --- /dev/null +++ b/talk/gclth1demo.tex @@ -0,0 +1,46 @@ +% \only<1>{ +% \begin{theorem}[Théorème 1: Hull \& Dobell 1962] +% La période de la suite produite par le générateur congruentiel linéaire est $m$ si %et seulement si +% \begin{enumerate} +% \item\label{item:gcl:c1} $c$ et $m$: premiers entre eux, +% \item\label{item:gcl:c2} $a \equiv 1 \mod p$ pour chaque facteur premier $p$ de $m$ et +% \item\label{item:gcl:c3} $a \equiv 1 \mod 4$ si $4$ divise $m$. +% \end{enumerate} +% \end{theorem} +% } + +\begin{theorem}[]%[Théorème 1: Hull \& Dobell 1962]\label{theo:hull:dobell} +La période de la suite produite par le générateur congruentiel linéaire est $m$ si %et seulement si +\begin{enumerate} +\item\label{item:gcl:c1} $c$ et $m$: premiers entre eux, +\item\label{item:gcl:c2} $a \equiv 1 \mod p$ pour chaque facteur premier $p$ de $m$ et +\item\label{item:gcl:c3} $a \equiv 1 \mod 4$ si $4$ divise $m$. +\end{enumerate} +\end{theorem} + + +\begin{enumerate} +\item {\sc Lemme}~1 $\Rightarrow$ la période est l'entier $n$ minimum qui établit +\begin{equation} +\tag{4} +\frac{a^n-1}{a-1}\equiv 0 \mod m. +\end{equation} + +\item \textbf{Cas $n=m = p^{\alpha}$, $p$ premier et $\alpha \geq 2$} ({\sc Lemme}~2): +\begin{enumerate} +\item binôme de Newton pour réécrire $a^n$; +\item analyse de $\frac{p^{\alpha}}{j}$ dans $(6)$ $\leadsto$ (4) établie. +\end{enumerate} +\item \textbf{Cas $n{ +% \begin{theorem}[Théorème 1: Hull \& Dobell 1962] +% La période de la suite produite par le générateur congruentiel linéaire est $m$ si %et seulement si +% \begin{enumerate} +% \item\label{item:gcl:c1} $c$ et $m$: premiers entre eux, +% \item\label{item:gcl:c2} $a \equiv 1 \mod p$ pour chaque facteur premier $p$ de $m$ et +% \item\label{item:gcl:c3} $a \equiv 1 \mod 4$ si $4$ divise $m$. +% \end{enumerate} +% \end{theorem} +% } + +\begin{theorem}[Théorème 3] +Soit $m$ un nombre premier. On peut toujours trouver $a$ d'ordre $m-1$, +$1 \le a \le m-1$. +\end{theorem} + + + +\begin{enumerate} +\item {\sc Lemme}~1 $\Rightarrow$ recherche de l'entier $n$ minimum qui établit +\begin{equation} +\tag{4} +\frac{a^n-1}{a-1}\equiv 0 \mod m. +\end{equation} + +\item \textbf{Cas $n=m = p^{\alpha}$, $p$ premier et $\alpha \geq 2$} ({\sc Lemme}~2): +\begin{enumerate} +\item binôme de Newton pour réécrire $a^n$; +\item analyse de $\frac{p^{\alpha}}{j}$ dans $(6)$ $\leadsto$ (4) établie. +\end{enumerate} +\item \textbf{Cas $n -- 72 +< -- 5 ++ -- 26 +- -- 25 +* -- 104 +:= -- 271 +add.period$ -- 13 +call.type$ -- 4 +change.case$ -- 24 +chr.to.int$ -- 4 +cite$ -- 4 +duplicate$ -- 67 +empty$ -- 104 +format.name$ -- 31 +if$ -- 312 +int.to.chr$ -- 1 +int.to.str$ -- 0 +missing$ -- 3 +newline$ -- 24 +num.names$ -- 12 +pop$ -- 27 +preamble$ -- 1 +purify$ -- 29 +quote$ -- 0 +skip$ -- 50 +stack$ -- 0 +substring$ -- 61 +swap$ -- 13 +text.length$ -- 5 +text.prefix$ -- 2 +top$ -- 0 +type$ -- 28 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defining Unicode char U+0141 (decimal 321) + defining Unicode char U+0142 (decimal 322) + defining Unicode char U+0143 (decimal 323) + defining Unicode char U+0144 (decimal 324) + defining Unicode char U+0145 (decimal 325) + defining Unicode char U+0146 (decimal 326) + defining Unicode char U+0147 (decimal 327) + defining Unicode char U+0148 (decimal 328) + defining Unicode char U+014A (decimal 330) + defining Unicode char U+014B (decimal 331) + defining Unicode char U+014C (decimal 332) + defining Unicode char U+014D (decimal 333) + defining Unicode char U+014E (decimal 334) + defining Unicode char U+014F (decimal 335) + defining Unicode char U+0150 (decimal 336) + defining Unicode char U+0151 (decimal 337) + defining Unicode char U+0152 (decimal 338) + defining Unicode char U+0153 (decimal 339) + defining Unicode char U+0154 (decimal 340) + defining Unicode char U+0155 (decimal 341) + defining Unicode char U+0156 (decimal 342) + defining Unicode char U+0157 (decimal 343) + defining 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+%\usepackage{ntheorem} +\deftranslation[to=french]{Theorem}{} + + + +% \theoremstyle{plain} +% \theoremsymbol{\ensuremath{\clubsuit}} +% \theoremseparator{.} +% \theoremprework{\hrulefill} +% \theorempostwork{\hrulefill\newline} +% \renewtheorem{theorem}[theorem]{} + + + +\newcommand{\inputFrameb}[2]{ +\frame{ +\frametitle{#1} +\begin{footnotesize} +\input{#2} +\end{footnotesize} +}} + +\newcommand{\vectornorm}[1]{\ensuremath{\left|\left|#1\right|\right|_2}} +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\Nats}[0]{\ensuremath{\mathbb{N}}} +\newcommand{\R}[0]{\ensuremath{\mathbb{R}}} +\newcommand{\K}[0]{\ensuremath{\mathbb{R}}} +\newcommand{\Z}[0]{\ensuremath{\mathbb{Z}}} +\newcommand{\Bool}[0]{\ensuremath{\mathds{B}}} +\newcommand{\rel}[0]{\ensuremath{{\mathcal{R}}}} +\newcommand{\Gall}[0]{\ensuremath{\mathcal{G}}} +\newcommand{\F}[0]{\ensuremath{\mathbb{F}}} +\newcommand{\StratSet}[0]{\ensuremath{\mathbb{S}}} +\newcommand{\class}[1]{\ensuremath{\langle #1\rangle}} + + + \newcommand{\eqNode}[0]{\ensuremath{{\mathcal{R}}}} + + +\setbeamertemplate{itemize/enumerate body begin}{\footnotesize} +\setbeamertemplate{itemize/enumerate subbody begin}{\footnotesize} + + +\newcommand{\fc}[1]{\ensuremath{\overline{#1}}} +%\renewcommand\footnotelayout{\tiny} +\let\oldfootnotesize\footnotesize +\renewcommand*{\footnotesize}{\oldfootnotesize\scriptsize} + +\usepackage{graphicx} +\DeclareGraphicsExtensions{.jpg, .png , .pdf, .bmp, .pdftex} +\begin{document} + + +\title[Modèles discrets pour la sécurité]{Modèles discrets pour la sécurité informatique: des méthodes itératives à l'analyse vectorielle.} + + + +\author[J.-F. Couchot]{Jean-Fran\c{c}ois {\sc Couchot}} +\date[HDR, le 30/01/16]{\vspace{-1.5em}Soutenance d'HDR: le 30/01/16} + + + +\institute[Institut FEMTO-ST]{FEMTO-ST Institut, Université de Franche-Comté, France} + + +% \newtheorem{theorem}{Théorème} +% \newtheorem{lemma}{Lemme} +% \newtheorem{corollary}{Corollaire} +% \newtheorem*{xpl}{Exemple} + +% \newtheorem{Def}{Définition} +% ____ _____ ____ _ _ _____ +% | _ \| ____| __ )| | | |_ _| +% | | | | _| | _ \| | | | | | +% | |_| | |___| |_) | |_| | | | +% |____/|_____|____/ \___/ |_| +% + +%\newcommand{\Bool}[0]{\ensuremath{\mathds{B}}} + +\setbeamertemplate{background}{\titrefemto} + +\begin{frame}[plain] +\titlepage +\end{frame} + +\setbeamertemplate{background}{\pagefemto} + +% \begin{frame}{Outline} +\setbeamertemplate{section in toc}[sections numbered] +% \tableofcontents +% \end{frame} + + + + +%\frame{\frametitle{Plan}\tableofcontents[hideallsubsections]} + +\section{Introduction: iterations de réseaux booléens} +\inputFrameb{Réseau booléen (définition)}{rb} +\inputFrameb{3 schémas $\leadsto$ 3 graphes d'itérations}{graphes} +\inputFrameb{Attracteurs}{attracteurs} +\inputFrameb{Dépendance entre éléments}{jacobienne} +\inputFrameb{Mode asynchrone}{asynchrone} +\inputFrameb{Un exemple motivant}{xplsddasync} +\inputFrameb{Problématique}{sddproblematique} + +\section{Réseaux booléens: des preuves de convergences}\label{sec:sdd} +\frame{\frametitle{Plan}\tableofcontents[currentsection]} +\inputFrameb{Un peu de Synchronisme}{synchronisons} +\inputFrameb{Mode mixte avec $g$}{mixteexample} +\inputFrameb{Convergence par la pratique}{preuveconvmotiv} +\inputFrameb{Du système booléen au modèle PROMELA}{sdd2promela} + + + +\section{Des systèmes dynamiques discrets au chaos}\label{sec:sddchaos} +\frame{\frametitle{Plan}\tableofcontents[currentsection]} + + +\section{Applications à la génération de nombres pseudo-aléatoires}\label{sec:prng} +\frame{\frametitle{Plan}\tableofcontents[currentsection]} + + + +\section{Application au masquage d'information}\label{sec:ih} +\frame{\frametitle{Plan}\tableofcontents[currentsection]} + +\section{Conclusion} +\frame{\frametitle{Plan}\tableofcontents[currentsection]} +\inputFrameb{Synthèse scientifique}{conclusion} + + + +% \section{Formalisation} +% \frame{\frametitle{Plan}\tableofcontents[currentsection,hideallsubsections]} +% \inputFrameb{Algèbre de Boole}{ab} +% \inputFrameb{Réseaux booléens (définition)}{rb} +% \inputFrameb{Trois modes $\leadsto$ trois graphes d'itérations}{graphes} +% \inputFrameb{Attracteurs}{attracteurs} +% \inputFrameb{Matrice Jacobienne $f'$}{jacobienne} + + +% \section{Quelques résultats théoriques} +% \frame{\frametitle{Plan}\tableofcontents[currentsection,hideallsubsections]} +% \subsection{Conditions de convergence} +% \inputFrameb{François Robert 1980}{robert80} +% \inputFrameb{François Robert 1995}{robert95} +% \subsection{Nombre d'attracteurs} +% \inputFrameb{Absence de cycles}{attracteursth} + +% \section{Application à l'étude du bacille de pyo} +% \frame{\frametitle{Plan}\tableofcontents[currentsection,hideallsubsections]} +% \inputFrameb{Retour aux réseaux de régulation de gènes}{pyo} + +% \section{Des questions?} +% \frame{\frametitle{Plan}\tableofcontents[currentsection,hideallsubsections]} +% \inputFrameb{Un résumé}{../abstract} +% \inputFrameb{Mettre en confiance le candidat}{confiance} +% \inputFrameb{Similarités}{similarites} +% \inputFrameb{Exemple d'itérations asynchrones}{asynchroneex} +% \inputFrameb{Montrer que $d$ (Sect. 3) est une distance}{dv} +% \inputFrameb{Application du théorème 2}{th2} +% \inputFrameb{Théorème 4: illustrez le dans $\Bool^2$}{th4} + + + +% \section{Conclusion et perspectives} +% \frame{\frametitle{Outline}\tableofcontents[currentsection,hideallsubsections]} + + +\bibliographystyle{alpha} +\bibliography{abbrev,biblioand} +\end{document} +% _____ ___ _ _ +% | ___|_ _| \ | | +% | |_ | || \| | +% | _| | || |\ | +% |_| |___|_| \_| +% diff --git a/talk/main.thm b/talk/main.thm new file mode 100644 index 0000000..5278c70 --- /dev/null +++ b/talk/main.thm @@ -0,0 +1 @@ +\contentsline {\let N}{{\translate {Theorem}}{1}{~\cite {BCVC10:ir}}} diff --git a/talk/main.toc b/talk/main.toc new file mode 100644 index 0000000..fbd10b8 --- /dev/null +++ b/talk/main.toc @@ -0,0 +1,8 @@ +\beamer@endinputifotherversion {3.36pt} +\select@language {french} +\beamer@sectionintoc {1}{Introduction: iterations de r\IeC {\'e}seaux bool\IeC {\'e}ens}{2}{0}{1} +\beamer@sectionintoc {2}{R\IeC {\'e}seaux bool\IeC {\'e}ens: des preuves de convergences}{9}{0}{2} +\beamer@sectionintoc {3}{Des syst\IeC {\`e}mes dynamiques discrets au chaos}{14}{0}{3} +\beamer@sectionintoc {4}{Applications \IeC {\`a} la g\IeC {\'e}n\IeC {\'e}ration de nombres pseudo-al\IeC {\'e}atoires}{15}{0}{4} +\beamer@sectionintoc {5}{Application au masquage d'information}{16}{0}{5} +\beamer@sectionintoc {6}{Conclusion}{17}{0}{6} diff --git a/talk/mixteexample.tex b/talk/mixteexample.tex new file mode 100644 index 0000000..0617945 --- /dev/null +++ b/talk/mixteexample.tex @@ -0,0 +1,19 @@ +\begin{exampleblock}{} +\begin{figure} +\subfloat[Parallèle synchrone]{ + \includegraphics[scale=0.2]{../images/eval_sync} +} + +\vspace{-1em} +\subfloat[Asynchrone]{ + \includegraphics[scale=0.2]{../images/eval_async} +} + +\vspace{-1em} +\subfloat[Mixte tq.\class{1} $=\{1,2\}$, \class{3} $=\{3\}$, +\class{4} $=\{4,5\}$.]{ + \includegraphics[scale=0.2]{../images/eval_mixte} +} +\caption{Graphes des itérations de $g$} +\end{figure} +\end{exampleblock} diff --git a/talk/preuveconvmotiv.tex b/talk/preuveconvmotiv.tex new file mode 100644 index 0000000..b3e38d4 --- /dev/null +++ b/talk/preuveconvmotiv.tex @@ -0,0 +1,18 @@ + +\begin{itemize} +\item Conditions suffisantes de convergence: facile à appliquer, + domaine restreint +\item Recherche d'une métrique decroissante minorée: difficile +\item Simulations: +\begin{itemize} +\item Non exhaustives pour les schémas généralisés et asynchrones. +\item Verdict $\leftrightarrow$ vérité ssi divergence (contre-exemple). +\end{itemize} +\item Souhait: exploiter un outil qui traiterait toutes les transitions +\begin{itemize} +\item explosion combinatoire: par abstraction, + quotientage, ordre partiel\ldots +\item Model checker: SPIN~\cite{Hol03}. +\item Correction et complétude de la démarche. + \end{itemize} +\end{itemize} diff --git a/talk/q2.tex b/talk/q2.tex new file mode 100644 index 0000000..8a2c11b --- /dev/null +++ b/talk/q2.tex @@ -0,0 +1,5 @@ +Dans quel(s) contexte(s) avez vous déjà rencontré des générateurs de nombres pseudo-aléatoires. + +Réponses: les probabilités, les statistiques, dans des simulations (méthodes de Monte-Carlo), +le chiffrage (algorithme RSA par exemple), notamment pour générer des grands nombres aléatoires +premiers, la localisation par GPS. \ No newline at end of file diff --git a/talk/q3.tex b/talk/q3.tex new file mode 100644 index 0000000..13b29b6 --- /dev/null +++ b/talk/q3.tex @@ -0,0 +1,6 @@ +A la ligne~30, il est écrit qu'\og un générateur congruentiel linéaire +produit une suite périodique dont la période ne peut pas excéder $m$\fg{}. Pourquoi? + +Réponse: En construisant $m+1$ termes successifs, il est nécessaire de rencontrer +au moins deux termes $x_i$ et $x_j$ égaux puisqu'il n'y a que $m$ termes différents dans +$\{0,\dots,m-1\}$. Quel que soit $k \in \Nats$, on a $x_{i+k} = x_{j+k}$. \ No newline at end of file diff --git a/talk/q4.tex b/talk/q4.tex new file mode 100644 index 0000000..b316811 --- /dev/null +++ b/talk/q4.tex @@ -0,0 +1,9 @@ +A la ligne~204, on prétend savoir calculer +\og l'inverse de $x_i-x_{i+1}$ modulo $m$\fg{}. Comment faire ceci? Ce problème a-t-il toujours une solution? + +Réponses: +Il s'agit d'appliquer l'algorithme d'Euclide étendu: on cherche $u$ (et $v$) +tels que $(x_i-x_{i+1}).u + m.v = 1$. Or ceci n'admet une solution que si +$(x_i-x_{i+1})$ et $m$ sont premiers entre-eux. +(Par exemple, impossible de trouver l'inverse de +2 modulo 6). \ No newline at end of file diff --git a/talk/q6.tex b/talk/q6.tex new file mode 100644 index 0000000..2988081 --- /dev/null +++ b/talk/q6.tex @@ -0,0 +1,13 @@ +Que dire des générateurs suivants: +\begin{itemize} +\item $x_i = (129x_{i-1} + 907633385) \mod 2^{32}$ (générateur du Turbo Pascal) +\item $x_i = 69069x_{i-1} \mod 2^{32}$ (générateur de Marsaglia) +\end{itemize} + +Réponses:\begin{itemize} +\item Le premier vérifie toutes les hypothèses du théorème~1 +($907633385$ et $2^{32}$ sont premiers entre eux, $2$ est le seul facteur premier de $2^{32}$ et comme 129 est impair, $129 \equiv 1 \mod 2$, et, enfin, $129 \equiv 1 \mod 4$). Sa période est donc $2^{32}$. +\item $2^{32}$ n'est pas premier. On ne peut pas directement appliquer +le théorème~2 et on ne peut donc rien conclure. +Remarque: cette condition n'est en effet pas nécessaire car la période de ce générateur est cependant $2^{32}$. +\end{itemize} diff --git a/talk/q8.tex b/talk/q8.tex new file mode 100644 index 0000000..8b2552a --- /dev/null +++ b/talk/q8.tex @@ -0,0 +1,9 @@ +Que démontre la section~2.4? Quelle est la portée de ce résultat? + +Réponse. Cette section montre qu'un générateur de nombres pseudo-aléatoires +peut certes produire des sorties qui vérifient les critères statistiques +de bonne distribution (période longues, sous-suites équitablement réparties), +mais une personne peut cependant deviner les paramètres et donc +\og casser \fg{} l'aléa. + +Dans une simulation physique, cela n'est pas nécessairement grave. Lorsqu'on génère des grands nombres pseudo aléatoirement pour de la cryptographie ou dans un casino, connaître l'algorithme sous-jacent peut être une faille de sécurité. diff --git a/talk/q9.tex b/talk/q9.tex new file mode 100644 index 0000000..e57eee3 --- /dev/null +++ b/talk/q9.tex @@ -0,0 +1,22 @@ +Supposons qu'on dispose d'une fonction qui génère un nombre pseudo-aléatoire +dont la sortie $j$ est uniformément distribuée sur l'intervalle $[0,1[$. +Comment peut-on faire +pour disposer d'une sortie uniformément distribuée sur l'ensemble des entiers +$\{0,\dots, m\}$. + +Réponse: On commence par diviser $[0,1[$ en $m+1$ intervalles de même +longueurs: +$I_0= [0, \dfrac{1}{m+1}[$, +$I_1= [\dfrac{1}{m+1}, \dfrac{2}{m+1} [$, \ldots +$I_m = [\dfrac{m}{m+1}, 1[$. +Si la sortie $j$ appartient à $I_0$, on retourne 0,\ldots. + +Ceci revient à d'abord multiplier $j$ par $m+1$ et à prendre ensuite sa partie entière, ce qui est fait dans le code Python suivant. +La fonction Python \verb+randint(a,b)+ génère un entier +entre $a$ et $b$, les deux nombres compris. + +\begin{lstlisting} +def genereAleaEntier(m): + j= random() + return int(j*(m+1)) +\end{lstlisting} \ No newline at end of file diff --git a/talk/rb.tex b/talk/rb.tex new file mode 100644 index 0000000..ad0d751 --- /dev/null +++ b/talk/rb.tex @@ -0,0 +1,45 @@ +\begin{itemize} +\item Une fonction +$ +f:\Bool^{\mathsf N}\to\Bool^{\mathsf N}$, $x=(x_1,\dots,x_{\mathsf N})\mapsto (f_1(x),\dots,f_{\mathsf N}(x)) +$ +\item Un schéma de mise à jour de la suite $(x^{t})^{t + \in \Nats}$ des configurations: +\begin{itemize} +\item \emph{Parallèle synchrone:} + $x^{t+1}=f(x^t)$. +\item \emph{Unaire:} à partir de la \emph{stratégie unaire} + $S = \left(s^t\right)^{t \in \mathds{N}}$, + modification de l'élément $s^t$ de $x^t$ + $$ + \begin{array}{l} + x^{t+1}= (x^{t+1}_1,\dots,x^{t+1}_n) \textrm{ où } + x^{t+1}_i = + \left\{ + \begin{array}{l} + f_i( x^{t}) \textrm{ si $i = s^t$} \\ + x^t_i \textrm{ sinon.} + \end{array} + \right. + \end{array} +$$ + +\item \emph{Généralisé:} à partir de la \emph{stratégie généralisée} + $\left(s^t\right)^{t \in \mathds{N}}$, + à chaque itération $t$, + modification des éléments de $x^t$ dans $s^t\subset\{1,\ldots,n\}$ + $$ + \begin{array}{l} + x^{t+1}= (x^{t+1}_1,\dots,x^{t+1}_n) \textrm{ où } + x^{t+1}_i = + \left\{ + \begin{array}{l} + f_i( x^{t}) \textrm{ si $i \alert{\in} s^t$} \\ + x^t_i \textrm{ sinon} + \end{array} + \right. + \end{array} +$$ + +\end{itemize} +\end{itemize} diff --git a/talk/regdef.tex b/talk/regdef.tex new file mode 100644 index 0000000..a88cc0c --- /dev/null +++ b/talk/regdef.tex @@ -0,0 +1,19 @@ +\begin{definition}[]%[A base de registres à décalage linéaire] +Soit $p$ un nombre premier. +\vspace{-2em} +\begin{center} +\includegraphics[scale=0.2]{regxpl.png} +\end{center} +\vspace{-2em} +\noindent Un générateur \emph{à base de registres à décalage linéaires} +produit $a_{k+r}$ de $\F_{p}$ selon +\begin{equation} +\tag{9} +a_{k+r} \equiv \sum\limits_{j=0}^{r-1}q_ja_{j+k} \mod p. +\label{eq:reg:def} +\end{equation} + +\end{definition} + + + diff --git a/talk/regth5.tex b/talk/regth5.tex new file mode 100644 index 0000000..1cdb543 --- /dev/null +++ b/talk/regth5.tex @@ -0,0 +1,29 @@ +\begin{theorem}[Théorème 5] +Soit $p$ premier, $q_0, \dots , q_{r-1} \in \F_p$ tels que +$Q(x) = x^r - q_{r-1} x^{r-1} - \dots - q_1 x - q_0$ est primitif +sur $\F_p$ et soit +$a_0, \dots, a_{r-1}$ une graine différente du vecteur nul. +Le générateur à base de registres à décalage linéaires engendre une suite de +période $p^r-1$. +\end{theorem} + + + +\begin{exampleblock}{Analyse de l'exemple de la Figure~5} +On a $p=3$, $r=4$, la graine $1,0,0,0$ n'est pas nulle. +Il suffi(rai)t de démontrer que + +\begin{enumerate} +\item $Q(x) = x^4 - 2 x^{3} - 1 \equiv x^4 + x^{3} + 2$ est irréductible +(facile) +\item chaque élément de +$ +(\F_{p})_{r-1}(x) +$ +peut s'écrire sous la forme $x^i$ modulo $Q(x)$, $0 \le i \le p^r-2$ (long, mais vrai) +\end{enumerate} + +La période d'un tel générateur est donc $3^4-1=80$. +\end{exampleblock} + + diff --git a/talk/regth5.tex.bak b/talk/regth5.tex.bak new file mode 100644 index 0000000..e9a2696 --- /dev/null +++ b/talk/regth5.tex.bak @@ -0,0 +1,29 @@ +\begin{theorem}[Théorème 5] +Soit $p$ premier, $q_0, \dots , q_{r-1} \in \F_p$ tels que +$Q(x) = x^r - q_{r-1} x^{r-1} - \dots - q_1 x - q_0$ est primitif +sur $\F_p$ et soit +$a_0, \dots, a_{r-1}$ une graine différente du vecteur nul. +Le générateur à base de registres à décalage linéaires engendre une suite de +période $p^r-1$. +\end{theorem} + + + +\begin{exampleblock}{Analyse de l'exemple de la Figure~5} +On a $p=3$, $r=4$, la graîne $1,0,0,0$ n'est pas nulle. +Il suffi(rai)t de démontrer que + +\begin{enumerate} +\item $Q(x) = x^4 - 2 x^{3} - 1 \equiv x^4 + x^{3} + 2$ est irréductible +(facile) +\item chaque élément de +$ +(\F_{p})_{r-1}(x) +$ +peut s'écrire sous la forme $x^i$ modulo $Q(x)$, $0 \le i \le p^r-2$ (long, mais vrai) +\end{enumerate} + +La période d'un tel générateur est donc $3^4-1=80$. +\end{exampleblock} + + diff --git a/talk/regth5demo.tex b/talk/regth5demo.tex new file mode 100644 index 0000000..5aa50f7 --- /dev/null +++ b/talk/regth5demo.tex @@ -0,0 +1,20 @@ +\begin{theorem}[Théorème 5] +Soit $p$ premier, $q_0, \dots , q_{r-1} \in \F_p$ tels que +$Q(x) = x^r - q_{r-1} x^{r-1} - \dots - q_1 x - q_0$ est primitif +sur $\F_p$ et soit +$a_0, \dots, a_{r-1}$ une graine différente du vecteur nul. +Le générateur à base de registres à décalage linéaires engendre une suite de +période $p^r-1$. +\end{theorem} + + +\begin{enumerate} +\item Construction d'un polynôme $b \in (\F_{p})_{r-1}(x)$ et d'une fonction +$T: (\F_{p})_{r-1}(x) \rightarrow \F_p$ t.q. $T(b.x^n)=a_n$ ({\sc Lemme}~6) +\item Comme $x^{p^r-1}\equiv 1 \mod Q(x)$, d'après 1. la période de $a_n$ est au plus $p^r-1$ (l.~290) +\begin{itemize} +\item $a_{n+p^r+1} \equiv T(b.x^{n+p^r+1}) \equiv T(b.x^{n}x^{p^r+1}) \equiv a_n$. +\end{itemize} +\item Si la période était inférieure à $p^r -1$, alors $Q(x)$ ne serait pas +primitif (l. 296) +\end{enumerate} diff --git a/talk/regth6.tex b/talk/regth6.tex new file mode 100644 index 0000000..f82225c --- /dev/null +++ b/talk/regth6.tex @@ -0,0 +1,15 @@ + +\begin{definition}[Fenêtre] +Sous les mêmes conditions que celles du théorème~5, +on appelle \emph{fenêtre} toute sous-suite de longueur +$p^r-1$. +\end{definition} + +\begin{theorem}[Théorème 6] +Sous les mêmes conditions que celle du théorème~5, +soit $k$ un entier, $k \le r$. Alors dans chaque fenêtre, toute +sous-suite de longueur $k$ apparaît $p^{r-k}$ fois, +sauf la sous-suite nulle qui, elle, apparaît $p^{r-k}-1$ fois. +\end{theorem} + + diff --git a/talk/regxpl.png b/talk/regxpl.png new file mode 100644 index 0000000..b5e56d4 Binary files /dev/null and b/talk/regxpl.png differ diff --git a/talk/sdd2promela.tex b/talk/sdd2promela.tex new file mode 100644 index 0000000..4676dfa --- /dev/null +++ b/talk/sdd2promela.tex @@ -0,0 +1,17 @@ +\begin{itemize} +\item Points clefs de la traduction: +\begin{itemize} +\item Stratégie: pseudo périodicité garantie par le choix indéterministe de + SPIN. +\item Délais(bornés par construction): oubli de certaines valeurs grâce à l'indéterminisme de SPIN. +\end{itemize} +\item Convergence universelle: $\diamond (\Box \verb+Xp+ = \verb+X+)$. +\end{itemize} + +\begin{theorem}[Correction et complétude de la traduction vers Promela~\cite{Cou10:ir}] + Soit $\phi$ un modèle de système dynamique discret et $\psi$ sa traduction PROMELA. + Les itérations de $\phi$ sont universellement convergentes si et + seulement si + $\psi$ vérifie + la propriété LTL sous hypothèse d'équité faible. +\end{theorem} diff --git a/talk/sddproblematique.tex b/talk/sddproblematique.tex new file mode 100644 index 0000000..960b079 --- /dev/null +++ b/talk/sddproblematique.tex @@ -0,0 +1,25 @@ +\begin{itemize} +\item Peut-on prédire le comportement de réseaux bouléens (Sec.~\ref{sec:sdd}) +\begin{itemize} +\item Théorique: \alert{conditions théoriques} + nécessaires/suffisantes de convergence/divergence? +\item Pratique: vérification par simulation (\alert{exhaustive?}) +\end{itemize} +\item Itérations divergentes \alert{$ \leftrightarrow$ comportement chaotique} (Sec.~\ref{sec:sddchaos})? +\begin{itemize} +\item Caractérisation des réseaux bouléens chaotiques. +\item Génération et prédiction. +\end{itemize} +\item Générateurs de nombres pseudo-aléatoires (Sec.~\ref{sec:prng}): +\begin{itemize} +\item Caractérisation d'un PRNG chaotique. +\item Générations et qualité. +\end{itemize} +\item Du Chaos au masquage d'information (Sec.\ref{sec:ih}). +\begin{itemize} +\item D'un point de vue chaotique. +\item D'un point de vue analyse vectorielle discrète. +\end{itemize} +\end{itemize} + + diff --git a/talk/sense.jpg b/talk/sense.jpg new file mode 100644 index 0000000..697d443 Binary files /dev/null and b/talk/sense.jpg differ diff --git a/talk/synchronisons.tex b/talk/synchronisons.tex new file mode 100644 index 0000000..7350d2a --- /dev/null +++ b/talk/synchronisons.tex @@ -0,0 +1,18 @@ +\begin{itemize} +\item \emph{Mode mixte}~\cite{abcvs05}: regroupement +des n{\oe}uds qui pourraient introduire des cycles. +\begin{itemize} +\item A l'intérieur de chaque groupe: mode synchrone. +\item A l'extérieur de chaque groupe: mode asynchrone. +\end{itemize} +\item Relation de synchronisation: $i \eqNode j$ + si $i$ et $j$ dans la même CFC du graphe des interactions. +\end{itemize} +\begin{theorem}[~\cite{BCVC10:ir}] + Soit $f$ possédant un unique point fixe $x^*$ et une stratégie + pseudo-périodique $s$. + Si les itérations synchrones convergent vers $x^*$ pour cette stratégie, + alors les itérations mixtes à délai uniforme convergent aussi vers $x^*$ + pour cette stratégie. +\end{theorem} + diff --git a/talk/wsn.png b/talk/wsn.png new file mode 100644 index 0000000..765a04f Binary files /dev/null and b/talk/wsn.png differ diff --git a/talk/xplsddasync.tex b/talk/xplsddasync.tex new file mode 100644 index 0000000..f7211bf --- /dev/null +++ b/talk/xplsddasync.tex @@ -0,0 +1,19 @@ +\begin{exampleblock}{ $g(x_1,x_2,x_3,x_4,x_5) = +(x_1.\overline{x_2} + \overline{x_1}.x_2,\overline{x_1 + x_2}, x_3.\overline{x_1}, x_5, \overline{x_3} + x_4)$} + \begin{figure} + \centering + \subfloat[$\textsc{gis}(g)$]{ + \includegraphics[scale=0.30]{../images/para_iterate_dec} + }\qquad + \subfloat[$\textsc{giu}(g)$ (extrait)]{ + \includegraphics[scale=0.30]{../images/chao_iterate_excerpt} + } +\caption{Graphes des itérations synchrones} + \begin{itemize} + \item Avec $D^t = t$ sauf $D^t_{12} = t-1$ pour $t$ impair, + $g$ oscille entre $(0,0,0,1,1)$ et $(0,1,0,1,1)$ + \item Schéma parallèle: converge en synchrone, diverge en asynchrone + \end{itemize} + +\end{figure} +\end{exampleblock}