--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/slides/bak-slides06.tex Mon Nov 10 05:57:10 2014 +0000
@@ -0,0 +1,1484 @@
+\documentclass[dvipsnames,14pt,t]{beamer}
+\usepackage{proof}
+\usepackage{beamerthemeplaincu}
+%\usepackage[T1]{fontenc}
+%\usepackage[latin1]{inputenc}
+\usepackage{mathpartir}
+\usepackage{isabelle}
+\usepackage{isabellesym}
+\usepackage[absolute,overlay]{textpos}
+\usepackage{ifthen}
+\usepackage{tikz}
+\usepackage{courier}
+\usepackage{listings}
+\usetikzlibrary{arrows}
+\usetikzlibrary{positioning}
+\usetikzlibrary{calc}
+\usetikzlibrary{shapes}
+\usepackage{graphicx}
+\setmonofont[Scale=MatchLowercase]{Consolas}
+
+\isabellestyle{rm}
+\renewcommand{\isastyle}{\rm}%
+\renewcommand{\isastyleminor}{\rm}%
+\renewcommand{\isastylescript}{\footnotesize\rm\slshape}%
+\renewcommand{\isatagproof}{}
+\renewcommand{\endisatagproof}{}
+\renewcommand{\isamarkupcmt}[1]{#1}
+
+% Isabelle characters
+\renewcommand{\isacharunderscore}{\_}
+\renewcommand{\isacharbar}{\isamath{\mid}}
+\renewcommand{\isasymiota}{}
+\renewcommand{\isacharbraceleft}{\{}
+\renewcommand{\isacharbraceright}{\}}
+\renewcommand{\isacharless}{$\langle$}
+\renewcommand{\isachargreater}{$\rangle$}
+\renewcommand{\isasymsharp}{\isamath{\#}}
+\renewcommand{\isasymdots}{\isamath{...}}
+\renewcommand{\isasymbullet}{\act}
+\newcommand{\isaliteral}[1]{}
+\newcommand{\isactrlisub}[1]{\emph{\isascriptstyle${}\sb{#1}$}}
+
+
+
+\definecolor{javared}{rgb}{0.6,0,0} % for strings
+\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments
+\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords
+\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc
+
+\lstset{language=Java,
+ basicstyle=\ttfamily,
+ keywordstyle=\color{javapurple}\bfseries,
+ stringstyle=\color{javagreen},
+ commentstyle=\color{javagreen},
+ morecomment=[s][\color{javadocblue}]{/**}{*/},
+ numbers=left,
+ numberstyle=\tiny\color{black},
+ stepnumber=1,
+ numbersep=10pt,
+ tabsize=2,
+ showspaces=false,
+ showstringspaces=false}
+
+\lstdefinelanguage{scala}{
+ morekeywords={abstract,case,catch,class,def,%
+ do,else,extends,false,final,finally,%
+ for,if,implicit,import,match,mixin,%
+ new,null,object,override,package,%
+ private,protected,requires,return,sealed,%
+ super,this,throw,trait,true,try,%
+ type,val,var,while,with,yield},
+ otherkeywords={=>,<-,<\%,<:,>:,\#,@},
+ sensitive=true,
+ morecomment=[l]{//},
+ morecomment=[n]{/*}{*/},
+ morestring=[b]",
+ morestring=[b]',
+ morestring=[b]"""
+}
+
+\lstset{language=Scala,
+ basicstyle=\ttfamily,
+ keywordstyle=\color{javapurple}\bfseries,
+ stringstyle=\color{javagreen},
+ commentstyle=\color{javagreen},
+ morecomment=[s][\color{javadocblue}]{/**}{*/},
+ numbers=left,
+ numberstyle=\tiny\color{black},
+ stepnumber=1,
+ numbersep=10pt,
+ tabsize=2,
+ showspaces=false,
+ showstringspaces=false}
+
+%sudoku
+\newcounter{row}
+\newcounter{col}
+
+\newcommand\setrow[9]{
+ \setcounter{col}{1}
+ \foreach \n in {#1, #2, #3, #4, #5, #6, #7, #8, #9} {
+ \edef\x{\value{col} - 0.5}
+ \edef\y{9.5 - \value{row}}
+ \node[anchor=center] at (\x, \y) {\n};
+ \stepcounter{col}
+ }
+ \stepcounter{row}
+}
+
+\newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions
+
+% beamer stuff
+\renewcommand{\slidecaption}{APP 06, King's College London, 12 November 2013}
+
+\newcommand{\bl}[1]{\textcolor{blue}{#1}}
+\begin{document}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}<1>[t]
+\frametitle{%
+ \begin{tabular}{@ {}c@ {}}
+ \\
+ \LARGE Access Control and \\[-3mm]
+ \LARGE Privacy Policies (6)\\[-6mm]
+ \end{tabular}}\bigskip\bigskip\bigskip
+
+ %\begin{center}
+ %\includegraphics[scale=1.3]{pics/barrier.jpg}
+ %\end{center}
+
+\normalsize
+ \begin{center}
+ \begin{tabular}{ll}
+ Email: & christian.urban at kcl.ac.uk\\
+ Office: & S1.27 (1st floor Strand Building)\\
+ Slides: & KEATS (also homework is there)\\
+ \end{tabular}
+ \end{center}
+
+
+\end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[t]
+ \frametitle{\Large\begin{tabular}{@ {}c@ {}}Access Control Logic\end{tabular}}
+
+ Formulas
+
+ \begin{itemize}
+ \item[]
+
+ \begin{center}\color{blue}
+ \begin{tabular}[t]{rcl@ {\hspace{10mm}}l}
+ \isa{F} & \isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}} & \isa{true} \\
+ & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{false} \\
+ & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{F\ {\isaliteral{5C3C616E643E}{\isasymand}}\ F} \\
+ & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{F\ {\isaliteral{5C3C6F723E}{\isasymor}}\ F} \\
+ & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{F\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F}\\
+ & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{p\ {\isaliteral{28}{\isacharparenleft}}t\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}t\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}} \\
+ & \isa{{\isaliteral{7C}{\isacharbar}}} & \alert{\isa{P\ says\ F}} & \textcolor{black}{``saying predicate''}\\
+ \end{tabular}
+ \end{center}
+
+ \end{itemize}
+
+Judgements
+
+\begin{itemize}
+\item[] \mbox{\hspace{9mm}}\bl{$\Gamma \vdash \text{F}$}
+\end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Judgements}
+
+\begin{center}
+\begin{tikzpicture}[scale=1]
+
+ \draw (0.0,0.0) node {\LARGE \bl{$\Gamma \vdash F$}};
+ \onslide<2->{
+ \draw (-1,-0.3) node (X) {};
+ \draw (-2.0,-2.0) node (Y) {};
+ \draw (0.7,-3) node {\begin{tabular}{l}Gamma\\stands for a collection of formulas\\(``assumptions'')\end{tabular}};
+ \draw[red, ->, line width = 2mm] (Y) -- (X);
+
+ \draw (1.2,-0.1) node (X1) {};
+ \draw (2.8,-0.1) node (Y1) {};
+ \draw (4.5,-0.1) node {\begin{tabular}{l}a single formula\end{tabular}};
+ \draw[red, ->, line width = 2mm] (Y1) -- (X1);
+
+ \draw (-0.1,0.1) node (X2) {};
+ \draw (0.5,1.5) node (Y2) {};
+ \draw (1,1.8) node {\begin{tabular}{l}entails sign\end{tabular}};
+ \draw[red, ->, line width = 2mm] (Y2) -- (X2);}
+
+ \end{tikzpicture}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Inference Rules}
+
+\begin{center}
+\begin{tikzpicture}[scale=1]
+
+ \draw (0.0,0.0) node
+ {\Large\bl{\infer{\Gamma \vdash F_1 \wedge F_2}{\Gamma \vdash F_1 & \Gamma \vdash F_2}}};
+
+ \draw (-0.1,-0.7) node (X) {};
+ \draw (-0.1,-1.9) node (Y) {};
+ \draw (-0.2,-2) node {\begin{tabular}{l}conclusion\end{tabular}};
+ \draw[red, ->, line width = 2mm] (Y) -- (X);
+
+ \draw (-1,0.6) node (X2) {};
+ \draw (0.0,1.6) node (Y2) {};
+ \draw (0,1.8) node {\begin{tabular}{l}premisses\end{tabular}};
+ \draw[red, ->, line width = 2mm] (Y2) -- (X2);
+ \draw (1,0.6) node (X3) {};
+ \draw (0.0,1.6) node (Y3) {};
+ \draw[red, ->, line width = 2mm] (Y3) -- (X3);
+ \end{tikzpicture}
+\end{center}
+
+\only<2>{
+\begin{textblock}{11}(1,13)
+\small
+\bl{$P \,\text{says}\, F \vdash Q\,\text{says}\, F\wedge P \,\text{says}\, G $}
+\end{textblock}}
+\only<3>{
+\begin{textblock}{11}(1,13)
+\small
+\bl{$\underbrace{P \,\text{says}\, F}_{\Gamma} \vdash \underbrace{Q\,\text{says}\, F}_{F_1} \,\wedge
+ \underbrace{P \,\text{says}\, G}_{F_2} $}
+\end{textblock}}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Inference Rules}
+
+\begin{center}
+\bl{\infer{\Gamma, F\vdash F}{}}\bigskip\\
+
+\bl{\infer{\Gamma \vdash F_2}{\Gamma \vdash F_1 \Rightarrow F_2 \quad \Gamma \vdash F_1}}
+\qquad
+\bl{\infer{\Gamma \vdash F_1 \Rightarrow F_2}{F_1, \Gamma \vdash F_2}}\bigskip\\
+
+\bl{\infer{\Gamma \vdash P\,\text{says}\, F}{\Gamma \vdash F}}\medskip\\
+
+\bl{\infer{\Gamma \vdash P \,\text{says}\, F_2}
+ {\Gamma \vdash P \,\text{says}\, (F_1\Rightarrow F_2) \quad
+ \Gamma \vdash P \,\text{says}\, F_1}}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Sending Messages}
+
+ \begin{itemize}
+ \item Alice sends a message \bl{$m$}
+ \begin{center}
+ \bl{Alice says $m$}
+ \end{center}\medskip\pause
+
+ \item Alice sends an encrypted message \bl{$m$} with key \bl{$K$}
+ (\bl{$\{m\}_K \dn K \Rightarrow m$})
+ \begin{center}
+ \bl{Alice says $\{m\}_K$}
+ \end{center}\medskip\pause
+
+ \item Decryption of Alice's message\smallskip
+ \begin{center}
+ \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;m}
+ {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_K & \Gamma \vdash \text{Alice}\,\text{says}\,K}}}
+ \end{center}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Proofs}
+
+\begin{center}
+\bl{
+\infer{\Gamma \vdash F}
+ {\infer{\hspace{1cm}:\hspace{1cm}}
+ {\infer{\hspace{1cm}:\hspace{1cm}}{:}
+ &
+ \infer{\hspace{1cm}:\hspace{1cm}}{:\quad :}
+ }}
+}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{The Access Control Problem}
+
+
+\begin{center}
+ \begin{tikzpicture}[scale=1]
+
+ \draw[line width=1mm] (-.3, -0.5) rectangle (1.5,2);
+ \draw (-2.7,1) node {\begin{tabular}{l}access\\request\\ (\bl{$F$})\end{tabular}};
+ \draw (4.2,1) node {\begin{tabular}{l}provable/\\not provable\end{tabular}};
+ \draw (0.6,0.8) node {\footnotesize \begin{tabular}{l}AC-\\ Checker:\\ applies\\ inference\\ rules\end{tabular}};
+
+ \draw[red, ->, line width = 2mm] (1.7,1) -- (2.7,1);
+ \draw[red,<-, line width = 2mm] (-0.6,1) -- (-1.6,1);
+ \draw[red, <-, line width = 3mm] (0.6,2.2) -- (0.6,3.2);
+
+ \draw (0.6,4) node {\begin{tabular}{l}\large Access Policy (\bl{$\Gamma$})\end{tabular}};
+
+ \end{tikzpicture}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Proofs}
+
+\begin{center}
+\includegraphics[scale=0.4]{pics/river-stones.jpg}
+\end{center}
+
+\begin{textblock}{5}(11.7,5)
+goal
+\end{textblock}
+
+\begin{textblock}{5}(11.7,14)
+start
+\end{textblock}
+
+\begin{textblock}{5}(0,7)
+\begin{center}
+\bl{\infer[\small\textcolor{black}{\text{axiom}}]{\quad\vdash\quad}{}}\\[8mm]
+\bl{\infer{\vdash}{\quad\vdash\quad}}\\[8mm]
+\bl{\infer{\vdash}{\quad\vdash\qquad\vdash\quad}}
+\end{center}
+\end{textblock}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Sudoku}
+
+\begin{tikzpicture}[scale=.5]
+ \begin{scope}
+ \draw (0, 0) grid (9, 9);
+ \draw[very thick, scale=3] (0, 0) grid (3, 3);
+
+ \setcounter{row}{1}
+ \setrow { }{2}{ } {5}{ }{1} { }{9}{ }
+ \setrow {8}{ }{ } {2}{ }{3} { }{ }{6}
+ \setrow { }{3}{ } { }{6}{ } { }{7}{ }
+
+ \setrow { }{ }{1} { }{ }{ } {6}{ }{ }
+ \setrow {5}{4}{ } { }{ }{ } { }{1}{9}
+ \setrow { }{ }{2} { }{ }{ } {7}{ }{ }
+
+ \setrow { }{9}{ } { }{3}{ } { }{8}{ }
+ \setrow {2}{ }{ } {8}{ }{4} { }{ }{7}
+ \setrow { }{1}{ } {9}{ }{7} { }{6}{ }
+
+ \fill[red, fill opacity=0.4] (4,0) rectangle (5,9);
+ \fill[red, fill opacity=0.4] (0,5) rectangle (9,6);
+ \fill[red!50, fill opacity=0.4] (3,3) rectangle (4,5);
+ \fill[red!50, fill opacity=0.4] (5,3) rectangle (6,5);
+ \node[gray, anchor=center] at (4.5, -0.5) {columns};
+ \node[gray, rotate=90, anchor=center] at (-0.6, 4.5, -0.5) {rows};
+ \node[gray, anchor=center] at (4.5, 4.5) {box};
+ \end{scope}
+ \end{tikzpicture}
+
+\small
+\begin{textblock}{7}(9,3)
+\begin{enumerate}
+\item {\bf Row-Column:} each cell, must contain exactly one number
+\item {\bf Row-Number:} each row must contain each number exactly once
+\item {\bf Column-Number:} each column must contain each number exactly once
+\item {\bf Box-Number:} each box must contain each number exactly once
+\end{enumerate}
+\end{textblock}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Solving Sudokus}
+
+\begin{tikzpicture}[scale=.5]
+ \begin{scope}
+ \draw (0, 0) grid (9, 9);
+ \draw[very thick, scale=3] (0, 0) grid (3, 3);
+
+ \setcounter{row}{1}
+ \setrow { }{ }{ } {7}{ }{ } { }{5}{8}
+ \setrow {}{5}{6} {2}{1}{8} {7}{9}{3}
+ \setrow { }{ }{ } { }{ }{ } {1}{ }{ }
+
+ \setrow { }{ }{ } { }{ }{ } { }{8}{1}
+ \setrow { }{ }{ } {3}{7}{6} { }{ }{ }
+ \setrow {9}{6}{ } { }{ }{ } { }{ }{ }
+
+ \setrow { }{ }{5} { }{ }{ } { }{ }{ }
+ \setrow { }{ }{4} { }{2}{1} {8}{3}{ }
+ \setrow {8}{7}{ } { }{ }{3} { }{ }{ }
+
+ \fill[red, fill opacity=0.4] (0,7) rectangle (1,8);
+
+ \end{scope}
+ \end{tikzpicture}
+
+\small
+\begin{textblock}{6}(9,6)
+{\bf single position rules}\\
+\begin{center}
+\bl{\infer{4\;\text{in empty position}}{\{1..9\} - \{4\}\;\text{in one row}}}
+\end{center}
+
+\onslide<2->{
+\begin{center}
+\bl{\infer{x\;\text{in empty position}}{\{1..9\} - \{x\}\;\text{in one column}}}\medskip\\
+\bl{\infer{x\;\text{in empty position}}{\{1..9\} - \{x\}\;\text{in one box}}}
+\end{center}}
+\end{textblock}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Solving Sudokus}
+
+\begin{tikzpicture}[scale=.5]
+ \begin{scope}
+ \draw (0, 0) grid (9, 9);
+ \draw[very thick, scale=3] (0, 0) grid (3, 3);
+
+ \setcounter{row}{1}
+ \setrow { }{ }{ } {7}{ }{ } {\alert{\footnotesize 2}}{5}{8}
+ \setrow {}{5}{6} {2}{1}{8} {7}{9}{3}
+ \setrow { }{ }{ } { }{ }{ } {1}{\alert{\footnotesize 2}}{\alert{\footnotesize 2}}
+
+ \setrow { }{ }{ } { }{ }{ } { }{8}{1}
+ \setrow { }{ }{ } {3}{7}{6} { }{ }{ }
+ \setrow {9}{6}{ } { }{ }{ } { }{ }{ }
+
+ \setrow { }{ }{5} { }{ }{ } { }{ }{ }
+ \setrow { }{ }{4} { }{2}{1} {8}{3}{ }
+ \setrow {8}{7}{ } { }{ }{3} { }{ }{ }
+
+ \end{scope}
+ \end{tikzpicture}
+
+\small
+\begin{textblock}{6}(7.5,6)
+{\bf candidate rules}\\
+\begin{center}
+\bl{\infer{x\;\text{candidate in empty positions}}{X - \{x\}\;\text{in one box} & X \subseteq \{1..9\}}}
+\end{center}
+\end{textblock}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Solving Sudokus}
+
+\begin{tikzpicture}[scale=.5]
+ \begin{scope}
+ \draw (0, 0) grid (9, 9);
+ \draw[very thick, scale=3] (0, 0) grid (3, 3);
+
+ \setcounter{row}{1}
+ \setrow { }{ }{ } {7}{ }{ } {\alert{\footnotesize 2}}{5}{8}
+ \setrow {\alert{4}}{5}{6} {2}{1}{8} {7}{9}{3}
+ \setrow { }{ }{ } { }{ }{ } {1}{\alert{\footnotesize 2}}{\alert{\footnotesize 2}}
+
+ \setrow { }{ }{ } { }{ }{ } { }{8}{1}
+ \setrow { }{ }{ } {3}{7}{6} { }{ }{ }
+ \setrow {9}{6}{ } { }{ }{ } { }{ }{ }
+
+ \setrow { }{ }{5} { }{ }{ } { }{ }{ }
+ \setrow { }{ }{4} { }{2}{1} {8}{3}{ }
+ \setrow {8}{7}{ } { }{ }{3} { }{ }{ }
+
+ \end{scope}
+ \end{tikzpicture}
+
+\small
+\begin{textblock}{6}(7.5,6)
+\begin{center}
+\bl{\infer{4\;\text{in empty position}}{\{1..9\} - \{4\}\;\text{in one row}}}\bigskip\\
+\bl{\infer{2\;\text{candidate in empty positions}}{X - \{2\}\;\text{in one box} & X \subseteq \{1..9\}}}
+\end{center}
+\end{textblock}
+
+
+\begin{textblock}{3}(13.5,6.8)
+ \begin{tikzpicture}
+ \onslide<1>{\node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{\mbox{\alert{a}}};}
+ \onslide<2>{\node at (0,0) [single arrow, shape border rotate=90, fill=red,text=white]{\mbox{\alert{a}}};}
+ \end{tikzpicture}
+\end{textblock}
+
+\begin{textblock}{3}(14.5,9.3)
+ \begin{tikzpicture}
+ \onslide<1>{\node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{\mbox{\alert{a}}};}
+ \onslide<2>{\node at (0,0) [single arrow, shape border rotate=90, fill=red,text=white]{\mbox{\alert{a}}};}
+ \end{tikzpicture}
+\end{textblock}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Solving Sudokus}
+
+\begin{tikzpicture}[scale=.5]
+ \begin{scope}
+ \draw (0, 0) grid (9, 9);
+ \draw[very thick, scale=3] (0, 0) grid (3, 3);
+
+ \setcounter{row}{1}
+ \setrow { }{ }{ } {7}{ }{ } { }{5}{8}
+ \setrow { }{5}{6} {2}{1}{8} {7}{9}{3}
+ \setrow { }{ }{ } { }{ }{ } {1}{ }{ }
+
+ \setrow { }{ }{ } { }{ }{ } { }{8}{1}
+ \setrow { }{ }{ } {3}{7}{6} { }{ }{ }
+ \setrow {9}{6}{ } { }{ }{ } { }{ }{ \alert{2}}
+
+ \setrow { }{ }{5} { }{ }{ } { }{ }{ }
+ \setrow { }{ }{4} { }{2}{1} {8}{3}{ }
+ \setrow {8}{7}{ } { }{ }{3} { }{ }{ }
+
+ \end{scope}
+ \end{tikzpicture}
+
+\small
+\begin{textblock}{6}(7.5,6)
+\begin{center}
+\bl{\infer{2\;\text{candidate}}{X - \{2\}\;\text{in one box} & X \subseteq \{1..9\}}}
+\end{center}
+\end{textblock}
+
+\begin{textblock}{3}(14.5,8.3)
+ \begin{tikzpicture}
+ \onslide<1>{\node at (0,0) [single arrow, shape border rotate=90, fill=red,text=white]{\mbox{\alert{a}}};}
+ \end{tikzpicture}
+\end{textblock}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{BTW}
+
+Are there sudokus that cannot be solved?\pause
+
+\begin{center}
+\begin{tikzpicture}[scale=.5]
+ \begin{scope}
+ \draw (0, 0) grid (9, 9);
+ \draw[very thick, scale=3] (0, 0) grid (3, 3);
+
+ \setcounter{row}{1}
+ \setrow {1}{2}{3} {4}{5}{6} {7}{8}{ }
+ \setrow { }{ }{ } { }{ }{ } { }{ }{2}
+ \setrow { }{ }{ } { }{ }{ } { }{ }{3}
+
+ \setrow { }{ }{ } { }{ }{ } { }{ }{4}
+ \setrow { }{ }{ } { }{ }{ } { }{ }{5}
+ \setrow { }{ }{ } { }{ }{ } { }{ }{6}
+
+ \setrow { }{ }{ } { }{ }{ } { }{ }{7}
+ \setrow { }{ }{ } { }{ }{ } { }{ }{8}
+ \setrow { }{ }{ } { }{ }{ } { }{ }{9}
+
+ \end{scope}
+ \end{tikzpicture}
+\end{center}
+
+Sometimes no rules apply at all....unsolvable sudoku.
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Example Proof}
+
+\begin{center}
+\bl{\infer{P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2 \wedge P\;\text{says}\;F_1}
+ {\raisebox{2mm}{\text{\LARGE $?$}}}}
+\end{center}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Example Proof}
+
+\begin{tabular}{@{\hspace{-6mm}}l}
+\begin{minipage}{1.1\textwidth}
+We have (by axiom)
+
+\begin{center}
+\begin{tabular}{@{}ll@{}}
+(1) & \bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2$}
+\end{tabular}
+\end{center}
+
+From (1) we get
+
+\begin{center}
+\begin{tabular}{@{}ll@{}}
+(2) & \bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash P\;\text{says}\;F_1$}\\
+(3) & \bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2$}\\
+\end{tabular}
+\end{center}
+
+From (3) and (2) we get
+
+\begin{center}
+\begin{tabular}{@{}ll@{}}
+\bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2 \wedge P\;\text{says}\;F_1$}
+\end{tabular}
+\end{center}
+
+Done.
+\end{minipage}
+\end{tabular}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Other Direction}
+
+\begin{tabular}{@{\hspace{-6mm}}l}
+\begin{minipage}{1.1\textwidth}
+We want to prove
+
+\begin{center}
+\begin{tabular}{@{}ll@{}}
+\bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2 \wedge P\;\text{says}\;F_1$}
+\end{tabular}
+\end{center}
+
+We better be able to prove:
+
+\begin{center}
+\begin{tabular}{@{}ll@{}}
+(1) & \bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2$}\\
+(2) & \bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash P\;\text{says}\;F_1$}\\
+\end{tabular}
+\end{center}
+
+For (1): If we can prove
+
+\begin{center}
+\begin{tabular}{@{}ll@{}}
+\bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2 \wedge P\;\text{says}\;F_1$}
+\end{tabular}
+\end{center}
+
+then (1) is fine. Similarly for (2).
+\end{minipage}
+\end{tabular}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+
+I want to prove
+
+\begin{center}
+\bl{$\Gamma \vdash \text{del\_file}$}
+\end{center}\pause
+
+There is an inference rule
+
+\begin{center}
+\bl{\infer{\Gamma \vdash P \,\text{says}\, F}{\Gamma \vdash F}}
+\end{center}\pause
+
+So I can derive \bl{$\Gamma \vdash \text{Alice} \,\text{says}\,\text{del\_file}$}.\bigskip\pause
+
+\bl{$\Gamma$} contains already \bl{$\text{Alice} \,\text{says}\,\text{del\_file}$}. \\
+So I can use the rule
+
+\begin{center}
+\bl{\infer{\Gamma, F \vdash F}{}}
+\end{center}
+
+\onslide<5>{\bf\alert{What is wrong with this?}}
+\hfill{\bf Done. Qed.}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Program}
+
+How to prove \bl{$\Gamma \vdash F$}?\bigskip\bigskip
+
+\begin{center}
+\Large \bl{\infer{\Gamma, F\vdash F}{}}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{center}
+\Large
+\bl{\infer{\Gamma \vdash F_1 \Rightarrow F_2}{F_1, \Gamma \vdash F_2}}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{center}
+\Large
+\bl{\infer{\Gamma \vdash P \,\text{says}\, F}{\Gamma \vdash F}}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{center}
+\Large
+\bl{\infer{\Gamma \vdash F_1 \vee F_2}{\Gamma \vdash F_1}}\qquad
+\bl{\infer{\Gamma \vdash F_1 \vee F_2}{\Gamma \vdash F_2}}\
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{center}
+\Large
+\bl{\infer{\Gamma \vdash F_1 \wedge F_2}{\Gamma \vdash F_1 \quad \Gamma \vdash F_2}}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[t]
+\frametitle{Program: \texttt{prove2}}
+
+I want to prove \bl{$\Gamma \vdash \text{Pred}$}\bigskip\bigskip\pause
+
+\begin{enumerate}
+\item I found that \bl{$\Gamma$} contains the assumption \bl{$F_1 \Rightarrow F_2$}\bigskip\pause
+\item If I can prove \bl{$\Gamma \vdash F_1$},\pause{} then I can prove
+\begin{center}
+\bl{$\Gamma \vdash F_2$}
+\end{center}\bigskip\pause
+
+\item So I am able to try to prove \bl{$\Gamma \vdash \text{Pred}$} with the additional assumption
+\bl{$F_2$}.\bigskip
+\begin{center}
+\bl{$F_2, \Gamma \vdash \text{Pred}$}
+\end{center}
+\end{enumerate}
+
+\only<4>{
+\begin{textblock}{11}(1,10.5)
+\bl{\infer{\Gamma\vdash F_2}{\Gamma\vdash F_1\Rightarrow F_2 & \Gamma\vdash F_1}}
+\end{textblock}}
+
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{}
+
+ Recall the following scenario:
+
+ \begin{itemize}
+ \item If \textcolor{blue}{Admin} says that \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{} {}}}
+ should be deleted, then this file must be deleted.
+ \item \textcolor{blue}{Admin} trusts \textcolor{blue}{Bob} to decide whether
+ \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}}} should be deleted.
+ \item \textcolor{blue}{Bob} wants to delete \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}}}.
+ \end{itemize}\bigskip
+
+ \small
+ \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{3D}{\isacharequal}}}\small\begin{tabular}{l}
+ \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}},\\
+ \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}},\\
+ \isa{Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}}\\
+ \end{tabular}}\medskip
+
+ \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}}}
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+
+\begin{itemize}
+\item \bl{$P \,\text{says}\, F$} means \bl{$P$} can send a ``signal'' \bl{$F$} through a wire, or
+can make a statement \bl{$F$}\bigskip
+
+\item \bl{$P$} is entitled to do \bl{$F$}\smallskip\\
+\bl{$P \,\text{controls}\, F \,\dn\, (P\,\text{says}\, F) \Rightarrow F$}\medskip
+
+\begin{center}
+\bl{\infer{\Gamma \vdash F}{\Gamma \vdash P\,\text{controls}\, F & \Gamma \vdash P\,\text{says}\,F}}
+\end{center}
+
+
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Trusted Third Party}
+
+Simple protocol for establishing a secure connection via a mutually
+trusted 3rd party (server):
+
+\begin{center}
+\begin{tabular}{@ {\hspace{-7mm}}l@{\hspace{2mm}}r@ {\hspace{1mm}}l}
+Message 1 & \bl{$A \rightarrow S :$} & \bl{$A, B$}\\
+Message 2 & \bl{$S \rightarrow A :$} & \bl{$\{K_{AB}\}_{K_{AS}}$} and \bl{$\{\{K_{AB}\}_{K_{BS}} \}_{K_{AS}}$}\\
+Message 3 & \bl{$A \rightarrow B :$} & \bl{$\{K_{AB}\}_{K_{BS}} $}\\
+Message 4 & \bl{$A \rightarrow B :$} & \bl{$\{m\}_{K_{AB}}$}\\
+\end{tabular}
+\end{center}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Sending Rule}
+
+ \bl{\begin{center}
+ \mbox{\infer{\Gamma \vdash Q \;\text{says}\; F}
+ {\Gamma \vdash P \;\text{says}\; F & \Gamma \vdash P \;\text{sends}\; Q : F}}
+ \end{center}}\bigskip\pause
+
+ \bl{$P \,\text{sends}\, Q : F \dn$}\\
+ \hspace{6mm}\bl{$(P \,\text{says}\, F) \Rightarrow (Q \,\text{says}\, F)$}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Trusted Third Party}
+
+ \begin{center}
+ \bl{\begin{tabular}{l}
+ $A$ sends $S$ : $\text{Connect}(A,B)$\\
+ \bl{$S \,\text{says}\, (\text{Connect}(A,B) \Rightarrow$}\\
+ \hspace{2.5cm}\bl{$\{K_{AB}\}_{K_{AS}} \wedge
+ \{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}})$}\\
+ $S$ sends $A$ : $\{K_{AB}\}_{K_{AS}}$ \bl{$\wedge$} $\{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}}$\\
+ $A$ sends $B$ : $\{K_{AB}\}_{K_{BS}}$\\
+ $A$ sends $B$ : $\{m\}_{K_{AB}}$
+ \end{tabular}}
+ \end{center}\bigskip\pause
+
+
+ \bl{$\Gamma \vdash B \,\text{says} \, m$}?
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Public/Private Keys}
+
+ \begin{itemize}
+ \item Bob has a private and public key: \bl{$K_{Bob}^{pub}$}, \bl{$K_{Bob}^{priv}$}\bigskip
+ \begin{center}
+ \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;m}
+ {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_{K_{Bob}^{pub}} &
+ \Gamma \vdash K_{Bob}^{priv}}}}
+ \end{center}\bigskip\pause
+
+ \item this is {\bf not} a derived rule!
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Security Levels}
+ \small
+
+ \begin{itemize}
+ \item Top secret (\bl{$T\!S$})
+ \item Secret (\bl{$S$})
+ \item Public (\bl{$P$})
+ \end{itemize}
+
+ \begin{center}
+ \bl{$slev(P) < slev(S) < slev(T\!S)$}\pause
+ \end{center}
+
+ \begin{itemize}
+ \item Bob has a clearance for ``secret''
+ \item Bob can read documents that are public or sectret, but not top secret
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Reading a File}
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ \begin{tabular}{@ {}l@ {}}
+ \only<2->{\textcolor{red}{$slev($File$)$ $<$ $slev($Bob$)$ $\Rightarrow$}}\\
+ \only<2->{\hspace{3cm}}Bob controls Permitted $($File, read$)$\\
+ Bob says Permitted $($File, read$)$\only<2->{\\}
+ \only<2>{\textcolor{red}{$slev($File$)$ $<$ $slev($Bob$)$}}%
+ \only<3>{\textcolor{red}{$slev($File$)$ $=$ $P$}\\}%
+ \only<3>{\textcolor{red}{$slev($Bob$)$ $=$ $S$}\\}%
+ \only<3>{\textcolor{red}{$slev(P)$ $<$ $slev(S)$}\\}%
+ \end{tabular}\\
+ \hline
+ Permitted $($File, read$)$
+ \end{tabular}
+ \end{center}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Substitution Rule}
+ \small
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ $\Gamma \vdash slev(P) = l_1$ \hspace{4mm} $\Gamma \vdash slev(Q) = l_2$
+ \hspace{4mm} $\Gamma \vdash l_1 < l_2$\\\hline
+ $\Gamma \vdash slev(P) < slev(Q)$
+ \end{tabular}
+ \end{center}}\bigskip\pause
+
+ \begin{itemize}
+ \item \bl{$slev($Bob$)$ $=$ $S$}
+ \item \bl{$slev($File$)$ $=$ $P$}
+ \item \bl{$slev(P) < slev(S)$}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Reading a File}
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ \begin{tabular}{@ {}l@ {}}
+ $slev($File$)$ $<$ $slev($Bob$)$ $\Rightarrow$\\
+ \hspace{3cm}Bob controls Permitted $($File, read$)$\\
+ Bob says Permitted $($File, read$)$\\
+ $slev($File$)$ $=$ $P$\\
+ $slev($Bob$)$ $=$ $T\!S$\\
+ \only<1>{\textcolor{red}{$?$}}%
+ \only<2>{\textcolor{red}{$slev(P) < slev(S)$}\\}%
+ \only<2>{\textcolor{red}{$slev(S) < slev(T\!S)$}}%
+ \end{tabular}\\
+ \hline
+ Permitted $($File, read$)$
+ \end{tabular}
+ \end{center}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Transitivity Rule}
+ \small
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ $\Gamma \vdash l_1 < l_2$
+ \hspace{4mm} $\Gamma \vdash l_2 < l_3$\\\hline
+ $\Gamma \vdash l_1 < l_3$
+ \end{tabular}
+ \end{center}}\bigskip
+
+ \begin{itemize}
+ \item \bl{$slev(P) < slev (S)$}
+ \item \bl{$slev(S) < slev (T\!S)$}
+ \item[] \bl{$slev(P) < slev (T\!S)$}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Reading Files}
+
+ \begin{itemize}
+ \item Access policy for reading
+ \end{itemize}
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ \begin{tabular}{@ {}l@ {}}
+ $\forall f.\;slev(f)$ \only<1>{$<$}\only<2>{\textcolor{red}{$\le$}} $slev($Bob$)$ $\Rightarrow$\\
+ \hspace{3cm}Bob controls Permitted $(f$, read$)$\\
+ Bob says Permitted $($File, read$)$\\
+ $slev($File$)$ $=$ \only<1>{$P$}\only<2>{\textcolor{red}{$T\!S$}}\\
+ $slev($Bob$)$ $=$ $T\!S$\\
+ $slev(P) < slev(S)$\\
+ $slev(S) < slev(T\!S)$
+ \end{tabular}\\
+ \hline
+ Permitted $($File, read$)$
+ \end{tabular}
+ \end{center}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Writing Files}
+
+ \begin{itemize}
+ \item Access policy for \underline{writing}
+ \end{itemize}
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ \begin{tabular}{@ {}l@ {}}
+ $\forall f.\;slev($Bob$)$ $\le$ $slev(f)$ $\Rightarrow$\\
+ \hspace{3cm}Bob controls Permitted $(f$, write$)$\\
+ Bob says Permitted $($File, write$)$\\
+ $slev($File$)$ $=$ $T\!S$\\
+ $slev($Bob$)$ $=$ $S$\\
+ $slev(P) < slev(S)$\\
+ $slev(S) < slev(T\!S)$
+ \end{tabular}\\
+ \hline
+ Permitted $($File, write$)$
+ \end{tabular}
+ \end{center}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+\end{document}
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Encryption}
+
+ \begin{itemize}
+ \item Encryption of a message\smallskip
+ \begin{center}
+ \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_K}
+ {\Gamma \vdash \text{Alice}\;\text{says}\;m & \Gamma \vdash \text{Alice}\,\text{says}\,K}}}
+ \end{center}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Public/Private Keys}
+
+ \begin{itemize}
+ \item Bob has a private and public key: \bl{$K_{Bob}^{pub}$}, \bl{$K_{Bob}^{priv}$}\bigskip
+ \begin{center}
+ \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;m}
+ {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_{K_{Bob}^{pub}} &
+ \Gamma \vdash K_{Bob}^{priv}}}}
+ \end{center}\bigskip\pause
+
+ \item this is {\bf not} a derived rule!
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Trusted Third Party}
+
+ \begin{itemize}
+ \item Alice calls Sam for a key to communicate with Bob
+ \item Sam responds with a key that Alice can read and a key Bob can read (pre-shared)
+ \item Alice sends the message encrypted with the key and the second key it recieved
+ \end{itemize}\bigskip
+
+ \begin{center}
+ \bl{\begin{tabular}{lcl}
+ $A$ sends $S$ &:& $\textit{Connect}(A,B)$\\
+ $S$ sends $A$ &:& $\{K_{AB}\}_{K_{AS}}$ \textcolor{black}{and} $\{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}}$\\
+ $A$ sends $B$ &:& $\{K_{AB}\}_{K_{BS}}$\\
+ $A$ sends $B$ &:& $\{m\}_{K_{AB}}$
+ \end{tabular}}
+ \end{center}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Controls}
+ \small
+
+ \begin{itemize}
+ \item \bl{\isa{P\ controls\ F\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{28}{\isacharparenleft}}P\ says\ F{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F}}
+
+ \item its meaning ``\bl{P} is entitled to do \bl{F}''
+ \item if \bl{P controls F} and \bl{P says F} then \bl{F}\pause
+
+ \begin{center}
+ \bl{\mbox{
+ \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}}}
+ {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ controls\ F}} & \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F}}}
+ }}
+ \end{center}\pause
+
+ \begin{center}
+ \bl{\mbox{
+ \infer{\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ F}}}
+ {\mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ {\isaliteral{28}{\isacharparenleft}}P\ says\ F{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F}} & \mbox{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ P\ says\ F}}}
+ }}
+ \end{center}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Security Levels}
+ \small
+
+ \begin{itemize}
+ \item Top secret (\bl{$T\!S$})
+ \item Secret (\bl{$S$})
+ \item Public (\bl{$P$})
+ \end{itemize}
+
+ \begin{center}
+ \bl{$slev(P) < slev(S) < slev(T\!S)$}\pause
+ \end{center}
+
+ \begin{itemize}
+ \item Bob has a clearance for ``secret''
+ \item Bob can read documents that are public or sectret, but not top secret
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Reading a File}
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ \begin{tabular}{@ {}l@ {}}
+ \only<2->{\textcolor{red}{$slev($File$)$ $<$ $slev($Bob$)$ $\Rightarrow$}}\\
+ \only<2->{\hspace{3cm}}Bob controls Permitted $($File, read$)$\\
+ Bob says Permitted $($File, read$)$\only<2->{\\}
+ \only<2>{\textcolor{red}{$slev($File$)$ $<$ $slev($Bob$)$}}%
+ \only<3>{\textcolor{red}{$slev($File$)$ $=$ $P$}\\}%
+ \only<3>{\textcolor{red}{$slev($Bob$)$ $=$ $S$}\\}%
+ \only<3>{\textcolor{red}{$slev(P)$ $<$ $slev(S)$}\\}%
+ \end{tabular}\\
+ \hline
+ Permitted $($File, read$)$
+ \end{tabular}
+ \end{center}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Substitution Rule}
+ \small
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ $\Gamma \vdash slev(P) = l_1$ \hspace{4mm} $\Gamma \vdash slev(Q) = l_2$
+ \hspace{4mm} $\Gamma \vdash l_1 < l_2$\\\hline
+ $\Gamma \vdash slev(P) < slev(Q)$
+ \end{tabular}
+ \end{center}}\bigskip\pause
+
+ \begin{itemize}
+ \item \bl{$slev($Bob$)$ $=$ $S$}
+ \item \bl{$slev($File$)$ $=$ $P$}
+ \item \bl{$slev(P) < slev(S)$}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Reading a File}
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ \begin{tabular}{@ {}l@ {}}
+ $slev($File$)$ $<$ $slev($Bob$)$ $\Rightarrow$\\
+ \hspace{3cm}Bob controls Permitted $($File, read$)$\\
+ Bob says Permitted $($File, read$)$\\
+ $slev($File$)$ $=$ $P$\\
+ $slev($Bob$)$ $=$ $T\!S$\\
+ \only<1>{\textcolor{red}{$?$}}%
+ \only<2>{\textcolor{red}{$slev(P) < slev(S)$}\\}%
+ \only<2>{\textcolor{red}{$slev(S) < slev(T\!S)$}}%
+ \end{tabular}\\
+ \hline
+ Permitted $($File, read$)$
+ \end{tabular}
+ \end{center}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Transitivity Rule}
+ \small
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ $\Gamma \vdash l_1 < l_2$
+ \hspace{4mm} $\Gamma \vdash l_2 < l_3$\\\hline
+ $\Gamma \vdash l_1 < l_3$
+ \end{tabular}
+ \end{center}}\bigskip
+
+ \begin{itemize}
+ \item \bl{$slev(P) < slev (S)$}
+ \item \bl{$slev(S) < slev (T\!S)$}
+ \item[] \bl{$slev(P) < slev (T\!S)$}
+ \end{itemize}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Reading Files}
+
+ \begin{itemize}
+ \item Access policy for reading
+ \end{itemize}
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ \begin{tabular}{@ {}l@ {}}
+ $\forall f.\;slev(f)$ \only<1>{$<$}\only<2>{\textcolor{red}{$\le$}} $slev($Bob$)$ $\Rightarrow$\\
+ \hspace{3cm}Bob controls Permitted $(f$, read$)$\\
+ Bob says Permitted $($File, read$)$\\
+ $slev($File$)$ $=$ \only<1>{$P$}\only<2>{\textcolor{red}{$T\!S$}}\\
+ $slev($Bob$)$ $=$ $T\!S$\\
+ $slev(P) < slev(S)$\\
+ $slev(S) < slev(T\!S)$
+ \end{tabular}\\
+ \hline
+ Permitted $($File, read$)$
+ \end{tabular}
+ \end{center}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Writing Files}
+
+ \begin{itemize}
+ \item Access policy for \underline{writing}
+ \end{itemize}
+
+ \bl{\begin{center}
+ \begin{tabular}{c}
+ \begin{tabular}{@ {}l@ {}}
+ $\forall f.\;slev($Bob$)$ $\le$ $slev(f)$ $\Rightarrow$\\
+ \hspace{3cm}Bob controls Permitted $(f$, write$)$\\
+ Bob says Permitted $($File, write$)$\\
+ $slev($File$)$ $=$ $T\!S$\\
+ $slev($Bob$)$ $=$ $S$\\
+ $slev(P) < slev(S)$\\
+ $slev(S) < slev(T\!S)$
+ \end{tabular}\\
+ \hline
+ Permitted $($File, write$)$
+ \end{tabular}
+ \end{center}}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Sending Rule}
+
+ \bl{\begin{center}
+ \mbox{\infer{\Gamma \vdash Q \;\textit{says}\; F}
+ {\Gamma \vdash P \;\textit{says}\; F & \Gamma \vdash P \;\textit{sends}\; Q : F}}
+ \end{center}}\bigskip\pause
+
+ \bl{$P \,\text{sends}\, Q : F \dn$}\\
+ \hspace{6mm}\bl{$(P \,\text{says}\, F) \Rightarrow (Q \,\text{says}\, F)$}
+
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+ \mode<presentation>{
+ \begin{frame}[c]
+ \frametitle{Trusted Third Party}
+
+ \begin{center}
+ \bl{\begin{tabular}{l}
+ $A$ sends $S$ : $\textit{Connect}(A,B)$\\
+ \bl{$S \,\text{says}\, (\textit{Connect}(A,B) \Rightarrow$}\\
+ \hspace{2.5cm}\bl{$\{K_{AB}\}_{K_{AS}} \wedge
+ \{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}})$}\\
+ $S$ sends $A$ : $\{K_{AB}\}_{K_{AS}}$ \bl{$\wedge$} $\{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}}$\\
+ $A$ sends $B$ : $\{K_{AB}\}_{K_{BS}}$\\
+ $A$ sends $B$ : $\{m\}_{K_{AB}}$
+ \end{tabular}}
+ \end{center}\bigskip\pause
+
+
+ \bl{$\Gamma \vdash B \,\text{says} \, m$}?
+ \end{frame}}
+ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End:
+