# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1415020560 0
# Node ID d6dc6f0e355608467501537f11a94c628a86bb0f
# Parent  d7109c6e721d87d093a0a4ab08b5fa23b0bbbea2
updated

diff -r d7109c6e721d -r d6dc6f0e3556 slides/slides06-bak.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/slides/slides06-bak.tex	Mon Nov 03 13:16:00 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: 
+
diff -r d7109c6e721d -r d6dc6f0e3556 slides/slides06.pdf
Binary file slides/slides06.pdf has changed
diff -r d7109c6e721d -r d6dc6f0e3556 slides/slides06.tex
--- a/slides/slides06.tex	Mon Nov 03 03:22:41 2014 +0000
+++ b/slides/slides06.tex	Mon Nov 03 13:16:00 2014 +0000
@@ -1,135 +1,28 @@
 \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
+\usepackage{../slides}
+\usepackage{../graphics}
 
-\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]"""
-}
+\setmonofont[Scale=.88]{Consolas}
+\newfontfamily{\consolas}{Consolas}
 
-\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
+\hfuzz=220pt 
 
 % beamer stuff 
-\renewcommand{\slidecaption}{APP 06, King's College London, 12 November 2013}
+\newcommand{\bl}[1]{\textcolor{blue}{#1}}  
+\renewcommand{\slidecaption}{APP 06, King's College London}
 
-\newcommand{\bl}[1]{\textcolor{blue}{#1}}
 \begin{document}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\mode<presentation>{
-\begin{frame}<1>[t]
+\begin{frame}[t]
 \frametitle{%
   \begin{tabular}{@ {}c@ {}}
   \\
   \LARGE Access Control and \\[-3mm] 
-  \LARGE Privacy Policies (6)\\[-6mm] 
+  \LARGE Privacy Policies (9)\\[-6mm] 
   \end{tabular}}\bigskip\bigskip\bigskip
 
-  %\begin{center}
-  %\includegraphics[scale=1.3]{pics/barrier.jpg}
-  %\end{center}
-
-\normalsize
+  \normalsize
   \begin{center}
   \begin{tabular}{ll}
   Email:  & christian.urban at kcl.ac.uk\\
@@ -138,518 +31,46 @@
   \end{tabular}
   \end{center}
 
-
-\end{frame}}
+\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{frame}[t]
+\frametitle{Checking Solutions}
 
-\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}{ }
+How can you check somebody's solution without revealing the solution?\pause\bigskip
 
-    \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}{ }{ }
+Alice and Bob solve crosswords. Alice knows the answer for 21D (folio) but doesn't 
+want to tell Bob.\medskip
 
-    \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}}
+You use an English  dictionary:
 
-    \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{itemize}
+\item folio \onslide<4->{$\stackrel{1}{\rightarrow}$ individual }
+                \onslide<5->{$\stackrel{2}{\rightarrow}$ human}
+                \onslide<6->{$\stackrel{3}{\rightarrow}$ or \ldots}
+\only<3>{
+\begin{quote}
+``an \alert{individual} leaf of paper or parchment, either loose as one of a series or 
+forming part of a bound volume, which is numbered on the recto or front side only.''	
+\end{quote}}
+\only<4>{
+\begin{quote}
+``a single \alert{human} being as distinct from a group''
+\end{quote}}
+\only<5>{
+\begin{quote}
+``relating to \alert{or} characteristic of humankind''
+\end{quote}}
+\end{itemize}\bigskip\bigskip
 
-\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}
-
+\only<7->{
+this is essentially a hash function...but Bob can only check once he has also found the solution
+}
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
@@ -657,122 +78,48 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \mode<presentation>{
 \begin{frame}[c]
-\frametitle{Example Proof}
-
-\begin{tabular}{@{\hspace{-6mm}}l}
-\begin{minipage}{1.1\textwidth}
-We have (by axiom)
+\frametitle{Zero-Knowledge Proofs}
 
-\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
+Two remarkable properties of \alert{Zero-Knowledge Proofs}:\bigskip
 
-\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}
+\begin{itemize}
+\item Alice only reveals the fact that she knows a secret, not the secret itself (meaning 
+she can convince Bob that she knows the secret).\bigskip
+\item Having been convinced, Bob cannot use the evidence in order to convince Carol 
+that Alice knows the secret.
+\end{itemize}
 
 \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{frame}[t]
+\frametitle{\begin{tabular}{@{}c@{}}The Idea \end{tabular}}
 
 \begin{center}
-\begin{tabular}{@{}ll@{}}
-\bl{$P\;\text{says}\;F_1 \wedge Q\;\text{says}\;F_2 \vdash Q\;\text{says}\;F_2 \wedge P\;\text{says}\;F_1$}
+\begin{tabular}{l@{\hspace{10mm}}r}
+\\[-10mm]
+\raisebox{10mm}{\large 1.} & \includegraphics[scale=0.1]{../pics/alibaba1.png}\\
+\raisebox{10mm}{\large 2.} & \includegraphics[scale=0.1]{../pics/alibaba2.png}\\
+\raisebox{10mm}{\large 3.} & \includegraphics[scale=0.1]{../pics/alibaba3.png}
 \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{textblock}{7}(1,2.5)
+The Alibaba cave:
+\end{textblock}
 
-\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}
+\small
+\only<2>{
+\begin{textblock}{12}(2,13.3)
+Even if Bob has a hidden camera, a recording will not be convincing to anyone else 
+(Alice and Bob could have made it all up).
+\end{textblock}}
+\only<3>{
+\begin{textblock}{12}(2,13.3)
+Even worse, an observer present at the experiment would not be convinced.
+\end{textblock}}
 
 \end{frame}}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
@@ -780,701 +127,294 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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]
+\frametitle{Applications of ZKPs}
 
-\begin{center}
-\Large 
-\bl{\infer{\Gamma \vdash P \,\text{says}\, F}{\Gamma \vdash F}}
-\end{center}
-
+\begin{itemize}
+\item authentication, where one party wants to prove its identity to a 
+second party via some secret information,  but doesn't want the second 
+party to learn anything about this secret\bigskip
+\item to enforce honest behaviour while maintaining privacy: the idea is to 
+force a user to prove, using a zero-knowledge proof, that its behaviour is 
+correct according to the protocol
+\end{itemize}
 \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}
+\frametitle{Central Properties}
 
-\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]
+Zero-knowledge proof protocols should satisfy:\bigskip
 
 \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}
-
-
+\item \alert{\bf Completeness} If Alice knows the secret, Bob accepts Alice ``proof'' for sure.\bigskip
+\item \alert{\bf Soundness} If Alice does not know the secret, Bob accepts her ``proof'' with a very 
+small probability.
 \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):
+\frametitle{Graph Isomorphism}
 
 \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}}$}\\
+\begin{tabular}{l@{\hspace{10mm}}r}
+\includegraphics[scale=0.8]{../pics/graphs.png}\\
 \end{tabular}
 \end{center}
 
+Finding an isomorphism between two graphs is an NP complete problem.
 \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{$(P \,\text{says}\, F) \Rightarrow (Q \,\text{says}\, F)$}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Graph Isomorphism Protocol}
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     
-  
-    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Trusted Third Party}
+Alice starts with knowing an isomorphism \bl{$\sigma$} between graphs \bl{$G_1$} and \bl{$G_2$}\medskip
 
-  \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}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      
+\begin{enumerate}
+\item Alice generates an isomorphic graph \bl{$H$} which she sends to Bob 
+\item Bob asks either for an isomorphism between \bl{$G_1$} and \bl{$H$}, or
+\bl{$G_2$} and \bl{$H$}	
+\item Alice and Bob repeat this procedure \bl{$n$} times	
+\end{enumerate}\pause
+
+these are called commitment algorithms
 
-   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Public/Private Keys}
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
+   
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Graph Isomorphism Protocol (2)}
 
-  \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
+If Alice knows the isomorphism, she can always calculate \bl{$\sigma$}.\bigskip
 
-  \item this is {\bf not} a derived rule! 
-  \end{itemize}
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      
-  
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Security Levels}
-  \small
+ If she doesn't, she can only correctly respond if Bob's 
+ choice of index is the same as the one she used to form \bl{$H$}. The probability 
+ of this happening is \bl{$\frac{1}{2}$}, so after \bl{$n$} rounds the probability of her 
+ always responding correctly is only \bl{$\frac{1}{2}^n$}.
 
-  \begin{itemize}
-  \item Top secret (\bl{$T\!S$})
-  \item Secret (\bl{$S$})
-  \item Public (\bl{$P$})
-  \end{itemize}
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
 
-  \begin{center}
-  \bl{$slev(P) < slev(S) < slev(T\!S)$}\pause
-  \end{center}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Graph Isomorphism Protocol (3)}
 
-  \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}
+Why is the GI-protocol zero-knowledge?\bigskip\pause
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Reading a File}
+A: We can generate a fake transcript of a conversation, which 
+cannot be distinguished from a ``real'' conversation.\bigskip
+
+Anything Bob can compute using the information obtained from 
+the transcript can be computed using only a forged transcript and 
+therefore participation in such a communication does not increase 
+Bob's capability  to perform any computation.
 
-  \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}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    
+   
+   
+   
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Non-Interactive ZKPs}
 
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \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
+\bigskip
+This is amazing: Alison can publish some data that contains no data about her secret,
+but this data can be used to convince anyone of the secret's existence.
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
-  \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}
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Non-Interactive ZKPs (2)}
+
+Alice starts with knowing an isomorphism \bl{$\sigma$} between graphs \bl{$G_1$} and \bl{$G_2$}\medskip
 
-  \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}}
+\begin{enumerate}
+\item Alice generates \bl{$n$} isomorphic graphs \bl{$H_{1..n}$} which she makes public 
+\item she feeds the \bl{$H_{1..n}$} into a hashing function (she has no control over what
+	the output will be)
+\item Alice takes the first \bl{$n$} bits of the output: whenever output is \bl{$0$}, she shows 
+an isomorphism with \bl{$G_1$} ; for \bl{$1$} she shows an isomorphism with \bl{$G_2$}
+\end{enumerate}
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
+\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
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Problems of ZKPs}
 
-  \begin{itemize}
-  \item \bl{$slev(P) < slev (S)$}
-  \item \bl{$slev(S) < slev (T\!S)$}
-  \item[] \bl{$slev(P) < slev (T\!S)$}
-  \end{itemize}
+\begin{itemize}
+\item ``grand chess master problem''\\ 
+(person in the middle)\bigskip
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
+\item Alice can have multiple identities; once she committed a fraud she stops using one
+\end{itemize}
+
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Reading Files}
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Other Methods for ZKPs}
 
-  \begin{itemize}
-  \item Access policy for reading
-  \end{itemize}
+Essentially every NP-problem can be used for ZKPs
 
-  \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}}
+\begin{itemize}
+\item modular logarithms: Alice chooses public \bl{$A$},  \bl{$B$}, \bl{$p$}; and private \bl{$x$}
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
+\begin{center}
+\bl{$A^x \equiv B\; mod\; p$}
+\end{center} 
+\end{itemize}
 
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Writing Files}
-
-  \begin{itemize}
-  \item Access policy for \underline{writing}
-  \end{itemize}
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Commitment Stage}
 
-  \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}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
+\begin{enumerate}
+\item Alice generates \bl{$z$} random numbers \bl{$r_1$}, ..., \bl{$r_z$}, all less than \bl{$p - 1$}.
+\item Alice sends Bob for all \bl{$1..z$} 
+\begin{center}
+\bl{$h_i = A^{r_i} \;mod\; p$}
+\end{center}
+\item Bob generates random bits   \bl{$b_1$}, ..., \bl{$b_z$} by flipping a coin
+\item For each bit \bl{$b_i$}, Alice sends Bob an \bl{$s_i$} where
 
-  
-\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}
+\begin{center}
+\begin{tabular}{ll}
+\bl{$b_i = 0$}: & \bl{$s_i = r_i$}\\
+\bl{$b_i = 1$}: & \bl{$s_i = (r_i - r_j) \;mod\; (p -1)$}\\
+\end{tabular}
+\end{center}
+where \bl{$r_j$} is the lowest \bl{$j$} where \bl{$b_j = 1$}
+ 
+\end{enumerate}
 
-  \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
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
-  \item this is {\bf not} a derived rule! 
-  \end{itemize}
-
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      
-  
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Trusted Third Party}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Confirmation Stage}
 
-  \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}
+\begin{enumerate}
+\item For each \bl{$b_i$} Bob checks whether \bl{$s_i$} conforms to the protocol
 
-  \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}}
+\begin{center}
+\begin{tabular}{ll}
+\bl{$b_i = 0$}: & \bl{$A^{s_i} \equiv B\;mod\;p$}\\
+\bl{$b_i = 1$}: & \bl{$A^{s_i}  \equiv h_i * h_j^{-1}  \;mod\; p$}\\
+\end{tabular}
+\end{center}\bigskip
 
-  \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
+Bob was send 
 
-  \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{$r_j - r _j$},  \bl{$r_m - r _j$}, \ldots, \bl{$r_p - r _j$ \;mod \;p} 
+\end{center}
 
-  \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}
+where the corresponding bits were 
+\bl{$1$}; Bob does not know \bl{$r_j$}, he does not know any \bl{$r_i$} where the bit was \bl{$1$}
+\end{enumerate}
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
+\end{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Security Levels}
-  \small
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Proving Stage}
 
-  \begin{itemize}
-  \item Top secret (\bl{$T\!S$})
-  \item Secret (\bl{$S$})
-  \item Public (\bl{$P$})
-  \end{itemize}
+\begin{enumerate}
+\item Alice proves she knows \bl{$x$}, the discrete log of \bl{$B$}\\
+she sends
 
-  \begin{center}
-  \bl{$slev(P) < slev(S) < slev(T\!S)$}\pause
-  \end{center}
+\begin{center}
+\bl{$s_{z+1} = (x - r_j)$}
+\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}
+\item Bob confirms
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Reading a File}
+\begin{center}
+\bl{$A^{s_{z+1}} \equiv B * h_j^{-1} \;mod \; p$}
+\end{center}
+\end{enumerate}\bigskip\pause
 
-  \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}}
+In order to cheat, Alice has to guess all bits in advance. She has only \bl{$1$} to \bl{$2^z$}
+chance. \\
+\small\hspace{7mm}\textcolor{gray}{(explanation $\rightarrow$ \url{http://goo.gl/irL9GK})}
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
+\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
+\mode<presentation>{
+\begin{frame}[c]
+\frametitle{Random Number Generators}
 
-  \begin{itemize}
-  \item \bl{$slev($Bob$)$ $=$ $S$}
-  \item \bl{$slev($File$)$ $=$ $P$}
-  \item \bl{$slev(P) < slev(S)$}
-  \end{itemize}
+\begin{itemize}
+\item Computers are deterministic. How do they generate random numbers?\bigskip\pause
 
-  \end{frame}}
-  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-  \mode<presentation>{
-  \begin{frame}[c]
-  \frametitle{Reading a File}
+\item The most popular method to generate random numbers between \bl{$0$} and \bl{$m$} is: choose
+three integers
 
-  \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}}
+\begin{center}
+\begin{tabular}{ll}
+\bl{$a$} & multiplier\\
+\bl{$c$} & increment\\
+\bl{$X_0$} & start value
+\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}
+and calculate
 
-  \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}
+\begin{center}
+\bl{$X_{n+1} = (a * X_n + c) \;mod\; m$}
+\end{center}
+\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}
+\only<3->{
+\begin{textblock}{7}(10,2)
+\begin{tikzpicture}
+\draw (0,0) node[inner sep=2mm,fill=cream, ultra thick, draw=red, rounded corners=2mm] 
+{\begin{minipage}{3.8cm}
+\begin{tabular}{ll|l}
+\bl{$m =$}    & \bl{$16$} & \bl{$16$}\\
+\bl{$X_0 =$} &  \bl{$1$} & \bl{$1$}\\
+\bl{$a = $}    & \bl{$5$} & \bl{$5$}\\
+\bl{$c =$}     & \bl{$1$} & \bl{$0$}\\
+\end{tabular} 
+\end{minipage}};
+\end{tikzpicture}
+\end{textblock}}
 
-  \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{frame}}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 \end{document}
 
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