551    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   552   \mode<presentation>{

   553   \begin{frame}[c]

   554   \frametitle{Public/Private Keys}

   555 

   556   \begin{itemize}

   557   \item Bob has a private and public key: \bl{$K_{Bob}^{pub}$}, \bl{$K_{Bob}^{priv}$}\bigskip

   558   \begin{center}

   559   \bl{\mbox{\infer{\Gamma \vdash \text{Alice}\;\text{says}\;m}

   560               {\Gamma \vdash \text{Alice}\;\text{says}\;\{m\}_{K_{Bob}^{pub}} & 

   561                \Gamma \vdash K_{Bob}^{priv}}}}

   562   \end{center}\bigskip\pause

   563 

   564   \item this is {\bf not} a derived rule! 

   565   \end{itemize}

   566 

   567   \end{frame}}

   568   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      

   569   

   570   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   571   \mode<presentation>{

   572   \begin{frame}[c]

   573   \frametitle{Trusted Third Party}

   574 

   575   \begin{itemize}

   576   \item Alice calls Sam for a key to communicate with Bob

   577   \item Sam responds with a key that Alice can read and a key Bob can read (pre-shared)

   578   \item Alice sends the message encrypted with the key and the second key it recieved

   579   \end{itemize}\bigskip

   580 

   581   \begin{center}

   582   \bl{\begin{tabular}{lcl}

   583   $A$ sends $S$ &:& $\textit{Connect}(A,B)$\\

   584   $S$ sends $A$ &:& $\{K_{AB}\}_{K_{AS}}$ \textcolor{black}{and} $\{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}}$\\

   585   $A$ sends $B$ &:& $\{K_{AB}\}_{K_{BS}}$\\

   586   $A$ sends $B$ &:& $\{m\}_{K_{AB}}$

   587   \end{tabular}}

   588   \end{center}

   589 

   590   \end{frame}}

   591   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%      

   592   

   593    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   594   \mode<presentation>{

   595   \begin{frame}[c]

   596   \frametitle{Sending Rule}

   597 

   598   \bl{\begin{center}

   599   \mbox{\infer{\Gamma \vdash Q \;\textit{says}\; F}

   600               {\Gamma \vdash P \;\textit{says}\; F & \Gamma \vdash P \;\textit{sends}\; Q : F}}

   601   \end{center}}\bigskip\pause

   602   

   603   \bl{$P \,\text{sends}\, Q : F \dn$}\\

   604   \hspace{6mm}\bl{$(P \,\text{says}\, F) \Rightarrow (Q \,\text{says}\, F)$}

   605 

   606   \end{frame}}

   607   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     

   608   

   609     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

   610   \mode<presentation>{

   611   \begin{frame}[c]

   612   \frametitle{Trusted Third Party}

   613 

   614   \begin{center}

   615   \bl{\begin{tabular}{l}

   616   $A$ sends $S$ : $\textit{Connect}(A,B)$\\  

   617   \bl{$S \,\text{says}\, (\textit{Connect}(A,B) \Rightarrow$}\\ 

   618   \hspace{2.5cm}\bl{$\{K_{AB}\}_{K_{AS}} \wedge 

   619   \{\{K_{AB}\}_{K_{BS}}\}\_{K_{AS}})$}\\

   620  $S$ sends $A$ : $\{K_{AB}\}_{K_{AS}}$ \bl{$\wedge$} $\{\{K_{AB}\}_{K_{BS}}\}_{K_{AS}}$\\

   621   $A$ sends $B$ : $\{K_{AB}\}_{K_{BS}}$\\

   622   $A$ sends $B$ : $\{m\}_{K_{AB}}$

   623   \end{tabular}}

   624   \end{center}\bigskip\pause

   625   

   626   

   627   \bl{$\Gamma \vdash B \,\text{says} \, m$}?

   628   \end{frame}}

   629   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%