added content to long presentation

presentation/191119_long/header.tex

@@ -24,6 +24,8 @@
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{textcomp}
+\usepackage{amssymb}
+\usepackage{mathtools}
 
 
 %properties for listings:

presentation/191119_long/long.tex

@@ -54,19 +54,86 @@
 
 \begin{frame}
 	\frametitle{Solution: Direct Anonymous Attestation (DAA)}
+	\begin{center}
+		\includegraphics[width=0.7\textwidth]{../../resources/daa}
+	\end{center}
+	
 	\begin{itemize}
 		\item based on group signatures
 		\item Zero Knowledge Proof to verify group membership
 		\item defines 3 Parties
 		\begin{itemize}
-			\item Issuer: provides public key for a group (e.g. all Biometric Sensors) and manages group memberships
-			\item Member: holds a group private key to sign messages (e.g. a Biometric Sensor)
-			\item Verifier: knows the group public key and is able to verify correctness of signature (e.g. Personal Agent)
+			\item \emph{Issuer}: provides public key for a group (e.g. all Biometric Sensors) and manages group memberships
+			\item \emph{Member}: holds a group private key to sign messages (e.g. a Biometric Sensor)
+			\item \emph{Verifier}: knows the group public key and is able to verify correctness of signature (e.g. Personal Agent)
+		\end{itemize}
+	\end{itemize}
+\end{frame}
+
+\begin{frame}
+	\frametitle{DAA Setup: Issuer creates Bilinear Group and Keys}
+	\begin{eqnarray*}
+		q,\mathbb{G}_1, \mathbb{G}_2, \mathbb{G}_T, g_1, g_2, e
+	\end{eqnarray*}
+	\begin{itemize}
+		\item $\mathbb{G}_1, \mathbb{G}_2, \mathbb{G}_T$: groups of prime order $q$
+		\item $g_1 \in \mathbb{G}_1$, $g_2 \in \mathbb{G}_2$: generator points
+		\item $e$: bilinear map with properties
+		\begin{itemize}
+			\item \emph{Bilinear}: For all $P \in \mathbb{G}_1, Q \in \mathbb{G}_2$, for all $a,b \in \mathbb{Z}$, $ e(P^a,Q^b) = e(P,Q)^{ab}$
+			\item \emph{Non-degenerate}: There exists some $P \in \mathbb{G}_1, Q \in \mathbb{G}_2$ such that $e(P,Q) \neq 1$, where 1 is the identity of $\mathbb{G}_T$
+			\item \emph{Efficient}: There exists an efficient algorithm for computing $e$
+		\end{itemize}
+		\item Choose secret key $x\leftarrow\mathbb{Z}_q$, $y\leftarrow\mathbb{Z}_q$
+		\item generate public key $X=g_2^x$, $Y=g_2^y$
 	\end{itemize}
-		\item used DAA is based on Elliptic Curves (ECDAA)
+\end{frame}
+
+\begin{frame}
+	\frametitle{Bilinear Maps}
+	Definition Bilinear Maps:
+	\begin{itemize}
+		\item \emph{Bilinear}: For all $P,Q \in G$, for all $a,b \in \mathbb{Z}$, $e(P^a,Q^b) = e(P,Q)^{ab}$
+		\item \emph{Non-degenerate}: There exists some $P,Q \in G$ such that $e(P,Q) \not 1$ where 1 is the identity of $\mathrm{G}$
 	\end{itemize}
 \end{frame}
 
+\begin{frame}
+	\frametitle{Bilinear Maps: Signatures}
+	\begin{itemize}
+		\item given message $m$ and random number $r leftarrow \mathbb{Z}_q$
+		
+	\end{itemize}
+\end{frame}
 
+\begin{frame}
+	\frametitle{Bilinear Maps: Zero Knowledge Proofs}
+	\begin{itemize}
+		\item Do the same as before, but choose to additional random variables $r$ and $r'$
+	\end{itemize}
+\end{frame}
 
+\begin{frame}
+	\frametitle{DAA Join: Member joins to Issuer's Group}
+	\begin{center}
+	\begin{footnotesize}
+	\begin{tabular}{|lclcl|}\hline
+		\multicolumn{1}{|c}{TPM}&&\multicolumn{1}{c}{Host}&&\multicolumn{1}{c|}{Issuer}\\\hline
+		&&&$\xrightarrow{\text{JOIN}}$&$n\leftarrow\{0,1\}^{ln}$\\
+		$gsk\leftarrow\mathbb{Z}$&$\xleftarrow{\makebox[5mm]{n}}$&&$\xleftarrow{\makebox[5mm]{n}}$&\\
+		$Q\leftarrow g_1^{gsk}$&&&&\\
+		$\pi_1\rightarrow SPK\{(\alpha):g_1^\alpha\}$&$\xrightarrow{Q,\pi_1}$&&$\xrightarrow{Q,\pi_1}$&verify $\pi_1$\\
+		&&&&$r\leftarrow\mathbb{Z}_q$\\
+		&&&&$a\leftarrow g_1^r$\\
+		&&&&$b\leftarrow a^{x+ym}$\\
+		&&&&$c\leftarrow a^x\cdot Q^{rxy}$\\
+		&&&&$d\leftarrow Q^{ry}$\\
+		&&&&$\pi_2\leftarrow SPK\{(t):$\\
+		&&$e(a,X)\cdot e(c,Y)$&$\xleftarrow{a,b,c,d,\pi_2}$&$ b=g_1^t \wedge d=Q^t\}$\\
+		verify $\pi_2$&$\xleftarrow{b,d,\pi_2}$&$\stackrel{?}{=}e(b,g_2)$&&\\
+		store($gsk, b, d$)&$\xrightarrow{JOINED}$&store($a,b,c,d$)&&\\\hline
+	\end{tabular}
+	\end{footnotesize}
+	\end{center}
+\end{frame}
 \end{document}

resources/daa.fig

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