diff --git a/ACNS/presentation/key_exchange.tex b/ACNS/presentation/key_exchange.tex
index eacdcb1019afe21f4d4ced4ab84bbee163e8f055..ea7cd14cc82958eb3a810a17412feef36142802c 100644
--- a/ACNS/presentation/key_exchange.tex
+++ b/ACNS/presentation/key_exchange.tex
@@ -10,14 +10,14 @@
\begin{definition}[Piecewise Verifiable Proof \cite{DBLP:conf/asiacrypt/BeullensKV19}]
A party proves knowledge of a polynomial \(f\) for a statement
\[x = \left(\left(E_0,E_1\right),s_1,\ldots, s_n\right),\]
- where \(E_1 = \left[f\paren* 0\right] E_0\) and \(s_i = f\paren* i\) for \(i=1,\ldots, n\).
+ where \(E_1 = \left[f\paren* 0\right] E_0\) and \(s_i = f\paren* i \in \ZZ_p\) for \(i=1,\ldots, n\).
\end{definition}
\end{frame}
\begin{frame}
\frametitle{What to do?}
\begin{itemize}
- \item Transfer PVP and ZK proof to the threshold setting
+ \item Transfer PVP proof to threshold setting
\item Integrate both to decapsulation protocol to achieve active security
\item Prove, that resulting protocol is at least as secure as original decapsulation
\end{itemize}
@@ -61,7 +61,7 @@
\frametitle{A Shareholder \(P_i\)'s Turn in the Decapsulation Protocol}
Let \(S^\ast\) be a superauthorised set of shareholders executing the decapsulation protocol.
\begin{enumerate}
- \item Ascertain \(E^{k-1} \in \mathcal E\), where \(E^{k-1}\) is previous shareholder's output.
+ \item Ascertain \(E^{k-1} \in \mathcal E\), where \(E^{k-1}\) is previous shareholder's output of \(E^0 = c\)
\item Sample \(R_k \sample \mathcal E\), compute \(R_k'\gets \left[L_{i,S^\ast}s_i\right] R_k\).
\item Compute and publish
\begin{align*}
diff --git a/ACNS/presentation/main.tex b/ACNS/presentation/main.tex
index a7e52bf6d11c8de2ead3ec49c058371e37eaede7..bcfd27d658cddd46218709a6a51ef1d6fddcd66b 100644
--- a/ACNS/presentation/main.tex
+++ b/ACNS/presentation/main.tex
@@ -28,6 +28,16 @@
\input{generalsss}
+\begin{frame}
+ \frametitle{Conclusion}
+ \begin{itemize}
+ \item Transfer PVP to threshold setting
+ \item Actively secure key exchange mechanism
+ \item Transformed into signature scheme
+ \item Define, which field of secret sharing schemes is compatible
+ \end{itemize}
+\end{frame}
+
\section{References}
\begin{frame}[allowframebreaks]
\bibliographystyle{amsalpha}
diff --git a/ACNS/presentation/motivation.tex b/ACNS/presentation/motivation.tex
index a32d71d356e339de8e3c28d4e2a0be5e8a15a418..f276bb2897abeff6ac705151f2607e93b90e117a 100644
--- a/ACNS/presentation/motivation.tex
+++ b/ACNS/presentation/motivation.tex
@@ -39,7 +39,7 @@
\begin{frame}
\frametitle{Secret Sharing Schemes}
\begin{itemize}
- \item Distribute a secret \(s\) among \(n\) shareholders via
+ \item Distribute a secret \(s\) among shareholders \(P_1,\ldots, P_n\) via
\[ \mathcal S.\share\paren* s\]
\item Reconstruct a shared secret via
\[ \SH.\rec\paren*{\set{s_i}_{P_i \in S'}} \]
@@ -53,27 +53,26 @@
\end{frame}
-\begin{frame}
+\begin{frame}%[allowframebreaks]
\frametitle{Key Exchange Mechanisms}
-
\begin{center}
\begin{tikzpicture}
\begin{scope}[minimum size = .7cm]
\node [alice] (alice) at (-3,0){Alice};
\node [bob] (bob) at (3,0){Bob};
\end{scope}
- \pause
+ %\pause
\node [left = .5 of alice] (pair) {$\left(\sk,\pk\right)$};
%\node [above = 3 of alice] (pk) {$\pk$};
%\draw [->] (alice) -- (pk);
- \pause
+ %\pause
\node [above = 2.5 of bob] (encaps) {$\encaps \paren* \pk$};
\node [above = 0.5 of bob] (keybob) {$\key$};
%\node [above = 2.5 of alice] (cipher) {$c$};
\node (cipher) at ($(encaps)+(-6,0)$) {$c$};
\draw [->] (encaps) -- (cipher);
\draw [->] (encaps) -- (keybob);
- \pause
+ %\pause
\node [above = 1.5 of alice] (decaps) {$\decaps\paren *{\sk,c}$};
\draw [->] (cipher) -- (decaps);
@@ -87,6 +86,30 @@
\end{center}
\end{frame}
+\begin{frame}
+ \frametitle{Key Exchange Mechanisms in a HHS}
+ \begin{pchstack}[center,space=2em]
+ \procedure[space=auto]{$\keygen()$}{
+ \sk \sample \ZZ_p\\
+ \pk \gets \left[\sk\right] E_0\\
+ \pcreturn \left(\sk,\pk\right)
+ }
+ \vspace{1em}
+ \procedure[space=auto]{$\encaps\paren* \pk$}{
+ b \sample \mathcal G\\
+ \key \gets b \ast \pk\\
+ c \gets b \ast E_0\\
+ \pcreturn \left(\key,c\right)
+ }
+ \vspace{1em}
+ \procedure[space=auto]{$\decaps\paren*{\sk,c}$}{
+ \key \gets \left[\sk\right] c\\
+ \pcreturn \key
+ }
+ \end{pchstack}
+
+\end{frame}
+
\begin{frame}
\frametitle{A Decapsulation Protocol with Shared Secret Key \cite{FeoM20}}
@@ -104,14 +127,14 @@
\draw [help lines] (sk) edge (sh1) edge (sh2) edge (sh3) edge (sh4);
\pause
- \node [right=of sh1] (cipher) {$E_0 = c$};
+ \node [right=of sh1] (cipher) {$E^0 = c$};
\coordinate [left = of sh2] (bla) ;
\draw [->] (cipher) -- (sh1);
\pause
- \draw [->, >=Stealth, bend right] (sh1) edge node [midway, above] {$ E^1 = \left[L_{1,S'} s_1\right] E_0$} (sh2) ;
- \draw [->, >=Stealth, bend right] (sh2) edge node [midway, left] {$ E^2 = \left[L_{2,S'} s_2\right] E_1$} (sh3) ;
- \draw [->, >=Stealth, bend right] (sh3) edge node [midway, below] {$ E^3 = \left[L_{3,S'} s_3\right] E_2$} (sh4) ;
+ \draw [->, >=Stealth, bend right] (sh1) edge node [midway, above] {$ E^1 = \left[L_{1,S'} s_1\right] E^0$} (sh2) ;
+ \draw [->, >=Stealth, bend right] (sh2) edge node [midway, left] {$ E^2 = \left[L_{2,S'} s_2\right] E^1$} (sh3) ;
+ \draw [->, >=Stealth, bend right] (sh3) edge node [midway, below] {$ E^3 = \left[L_{3,S'} s_3\right] E^2$} (sh4) ;
\node [right = of sh4] (key) {$\key = \left[L_{4,S'} s_4\right] E^3$};
\draw [->] (sh4) -- (key);
@@ -121,6 +144,9 @@
\begin{frame}
\frametitle{Features of the Protocol}
+ \begin{block}{Threshold Group Action}
+ \[E^{\# S'} = \left[L_{j,S'}s_j\right] \left(\left[\ldots\right]E^{0}\right) = \left[\sum_{P_i\in S'} L_{i,S'}s_i\right] E^0 = \left[s\right] c.\]
+ \end{block}
\begin{block}{Advantages}
\begin{itemize}
\item Simulatable
@@ -150,14 +176,14 @@
\draw [help lines] (sk) edge (sh1) edge (sh2) edge (sh3) edge (sh4);
%\pause
- \node [right=of sh1] (cipher) {$E_0 = c$};
+ \node [right=of sh1] (cipher) {$E^0 = c$};
\coordinate [left = of sh2] (bla) ;
\draw [->] (cipher) -- (sh1);
%\pause
- \draw [->, >=Stealth, bend right] (sh1) edge node [midway, above] {$ E^1 = \left[L_{1,S'} s_1\right] E_0$} (sh2) ;
- \draw [->, >=Stealth, bend right] (sh2) edge node [midway, left, color = red] {$ {E^2} \neq \left[L_{2,S'} s_2\right] E_1$} (sh3) ;
- \draw [->, >=Stealth, bend right] (sh3) edge node [midway, below] {$ E^3 = \left[L_{3,S'} s_3\right] E_2$} (sh4) ;
+ \draw [->, >=Stealth, bend right] (sh1) edge node [midway, above] {$ E^1 = \left[L_{1,S'} s_1\right] E^0$} (sh2) ;
+ \draw [->, >=Stealth, bend right] (sh2) edge node [midway, left, color = red] {$ {E^2} \neq \left[L_{2,S'} s_2\right] E^1$} (sh3) ;
+ \draw [->, >=Stealth, bend right] (sh3) edge node [midway, below] {$ E^3 = \left[L_{3,S'} s_3\right] E^2$} (sh4) ;
\node [right = of sh4] (key) {$\key = \left[L_{4,S'} s_4\right] E^3$};
\draw [->] (sh4) -- (key);