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authorAndreas Müller <andreas.mueller@ost.ch>2021-04-15 12:16:19 +0200
committerAndreas Müller <andreas.mueller@ost.ch>2021-04-15 12:16:19 +0200
commitd83ad723f1f7e5fc30f5e0e4f87a77668aac0918 (patch)
tree12ac7f24a4447158e070beb48c5a79ec2c9ddee8 /vorlesungen/slides/a/ecc
parentadd MSE presentation (diff)
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SeminarMatrizen-d83ad723f1f7e5fc30f5e0e4f87a77668aac0918.zip
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Diffstat (limited to '')
-rw-r--r--vorlesungen/slides/a/ecc/gruppendh.tex51
-rw-r--r--vorlesungen/slides/a/ecc/inverse.tex48
-rw-r--r--vorlesungen/slides/a/ecc/kurve.tex56
3 files changed, 155 insertions, 0 deletions
diff --git a/vorlesungen/slides/a/ecc/gruppendh.tex b/vorlesungen/slides/a/ecc/gruppendh.tex
new file mode 100644
index 0000000..13d85c8
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/gruppendh.tex
@@ -0,0 +1,51 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Diffie-Hellmann verallgemeinern}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Diffie-Hellman in $\mathbb{F}_p$\strut}
+\begin{enumerate}
+\item<2-> Parteien einigen sich auf $g\in \mathbb{F}_p$, $g\ne 0$, $g\ne 1$
+\item<3-> $A$ und $B$ wählen Exponenten $a,b\in \mathbb{N}$
+\item<4-> Parteien tauschen $u=g^a$ und $v=g^b$ aus
+\item<5-> Parteien berechnen $v^a$ und $u^b$
+\[
+v^a = (g^b)^a = g^{ab} =(g^a)^b = u^b
+\]
+gemeinsamer privater Schlüssel
+\end{enumerate}
+\end{block}
+\uncover<11->{%
+{\usebeamercolor[fg]{title}Spezialfall:} $G=\mathbb{F}_p^*$
+}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Diffie-Hellmann in $G$\strut}
+\begin{enumerate}
+\item<7-> Parteien einigen sich auf $g\in G$, $g\ne e$
+\item<8-> $A$ und $B$ wählen Exponenten $a,b\in \mathbb{N}$
+\item<9-> Parteien tauschen $u=g^a$ und $v=g^b$ aus
+\item<10-> Parteien berechnen $v^a$ und $u^b$
+\[
+v^a = (g^b)^a = g^{ab} =(g^a)^b = u^b
+\]
+gemeinsamer privater Schlüssel
+\end{enumerate}
+\end{block}}
+\uncover<12->{%
+{\usebeamercolor[fg]{title}Idee:} Wähle effizient zu berechnende, ``grosse''
+Gruppen, mit ``komplizierter'' Multiplikation
+}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/ecc/inverse.tex b/vorlesungen/slides/a/ecc/inverse.tex
new file mode 100644
index 0000000..f66101d
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/inverse.tex
@@ -0,0 +1,48 @@
+%
+% inverse.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Involution/Inverse}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/90-crypto/images/elliptic.pdf}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{In speziellen Koordinaten}
+\vspace{-12pt}
+\[
+v^2 = u^3+Au+B
+\]
+\uncover<2->{invariant unter $v\mapsto -v$}%
+\\
+\uncover<3->{{\color{red}geht nicht in $\mathbb{F}_2$}}
+\end{block}
+\uncover<4->{%
+\begin{block}{Allgemein}
+\vspace{-12pt}
+\begin{align*}
+Y^2+XY &= X^3 + aX+b
+\\
+\uncover<5->{%
+Y(Y+X) &= X^3 + aX + b}
+\end{align*}
+\uncover<6->{invariant unter}
+\begin{align*}
+\uncover<7->{X&\mapsto X,& Y&\mapsto -X-Y}
+\\
+\uncover<8->{&&\Rightarrow X+Y&\mapsto -Y}
+\end{align*}
+Spezialfall $\mathbb{F}_2$: $Y\leftrightarrow X+Y$
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/a/ecc/kurve.tex b/vorlesungen/slides/a/ecc/kurve.tex
new file mode 100644
index 0000000..9cf1aa2
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/kurve.tex
@@ -0,0 +1,56 @@
+%
+% kurve.tex -- elliptische Kurven
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kubische Kurven}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\uncover<5->{%
+\includegraphics[width=\textwidth]{../../buch/chapters/90-crypto/images/elliptic.pdf}
+}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Allgemein}
+mit $a,b\in\Bbbk$
+\[
+Y^2 + XY = X^3 + aX + b
+\]
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Spezielle Parametrisierung}
+\vspace{-10pt}
+\begin{align*}
+Y^2 + XY + \frac14X^2
+&=
+X^3 + \frac14X^2 + aX + b
+\\
+\uncover<3->{
+(Y+\frac12X)^2
+&=
+X^3 + \frac14X^2 + aX + b
+}\\
+\uncover<4->{
+v^2
+&=
+u^3+Au+B}
+\end{align*}
+\uncover<4->{mit
+\[
+v=Y+{\textstyle\frac12}X,
+\qquad
+u=X+\frac1{12}
+\]}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup