From 91284841f585ad2e5bf5002ce10ee4f3baa93b95 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Thu, 15 Apr 2021 16:43:09 +0200
Subject: add oakley groups

---
 vorlesungen/slides/a/ecc/operation.tex | 68 ++++++++++++++++++++++++++++++++++
 1 file changed, 68 insertions(+)
 create mode 100644 vorlesungen/slides/a/ecc/operation.tex

(limited to 'vorlesungen/slides/a/ecc/operation.tex')

diff --git a/vorlesungen/slides/a/ecc/operation.tex b/vorlesungen/slides/a/ecc/operation.tex
new file mode 100644
index 0000000..61ef95d
--- /dev/null
+++ b/vorlesungen/slides/a/ecc/operation.tex
@@ -0,0 +1,68 @@
+%
+% operation.tex -- Gruppen-Operation auf einer elliptischen Kurve
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Gruppenoperation}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{center}
+\includegraphics[width=\textwidth]{../../buch/chapters/90-crypto/images/elliptic.pdf}
+\end{center}
+\vspace{-23pt}
+\uncover<8->{%
+\begin{block}{Verifizieren}
+\begin{enumerate}
+\item<9-> Assoziativ?
+\item<10-> Neutrales Element $\mathstrut=\infty$
+\item<11-> Involution = Inverse?
+\end{enumerate}
+\end{block}}
+\end{column}
+\begin{column}{0.56\textwidth}
+\begin{block}{Gerade}
+$g_1,g_2\in G$, $t\in \Bbbk$
+\begin{align*}
+g(t)
+&=
+tg_1+(1-t)g_2
+\\
+\uncover<2->{
+\begin{pmatrix}X(t)\\Y(t)\end{pmatrix}
+&=
+t\begin{pmatrix}x_1\\y_1\end{pmatrix}
++
+(1-t)\begin{pmatrix}x_2\\y_2\end{pmatrix}
+\in\Bbbk^2
+}
+\end{align*}
+\end{block}
+\vspace{-13pt}
+\uncover<3->{%
+\begin{block}{3. Schnittpunkt}
+$g(t)$ einsetzen in die elliptische Kurve
+\[
+p(t)
+=
+Y(t)^2+X(t)Y(t)-X(t)^3-aX(t)-b=0
+\]
+\vspace{-12pt}
+\begin{enumerate}
+\item<4->
+kubisches Polynom mit Nullstellen $t=0,1$
+\item<5->
+$p(t) $ ist durch $t(t-1)$ teilbar
+\item<6->
+$p(t) = t(t-1)(Jt+K)=0
+\uncover<7->{\Rightarrow t=-K/J$}
+\end{enumerate}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
-- 
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