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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 |