%
% oakley.tex -- Oakley Gruppen
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\bgroup
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Oakley-Gruppen}
\only<1>{%
\small
\verbatiminput{../slides/a/ecc/oakley1.txt}
$\approx 1.55252\cdot 10^{231}$
}
\only<2>{%
\begin{block}{$\mathbb{F}_p$}
Endlicher Körper mit $p = $
\verbatiminput{../slides/a/ecc/prime1.txt}
\end{block}
}
\only<3>{%
\small
\verbatiminput{../slides/a/ecc/oakley2.txt}
}
\only<4>{%
\begin{block}{$\mathbb{F}_p$}
Endlicher Körper mit $p = $
\verbatiminput{../slides/a/ecc/prime2.txt}
$\approx 1.7977\cdot 10^{308}$
\end{block}
}
\only<5>{%
\small
\verbatiminput{../slides/a/ecc/oakley3.txt}
}
\only<6>{%
\begin{block}{Oakley Gruppe 3}
\begin{align*}
m(x) &= x^{155} + x^{62} + 1
\\
a    &= 0
\\
b    &= \texttt{0x07338f}
\\
g_x  &= 0x7b = x^6 + x^5 + x^4 + x^3 + x + 1
\\
&=
x^{18}+x^{17}+x^{16}
+
x^{13}+x^{12}
+
x^{9}+x^{8}+x^{7}
+
x^{3}+x^{1}+x^{1}+1
\\
|G|&=45671926166590716193865565914344635196769237316 = 4.5672\cdot 10^{46}
\\
\log_2|G|&=155\,\text{bit}
\end{align*}
\end{block}}
\only<7>{%
\small
\verbatiminput{../slides/a/ecc/oakley4.txt}
}
\only<8>{%
\begin{block}{Oakley Gruppe 4}
\begin{align*}
m(x) &= x^{185} + x^{69} + 1
\\
a    &= 0
\\
b    &= \texttt{0x1ee9} = x^{12} + x^{11}+x^{10}+x^9 + x^7+x^6+x^5 + x^3+1
\\
g_x  &= \texttt{0x18} = x^4+x^3
\\
|G| &= 49039857307708443467467104857652682248052385001045053116
\\
&= 4.9040\cdot 10^{55}
\\
\log_2|G| &= 185
\end{align*}
\end{block}}
\end{frame}
\egroup
