%
% euklidpoly.tex
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Euklidischer Algorithmus in $\mathbb{F}_2[X]$}
Gegeben: $m(X)=X^4+X+1$, $b(X) = {\color{blue}X^2+1}$
\\
\uncover<2->{Berechne $s,t\in\mathbb{F}_2[X]$ derart, dass $sm+tb=1$}
\uncover<3->{%
\begin{center}
\begin{tabular}{|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|>{$}c<{$}|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|}
\hline
k&    a_k&  b_k&  q_k&r_k&  c_k&    d_k\\
\hline
 &       &     &     &   &    1&      0\\
0&X^4+X+1&{\color{blue}X^2+1}&\uncover<4->{X^2+1}&\uncover<4->{X}&    0&      1\\
1&\uncover<5->{X^2+1 }&\uncover<5->{X}&\uncover<5->{X}&\uncover<5->{1}&\uncover<5->{1}&\uncover<5->{X^2+1}\\
2&\uncover<6->{X     }&\uncover<6->{1}&\uncover<6->{X}&\uncover<6->{0}&\uncover<6->{{\color{red}X}}&\uncover<6->{{\color{red}X^3+X+1}}\\
3&\uncover<7->{1     }&\uncover<7->{0}&&&\uncover<7->{X^2+1}&\uncover<7->{X^4+X+1} \\
\hline
\end{tabular}
\end{center}}
\ifthenelse{\boolean{presentation}}{
\only<8->{%
\begin{block}{Kontrolle}
\vspace{-10pt}
\begin{align*}
{\color{red}X}\cdot (X^4+X+1) + ({\color{red}X^3+X+1})({\color{blue}X^2+1})
&\uncover<9->{=
(X^5+X^2+X)}\\
&\qquad \uncover<10->{+ (X^5+X^3+X^2+X^3+X+1)}
\\
&\uncover<11->{=(X^5+X^2+X) + (X^5+X^2+X+1)}
\\
&\uncover<12->{=1}
\end{align*}
\end{block}}}{}
\begin{block}{Rechenregeln in $\mathbb{F}_2$}
$1+1=0$,
$2=0$, $+1=-1$.
\end{block}

\end{frame}