diff options
Diffstat (limited to 'vorlesungen/slides/4')
| -rw-r--r-- | vorlesungen/slides/4/Makefile.inc | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/4/chapter.tex | 1 | ||||
| -rw-r--r-- | vorlesungen/slides/4/euklidpoly.tex | 47 |
3 files changed, 49 insertions, 0 deletions
diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc index 13de58c..2329e4a 100644 --- a/vorlesungen/slides/4/Makefile.inc +++ b/vorlesungen/slides/4/Makefile.inc @@ -14,6 +14,7 @@ chapter4 = \ ../slides/4/gauss.tex \ ../slides/4/dh.tex \ ../slides/4/divisionpoly.tex \ + ../slides/4/euklidpoly.tex \ ../slides/4/polynomefp.tex \ ../slides/4/chapter.tex diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex index 84b1f8f..830537e 100644 --- a/vorlesungen/slides/4/chapter.tex +++ b/vorlesungen/slides/4/chapter.tex @@ -12,4 +12,5 @@ \folie{4/gauss.tex} \folie{4/dh.tex} \folie{4/divisionpoly.tex} +\folie{4/euklidpoly.tex} \folie{4/polynomefp.tex} diff --git a/vorlesungen/slides/4/euklidpoly.tex b/vorlesungen/slides/4/euklidpoly.tex new file mode 100644 index 0000000..432b6b4 --- /dev/null +++ b/vorlesungen/slides/4/euklidpoly.tex @@ -0,0 +1,47 @@ +% +% 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} |