Merge pull request #1 from AndreasFMueller/master

update

15 files changed, 1142 insertions, 0 deletions

diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
new file mode 100644
index 0000000..ad1081e
--- /dev/null
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -0,0 +1,22 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter4 =								\
+	../slides/4/ggt.tex						\
+	../slides/4/euklidmatrix.tex					\
+	../slides/4/euklidbeispiel.tex					\
+	../slides/4/euklidtabelle.tex					\
+	../slides/4/fp.tex						\
+	../slides/4/division.tex					\
+	../slides/4/gauss.tex						\
+	../slides/4/dh.tex						\
+	../slides/4/divisionpoly.tex					\
+	../slides/4/euklidpoly.tex					\
+	../slides/4/polynomefp.tex					\
+	../slides/4/schieberegister.tex					\
+	../slides/4/alpha.tex						\
+	../slides/4/chapter.tex
+
diff --git a/vorlesungen/slides/4/alpha.tex b/vorlesungen/slides/4/alpha.tex
new file mode 100644
index 0000000..3cd54c0
--- /dev/null
+++ b/vorlesungen/slides/4/alpha.tex
@@ -0,0 +1,54 @@
+%
+% alpha.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\frametitle{Was ist $\alpha$?}
+$m(X)$ ein irreduzibles Polynome in $\mathbb{F}_2[X]$
+
+Beispiel: $m(X) = X^8{\color{red}\mathstrut+X^4+X^3+X^2+1}\in\mathbb{F}_2[X]$
+\begin{columns}[t]
+\begin{column}{0.40\textwidth}
+\uncover<2->{%
+\begin{block}{Abstrakt}
+$\alpha$ ist ein ``imaginäres''
+Objekt mit der Rechenregel $m(\alpha)=0$
+\begin{align*}
+\alpha^8 &= {\color{red}\alpha^4+\alpha^3+\alpha^2+1}\\
+\uncover<3->{
+\alpha^9 &= \alpha^5+\alpha^4+\alpha^3+\alpha}\\
+\uncover<4->{
+\alpha^{10}&= \alpha^6+\alpha^5+\alpha^4+\alpha^2}\\
+\uncover<5->{
+\alpha^{11}&= \alpha^7+\alpha^6+\alpha^5+\alpha^3}\\
+\uncover<6->{
+\alpha &= \alpha^7+\alpha^3+\alpha^2+\alpha}
+\\
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.54\textwidth}
+\uncover<7->{%
+\begin{block}{Matrix}
+Eine konkretes Element in $M_n(\mathbb{F}_2)$
+\[
+\alpha
+=
+\begin{pmatrix}
+0& 0& 0& 0& 0& 0& 0& {\color{red}1}\\
+1& 0& 0& 0& 0& 0& 0& {\color{red}0}\\
+0& 1& 0& 0& 0& 0& 0& {\color{red}1}\\
+0& 0& 1& 0& 0& 0& 0& {\color{red}1}\\
+0& 0& 0& 1& 0& 0& 0& {\color{red}1}\\
+0& 0& 0& 0& 1& 0& 0& {\color{red}0}\\
+0& 0& 0& 0& 0& 1& 0& {\color{red}0}\\
+0& 0& 0& 0& 0& 0& 1& {\color{red}0}
+\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+
+\end{frame}
diff --git a/vorlesungen/slides/4/chapter.tex b/vorlesungen/slides/4/chapter.tex
new file mode 100644
index 0000000..a10712a
--- /dev/null
+++ b/vorlesungen/slides/4/chapter.tex
@@ -0,0 +1,18 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{4/ggt.tex}
+\folie{4/euklidmatrix.tex}
+\folie{4/euklidbeispiel.tex}
+\folie{4/euklidtabelle.tex}
+\folie{4/fp.tex}
+\folie{4/division.tex}
+\folie{4/gauss.tex}
+\folie{4/dh.tex}
+\folie{4/divisionpoly.tex}
+\folie{4/euklidpoly.tex}
+\folie{4/polynomefp.tex}
+\folie{4/alpha.tex}
+\folie{4/schieberegister.tex}
diff --git a/vorlesungen/slides/4/dh.tex b/vorlesungen/slides/4/dh.tex
new file mode 100644
index 0000000..b0a88e5
--- /dev/null
+++ b/vorlesungen/slides/4/dh.tex
@@ -0,0 +1,62 @@
+%
+% dh.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Diffie-Hellmann Schlüsselaushandlung}
+
+\begin{center}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\skala{0.95}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+\def\l{2.5}
+\fill[color=blue!20] (-7,-6.5) rectangle (7,0.5);
+\fill[color=red!20] (-\l,-6.5) rectangle (\l,0.501);
+\node[color=red] at (0,-1.5) {öffentliches Netzwerk};
+\node[color=blue] at (-7,0.2) [right] {privat};
+\node[color=blue] at (7,0.2) [left] {privat};
+\coordinate (A) at (-\l,-2.5);
+\coordinate (C) at (\l,-5.0);
+\coordinate (B) at (\l,-2.5);
+\coordinate (D) at (-\l,-5.0);
+\node at (0,0) {$p\in\mathbb{N},g\in\mathbb{F}_p$ aushandeln};
+\fill[color=white] (-\l,-0.7) circle[radius=0.3];
+\draw (-\l,-0.7) circle[radius=0.3];
+\fill[color=white] (\l,-0.7) circle[radius=0.3];
+\draw (\l,-0.7) circle[radius=0.3];
+\node at (-\l,-0.7) {$A$};
+\node at (\l,-0.7) {$B$};
+\uncover<2->{
+	\node at (-\l,-1.5) [left] {$a$ auswählen\strut};
+	\node at (-\l,-2.0) [left] {$x=g^a\in\mathbb{F}_p$\strut};
+	\node at (\l,-1.5) [right] {$b$ auswählen\strut};
+	\node at (\l,-2.0) [right] {$y=g^b\in\mathbb{F}_p$\strut};
+}
+\draw[->] (-\l,-1) -- (-\l,-6);
+\draw[->] (\l,-1) -- (\l,-6);
+\uncover<3->{
+	\draw[->] (A) -- (C);
+	\draw[->] (B) -- (D);
+	\fill (A) circle[radius=0.08];
+	\fill (B) circle[radius=0.08];
+	\node at ($0.8*(A)+0.2*(C)+(-0.4,0)$) [above right] {$x=g^a$};
+	\node at ($0.8*(B)+0.2*(D)+(0.4,0)$) [above left] {$y=g^b$};
+}
+\uncover<4->{
+	\node at (-\l,-5.0) [left] {$s=g^{ab}=y^a\in\mathbb{F}_p$};
+	\node at (-\l,-5.5) [left] {ausrechnen};
+	\node at (\l,-5.0) [right] {$s=g^{ab}=x^b\in\mathbb{F}_p$};
+	\node at (\l,-5.5) [right] {ausrechnen};
+}
+\uncover<5->{
+	\fill[rounded corners=0.3cm,color=darkgreen!20]
+		({-\l-1.7},-7) rectangle ({\l+1.7},-6);
+	\draw[rounded corners=0.3cm] ({-\l-1.7},-7) rectangle ({\l+1.7},-6);
+	\node at (0,-6.5) {$A$ und $B$ haben den gemeinsamen Schlüssel $s$};
+}
+\end{tikzpicture}
+
+\end{center}
+
+\end{frame}
diff --git a/vorlesungen/slides/4/division.tex b/vorlesungen/slides/4/division.tex
new file mode 100644
index 0000000..846738f
--- /dev/null
+++ b/vorlesungen/slides/4/division.tex
@@ -0,0 +1,65 @@
+%
+% division.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Division in $\mathbb{F}_p$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Inverse {\bf berechnen}}
+Gegeben $a\in\mathbb{F}_p$, finde $b=a^{-1}\in\mathbb{F}_p$
+\begin{align*}
+\uncover<2->{&& a{\color{blue}b} &\equiv 1 \mod p}
+\\
+\uncover<3->{&\Leftrightarrow& a{\color{blue}b}&=1 + {\color{blue}n}p}
+\\
+\uncover<4->{&&a{\color{blue}b}-{\color{blue}n}p&=1}
+\end{align*}
+\uncover<5->{Wegen
+$\operatorname{ggT}(a,p)=1$ gibt es
+$s$ und $t$ mit
+\[
+{\color{red}s}a+{\color{red}t}p=1
+\Rightarrow
+{\color{blue}b}={\color{red}s},\;
+{\color{blue}n}=-{\color{red}t}
+\]}
+\uncover<6->{%
+$\Rightarrow$ Die Inverse kann mit dem euklidischen Algorithmus
+berechnet werden}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{Beispiel in $\mathbb{F}_{1291}$}
+Finde $47^{-1}\in\mathbb{F}_{1291}$
+%\vspace{-10pt}
+\begin{center}
+\begin{tabular}{|>{$}r<{$}|>{$}r<{$}>{$}r<{$}|>{$}r<{$}|>{$}r<{$}>{$}r<{$}|}
+\hline
+k& a_k& b_k&q_k&  c_k& d_k\\
+\hline
+ &    &    &   &    1&   0\\
+0&  47&1291&\uncover<8->{  0}&    0&   1\\
+1&\uncover<9->{ 1291&  47}&\uncover<11->{ 27}&\uncover<10->{    1&   0}\\
+2&\uncover<12->{  47&  22}&\uncover<14->{  2}&\uncover<13->{  -27&   1}\\
+3&\uncover<15->{  22&   3}&\uncover<17->{  7}&\uncover<16->{   55&  -2}\\
+4&\uncover<18->{   3&   1}&\uncover<20->{  3}&\uncover<19->{{\color{red}-412}&{\color{red}15}}\\
+5&\uncover<21->{   1&   0}&   &\uncover<22->{ 1291& -47}\\
+\hline
+\end{tabular}
+\end{center}
+\uncover<23->{%
+\[
+{\color{red}-412}\cdot 47 +{\color{red}15}\cdot 1291 = 1
+\uncover<24->{\;\Rightarrow\;
+47^{-1}={\color{red}879}}
+\]}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/divisionpoly.tex b/vorlesungen/slides/4/divisionpoly.tex
new file mode 100644
index 0000000..5e71c95
--- /dev/null
+++ b/vorlesungen/slides/4/divisionpoly.tex
@@ -0,0 +1,37 @@
+%
+% divisionpoly.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Polynomdivision in $\mathbb{F}_3[X]$}
+Rechenregeln in $\mathbb{F}_3$: $1+2=0$, $2\cdot 2 = 1$
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcrcrcrcrcrc}
+\llap{$ ($}X^4&+&X^3&+& X^2&+& X&+&1\rlap{$)$}&\;\;:&(X^2&+&X&+&2)&=&\uncover<2->{X^2}&\uncover<5->{+&2=q}\\
+\uncover<3->{\llap{$-($}X^4&+&X^3&+&2X^2\rlap{$)$}}& & & &           &     &    & & & &  & &   & &  &   \\
+\uncover<4->{              & &   & &2X^2&+& X&+& 1}        &     &    & & & &  & &   & &   \\
+\uncover<6->{              & &   & &\llap{$-($}2X^2&+&2X&+&          2\rlap{$)$}}&     &    & & & &  & &   & &   \\
+\uncover<7->{              & &   & &    & &2X&+&2\rlap{$\mathstrut=r$}&     &    & & & &  & &  &  &}
+\end{array}
+\]
+\uncover<8->{%
+Kontrolle:
+\[
+\arraycolsep=1.4pt
+\begin{array}{rclcrcr}
+(\underbrace{X^2+2}_{\displaystyle=q})
+(X^2+X+2)
+              &=&\rlap{$\uncover<9->{X^4+X^3+2X^2}\uncover<10->{ + 2X^2+2X+2}$}
+\\
+\uncover<11->{&=& X^4+X^3+X^2&+&2X&+&2}
+\\
+\uncover<12->{& &            &&\llap{$r=\mathstrut$}2X&+&2}
+\\
+\uncover<13->{&=& X^4+X^3+X^2&+&1X&+&1}
+\end{array}
+\]
+}
+
+\end{frame}
diff --git a/vorlesungen/slides/4/euklidbeispiel.tex b/vorlesungen/slides/4/euklidbeispiel.tex
new file mode 100644
index 0000000..366a7a6
--- /dev/null
+++ b/vorlesungen/slides/4/euklidbeispiel.tex
@@ -0,0 +1,78 @@
+%
+% euklidmatrix.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Euklidischer Algorithmus: Beispiel}
+\setlength{\abovedisplayskip}{0pt}
+\setlength{\belowdisplayskip}{0pt}
+\vspace{-0pt}
+\begin{block}{Finde $\operatorname{ggT}(25,15)$}
+\vspace{-12pt}
+\begin{align*}
+a_0&=25 & b_0 &= 15 &\uncover<2->{25&=15 \cdot {\color{orange}      1} + 10 &q_0 &= {\color{orange}1} & r_0 &= 10}\\
+\uncover<3->{a_1&=15 & b_1 &= 10}&\uncover<4->{15&=10 \cdot {\color{darkgreen}1} + \phantom{0}5  &q_1 &= {\color{darkgreen}1} & r_1 &= \phantom{0}5}\\
+\uncover<5->{a_2&=10 & b_2 &= \phantom{0}5}&\uncover<6->{10&=\phantom{0}5  \cdot {\color{blue}     2} + \phantom{0}0  &q_2 &= {\color{blue}2} & r_2 &= \phantom{0}0 }
+\end{align*}
+\end{block}
+\vspace{-5pt}
+\uncover<7->{%
+\begin{block}{Matrix-Operationen}
+\begin{align*}
+Q
+&=
+\uncover<9->{Q({\color{blue}2})}
+\uncover<8->{Q({\color{darkgreen}1})}
+Q({\color{orange}1})
+=
+\uncover<9->{
+\begin{pmatrix*}[r]0&1\\1&-{\color{blue}2}\end{pmatrix*}
+}
+\uncover<8->{
+\begin{pmatrix*}[r]0&1\\1&-{\color{darkgreen}1}\end{pmatrix*}
+}
+\begin{pmatrix*}[r]0&1\\1&-{\color{orange}1}\end{pmatrix*}
+=
+\ifthenelse{\boolean{presentation}}{
+\only<7>{
+\begin{pmatrix*}[r]\phantom{-}0&1\\1&-1\end{pmatrix*}
+}
+\only<8>{
+\begin{pmatrix*}[r]
+1&-1\\-1&2
+\end{pmatrix*}
+}
+}{}
+\only<9->{
+\begin{pmatrix*}[r]
+{\color{red}-1}&{\color{red}2}\\3&-5
+\end{pmatrix*}}
+\end{align*}
+\end{block}}
+\vspace{-5pt}
+\uncover<10->{%
+\begin{block}{Relationen ablesen}
+\[
+\begin{pmatrix}
+\operatorname{ggT}(a,b)\\0
+\end{pmatrix}
+=
+Q
+\begin{pmatrix}a\\b\end{pmatrix}
+\uncover<11->{%
+\quad
+\Rightarrow\quad
+\left\{
+\begin{aligned}
+\operatorname{ggT}({\usebeamercolor[fg]{title}25},{\usebeamercolor[fg]{title}15}) &= 5 =
+{\color{red}-1}\cdot {\usebeamercolor[fg]{title}25} + {\color{red}2}\cdot {\usebeamercolor[fg]{title}15} \\
+ 0                        &= \phantom{5=-}3\cdot {\usebeamercolor[fg]{title}25} -5\cdot {\usebeamercolor[fg]{title}15}
+\end{aligned}
+\right.}
+\]
+\end{block}}
+
+\end{frame}
diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex
new file mode 100644
index 0000000..be5b3ca
--- /dev/null
+++ b/vorlesungen/slides/4/euklidmatrix.tex
@@ -0,0 +1,108 @@
+%
+% euklidmatrix.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Matrixform des euklidischen Algorithmus}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.52\textwidth}
+\begin{block}{Einzelschritt}
+\vspace{-10pt}
+\[
+a_k = b_kq_k + r_k
+\uncover<2->{
+\;\Rightarrow\;
+\left\{
+\begin{aligned}
+a_{k+1} &= b_k = \phantom{a_k-q_k}\llap{$-\mathstrut$}b_k \\
+b_{k+1} &= \phantom{b_k}\llap{$r_k$} = a_k - q_kb_k 
+\end{aligned}
+\right.}
+\]
+\end{block}
+\end{column}
+\begin{column}{0.44\textwidth}
+\uncover<3->{%
+\begin{block}{Matrixschreibweise}
+\vspace{-10pt}
+\begin{align*}
+\begin{pmatrix}
+a_{k+1}\\
+b_{k+1}
+\end{pmatrix}
+&=
+\begin{pmatrix}
+b_k\\r_k
+\end{pmatrix}
+=
+\uncover<4->{
+\underbrace{\begin{pmatrix}
+\uncover<5->{0&1}\\
+\uncover<6->{1&-q_k}
+\end{pmatrix}}_{\uncover<7->{\displaystyle =Q(q_k)}}
+}
+\begin{pmatrix}
+a_k\\b_k
+\end{pmatrix}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\uncover<8->{%
+\begin{block}{Ende des Algorithmus}
+\vspace{-10pt}
+\begin{align*}
+\uncover<9->{
+\begin{pmatrix}
+a_{n+1}\\
+b_{n+1}\\
+\end{pmatrix}
+&=}
+\begin{pmatrix}
+r_{n-1}\\
+r_{n}
+\end{pmatrix}
+=
+\begin{pmatrix}
+\operatorname{ggT}(a,b) \\
+0
+\end{pmatrix}
+\uncover<11->{
+=
+\underbrace{\uncover<15->{Q(q_n)}
+\uncover<14->{\dots}
+\uncover<13->{Q(q_1)}
+\uncover<12->{Q(q_0)}}_{\displaystyle =Q}}
+\uncover<10->{
+\begin{pmatrix} a_0\\ b_0\end{pmatrix}
+\uncover<6->{
+=
+Q\begin{pmatrix}a\\b\end{pmatrix}
+}
+}
+\end{align*}
+\end{block}}
+\uncover<16->{%
+\begin{block}{Konsequenzen}
+\[
+Q=\begin{pmatrix}
+q_{11}&q_{12}\\
+q_{21}&q_{22}
+\end{pmatrix}
+\quad\Rightarrow\quad
+\left\{
+\quad
+\begin{aligned}
+\operatorname{ggT}(a,b) &= q_{11}a + q_{12}b = {\color{red}s}a+{\color{red}t}b\\
+                      0 &= q_{21}a + q_{22}b
+\end{aligned}
+\right.
+\]
+\end{block}}
+
+\end{frame}
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}
diff --git a/vorlesungen/slides/4/euklidtabelle.tex b/vorlesungen/slides/4/euklidtabelle.tex
new file mode 100644
index 0000000..3f1b8d7
--- /dev/null
+++ b/vorlesungen/slides/4/euklidtabelle.tex
@@ -0,0 +1,73 @@
+%
+% euklidtabelle.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Durchführung des euklidischen Algorithmus}
+Problem: Berechnung der Produkte $Q(q_k)\cdots Q(q_1)Q(q_0)$ für $k=0,1,\dots,n$
+\uncover<2->{%
+\begin{block}{Multiplikation mit $Q(q_k)$}
+\vspace{-12pt}
+\begin{align*}
+Q(q_k)
+\ifthenelse{\boolean{presentation}}{
+\only<-3>{
+\begin{pmatrix}
+u&v\\c&d
+\end{pmatrix}
+=\begin{pmatrix}
+0&1\\1&-q_k
+\end{pmatrix}
+}}{}
+\begin{pmatrix}
+u&v\\c&d
+\end{pmatrix}
+&\uncover<3->{=
+\begin{pmatrix}
+c&d\\
+u-q_kc&v-q_kd
+\end{pmatrix}}
+&&\uncover<5->{\Rightarrow&
+\begin{pmatrix}
+c_k&d_k\\c_{k+1}&d_{k+1}
+\end{pmatrix}
+&=
+Q(q_k)
+%\begin{pmatrix}
+%0&1\\1&-q_k
+%\end{pmatrix}
+\begin{pmatrix}
+c_{k-1}&d_{k-1}\\c_{k}&d_{k}
+\end{pmatrix}}
+\end{align*}
+\end{block}}
+\vspace{-10pt}
+\uncover<6->{%
+\begin{equation*}
+\begin{tabular}{|>{\tiny$}r<{$}|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|}
+\hline
+k  &q_k    &                     c_k    &                     d_k    \\
+\hline
+-1 &       &                     1      &                     0      \\
+ 0 &\uncover<7->{q_0    }&                     0      &                     1      \\
+ 1 &\uncover<9->{q_1    }&\uncover<8->{c_{-1} -q_0    \cdot c_0    &d_{-1} -q_0    \cdot d_0    }\\
+ 2 &\uncover<11->{q_2    }&\uncover<10->{c_0    -q_1    \cdot c_1    &d_0    -q_1    \cdot d_1    }\\
+\vdots&\uncover<12->{\vdots}&\uncover<12->{\vdots                    &\vdots                      }\\
+ n &\uncover<14->{q_n    }&\uncover<13->{{\color{red}c_{n-2}-q_{n-1}\cdot c_{n-1}}&{\color{red}d_{n-2}-q_{n-1}\cdot d_{n-1}}}\\
+n+1&       &\uncover<15->{c_{n-1}-q_{n}  \cdot c_{n}  &d_{n-1}-q_{n}  \cdot d_{n}  }\\
+\hline
+\end{tabular}
+\uncover<16->{
+\Rightarrow
+\left\{
+\begin{aligned}
+\rlap{${\color{red}c_{n}}$}\phantom{c_{n+1}} a + \rlap{${\color{red}d_n}$}\phantom{d_{n+1}}b &= \operatorname{ggT}(a,b)
+\\
+c_{n+1} a + d_{n+1} b &= 0
+\end{aligned}
+\right.}
+\end{equation*}}
+\end{frame}
diff --git a/vorlesungen/slides/4/fp.tex b/vorlesungen/slides/4/fp.tex
new file mode 100644
index 0000000..968b777
--- /dev/null
+++ b/vorlesungen/slides/4/fp.tex
@@ -0,0 +1,178 @@
+%
+% fp.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\feld#1#2#3{
+	\node at ({#1},{5-#2}) {$#3$};
+}
+\def\geld#1#2#3{
+	\node at ({#1},{6-#2}) {$#3$};
+}
+\def\rot#1#2{
+	\fill[color=red!20] ({#1-0.5},{5-#2-0.5}) rectangle ({#1+0.5},{5-#2+0.5});
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\gruen#1#2{
+	\fill[color=darkgreen!20] ({#1-0.5},{6-#2-0.5}) rectangle ({#1+0.5},{6-#2+0.5});
+}
+\def\inverse#1#2{
+	\node at (9,{6-#1}) {$#1^{-1}=#2\mathstrut$};
+}
+\begin{frame}[t]
+\frametitle{Galois-Körper}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Restklassenring$\mathstrut$}
+$\mathbb{Z}/n\mathbb{Z}
+=\{ \llbracket r\rrbracket\;|\; 0\le r < n \} \mathstrut$
+ist ein Ring
+\end{block}
+\uncover<2->{%
+\begin{block}{Nullteiler}
+Falls $n=n_1n_2$, dann sind $\llbracket n_1\rrbracket$ und
+$\llbracket n_2\rrbracket$ Nullteiler in $\mathbb{Z}/n\mathbb{Z}$:
+\[
+\llbracket n_1\rrbracket
+\llbracket n_2\rrbracket
+=
+\llbracket n_1n_2 \rrbracket
+=
+\llbracket n\rrbracket
+=
+\llbracket 0 \rrbracket
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Galois-Körper $\mathbb{F}_p\mathstrut$}
+$\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}\mathstrut$
+\end{block}}
+\uncover<4->{%
+\begin{block}{$n$ prim}
+Für $n=p$ prim ist $\mathbb{Z}/n\mathbb{Z}$ nullteilerfrei
+\medskip
+
+\uncover<5->{
+$\Rightarrow \quad \mathbb{F}_p$ ist ein Körper
+}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.45]
+\fill[color=white] (-12,0) circle[radius=0.1];
+\fill[color=white] (12,0) circle[radius=0.1];
+\uncover<3->{
+\begin{scope}[xshift=-8cm]
+\rot{2}{3}
+\rot{4}{3}
+\rot{3}{2}
+\rot{3}{4}
+\fill[color=gray!40] (-0.5,5.5) rectangle (5.5,6.5);
+\fill[color=gray!40] (-1.5,-0.5) rectangle (-0.5,5.5);
+\foreach \x in {-0.5,5.5}{
+	\draw (\x,-0.5) -- (\x,6.5);
+}
+\foreach \x in {0.5,...,4.5}{
+	\draw[line width=0.3pt] (\x,-0.5) -- (\x,6.5);
+}
+\foreach \y in {0.5,...,5.5}{
+	\draw[line width=0.3pt] (-1.5,\y) -- (5.5,\y);
+}
+\foreach \y in {-0.5,5.5}{
+	\draw (-1.5,\y) -- (5.5,\y);
+}
+\draw (-1.5,-0.5) -- (-1.5,5.5);
+\draw (-0.5,6.5) -- (5.5,6.5);
+\foreach \x in {0,...,5}{
+	\node at (\x,6) {$\x$};
+	\node at (-1,{5-\x}) {$\x$};
+}
+\foreach \x in {0,...,5}{
+	\feld{\x}{0}{0}
+	\feld{0}{\x}{0}
+}
+\foreach \x in {2,...,5}{
+	\feld{\x}{1}{\x}
+	\feld{1}{\x}{\x}
+}
+\feld{1}{1}{1}
+\feld{2}{2}{4}
+\feld{2}{3}{0} \feld{3}{2}{0}
+\feld{2}{4}{2} \feld{4}{2}{2}
+\feld{2}{5}{4} \feld{5}{2}{4}
+\feld{3}{3}{3}
+\feld{4}{3}{0} \feld{3}{4}{0}
+\feld{5}{3}{3} \feld{3}{5}{3}
+\feld{4}{4}{4}
+\feld{4}{5}{2} \feld{5}{4}{2}
+\feld{5}{5}{1}
+\end{scope}}
+\uncover<6->{
+\begin{scope}[xshift=6cm]
+\uncover<7->{ \gruen{1}{1} }
+\uncover<8->{ \gruen{4}{2} }
+\uncover<9->{ \gruen{5}{3} }
+\uncover<10->{ \gruen{2}{4} }
+\uncover<11->{ \gruen{3}{5} }
+\uncover<12->{ \gruen{6}{6} }
+\fill[color=gray!40] (-0.5,6.5) rectangle (6.5,7.5);
+\fill[color=gray!40] (-1.5,-0.5) rectangle (-0.5,6.5);
+\foreach \x in {-0.5,6.5}{
+	\draw (\x,-0.5) -- (\x,7.5);
+}
+\foreach \x in {0.5,...,5.5}{
+	\draw[line width=0.3pt] (\x,-0.5) -- (\x,7.5);
+}
+\foreach \y in {0.5,...,6.5}{
+	\draw[line width=0.3pt] (-1.5,\y) -- (6.5,\y);
+}
+\foreach \y in {-0.5,6.5}{
+	\draw (-1.5,\y) -- (6.5,\y);
+}
+\draw (-1.5,-0.5) -- (-1.5,6.5);
+\draw (-0.5,7.5) -- (6.5,7.5);
+\foreach \x in {0,...,6}{
+	\node at (\x,7) {$\x$};
+	\node at (-1,{6-\x}) {$\x$};
+}
+\foreach \x in {0,...,6}{
+	\geld{\x}{0}{0}
+	\geld{0}{\x}{0}
+}
+\foreach \x in {2,...,6}{
+	\geld{\x}{1}{\x}
+	\geld{1}{\x}{\x}
+}
+\geld{1}{1}{1}
+\geld{2}{2}{4}
+\geld{2}{3}{6} \geld{3}{2}{6}
+\geld{2}{4}{1} \geld{4}{2}{1}
+\geld{2}{5}{3} \geld{5}{2}{3}
+\geld{2}{6}{5} \geld{6}{2}{5}
+\geld{3}{3}{2}
+\geld{4}{3}{5} \geld{3}{4}{5}
+\geld{5}{3}{1} \geld{3}{5}{1}
+\geld{6}{3}{4} \geld{3}{6}{4}
+\geld{4}{4}{2}
+\geld{5}{4}{6} \geld{4}{5}{6}
+\geld{6}{4}{3} \geld{4}{6}{3}
+\geld{5}{5}{4}
+\geld{6}{5}{2} \geld{5}{6}{2}
+\geld{6}{6}{1}
+\uncover<7->{ \inverse{1}{1} }
+\uncover<8->{ \inverse{2}{4} }
+\uncover<9->{ \inverse{3}{5} }
+\uncover<10->{ \inverse{4}{2} }
+\uncover<11->{ \inverse{5}{3} }
+\uncover<12->{ \inverse{6}{6} }
+\end{scope}}
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/4/gauss.tex b/vorlesungen/slides/4/gauss.tex
new file mode 100644
index 0000000..23cdfee
--- /dev/null
+++ b/vorlesungen/slides/4/gauss.tex
@@ -0,0 +1,143 @@
+%
+% gauss.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\bgroup
+\def\ds{0.5}
+\def\punkt#1#2{({(#1)*\ds},{-(#2)*\ds})}
+\def\tabelle{
+	\foreach \x in {-0.5,0.5,3.5}{
+		\draw \punkt{\x}{-0.5} -- \punkt{\x}{3.5};
+		\draw \punkt{-0.5}{\x} -- \punkt{3.5}{\x};
+	}
+	\node at \punkt{0}{1} {$0$};
+	\node at \punkt{0}{2} {$1$};
+	\node at \punkt{0}{3} {$2$};
+	\node at \punkt{1}{0} {$0$};
+	\node at \punkt{2}{0} {$1$};
+	\node at \punkt{3}{0} {$2$};
+}
+\begin{frame}[t]
+\frametitle{Gauss-Algorithmus in $\mathbb{F}_3$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\begin{block}{Additions-/Multiplikationstabelle}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}[xshift=-1.6cm]
+\tabelle
+\node at \punkt{0}{0} {$+$};
+\node at \punkt{1}{1} {$0$};
+\node at \punkt{1}{2} {$1$};
+\node at \punkt{1}{3} {$2$};
+\node at \punkt{2}{1} {$1$};
+\node at \punkt{2}{2} {$2$};
+\node at \punkt{2}{3} {$0$};
+\node at \punkt{3}{1} {$2$};
+\node at \punkt{3}{2} {$0$};
+\node at \punkt{3}{3} {$1$};
+\end{scope}
+\begin{scope}[xshift=1.6cm]
+\tabelle
+\node at \punkt{0}{0} {$\cdot$};
+\node at \punkt{1}{1} {$0$};
+\node at \punkt{1}{2} {$0$};
+\node at \punkt{1}{3} {$0$};
+\node at \punkt{2}{1} {$0$};
+\node at \punkt{2}{2} {$1$};
+\node at \punkt{2}{3} {$2$};
+\node at \punkt{3}{1} {$0$};
+\node at \punkt{3}{2} {$2$};
+\node at \punkt{3}{3} {$1$};
+\end{scope}
+\end{tikzpicture}
+\end{center}
+
+\end{block}
+\end{column}
+\begin{column}{0.52\textwidth}
+\uncover<2->{%
+\begin{block}{Gleichungssystem\uncover<9->{/Lösung}}
+\[
+\left.
+\begin{array}{rcrcrcrcr}
+ x&+&y&+2z&=&1\\
+2x& & &+ z&=&2\\
+ x&+&y&   &=&2
+\end{array}
+\uncover<9->{
+\right\}
+\Rightarrow
+\left\{
+\begin{aligned}
+x&=2\\
+y&=0\\
+z&=1
+\end{aligned}
+\right.}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Gauss-Algorithmus}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\node at (0,0) {\begin{minipage}{13cm}%
+\[
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 1 & 2 & 1 \\
+ 2 & 0 & 1 & 2 \\
+ 1 & 1 & 0 & 2 \\
+\hline
+\end{tabular}
+\uncover<5->{%
+\to
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 1 & 2 & 1 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 1 \\
+\hline
+\end{tabular}}
+\uncover<7->{%
+\to
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 1 & 0 & 2 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 1 \\
+\hline
+\end{tabular}}
+\uncover<9->{%
+\to
+\begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
+\hline
+ 1 & 0 & 0 & 2 \\
+ 0 & 1 & 0 & 0 \\
+ 0 & 0 & 1 & 1 \\
+\hline
+\end{tabular}}
+\]
+\end{minipage}};
+\begin{scope}[yshift=0.2cm]
+\uncover<4->{
+\draw[color=red] (-5.6,0.3) circle[radius=0.2];
+\draw[color=blue] (-5.4,-0.8) -- (-5.4,-0.2) arc (0:180:0.2) -- (-5.8,-0.8);
+}
+\uncover<6->{
+\draw[color=blue] (-1.45,0.5) -- (-1.45,-0.2) arc (180:360:0.2) -- (-1.05,0.5);
+}
+\uncover<8->{
+\draw[color=blue] (1.05,0.5) -- (1.05,0.2) arc (180:360:0.2) -- (1.45,0.5);
+}
+\end{scope}
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/4/ggt.tex b/vorlesungen/slides/4/ggt.tex
new file mode 100644
index 0000000..ef97182
--- /dev/null
+++ b/vorlesungen/slides/4/ggt.tex
@@ -0,0 +1,75 @@
+%
+% ggt.tex -- GGT, Definition und Algorithmus
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschuöe
+%
+\begin{frame}[t]
+\frametitle{Grösster gemeinsamer Teiler}
+\vspace{-15pt}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Gegeben: $a,b\in\mathbb Z$
+\\
+Gesucht: grösster gemeinsamer Teiler $\operatorname{ggT}(a,b)$
+\end{block}
+\uncover<4->{%
+\begin{block}{Euklidischer Algorithmus}
+$a_0 = a$, $b_0=b$
+\begin{align*}
+\uncover<5->{
+a_0&=b_0q_0 + r_0 & a_1 &=b_0 & b_1&=r_0}\\
+\uncover<6->{
+a_1&=b_1q_1 + r_1 & a_2 &=b_1 & b_2&=r_1}\\
+\uncover<7->{
+a_2&=b_2q_2 + r_2 & a_3 &=b_2 & b_3&=r_2}\\
+\uncover<8->{
+   &\;\vdots      &     &     &    &    }\\
+\uncover<9->{
+a_n&=b_nq_n + r_n & r_n &= 0  & r_{n-1}&=\operatorname{ggT}(a,b)}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{$\operatorname{ggT}(15,25) = 5$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.09]
+\draw[->] (-1,0) -- (65,0) coordinate[label={$a$}];
+\draw[->] (0,-1) -- (0,65) coordinate[label={right:$b$}];
+\begin{scope}
+\clip (-1,-1) rectangle (65,65);
+\foreach \x in {0,...,4}{
+	\draw[line width=0.2pt] ({\x*15},-2) -- ({\x*15},65);
+}
+\foreach \y in {0,...,2}{
+	\draw[line width=0.2pt] (-2,{\y*25}) -- (65,{\y*25});
+}
+\uncover<3->{
+	\foreach \x in {0,5,...,120}{
+		\draw[color=blue] ({\x+2},-2) -- ({\x+2-70},{-2+70});
+		\node[color=blue] at ({0.5*\x-0.5},{0.5*\x-0.5})
+			[rotate=-45,above] {\tiny $\x$};
+	}
+}
+\uncover<2->{
+	\foreach \x in {0,...,4}{
+		\foreach \y in {0,...,2}{
+			\fill[color=red] ({\x*15},{\y*25}) circle[radius=0.8];
+		}
+	}
+}
+\uncover<3->{
+	\foreach \x in {0,5,...,60}{
+		\fill[color=blue] (\x,0) circle[radius=0.5];
+		\node at (\x,0) [below] {\tiny $\x$};
+	}
+}
+\end{scope}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/polynomefp.tex b/vorlesungen/slides/4/polynomefp.tex
new file mode 100644
index 0000000..1db50e1
--- /dev/null
+++ b/vorlesungen/slides/4/polynomefp.tex
@@ -0,0 +1,62 @@
+%
+% polynomefp.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Polynome über $\mathbb{F}_p[X]$}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Polynomring}
+$\mathbb{F}_p[X]$ sind Polynome
+\[
+p(X)
+=
+a_0+a_1X+\dots+a_nX^n
+\]
+mit $a_i\in\mathbb{F}_p$.
+\uncover<2->{ObdA: $a_n=1$}%
+
+\end{block}
+\uncover<3->{%
+\begin{block}{Irreduzible Polynome}
+$m(X)$ ist irreduzibel, wenn es keine Faktorisierung
+$m(X)=p(X)q(X)$ mit $p,q\in\mathbb{F}_p[X]$ gibt
+\end{block}}
+\uncover<4->{%
+\begin{block}{Rest modulo $m(X)$}
+$X^{n+k}$ kann immer reduziert werden:
+\[
+X^{n+k} = -(a_0+a_1X+\dots+a_{n-1}X^{n-1})X^k
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Körper $\mathbb{F}_p/(m(X))$}
+Wenn $m(X)$ irreduzibel ist, dann ist
+$\mathbb{F}_p[X]$ nullteilerfrei.
+\medskip
+
+\uncover<6->{$a\in \mathbb{F}_p[X]$ mit $\deg a < \deg m$, dann ist}
+\begin{enumerate}
+\item<7->
+$\operatorname{ggT}(a,m) = 1$
+\item<8->
+Es gibt $s,t\in\mathbb{F}_p[X]$ mit
+\[
+s(X)m(X)+t(X)a(X) = 1
+\]
+(aus dem euklidischen Algorithmus)
+\item<9->
+$a^{-1} = t(X)$
+\end{enumerate}
+\uncover<9->{$\Rightarrow$ $\mathbb{F}_p[X]/(m(X))$ ist ein Körper
+mit genau $p^n$ Elementen}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/4/schieberegister.tex b/vorlesungen/slides/4/schieberegister.tex
new file mode 100644
index 0000000..f349337
--- /dev/null
+++ b/vorlesungen/slides/4/schieberegister.tex
@@ -0,0 +1,120 @@
+%
+% schieberegister.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\ds{0.7}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\punkt#1#2{({(#1)*\ds},{(#2)*\ds})}
+\def\rahmen{
+	\draw ({-0.5*\ds},{-0.5*\ds}) rectangle ({7.5*\ds},{0.5*\ds});
+	\foreach \x in {0.5,1.5,...,6.5}{
+		\draw ({\x*\ds},{-0.5*\ds}) rectangle ({\x*\ds},{0.5*\ds});
+	}
+}
+\def\polynom#1#2#3#4#5#6#7#8{
+	\node at \punkt{0}{0} {$#1$};
+	\node at \punkt{1}{0} {$#2$};
+	\node at \punkt{2}{0} {$#3$};
+	\node at \punkt{3}{0} {$#4$};
+	\node at \punkt{4}{0} {$#5$};
+	\node at \punkt{5}{0} {$#6$};
+	\node at \punkt{6}{0} {$#7$};
+	\node at \punkt{7}{0} {$#8$};
+}
+\begin{frame}[t]
+\frametitle{Implementation der Multiplikation in $\mathbb{F}_2(\alpha)$\uncover<10->{: Schieberegister}}
+Rechnen in $\mathbb{F}_2[X]$\only<5->{ und $\mathbb{F}_2(\alpha)$}
+ist speziell einfach
+\\
+Minimalpolynom von $\alpha$: ${\color{darkgreen}m(X) = X^8 + X^4+X^3+X+1}$
+(aus dem AES Standard)
+
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\uncover<4->{
+	\fill[color=blue!20]
+		\punkt{-0.5}{-0.5} rectangle \punkt{7.5}{0.5};
+}
+
+\uncover<2->{
+\begin{scope}
+	\rahmen
+	\node at \punkt{-0.5}{1} [left] {$p(X)=\mathstrut$};
+	\node at \punkt{0}{1} {$X^7$\strut};
+	\node at \punkt{2.5}{1}{$+$\strut};
+	\node at \punkt{3}{1} {$X^4$\strut};
+	\node at \punkt{4.5}{1}{$+$\strut};
+	\node at \punkt{5}{1} {$X^2$\strut};
+	\node at \punkt{6.5}{1}{$+$\strut};
+	\node at \punkt{7}{1} {$1$\strut};
+	\polynom10010101
+\end{scope}}
+
+\uncover<3->{
+	\draw[->] ({7.7*\ds},-0.2) to[out=-45,in=45] ({7.7*\ds},-1.8);
+	\node at ({8*\ds},-1) [right] {$\mathstrut\cdot X = \text{Shift}$};
+}
+\uncover<4->{
+	\foreach \x in {0,...,7}{
+		\draw[->,color=blue!40]
+			({\x*\ds},{-0.6*\ds}) -- ({(\x-1)*\ds},{-2+0.6*\ds});
+	}
+}
+
+\fill[color=white] (-4.65,0) circle[radius=0.01];
+
+\uncover<3->{
+	\begin{scope}[yshift=-2cm]
+		\uncover<4->{
+			\fill[color=blue!20]
+				\punkt{-1.5}{-0.5} rectangle \punkt{6.5}{0.5};
+			\rahmen
+			\polynom00101010
+		}
+		\node at \punkt{2}{1} {$X^5$\strut};
+		\node at \punkt{3.5}{1}{$+$\strut};
+		\node at \punkt{4}{1} {$X^3$\strut};
+		\node at \punkt{5.5}{1}{$+$\strut};
+		\node at \punkt{6}{1} {$X$\strut};
+		\begin{scope}[xshift=0.4cm]
+			\node at \punkt{-1}{1} [left]
+				{$\uncover<5->{{\color{darkgreen}\alpha^4+\alpha^3+\alpha+1=\alpha^8}}\only<-4>{X^8}$\strut};
+		\end{scope}
+		\node at \punkt{-1}{0} {$1$\strut};
+	\end{scope}
+}
+
+\uncover<6->{
+	{\color<8->{red}
+		\draw[->] (-2.5,-1.5) to[out=-90,in=180] (-0.5,-2.7);
+	}
+	\begin{scope}[yshift=-2.7cm]
+		\rahmen
+		{\color{darkgreen}
+			\polynom00011011
+		}
+	\end{scope}
+}
+
+\uncover<7->{
+	\node at ({3.5*\ds},-3.45) {$\|$};
+
+	\begin{scope}[yshift=-4.2cm]
+	\rahmen
+	\polynom00110001
+	\node at \punkt{7.6}{0} [right] {$\mathstrut=\alpha\cdot p(\alpha)$};
+	\end{scope}
+}
+
+\uncover<8->{
+	\node[color=red] at (-3.0,-2.5) {Feedback};
+}
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
+\egroup