final preparation

13 files changed, 357 insertions, 114 deletions

diff --git a/vorlesungen/slides/4/Makefile.inc b/vorlesungen/slides/4/Makefile.inc
index 2329e4a..ad1081e 100644
--- a/vorlesungen/slides/4/Makefile.inc
+++ b/vorlesungen/slides/4/Makefile.inc
@@ -16,5 +16,7 @@ chapter4 =								\
 	../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
index 830537e..a10712a 100644
--- a/vorlesungen/slides/4/chapter.tex
+++ b/vorlesungen/slides/4/chapter.tex
@@ -14,3 +14,5 @@
 \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/division.tex b/vorlesungen/slides/4/division.tex
index aeab1e3..846738f 100644
--- a/vorlesungen/slides/4/division.tex
+++ b/vorlesungen/slides/4/division.tex
@@ -13,26 +13,28 @@
 \begin{block}{Inverse {\bf berechnen}}
 Gegeben $a\in\mathbb{F}_p$, finde $b=a^{-1}\in\mathbb{F}_p$
 \begin{align*}
-&& a{\color{red}b} &\equiv 1 \mod p
+\uncover<2->{&& a{\color{blue}b} &\equiv 1 \mod p}
 \\
-&\Leftrightarrow& a{\color{red}b}&=1 + {\color{red}n}p
+\uncover<3->{&\Leftrightarrow& a{\color{blue}b}&=1 + {\color{blue}n}p}
 \\
-&&a{\color{red}b}-{\color{red}n}p&=1
+\uncover<4->{&&a{\color{blue}b}-{\color{blue}n}p&=1}
 \end{align*}
-Wegen
+\uncover<5->{Wegen
 $\operatorname{ggT}(a,p)=1$ gibt es
 $s$ und $t$ mit
 \[
-sa+tb=1
+{\color{red}s}a+{\color{red}t}p=1
 \Rightarrow
-{\color{red}b}=s,\;
-{\color{red}n}=-t
-\]
+{\color{blue}b}={\color{red}s},\;
+{\color{blue}n}=-{\color{red}t}
+\]}
+\uncover<6->{%
 $\Rightarrow$ Die Inverse kann mit dem euklidischen Algorithmus
-berechnet werden
+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}
@@ -42,21 +44,22 @@ Finde $47^{-1}\in\mathbb{F}_{1291}$
 k& a_k& b_k&q_k&  c_k& d_k\\
 \hline
  &    &    &   &    1&   0\\
-0&  47&1291&  0&    0&   1\\
-1&1291&  47& 27&    1&   0\\
-2&  47&  22&  2&  -27&   1\\
-3&  22&   3&  7&   55&  -2\\
-4&   3&   1&  3&{\color{red}-412}&{\color{red}15}\\
-5&   1&   0&   & 1291& -47\\
+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->{%
 \[
 {\color{red}-412}\cdot 47 +{\color{red}15}\cdot 1291 = 1
-\;\Rightarrow\;
-47^{-1}={\color{red}879}
-\]
-\end{block}
+\uncover<24->{\;\Rightarrow\;
+47^{-1}={\color{red}879}}
+\]}
+\end{block}}
 \end{column}
 \end{columns}
 \end{frame}
diff --git a/vorlesungen/slides/4/euklidbeispiel.tex b/vorlesungen/slides/4/euklidbeispiel.tex
index cbc3137..366a7a6 100644
--- a/vorlesungen/slides/4/euklidbeispiel.tex
+++ b/vorlesungen/slides/4/euklidbeispiel.tex
@@ -6,39 +6,73 @@
 \bgroup
 \definecolor{darkgreen}{rgb}{0,0.6,0}
 \begin{frame}[t]
-\frametitle{Beispiel}
+\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 &25&=15 \cdot {\color{red}      1} + 10 &q_0 &= {\color{red}1} & r_0 &= 10\\
-a_1&=15 & b_1 &= 10 &15&=10 \cdot {\color{darkgreen}1} + \phantom{0}5  &q_1 &= {\color{darkgreen}1} & r_1 &= \phantom{0}5 \\
-a_2&=10 & b_2 &= \phantom{0}5 &10&=\phantom{0}5  \cdot {\color{blue}     2} + \phantom{0}0  &q_2 &= {\color{blue}2} & r_2 &= \phantom{0}0 
+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
 &=
-Q({\color{blue}2}) Q({\color{darkgreen}1}) Q({\color{red}1})
+\uncover<9->{Q({\color{blue}2})}
+\uncover<8->{Q({\color{darkgreen}1})}
+Q({\color{orange}1})
 =
-\begin{pmatrix}0&1\\1&-{\color{blue}2}\end{pmatrix}
-\begin{pmatrix}0&1\\1&-{\color{darkgreen}1}\end{pmatrix}
-\begin{pmatrix}0&1\\1&-{\color{red}1}\end{pmatrix}
-=\begin{pmatrix}
--1&2\\3&-5
-\end{pmatrix}
+\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}
+\end{block}}
 \vspace{-5pt}
+\uncover<10->{%
 \begin{block}{Relationen ablesen}
-\begin{align*}
-\operatorname{ggT}({\usebeamercolor[fg]{title}25},{\usebeamercolor[fg]{title}15}) &= 5 = -1\cdot {\usebeamercolor[fg]{title}25} + 2\cdot {\usebeamercolor[fg]{title}15} \\
+\[
+\begin{pmatrix}
+\operatorname{ggT}(a,b)\\0
+\end{pmatrix}
+=
+Q
+\begin{pmatrix}a\\b\end{pmatrix}
+\uncover<11->{%
+\quad
+\Rightarrow\quad
+\left\{
+\begin{aligned}
+\operatorname{ggT}({\usebeamercolor[fg]{title}25},{\usebeamercolor[fg]{title}15}) &= 5 =
+{\color{red}-1}\cdot {\usebeamercolor[fg]{title}25} + {\color{red}2}\cdot {\usebeamercolor[fg]{title}15} \\
  0                        &= \phantom{5=-}3\cdot {\usebeamercolor[fg]{title}25} -5\cdot {\usebeamercolor[fg]{title}15}
-\end{align*}
-\end{block}
+\end{aligned}
+\right.}
+\]
+\end{block}}
 
 \end{frame}
diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex
index 6ffa4c2..be5b3ca 100644
--- a/vorlesungen/slides/4/euklidmatrix.tex
+++ b/vorlesungen/slides/4/euklidmatrix.tex
@@ -14,17 +14,19 @@
 \vspace{-10pt}
 \[
 a_k = b_kq_k + r_k
+\uncover<2->{
 \;\Rightarrow\;
 \left\{
 \begin{aligned}
 a_{k+1} &= b_k = \phantom{a_k-q_k}\llap{$-\mathstrut$}b_k \\
 b_{k+1} &= \phantom{b_k}\llap{$r_k$} = a_k - q_kb_k 
 \end{aligned}
-\right.
+\right.}
 \]
 \end{block}
 \end{column}
 \begin{column}{0.44\textwidth}
+\uncover<3->{%
 \begin{block}{Matrixschreibweise}
 \vspace{-10pt}
 \begin{align*}
@@ -37,23 +39,30 @@ b_{k+1}
 b_k\\r_k
 \end{pmatrix}
 =
-\underbrace{\begin{pmatrix}0&1\\1&-q_k\end{pmatrix}}_{\displaystyle =Q(q_k)}
+\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{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}
@@ -63,31 +72,37 @@ r_{n}
 \operatorname{ggT}(a,b) \\
 0
 \end{pmatrix}
+\uncover<11->{
 =
-\underbrace{Q(q_n)
-\dots
-Q(q_1)
-Q(q_0)}_{\displaystyle =Q}
+\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}
+\end{block}}
+\uncover<16->{%
 \begin{block}{Konsequenzen}
 \[
 Q=\begin{pmatrix}
 q_{11}&q_{12}\\
-a_{21}&q_{22}
+q_{21}&q_{22}
 \end{pmatrix}
 \quad\Rightarrow\quad
 \left\{
 \quad
 \begin{aligned}
-\operatorname{ggT}(a,b) &= q_{11}a + q_{12}b \\
+\operatorname{ggT}(a,b) &= q_{11}a + q_{12}b = {\color{red}s}a+{\color{red}t}b\\
                       0 &= q_{21}a + q_{22}b
 \end{aligned}
 \right.
 \]
-\end{block}
+\end{block}}
 
 \end{frame}
diff --git a/vorlesungen/slides/4/euklidtabelle.tex b/vorlesungen/slides/4/euklidtabelle.tex
index 2d67823..3f1b8d7 100644
--- a/vorlesungen/slides/4/euklidtabelle.tex
+++ b/vorlesungen/slides/4/euklidtabelle.tex
@@ -8,22 +8,29 @@
 \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)
-%\begin{pmatrix}
-%0&1\\1&-q_k
-%\end{pmatrix}
+\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}
-&&\Rightarrow&
+\end{pmatrix}}
+&&\uncover<5->{\Rightarrow&
 \begin{pmatrix}
 c_k&d_k\\c_{k+1}&d_{k+1}
 \end{pmatrix}
@@ -34,31 +41,33 @@ Q(q_k)
 %\end{pmatrix}
 \begin{pmatrix}
 c_{k-1}&d_{k-1}\\c_{k}&d_{k}
-\end{pmatrix}
+\end{pmatrix}}
 \end{align*}
-\end{block}
+\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 &q_0    &                     0      &                     1      \\
- 1 &q_1    &c_{-1} -q_0    \cdot c_0    &d_{-1} -q_0    \cdot d_0    \\
- 2 &q_2    &c_0    -q_1    \cdot c_1    &d_0    -q_1    \cdot d_1    \\
-\vdots&\vdots&\vdots                    &\vdots                      \\
- n &q_n    &c_{n-2}-q_{n-1}\cdot c_{n-1}&d_{n-2}-q_{n-1}\cdot d_{n-1}\\
-n+1&       &c_{n-1}-q_{n}  \cdot c_{n}  &d_{n-1}-q_{n}  \cdot d_{n}  \\
+ 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{$c_{n}$}\phantom{c_{n+1}} a + \rlap{$d_n$}\phantom{d_{n+1}}b &= \operatorname{ggT}(a,b)
+\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*}
+\right.}
+\end{equation*}}
 \end{frame}
diff --git a/vorlesungen/slides/4/fp.tex b/vorlesungen/slides/4/fp.tex
index caa6ceb..968b777 100644
--- a/vorlesungen/slides/4/fp.tex
+++ b/vorlesungen/slides/4/fp.tex
@@ -30,6 +30,7 @@ $\mathbb{Z}/n\mathbb{Z}
 =\{ \llbracket r\rrbracket\;|\; 0\le r < n \} \mathstrut$
 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 $\mathbb{Z}/n\mathbb{Z}$:
@@ -43,18 +44,22 @@ $\llbracket n_2\rrbracket$ Nullteiler in $\mathbb{Z}/n\mathbb{Z}$:
 =
 \llbracket 0 \rrbracket
 \]
-\end{block}
+\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}
+\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{block}}
 \end{column}
 \end{columns}
 \vspace{-20pt}
@@ -62,6 +67,7 @@ $\Rightarrow \quad \mathbb{F}_p$ ist ein Körper
 \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}
@@ -106,14 +112,15 @@ $\Rightarrow \quad \mathbb{F}_p$ ist ein Körper
 \feld{4}{4}{4}
 \feld{4}{5}{2} \feld{5}{4}{2}
 \feld{5}{5}{1}
-\end{scope}
+\end{scope}}
+\uncover<6->{
 \begin{scope}[xshift=6cm]
-\gruen{1}{1}
-\gruen{2}{4}
-\gruen{3}{5}
-\gruen{4}{2}
-\gruen{5}{3}
-\gruen{6}{6}
+\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}{
@@ -158,13 +165,13 @@ $\Rightarrow \quad \mathbb{F}_p$ ist ein Körper
 \geld{5}{5}{4}
 \geld{6}{5}{2} \geld{5}{6}{2}
 \geld{6}{6}{1}
-\inverse{1}{1}
-\inverse{2}{4}
-\inverse{3}{5}
-\inverse{4}{2}
-\inverse{5}{3}
-\inverse{6}{6}
-\end{scope}
+\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}
diff --git a/vorlesungen/slides/4/gauss.tex b/vorlesungen/slides/4/gauss.tex
index 960e8e1..23cdfee 100644
--- a/vorlesungen/slides/4/gauss.tex
+++ b/vorlesungen/slides/4/gauss.tex
@@ -95,7 +95,7 @@ z&=1
  1 & 1 & 0 & 2 \\
 \hline
 \end{tabular}
-\uncover<4->{%
+\uncover<5->{%
 \to
 \begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
 \hline
@@ -104,7 +104,7 @@ z&=1
  0 & 0 & 1 & 1 \\
 \hline
 \end{tabular}}
-\uncover<6->{%
+\uncover<7->{%
 \to
 \begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
 \hline
@@ -113,7 +113,7 @@ z&=1
  0 & 0 & 1 & 1 \\
 \hline
 \end{tabular}}
-\uncover<8->{%
+\uncover<9->{%
 \to
 \begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
 \hline
@@ -125,14 +125,14 @@ z&=1
 \]
 \end{minipage}};
 \begin{scope}[yshift=0.2cm]
-\uncover<3->{
+\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<5->{
+\uncover<6->{
 \draw[color=blue] (-1.45,0.5) -- (-1.45,-0.2) arc (180:360:0.2) -- (-1.05,0.5);
 }
-\uncover<7->{
+\uncover<8->{
 \draw[color=blue] (1.05,0.5) -- (1.05,0.2) arc (180:360:0.2) -- (1.45,0.5);
 }
 \end{scope}
diff --git a/vorlesungen/slides/4/ggt.tex b/vorlesungen/slides/4/ggt.tex
index e3c55e6..ef97182 100644
--- a/vorlesungen/slides/4/ggt.tex
+++ b/vorlesungen/slides/4/ggt.tex
@@ -15,16 +15,22 @@ 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*}
-a_0&=b_0q_0 + r_0 & a_1 &=b_0 & b_1&=r_0 \\
-a_1&=b_1q_1 + r_1 & a_2 &=b_1 & b_2&=r_1 \\
-a_2&=b_2q_2 + r_2 & a_3 &=b_2 & b_3&=r_2 \\
-   &\;\vdots      &     &     &    &     \\
-a_n&=b_nq_n + r_n & r_n &= 0  & r_{n-1}&=\operatorname{ggT}(a,b)
+\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{block}}
 \end{column}
 \begin{column}{0.48\textwidth}
 \begin{block}{$\operatorname{ggT}(15,25) = 5$}
@@ -40,18 +46,25 @@ a_n&=b_nq_n + r_n & r_n &= 0  & r_{n-1}&=\operatorname{ggT}(a,b)
 \foreach \y in {0,...,2}{
 	\draw[line width=0.2pt] (-2,{\y*25}) -- (65,{\y*25});
 }
-\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<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 $\x$};
+	}
 }
-\foreach \x in {0,...,4}{
-	\foreach \y in {0,...,2}{
-		\fill[color=red] ({\x*15},{\y*25}) circle[radius=0.8];
+\uncover<2->{
+	\foreach \x in {0,...,4}{
+		\foreach \y in {0,...,2}{
+			\fill[color=red] ({\x*15},{\y*25}) circle[radius=0.8];
+		}
 	}
 }
-\foreach \x in {0,5,...,60}{
-	\fill[color=blue] (\x,0) circle[radius=0.5];
-	\node at (\x,0) [below] {\tiny $\x$};
+\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}
diff --git a/vorlesungen/slides/4/polynomefp.tex b/vorlesungen/slides/4/polynomefp.tex
index fe514dd..1db50e1 100644
--- a/vorlesungen/slides/4/polynomefp.tex
+++ b/vorlesungen/slides/4/polynomefp.tex
@@ -18,40 +18,45 @@ p(X)
 a_0+a_1X+\dots+a_nX^n
 \]
 mit $a_i\in\mathbb{F}_p$.
-ObdA: $a_n=1$
+\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}
+\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{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
 
-$a\in \mathbb{F}_p[X]$ mit $\deg a < \deg m$, dann ist
+\uncover<6->{$a\in \mathbb{F}_p[X]$ mit $\deg a < \deg m$, dann ist}
 \begin{enumerate}
-\item
+\item<7->
 $\operatorname{ggT}(a,m) = 1$
-\item
+\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
+\item<9->
 $a^{-1} = t(X)$
 \end{enumerate}
-\end{block}
+\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..6914c79
--- /dev/null
+++ b/vorlesungen/slides/4/schieberegister.tex
@@ -0,0 +1,98 @@
+%
+% schieberegister.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\ds{0.7}
+\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{Schieberegister}
+Rechnen mit Polynomen in $\mathbb{F}_2(\alpha)$ ist speziell einfach
+\\
+Minimalpolynom von $\alpha$: $m(X) = X^8 + X^4+X^3+X+1$ (aus dem AES Standard)
+
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\uncover<2->{
+\begin{scope}
+	\rahmen
+	\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<3->{
+	\begin{scope}[yshift=-2cm]
+		\uncover<4->{
+			\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]
+				{$\uncover<5->{X^4+X^3+X+1=}X^8$\strut};
+		\end{scope}
+		\node at \punkt{-1}{0} {$1$\strut};
+	\end{scope}
+}
+
+\uncover<6->{
+	{\color<8->{red}
+		\draw[->] (-3,-1.5) to[out=-90,in=180] (-0.5,-2.7);
+	}
+	\begin{scope}[yshift=-2.7cm]
+		\rahmen
+		\polynom00011011
+	\end{scope}
+}
+
+\uncover<7->{
+	\node at ({3.5*\ds},-3.45) {$\|$};
+
+	\begin{scope}[yshift=-4.2cm]
+	\rahmen
+	\polynom00110111
+	\end{scope}
+}
+
+\uncover<8->{
+	\node[color=red] at (-3.5,-2.7) {Feedback};
+}
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/test.tex b/vorlesungen/slides/test.tex
index b7b696a..659be92 100644
--- a/vorlesungen/slides/test.tex
+++ b/vorlesungen/slides/test.tex
@@ -26,8 +26,8 @@
 
 %\folie{4/ggt.tex}
 %\folie{4/euklidmatrix.tex}
-\folie{4/euklidbeispiel.tex}
-\folie{4/euklidtabelle.tex}
+%\folie{4/euklidbeispiel.tex}
+%\folie{4/euklidtabelle.tex}
 %\folie{4/fp.tex}
 %\folie{4/division.tex}
 %\folie{4/gauss.tex}
@@ -35,11 +35,12 @@
 % XXX \folie{4/frobenius.tex}
 
 %\folie{4/divisionpoly.tex}
-\folie{4/euklidpoly.tex}
+%\folie{4/euklidpoly.tex}
 %\folie{4/polynomefp.tex}
+%\folie{4/alpha.tex}
 
 % XXX \folie{4/f2.tex}
-% XXX \folie{4/schieberegister.tex}
+\folie{4/schieberegister.tex}
 
 % XXX Idee der elliptischen Kurve
 % XXX \folie{4/ecidee.tex}