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authorAndreas Müller <andreas.mueller@ost.ch>2021-03-08 09:40:32 +0100
committerAndreas Müller <andreas.mueller@ost.ch>2021-03-08 09:40:32 +0100
commitf2454006fa4e2a0b4093507300fab8a29e3b5901 (patch)
tree407df3e48376af0962cebaedad6db75fe886ebb0 /vorlesungen/slides/4/division.tex
parenteuklidslide (diff)
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final preparation
Diffstat (limited to 'vorlesungen/slides/4/division.tex')
-rw-r--r--vorlesungen/slides/4/division.tex41
1 files changed, 22 insertions, 19 deletions
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}