first version of chapter "Modulare Arithmetik" finished
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@ -18,8 +18,18 @@
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\contentsline {section}{\numberline {2.1}Exkurs: Division mit Rest}{5}%
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\contentsline {section}{\numberline {2.2}Der Ring $\mathbb {Z}_n$}{5}%
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\end{aligned}
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\end{equation}
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\subsection{Inverse bezüglich der Addition}
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jedes $a\in \mathbb{Z}$ hat ein Inverses:
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$$ -a :=
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\begin{cases}
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0 &\text{für }a=0 \\
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n-a &\text{sonst}
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\end{cases}$$
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\subsubsection{Inverse bezüglich der Addition}
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jedes $a\in \mathbb{Z}$ hat ein Inverses:
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$$ -a :=
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\begin{cases}
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0 &\text{für }a=0 \\
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n-a &\text{sonst}
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\end{cases}$$
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\subsubsection{Inverse bezüglich der Multiplikation}
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ein Element $a\in \mathbb{Z}_n$ ist \textit{(multiplikativ) invertierbar}, falls es ein Element $b\in \mathbb{Z}_n$ gibt, für das gilt:
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$$a\cdot b = 1$$
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man schreibt auch:
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$$a^{-1}:=b$$
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Die Menge der invertierbaren Elemente in $\mathbb{Z}_n$ wird als $\mathbb{Z}_n^*$ bezeichnet:
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$$\mathbb{Z}_n^* = \{a\in\mathbb{Z}_n\mid a\cdot b=1 \text{ für ein }b\in\mathbb{Z}n\}$$
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Zudem gilt, dass ein Element nur dann invertierbar ist, falls $ggT(a,n)=1$:
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$$\mathbb{Z}_n^* = \{a\in\mathbb{Z}_n\mid ggT(a,n)=1\}$$
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\subsection{Subtraktion}
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Eine Subtraktion entspricht einer Addition mit der Inverse:
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\subsection{Kongruenz}
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$a,b\in \mathbb{Z}$ sind \textit{kongruent modulo n}, falls $n\in \mathbb{N}|(a-b)$.
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Man schreibt auch $a\equiv b$
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\subsection{Matrizen}
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\subsubsection{Determinantenberechnung}
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Die Determinante $\det(A)$ der ($N,N$)-Matrix $A=(a_{ij})_{1\le i,j \le N}$ (mit ganzzahligen Einträgen) über $\mathbb{Z}_n$ wird definiert durch:
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$$\det(A) \mod n = det((a_{i,j} \mod n)_{1\le i,j\le N})$$
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Zudem gilt für die Matrizen $A=(a_{ij})_{1\le i,j \le N}$ und $B=(b_{ij})_{1\le i,j \le N}$ (mit ganzzahligen Einträgen):
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$$\begin{aligned}
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\det(A\cdot B) \mod n &= (det(A)\cdot \det(B)) \mod n \\
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&= ((\det(A) \mod n)\cdot(\det(B) \mod n)) \mod n\\
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&= (\det(A) \mod n)\cdot_{\mathbb{Z}_n}(\det(B) \mod n)
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\end{aligned}$$
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\subsubsection{Inverse Matrix}
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Die Inverse einer quadratischen Matrix $A$ über $\mathbb{Z}_n$ lässt sich mithilfe der Adjunkten berechnen:
|
||||
$$A^{-1} = (\det(A))^{-1} \cdot adj(A)$$
|
||||
Die Adjunkte lässt sich über $\mathbb{Z}_n$ berechnen, da lediglich Summen und Differenzen von Produkten berechnet werden müssen.
|
||||
|
||||
\section{Der erweiterte Euklid'sche Algorithmus}
|
||||
Der Euklid'sche Algorithmus ist ein sehr effizienter Weg den ggT zweier Zahlen zu ermitteln.
|
||||
Der Euklid'sche Algorithmus lässt sich auch über $\mathbb{Z}_n$ verwenden.
|
||||
Man spricht dann von dem erweiterten Euklid'schen Algorithmus.
|
||||
|
||||
\subsection{Euklid'scher Algorithmus}
|
||||
gegeben: $a_0,b_0\in\mathbb{Z}$
|
||||
\begin{enumerate}
|
||||
\item $a:=a_0$ und $b:=b_0$
|
||||
\item falls $b=0$ gebe $|a|$ aus und beende
|
||||
\item $r:=a \mod b$
|
||||
\item $a:=b$
|
||||
\item $b:=r$
|
||||
\item goto 2.
|
||||
\end{enumerate}
|
||||
|
||||
\subsection{erweiterter Euklid'scher Algorithmus}
|
||||
gegeben: $a_0,b_0\in\mathbb{N}_0$\\
|
||||
gesucht: $\alpha\cdot a_0+\beta \cdot b_0 = g = ggT(a_0,b_0)$
|
||||
\begin{enumerate}
|
||||
\item $a:=a_0$, $\alpha_a=1$, $\beta_b=0$, $b := b_0$, $\alpha_b := 0$, $\beta_b:=1$
|
||||
\item falls $b=0$ gebe $g:=a$, $\alpha:=\alpha_a$ und $\beta:=\beta_a$ aus
|
||||
\item $q:=a/_\mathbb{Z} b$
|
||||
\item $r:=a-q\cdot b$,$\alpha_r := \alpha_a-q\cdot \alpha_b$, $\beta_r := \beta_a-q\cdot \beta_b$
|
||||
\item $a:=b$, $\alpha_a:=\alpha_b$, $\beta_a := \beta_b$
|
||||
\item $b:=r$, $\alpha_b :=\alpha_r$ $\beta_b := \beta_r$
|
||||
\item goto 2.
|
||||
\end{enumerate}
|
||||
|
||||
\subsubsection{Beispiel}
|
||||
Eingabe: $$a_0 = 1224\text{ und } b_0 = 156 $$
|
||||
|
||||
Berechnung:\\
|
||||
\begin{center}
|
||||
\begin{tabular}{c|c|c|c}
|
||||
1224 & 156 & a,b & q\\
|
||||
\hline
|
||||
1 & 0 & 1224 \\
|
||||
0 & 1 & 156 & 7\\
|
||||
1 & -7 & 132 & 1\\
|
||||
-1 & 8 & 24 & 5\\
|
||||
6 & -47 & 12 & 2 \\
|
||||
& & 0\\
|
||||
\end{tabular}\\
|
||||
\end{center}
|
||||
|
||||
Ergebnis: $$6\cdot 1224 + (-47)\cdot 156 = 12$$
|
||||
|
||||
\section{Euler'sche $\varphi$-Funktion}
|
||||
Die Euler'sche $\varphi$-Funktion bezeichnet die Anzahl invertierbarer Elemente in $\mathbb{Z}_n$
|
||||
$$\varphi(n):=\begin{cases}
|
||||
|\mathbb{Z}_n^*| &\text{für } n\in\mathbb{N},n\ge 2\\
|
||||
1 &\text{für } n = 1
|
||||
\end{cases}$$
|
||||
|
||||
Für $a,b\in\mathbb{N}$ mit $ggT(a,b)=1$ gilt:
|
||||
$$\varphi(a\cdot b) = \varphi(a) \cdot \varphi(b)$$
|
||||
|
||||
Zudem gilt für ein $n\in\mathbb{N}$ dessen Primzahlzerlegung $n = p_1^{e_1} \cdots p_r^{e_r}$:
|
||||
$$\varphi(n) = n\cdot \prod_{i=1}^r \big( 1-\frac{a}{p_i} \big)$$
|
||||
|
||||
\subsection{$\varphi$-Funktion und Primzahlen}
|
||||
für eine Primzahl $p$ gilt:
|
||||
$$\varphi(p)=p-1$$
|
||||
Für Primzahlpotenzen gilt zudem:
|
||||
$$\varphi(p^e)=p^{e-1}(p-1)$$
|
||||
|
||||
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Reference in New Issue
Block a user