From a690196fe5dc7450430b8c86c55c80834f09bee9 Mon Sep 17 00:00:00 2001 From: Paul Date: Sat, 19 Sep 2020 17:44:22 +0200 Subject: [PATCH] provisional version of Modulare Arithmetik - Teil 2 finished --- Inhalt.tex | 19 + Packages.tex | 35 + Zusammenfassung_DC.aux | 305 +++++---- Zusammenfassung_DC.dvi | Bin 0 -> 36440 bytes Zusammenfassung_DC.fdb_latexmk | 118 +++- Zusammenfassung_DC.fls | 178 ++++- Zusammenfassung_DC.log | 692 +++++++++++++++++--- Zusammenfassung_DC.out | 105 +++ Zusammenfassung_DC.pdf | Bin 2637795 -> 2678272 bytes Zusammenfassung_DC.synctex.gz | Bin 227817 -> 262329 bytes Zusammenfassung_DC.tex | 47 +- Zusammenfassung_DC.toc | 236 +++---- chapters/Blockverschlüsselungsverfahren.tex | 2 +- chapters/Endliche Körper.tex | 2 +- chapters/Modulare Arithmetik - Teil 2.tex | 128 ++++ chapters/Modulare Arithmetik.tex | 17 + 16 files changed, 1506 insertions(+), 378 deletions(-) create mode 100644 Inhalt.tex create mode 100644 Packages.tex create mode 100644 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z39p+nb1qiZ>!Lqu5wXXWN)^?%xEdI9f%Y8GS+u7d9KOhTn?kqGyw)B&d&vw&i!19n2PrJ7VYbu&3c)4VW>i(WKyP9NYzM_tRxviKQMs6JJC9YAQ z|AB9|ig&{GW|IsrK!JC<7IS9QxIE}o?C0KU>DzTywcmnlIGnUt09`jWRiJ7GlRR&I zC1Ugj+bn|rj2fa2KTARXmOKY4C?ONCuq0U7 z+i6olwJ(uA3pv@FI_6~D&K{sC6`!TXz3uNL%zpifZwCz%psE(G4}^o`r{2)<0B!!*TgiYa#ZSHh*AJd{ON z>0vyp?|!dm1T1hP*gp*@@NB7q`O|+O4@eK$4!S;U>hhY%=>s(ItYFHu=ecnk z;yiTV8`?nGR&*6#%|tWIjQmXJMhjWJ`{jDCr!k`c+T@Ga&DT#|W5BRYvH-BKsZlwI zGk1sj_~V{tMf(SwaIWX$?OBm1&-B~x-=B^>A0MCF0dEhjz;h^9QsZ;xac)p3r$Ph`M9cLxmp|D10N`oA5%PW=Iy NCtA0$ mithilfe der Polynome aus $\mathbb{F}_p$ vom Grad $:=\{a^i\mid i\in\mathbb{Z}\}$$ + Die Anzahl der verschiedenen Potenzen von $a$ wird als Ordnung von $a$ in $\mathbb{Z}_n^*$ bezeichnet: + $$o_n(a):=||$$ + Hierbei ergibt sich $a^{o(a)}=1$ und $=\{1,a,a^2,a^3,...,a^{o(a)-1}\}$.\\ + \begin{mybox} + für $n\in\mathbb{N}(n\ge 2);a,b\in \mathbb{Z}_n^*$ und $i\in\mathbb{Z}$ gilt: + \begin{enumerate}[(i)] + \item $a^i=1 \Longrightarrow o(a)|i$ + \item $o(a^i)|o(a)$ + \item $i|o(a) \Longrightarrow o(a^i)=\frac{o(a)}{|i|}$ + \item $\text{ggT}(o(a),i)=1 \Longrightarrow o(a^i)=o(a)$ + \item $o(ab)|o(a)\cdot o(b)$ + \item $\text{ggT}(o(a),o(b))=1\Longrightarrow o(a\cdot b)=o(a)\cdot o(b)$ + \end{enumerate} + \end{mybox} + \begin{mybox} + Wenn $\mu_n$ für $n\in\mathbb{N},n\ge2$ die maximale Ordnung eines Elementes aus $\mathbb{Z}_n^*$ ($\mu_n:=\max\{o(a)\mid a\in\mathbb{Z}_n^*\}$), + ist jede Ordnung von $a$ ein Teiler von $\mu_n$ + \end{mybox} + + \subsection{Faktorenzerlegung} + Falls es für ein Polynom $p(x) \in \mathbb{Z}_n[x]$ ($n\in\mathbb{N}$) vom Grad $k$ eine Nullstelle $a\in\mathbb{Z}_n$ gibt, + muss es ein Polynom $q(x) \in \mathbb{Z}_n[x]$ vom Grad $k-1$ geben, für das gilt: + $$p(x)=(x-a)\cdot q(x)$$ + + \subsection{Kleiner Satz von Fermat} + Für eine Primzahl $p$ gilt für alle $a\in\mathbb{Z}$ mit $\text{ggT}(a,p)=1$: + $$a^{p-1}\equiv 1 \hspace{10mm}(\mod p)$$ + + \section{Exkurs: Einheitengruppe $\mathbb{Z}_{p^e}^*$} + Für eine Primzahl $p$ gibt es ein Element $g\in\mathbb{Z}_p^*$ mit $o(g)=p-1$ ($\mathbb{Z}_p^*==\{1,g,g^2,...,g^{p-2}\}$). + Diese Elemente $g$ werden als \textbf{primitive Elemente} bezeichnet.\\ + Mithilfe der Potenzen der primitiven Elemente lassen sich alle Elemente von $\mathbb{Z}_p^*$ generieren: + $$=\mathbb{Z}_p^*$$ + Diese Elemente werden daher auch als \textbf{Generatoren} von $\mathbb{Z}_p^*$ bezeichnet. + Auch für Einheitengruppen $\mathbb{Z}_{p^e}$ für Primzahlpotenzen mit $e>1$ gibt es solche zyklische Generatoren: + \begin{mybox} + für eine Primzahl $p\ne2$ und $e\in\mathbb{N}$ gibt es ein $g\in\mathbb{Z}_{p^e}^*$ für das gilt: + $$\begin{aligned} + o(g)=p^{e-1}(p-1) &= \varphi(p^e)=|\mathbb{Z}_{p^e}^*\\ + \Leftrightarrow\hspace{10mm}&=\mathbb{Z}_{p^e}^* + \end{aligned}$$ + \end{mybox} + Hierbei gilt: + \begin{mybox} + Für ein primitives Element $g\in\mathbb{Z}_p^*$ gilt mit $a:=g^{\frac{o_{p^e}(g)}{p-1}}$: + $$\begin{aligned} + \mathbb{Z}_{p^e}^*&=\cdot <1+p>\\ + :&=\{a^i\cdot (1+p)^j\mid i=0,1,...,p-2; j=0,1,...p^{e-1}-1\} + \end{aligned}$$ + \end{mybox} + + \subsection{$\mathbb{Z}_{2^e}^*$} + Die Primzahl 2 stellt eine besonderheit dar, da $\mathbb{Z}_{2^e}^*$ für $e\ge3$ nicht zyklisch ist. + So gilt $o(5)=2^{e-2}$ und allgemein: + $$\begin{aligned} + \mathbb{Z}_{2^e}^*&=<2^e-1>\cdot <5>\\ + &= \{(2^e-1)^i\cdot 5^j\mid i=0,1;j=0,1,...,2^{e-2}-1\} + \end{aligned}$$ + + \section{Der chinesische Restsatz} + Für eine Menge teilerfremder Zahlen $\{(n_1,...,n_r)\mid n_i\in\mathbb{N}\}$ bildet einen Menge $\{(a_1,...,a_r)\mid a_i\in\mathbb{Z}\}$ ein System aus Kongruenzen: + $$\begin{aligned} + x\equiv&a_1\hspace{10mm} &(\textbf{mod }n_1)\\ + x\equiv&a_2 &(\textbf{mod } n_2)\\ + \vdots\\ + x\equiv&a_r &(\textbf{mod } n_r)\\ + \end{aligned}$$ + Die Menge der ganzzahligen Lösungen ist gegeben durch: + $$\mathbb{L}=\{a+v\cdot n_1n_2\cdots n_r\mid v\in\mathbb{Z}\}$$ + Hierbei ist $0\le a < n_1n_2\cdots n_r$ eindeutig. + $a$ wird im folgenden Beispiel mithilfe des erweiterten Euklid'schen Algorithmus (siehe \ref{erweiterter Euklid}) berechnet. + Aus dem Restsatz leitet sich folgender Satz ab: + \begin{mybox} + Sind $n_1,n_2,...,n_r$ paarweise teilerfremde Zahlen gibt es die folgende Isomorphie (\ref{Isomorphie}): + $$\mathbb{Z}_{n_1n_2\cdots n_r}\cong \mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times\cdots\times\mathbb{Z}_{n_r}$$ + \end{mybox} + + \subsection{Beispiel} + folgende Kongruenzen sind gegeben: + $$\begin{aligned} + x\equiv&2\hspace{10mm}&(\textbf{mod }5)\\ + x\equiv&7 &(\textbf{mod }2)\\ + x\equiv&2 &(\textbf{mod }5)\\ + \end{aligned}$$ + \begin{itemize} + \item \textbf{Bestimmung von} $e_1$\\ + für $n_1=5$ und $m_1=21\cdot 11=231$ liefert der erweiterte Euklid'sche Algorithmus (\ref{erweiterter Euklid}): + $$(-46)\cdot 5+1\cdot 231 = 1$$ + Daher ist $e_1 = 1\cdot 231$ + \item \textbf{Bestimmung von} $e_2$\\ + für $n_2=21$ und $m_2=5\cdot 11=55$ liefert der erweiterte Euklid'sche Algorithmus (\ref{erweiterter Euklid}): + $$(-21)\cdot 21+(-8)\cdot 55 = 1$$ + Daher ist $e_2 = (-8)\cdot 55=-440$ + \item \textbf{Bestimmung von} $e_3$\\ + für $n_3=11$ und $m_3=5\cdot 21=105$ liefert der erweiterte Euklid'sche Algorithmus (\ref{erweiterter Euklid}): + $$(-19)\cdot 11+2\cdot 105 = 1$$ + Daher ist $e_3 = 2\cdot 105=210$ + \item \textbf{Berechnung der Lösung}\\ + $n=5\cdot 21\cdot 11=1155$: + $$\begin{aligned} + a=&(2\cdot e_1+7\cdot e_2+6\cdot e_3)\mod 1155\\ + =&(2\cdot 231-7\cdot 440+6\cdot 210)\mod 1155\\ + =&-1358\mod 1155\\ + =952 + \end{aligned}$$ + \end{itemize} + + \section{Elemente gerader und ungerader Ordnung in $\mathbb{Z}_n$**} + siehe Skript 3 Kapitel 1.4 auf Seite 17(23). \ No newline at end of file diff --git a/chapters/Modulare Arithmetik.tex b/chapters/Modulare Arithmetik.tex index 3d5e8e2..62ecb1b 100644 --- a/chapters/Modulare Arithmetik.tex +++ b/chapters/Modulare Arithmetik.tex @@ -72,6 +72,23 @@ Die Inverse einer quadratischen Matrix $A$ über $\mathbb{Z}_n$ lässt sich mithilfe der Adjunkten berechnen: $$A^{-1} = (\det(A))^{-1} \cdot adj(A)$$ Die Adjunkte lässt sich über $\mathbb{Z}_n$ berechnen, da lediglich Summen und Differenzen von Produkten berechnet werden müssen. + + \subsection{Kartesisches Produkt von Ringen} + Sind $R_1,R_2,...,R_r$ Ringe, so bildet das kartesische Produkt ($R_1\times R_2\times\cdots\times R_r$)ebenfalls einen Ring mit den folgenden Verknüpfungen: + $$\begin{aligned} + (a_1,...,a_r)+(b_1,...,b_r)&=(a_1+b_1,...,a_r+b_r)\\ + (a_1,...,a_r)\cdot(b_1,...,b_r)&=(a_1\cdot b_1,...,a_r\cdot b_r)\\ + \end{aligned}$$ + Auch das kartesische Produkt der Einheitengruppen bildet einen Ring: + $$R^*=R_1^*\times\cdots\times R_r^*$$ + + \subsection{Isomorphie von Ringen}\label{Isomorphie} + Eine bijektive Abbildung $f:R_1\rightarrow R_2$ wird Isomorphismus genannt, falls für alle $a,b\in R_1$ gilt: + $$\begin{aligned} + f(a+b)&=f(a)+f(b)\\ + f(a\cdot b)&=f(a)\cdot f(b)\\ + \end{aligned}$$ + Zwei isomorphe Ringe werden als $R_1\cong R_2$ geschrieben. \section{Der erweiterte Euklid'sche Algorithmus} Der Euklid'sche Algorithmus ist ein sehr effizienter Weg den ggT zweier Zahlen zu ermitteln.

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