From 41f76bf2418f29b0889ea39c7c1650be89a259a9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Paul=20L=C3=B6dige?= Date: Wed, 16 Sep 2020 21:57:59 +0200 Subject: [PATCH] finished F_2 until the examples. continue Skript 2 Page 49(55) --- Zusammenfassung_DC.aux | 7 +++++ Zusammenfassung_DC.fdb_latexmk | 21 +++++++------ Zusammenfassung_DC.fls | 4 +++ Zusammenfassung_DC.log | 26 ++++++++-------- Zusammenfassung_DC.pdf | Bin 2599251 -> 2603298 bytes Zusammenfassung_DC.synctex.gz | Bin 181403 -> 193618 bytes Zusammenfassung_DC.toc | 4 +++ chapters/Endliche Körper.tex | 51 ++++++++++++++++++++++++++++++- chapters/Modulare Arithmetik.tex | 6 ++-- 9 files changed, 92 insertions(+), 27 deletions(-) diff --git a/Zusammenfassung_DC.aux b/Zusammenfassung_DC.aux index 7b26755..02fe797 100644 --- a/Zusammenfassung_DC.aux +++ b/Zusammenfassung_DC.aux @@ -27,6 +27,7 @@ \@writefile{lot}{\addvspace {10\p@ }} \@writefile{toc}{\contentsline {section}{\numberline {2.1}Exkurs: Division mit Rest}{6}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {2.2}Der Ring $\mathbb {Z}_n$}{6}\protected@file@percent } +\newlabel{Der Ring Z}{{2.2}{6}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2.1}Addition und Multiplikation}{6}\protected@file@percent } \newlabel{modulare_addition}{{2.2.1}{6}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {2.2.1.1}Inverse bezüglich der Addition}{6}\protected@file@percent } @@ -40,7 +41,9 @@ \@writefile{toc}{\contentsline {subsubsection}{\numberline {2.2.5.2}Inverse Matrix}{7}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {2.3}Der erweiterte Euklid'sche Algorithmus}{7}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {2.3.1}Euklid'scher Algorithmus}{8}\protected@file@percent } +\newlabel{euklid}{{2.3.1}{8}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.3.2}erweiterter Euklid'scher Algorithmus}{8}\protected@file@percent } +\newlabel{erweiterter Euklid}{{2.3.2}{8}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {2.3.2.1}Beispiel}{8}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\numberline {2.4}Euler'sche $\varphi $-Funktion}{9}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {2.4.1}$\varphi $-Funktion und Primzahlen}{9}\protected@file@percent } @@ -127,3 +130,7 @@ \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} \newlabel{endliche Körper}{{7}{31}} +\@writefile{toc}{\contentsline {section}{\numberline {7.1}Polynomdivision mit Rest}{31}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {7.2}Berechnung der Inversen}{31}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {7.3}$\mathbb {F}_2$}{31}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {7.3.1}Irreduzible Polynome über $\mathbb {F}_2$}{31}\protected@file@percent } diff --git a/Zusammenfassung_DC.fdb_latexmk b/Zusammenfassung_DC.fdb_latexmk index be35db5..b675c63 100644 --- a/Zusammenfassung_DC.fdb_latexmk +++ b/Zusammenfassung_DC.fdb_latexmk @@ -1,7 +1,7 @@ # Fdb version 3 -["pdflatex"] 1600272631 "/home/paul/Documents/TH/Datensicherheit (DC)/Zusammenfassung/Zusammenfassung_DC.tex" "/home/paul/Documents/TH/Datensicherheit (DC)/Zusammenfassung/Zusammenfassung_DC.pdf" "Zusammenfassung_DC" 1600272632 +["pdflatex"] 1600286191 "/home/paul/Documents/TH/Datensicherheit (DC)/Zusammenfassung/Zusammenfassung_DC.tex" "/home/paul/Documents/TH/Datensicherheit (DC)/Zusammenfassung/Zusammenfassung_DC.pdf" "Zusammenfassung_DC" 1600286193 "./images/Skytale.png" 1599914207 1414106 7498f70ad27e17e63eb468c2321baa5c "" - 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{7.1}Polynomdivision mit Rest}{31}% +\contentsline {section}{\numberline {7.2}Berechnung der Inversen}{31}% +\contentsline {section}{\numberline {7.3}$\mathbb {F}_2$}{31}% +\contentsline {subsection}{\numberline {7.3.1}Irreduzible Polynome über $\mathbb {F}_2$}{31}% diff --git a/chapters/Endliche Körper.tex b/chapters/Endliche Körper.tex index 787ae9e..ccf5111 100644 --- a/chapters/Endliche Körper.tex +++ b/chapters/Endliche Körper.tex @@ -1 +1,50 @@ -\chapter{Endliche Körper $\mathbb{F}_{p^n}$} \label{endliche Körper} \ No newline at end of file +\chapter{Endliche Körper $\mathbb{F}_{p^n}$} \label{endliche Körper} + $\mathbb{F}_p:=\mathbb{Z}_p$ ($p$ ist eine Primzahl) ist ein Körper. + Zudem können Körper auch für beliebige Primzahlpotenzen $p^n$ mit $n\in\mathbb{N},n>0$ mithilfe der Polynome aus $\mathbb{F}_p$ vom Grad $n$. + Hierbei lässt sich allerdings analog zu $\mathbb{Z}$ (siehe \ref{Der Ring Z}) durch eine Polynomdivision mit Rest durch $p(x)$ ein Wert in $\mathbb{F}_{p^n}$ erzeugen. + + \section{Polynomdivision mit Rest} + Ist $\mathbb{F}$ ein Körper und $a(x),b(x)\in\mathbb{F}[x],b(x)\ne0$, so gibt es eindeutig bestimmte Polynome $q(x),r(x)\in\mathbb{F}$ mit: + $$a(x)=b(x)\cdot q(x)+r(x)\text{ und } 0\le\text{grad}(r(x))<\text{grad}(b(x))$$ + Alternativ wird für den Rest $r(x)$ auch \say{$a(x) \mod b(x)=r(x)$} geschrieben. + + \section{Berechnung der Inversen} + Der größte gemeinsame Teiler (das Polynom mit den höchsten Grad, dass $a(x)$ und $b(x)$ teilt) kann mithilfe des Euklid'schen Algorithmus (siehe \ref{euklid}) berechnet werden. + Ist $\mathbb{F}_{p^n}$ bezüglich eines irreduziblen Polynoms $p(x)$ definiert, + lässt sich mithilfe der Anwendung des erweiterten Euklid'schen Algorithmus (siehe \ref{erweiterter Euklid}) auf die Polynome $a(x)$ und $b(x)$ eine Inverse von $a(x)$ in $\mathbb{F}_{p^n}$ bestimmen. + + \section{$\mathbb{F}_2$} + + \subsection{Irreduzible Polynome über $\mathbb{F}_2$} + \begin{itemize} + \item Polynome von Grad 1 (lineare Polynome) sind immer irreduziblen. + Für $\mathbb{F}_2$ sind dies: + $$\begin{aligned} + p_{1,1}(x)=&x\\ + p_{1,2}(x)=&x+1 + \end{aligned}$$ + + \item Es gibt in $\mathbb{F}_2$ 4 paarweise verschiedene Polynome von Grad 2. + 3 davon haben eine Nullstelle und lassen sích daher in Linearfaktoren zerlegen. + Das einzige irreduzible Polynom von Grad 2 ist: + $$p_2(x)=x^2+x+1$$ + \item Ein Polynom von Grad 3 ist genau dann irreduzible, falls $p(x)$ keine Nullstellen hat. + Dies trifft für die folgenden zwei Polynome zu: + $$\begin{aligned} + p_{3,1}(x)=&x^3+x+1\\ + p_{3,2}(x)=&x^3+x^2+1\\ + \end{aligned}$$ + \item Ein Polynom von Grad 4 ist genau dann irreduzible, wenn $p(x)$ keine Nullstelle hat und auch nicht das Quadrat von $p_2(x)$ ($x^4+x^2+1$) ist: + $$\begin{aligned} + p_{4,1}(x)=&x^4+x+1\\ + p_{4,1}(x)=&x^4+x^3+1\\ + p_{4,1}(x)=&x^4+x^3+x^2+x+1\\ + \end{aligned}$$ + \end{itemize} \ No newline at end of file diff --git a/chapters/Modulare Arithmetik.tex b/chapters/Modulare Arithmetik.tex index 6f2d0bc..3d5e8e2 100644 --- a/chapters/Modulare Arithmetik.tex +++ b/chapters/Modulare Arithmetik.tex @@ -7,7 +7,7 @@ a\mod b :&= r \\ \end{aligned}$$ - \section{Der Ring $\mathbb{Z}_n$} + \section{Der Ring $\mathbb{Z}_n$}\label{Der Ring Z} Ein Ring $\mathbb{Z}_n$ ist definiert durch: $$\mathbb{Z}_n := {0,1,...,n-1}$$ \subsection{Addition und Multiplikation}\label{modulare_addition} @@ -78,7 +78,7 @@ 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} + \subsection{Euklid'scher Algorithmus}\label{euklid} gegeben: $a_0,b_0\in\mathbb{Z}$ \begin{enumerate} \item $a:=a_0$ und $b:=b_0$ @@ -89,7 +89,7 @@ \item goto 2. \end{enumerate} - \subsection{erweiterter Euklid'scher Algorithmus} + \subsection{erweiterter Euklid'scher Algorithmus}\label{erweiterter Euklid} 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}