From e2c3343168894f3459cfccb9af79bc9a1032808e Mon Sep 17 00:00:00 2001 From: Paul Date: Sat, 12 Sep 2020 21:10:04 +0200 Subject: [PATCH] first part of modulare arithmetik finished --- Zusammenfassung_DC.aux | 11 ++++++++ Zusammenfassung_DC.fdb_latexmk | 17 +++++++---- Zusammenfassung_DC.fls | 19 +++++++++++++ Zusammenfassung_DC.log | 32 ++++++++++++--------- Zusammenfassung_DC.pdf | Bin 2170741 -> 2206985 bytes Zusammenfassung_DC.synctex.gz | Bin 21606 -> 31064 bytes Zusammenfassung_DC.tex | 2 ++ Zusammenfassung_DC.toc | 9 ++++++ chapters/Modulare Arithmetik.tex | 47 +++++++++++++++++++++++++++++++ 9 files changed, 117 insertions(+), 20 deletions(-) create mode 100644 chapters/Modulare Arithmetik.tex diff --git a/Zusammenfassung_DC.aux b/Zusammenfassung_DC.aux index d3af4a6..981e03b 100644 --- a/Zusammenfassung_DC.aux +++ b/Zusammenfassung_DC.aux @@ -22,3 +22,14 @@ \@writefile{toc}{\contentsline {subsection}{\numberline {1.4.1}Hill-Verfahren}{4}\protected@file@percent } \@writefile{toc}{\contentsline {subsubsection}{\numberline {1.4.1.1}Verschlüsselung}{4}\protected@file@percent } \@writefile{toc}{\contentsline {subsubsection}{\numberline {1.4.1.2}Entschlüsselung}{4}\protected@file@percent } +\@writefile{toc}{\contentsline {chapter}{\numberline {2}Modulare Arithmetik}{5}\protected@file@percent } +\@writefile{lof}{\addvspace {10\p@ }} +\@writefile{lot}{\addvspace {10\p@ }} +\@writefile{toc}{\contentsline {section}{\numberline {2.1}Exkurs: Division mit Rest}{5}\protected@file@percent } +\@writefile{toc}{\contentsline {section}{\numberline {2.2}Der Ring $\mathbb {Z}_n$}{5}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.1}Addition und Multiplikation}{5}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.2}Inverse bezüglich der Addition}{5}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.3}Subtraktion}{5}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.4}Teiler, Vielfache}{5}\protected@file@percent } +\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.2.4.1}Teilerregeln}{5}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2.5}Kongruenz}{6}\protected@file@percent } diff --git a/Zusammenfassung_DC.fdb_latexmk b/Zusammenfassung_DC.fdb_latexmk index c7d3d9c..adb8e31 100644 --- a/Zusammenfassung_DC.fdb_latexmk +++ b/Zusammenfassung_DC.fdb_latexmk @@ -1,8 +1,9 @@ # Fdb version 3 -["pdflatex"] 1599932297 "/home/paul/Documents/TH/Datensicherheit (DC)/Zusammenfassung/Zusammenfassung_DC.tex" "/home/paul/Documents/TH/Datensicherheit (DC)/Zusammenfassung/Zusammenfassung_DC.pdf" "Zusammenfassung_DC" 1599932298 - 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zNG!j-eSZG>>0$o;KY#i2pWc4`hMeI3=Kucu{POnWufN~l-Te9U*H16_25U)!+$eu| Xe)sg@%gfh~cQ^kVKwSaB4~+o;rM=v) diff --git a/Zusammenfassung_DC.tex b/Zusammenfassung_DC.tex index 3fb949b..eff2a8a 100644 --- a/Zusammenfassung_DC.tex +++ b/Zusammenfassung_DC.tex @@ -36,4 +36,6 @@ \input{chapters/Substitutionsverfahren.tex} +\input{chapters/Modulare Arithmetik.tex} + \end{document} diff --git a/Zusammenfassung_DC.toc b/Zusammenfassung_DC.toc index 137b5c5..d3f4784 100644 --- a/Zusammenfassung_DC.toc +++ b/Zusammenfassung_DC.toc @@ -14,3 +14,12 @@ \contentsline {subsection}{\numberline {1.4.1}Hill-Verfahren}{4}% \contentsline {subsubsection}{\numberline {1.4.1.1}Verschlüsselung}{4}% \contentsline {subsubsection}{\numberline {1.4.1.2}Entschlüsselung}{4}% +\contentsline {chapter}{\numberline {2}Modulare Arithmetik}{5}% +\contentsline {section}{\numberline {2.1}Exkurs: Division mit Rest}{5}% +\contentsline {section}{\numberline {2.2}Der Ring $\mathbb {Z}_n$}{5}% +\contentsline {subsection}{\numberline {2.2.1}Addition und Multiplikation}{5}% +\contentsline {subsection}{\numberline {2.2.2}Inverse bezüglich der Addition}{5}% +\contentsline {subsection}{\numberline {2.2.3}Subtraktion}{5}% +\contentsline {subsection}{\numberline {2.2.4}Teiler, Vielfache}{5}% +\contentsline {subsubsection}{\numberline {2.2.4.1}Teilerregeln}{5}% +\contentsline {subsection}{\numberline {2.2.5}Kongruenz}{6}% diff --git a/chapters/Modulare Arithmetik.tex b/chapters/Modulare Arithmetik.tex new file mode 100644 index 0000000..300a417 --- /dev/null +++ b/chapters/Modulare Arithmetik.tex @@ -0,0 +1,47 @@ +\chapter{Modulare Arithmetik} + \section{Exkurs: Division mit Rest} + Für $a,b\in \mathbb{Z},b\ne 0$ gibt es eindeutig bestimmte Element $q,r\in\mathbb{Z},0\le r<|b|$: + $$\begin{aligned} + a =b\cdot q+r\\ + a /_\mathbb{Z} b :&= q \\ + a\mod b :&= r \\ + \end{aligned}$$ + + \section{Der Ring $\mathbb{Z}_n$} + Ein Ring $\mathbb{Z}_n$ ist definiert durch: $$\mathbb{Z}_n := {0,1,...,n-1}$$ + + \subsection{Addition und Multiplikation} + \begin{equation} + \begin{aligned} + a +_{\mathbb{Z}_n} b :&= (a+b) \mod n\\ + a \cdot_{\mathbb{Z}_n} b :&= (a\cdot b) \mod n\\ + \end{aligned} + \end{equation} + + \subsection{Inverse bezüglich der Addition} + jedes $a\in \mathbb{Z}$ hat ein Inverses: + $$ -a := + \begin{cases} + 0 &\text{für }a=0 \\ + n-a &\text{sonst} + \end{cases}$$ + + \subsection{Subtraktion} + Eine Subtraktion entspricht einer Addition mit der Inverse: + $$a-_{\mathbb{Z}_n}b := a+_{\mathbb{Z}_n}(-b) \mod n$$ + + \subsection{Teiler, Vielfache} + $b\in \mathbb{Z}$ teilt $a\in \mathbb{Z}$ falls ein $q\in \mathbb{Z}$ existiert mit: + $$ a = b\cdot q$$ + man schreibt auch $b|a$ + + \subsubsection{Teilerregeln} + \begin{enumerate} + \item $a|0$ $\forall a\in \mathbb{Z}$ + \item $a|b \Leftrightarrow a|(-b)$ + \item $a|b \text{ und } a|c \Rightarrow a|(b+c)$ + \end{enumerate} + + \subsection{Kongruenz} + $a,b\in \mathbb{Z}$ sind \textit{kongruent modulo n}, falls $n\in \mathbb{N}|(a-b)$. + Man schreibt auch $a\equiv b$ \ No newline at end of file