From f061af49ba12ca0d7f41b7f083e101a6ea1a4d4e Mon Sep 17 00:00:00 2001 From: Peter Schaefer Date: Tue, 3 Jul 2012 12:10:45 +0200 Subject: [PATCH] Inhaltsstrucktur eingebettet --- UE/template.sty | 2 ++ Vorlesung.pdf | Bin 199608 -> 203245 bytes Vorlesung.tex | 50 ++++++++++++++++++++++++++++++++++++------------ 3 files changed, 40 insertions(+), 12 deletions(-) diff --git a/UE/template.sty b/UE/template.sty index 0836796..e44cad4 100644 --- a/UE/template.sty +++ b/UE/template.sty @@ -107,6 +107,8 @@ \noindent } +\renewcommand{\thechapter}{\Roman{chapter}} + %% Imported from python.def diff --git a/Vorlesung.pdf b/Vorlesung.pdf index 59f816b305d964fe12896bfe220e9cab19e2d4d2..7314b5e4592a9d4fd49b1a396b976b120aa0b707 100644 GIT binary patch delta 108082 zcmaIdQ*fqH`!4v{HaoV_v2Av2y|K+FPCD+`w$X7qwrx8dCmn0Pz30DczL|qLyK2>{ zb$r$DzE@is(q$7;y#hEEdz)?eeYImcwBq_ji%4J_?(ee@qIf&-#vi9@ke{zdNF%kE4jqX9WQkmk zZxRHsX1MTYx_kDtcAh4(HefYCq#5Yb7886;lF9~cxUzdtcCg8v`@z^fdFVQ&FFq-U 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z&fb%h5P63>WoUYtCSPcr6m8t=H4MiCS?8b8`-)f02H*#ZT`%KF$s1NG_#LIr@QowN zAqg?#s~_H6D?bgWEoq?N6v^O>V6X;eOcSniKKXFCHx+!CdU|=;pVHw*|0ML;#rKzZ zha8ySJzhV1)TkKFyz7aaot6Bliqv-21jB&v_;+%8y#9Zc=j-F+@#XdT)%8#K`e$Sm Pd(-jg!GqH;&c=TMwonzN diff --git a/Vorlesung.tex b/Vorlesung.tex index 05ab12f..b2c3529 100644 --- a/Vorlesung.tex +++ b/Vorlesung.tex @@ -1,11 +1,13 @@ \documentclass[a4paper,10pt,fleqn]{scrreprt} \usepackage{UE/template} + \author{} \title{Mitschrift zur Vorlesung \\ AKDIS: Zahlentheorie und Anwendungen,\\ gehalten von Prof. J. Wiesenbauer, Sommersemester 2012} + \pagestyle{plain} -\frenchspacing - \makeindex -\begin{document} +\makeindex + +\begin{document} \begin{titlepage} \maketitle @@ -13,20 +15,26 @@ \setcounter{secnumdepth}{-1} - \section{Kurzzusammenfassung} -Das ist die "ge {\TeX} te" Mitschrift zur Vorlesung $118 \dot 186$ \emph{AKDIS Zahlentheorie und Anwendungen} vom Sommersemester 2012, gehalten von Prof. J. Wiesenbauer. +Das ist die "ge {\TeX} te" Mitschrift zur Vorlesung $118 \dot 186$ \emph{AKDIS Zahlentheorie und Anwendungen} vom Sommersemester 2012, gehalten von Prof. J. Wiesenbauer.\bigskip + +\noindent +Für weitere Informationen siehe HomePage \emph{www.algebra.ac.at} + +\subsection{Übung} +Für die Übung sind $60\%$ der Beispiele, sowie eine Tafelleistung Pflicht. +Lösungen zu einigen Beispielen befinden sich im Anhang (Seite \pageref{UE}) + +\subsection{Prüfung} +In der Prüfung werden zwei Beweise sowie zwei Übersichtsfragen gestellt. Es gibt eine schriftliche und mündliche Prüfung. \newpage \tableofcontents \newpage \setcounter{secnumdepth}{2} -\section{Einleitung} -\chapter{Teil 1} -\begin{bem} -Prf: $4$ Bsp: 2 Beweise, 2 Überblicksfragen + schreiben -\end{bem} +\chapter{Zahlentheorethische Grundlagen} +\section{Teilbarkeit im Ring der ganzen Zahlen} \begin{bem} Die natürlichen Zahlen sind in dieser Vorlesungen inklusive $0$ zu verstehen, d.h. $\N = \lbrace 0,1,2,\ldots \rbrace$. @@ -368,8 +376,8 @@ Weiters gilt nach R. Koch: \end{equation} Es existiert eine Abschätzung für die maximale Anzahl von Nullstellen, man zeigt, dass die Anzahl der Nullstellen auf der Mittelgeraden mit der maximalen Abschätzung übereinstimmt $\implies$ es gibt keine weiteren. -\begin{bem}Mögliche Prüfungsfrage: Wozu braucht man Riemann'sche Vermutung? Für Primzahltests und Primzahlverteilung. -\end{bem} +\begin{bem}Mögliche Prüfungsfrage: Wozu braucht man Riemann'sche Vermutung? Für Primzahltests und Primzahlverteilung.\end{bem} +\section{Kongruenzen auf $\Z$} \subsection*{Beweis 2.2} $a \equiv b \mod m \land c \equiv d \mod m \Rightarrow m \mid a-b \wedge m\mid c-d$ \begin{align} @@ -513,6 +521,8 @@ Setzen $e:= \ord_m(a)$ \item \end{enumerate} +\section{Quadratische Reste}adb + % $\Rightarrow \ord_(a^k) = \frac e {\ggT(k,e)}$ % % \section*{Vorlesung 25.4.12} @@ -642,6 +652,7 @@ Definition $3.7$ ist eine beliebte Prüfungsfrage. Gilt $\left( \frac{x}{y} \right)_{J} = -1$, so ist $x$ sicher \emph{kein} quadratischer Rest $\mod y$. Aus $\left( \frac{x}{y} \right)_{J}=1$ lässt sich allerdings keine Aussage mehr darüber treffen. \end{bem} %Vorlesung 2.5.2012 +\section{Lucasfolgen und ihre Eigenschaften} Zu Lucas-Folgen kann man als Verallgemeinerung von Fibonacci-Folgen interpretieren, sind beispielsweise wichtig für Primzahltests. \\ \begin{proof}[Beweis zu Satz $4.2$, Seite $13$] Wegen $(x-\alpha)(x-\beta)=x^{2} - Px + Q = 0$ erhält man durch einen Koeffizientenvergleich @@ -822,6 +833,8 @@ Der Baillie-Wagstaff-Test ist ein starker Lucas-Test mit den oben genannten Para \end{itemize} \end{bem} +\chapter{Primzahltests und Faktorisierungsalgorithmen (mit Anwendungen in der Kryptographie)} +\section{Public Key Kryptosystem: Grundprinzip und Beispiele} \begin{bem}[Signaturen, Seite $15$] Angenommen Alice und Bob kommunizieren, wählen eine Hashfunktion $h$. Bob schickt nun an Alice eine Nachricht m und eine Signatur s: \begin{equation} @@ -963,6 +976,8 @@ Bei $G=E(\Z_{p}),p \in \P$ reicht schon $p \approx 2^{160}$, d.h. ein weitaus kl Problem: Faktorielle wachsen stark. \end{bem} +\section{Primzahltests} + \begin{proof}[Beweis zu Satz $2.2$] Sei \begin{equation} @@ -1188,6 +1203,8 @@ s_{4}=64-2=62 \equiv 0 \mod 31 \Rightarrow s_{p-1}=s_{4} \equiv 0 \mod 31 Daher gilt $M_{5} \in \P$ nach dem Lucas-Lehmer-Test. \end{bem} +\section{Faktorisierungsalgorithmen} + \begin{proof}[Beweis zu Satz $3.1$, Seite $27$] Zunächst gilt $p \nmid a \land p \nmid b$ (denn teilt $p$ eine der Zahlen $a$ oder $b$, dann wegen $p \mid a^{n}+b^{n}$ auch die andere, also $\gcd(a,b)=1$ WS, denn $1$ hat keine Primteiler).\\ Es $\exists b^{\prime} \in \Z: b \cdot b^{\prime} \equiv 1 \mod p$. Setzt man nun $c:=ab^{\prime}$, sog gilt: @@ -1482,4 +1499,13 @@ Man kennt $p$ nicht, wie findet man $r$? Wähle eine ``möglichst zusammengesetz r=\lcm(1,2,\ldots,B) \end{equation} \end{bem} + +%\chapter{Weitere Anwendungen der Zahlentheorie} +%\section{Darstellung von Zahlen in anderen Basen} +%\section{Zahlentheorethische Methoden bei der Erzeugung von Zufallszahlen} +\refstepcounter{chapter} % Chapter auslassen da es nicht behandelt wurde + +\chapter{Übung}\label{UE} +Übungen sollen hier angehängt werden. + \end{document} -- 2.47.3