246 lines
7.5 KiB
TeX
246 lines
7.5 KiB
TeX
|
\documentclass[12pt,a4paper,german]{article}
|
|||
|
|
\usepackage{url}
|
|||
|
|
%\usepackage{graphics}
|
|||
|
|
\usepackage{times}
|
|||
|
|
\usepackage[T1]{fontenc}
|
|||
|
|
\usepackage{pifont}
|
|||
|
|
\usepackage{ngerman}
|
|||
|
|
\usepackage{float}
|
|||
|
|
\usepackage{diagbox}
|
|||
|
|
\usepackage[latin1]{inputenc}
|
|||
|
|
\usepackage{geometry}
|
|||
|
|
\usepackage{amsfonts}
|
|||
|
|
\usepackage{amsmath}
|
|||
|
|
\usepackage{delarray}
|
|||
|
|
% \usepackage{minted}
|
|||
|
|
\usepackage{csquotes}
|
|||
|
|
\usepackage{graphicx}
|
|||
|
|
\usepackage{epsfig}
|
|||
|
|
\usepackage{longtable}
|
|||
|
|
\usepackage{paralist}
|
|||
|
|
\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
|
|||
|
|
|
|||
|
|
\graphicspath{.}
|
|||
|
|
|
|||
|
|
%%%%%%%%%% Fill out the the definitions %%%%%%%%%
|
|||
|
|
\def \name {Valentin Brandl} %
|
|||
|
|
\def \matrikel {108018274494} %
|
|||
|
|
% \def \pname {Vorname2 Nachname2} %
|
|||
|
|
% \def \pmatrikel {Matrikelnummer2} %
|
|||
|
|
\def \gruppe {Gruppe 193} %
|
|||
|
|
\def \uebung {7} %
|
|||
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|||
|
|
|
|||
|
|
% DO NOT MODIFY THIS HEADER
|
|||
|
|
\newcommand{\hwsol}{
|
|||
|
|
\vspace*{-2cm}
|
|||
|
|
\noindent \matrikel \quad \name \hfill Gruppe:\gruppe \\
|
|||
|
|
% \noindent \pmatrikel \quad \pname \\
|
|||
|
|
\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
\newcommand{\cmark}{\ding{51}}%
|
|||
|
|
\newcommand{\xmark}{\ding{55}}%
|
|||
|
|
\newcommand{\csquare}{\text{\rlap{$\checkmark$}}\square}%
|
|||
|
|
|
|||
|
|
\begin{document}
|
|||
|
|
%Import header
|
|||
|
|
\hwsol
|
|||
|
|
|
|||
|
|
\section*{Aufgabe 1}
|
|||
|
|
|
|||
|
|
$P(x) = x^4 + x + 1$ und $grad(P(X)) = 4$
|
|||
|
|
|
|||
|
|
\begin{enumerate}[a)]
|
|||
|
|
|
|||
|
|
\item $GF(2) = GF(2^1)$. Maximaler Grad von $G(p^m)$ ist $(m-1)$, maximaler Grad von $GF(2^1)$ ist also $1-1=0$, es
|
|||
|
|
sind also nur die polynome $a_1(x) = 0$ und $a_2(x) = 1$ m<>glich. Beide haben einen Grad $< 4$, also gibt es $2$
|
|||
|
|
Polynome in $GF(2)$, deren Grad kleiner ist, als $grad(P(x))$.
|
|||
|
|
|
|||
|
|
\item $GF(p^n)$: Alle $a_i$ des Polynoms sind Elemente der Menge $A = \{0,1,...,p-1\} \Rightarrow |A| = p$. Jedes
|
|||
|
|
Polynom besteht aus $n$ Koeffizienten, daher hat hat $GF(p^n)$ genau $p^n$ m<>gliche Polynome. Da der Grad der
|
|||
|
|
Polynome $< 4$ sein soll, muss $n$ also $\leq 4$ sein. Abh<62>ngig von $p$ sind also $4$ m<>gliche Werte f<>r $n$
|
|||
|
|
denkbar, sodass der Grad $< 4$ bleibt
|
|||
|
|
|
|||
|
|
\begin{tabular}{|c|c|}
|
|||
|
|
\hline
|
|||
|
|
$n$ & \# Polynome \\\hline
|
|||
|
|
1 & $p$ \\\hline
|
|||
|
|
2 & $p^2$ \\\hline
|
|||
|
|
3 & $p^3$ \\\hline
|
|||
|
|
4 & $p^4$ \\\hline
|
|||
|
|
\end{tabular}
|
|||
|
|
|
|||
|
|
\end{enumerate}
|
|||
|
|
|
|||
|
|
|
|||
|
|
\section*{Aufgabe 2}
|
|||
|
|
|
|||
|
|
\begin{tabular}{|c||c|c|c|c|c|c|c|c|}
|
|||
|
|
\hline
|
|||
|
|
$\cdot$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline\hline
|
|||
|
|
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline
|
|||
|
|
1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline
|
|||
|
|
2 & 0 & 2 & 4 & 6 & 5 & 7 & 1 & 3 \\\hline
|
|||
|
|
3 & 0 & 3 & 6 & 5 & 1 & 2 & 7 & 4 \\\hline
|
|||
|
|
4 & 0 & 4 & 5 & 1 & 7 & 3 & 2 & 5 \\\hline
|
|||
|
|
5 & 0 & 5 & 7 & 2 & 3 & 6 & 4 & 3 \\\hline
|
|||
|
|
6 & 0 & 6 & 1 & 7 & 2 & 4 & 6 & 5 \\\hline
|
|||
|
|
7 & 0 & 7 & 3 & 4 & 5 & 3 & 5 & 2 \\\hline
|
|||
|
|
\end{tabular}
|
|||
|
|
|
|||
|
|
|
|||
|
|
\section*{Aufgabe 3}
|
|||
|
|
|
|||
|
|
$P(x) = x^8 + x^4 + x^3 + x + 1$
|
|||
|
|
|
|||
|
|
\begin{enumerate}[a)]
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{align*}
|
|||
|
|
A(x) &= x^7 + x^6 + x^4 + x^2 + x + 1 \\
|
|||
|
|
B(x) &= x^2 + x \\
|
|||
|
|
A(x) * B(x) &= x^9+x^7+x^5+x^4+x \equiv x^7+x^2+1 &\mod P(x) \\
|
|||
|
|
&= 0x85
|
|||
|
|
\end{align*}
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{align*}
|
|||
|
|
A(x) &= x^4 + x^2 + x + 1 \\
|
|||
|
|
B(x) &= x^4 + 1 \\
|
|||
|
|
A(x) * B(x) &= x^8 + x^6 + x^5 + x^2 + x + 1 \equiv x^6 + x^5 + x^4 + x^3 + x^2 &\mod P(x) \\
|
|||
|
|
&= 0x7C
|
|||
|
|
\end{align*}
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{align*}
|
|||
|
|
A(x) &= x^6 + x^5 + x^2 + x \\
|
|||
|
|
B(x) &= x^2 + x + 1 \\
|
|||
|
|
A(x) * B(x) &= x^5 + x^4 + x &\mod P(x) \\
|
|||
|
|
&= 0x32
|
|||
|
|
\end{align*}
|
|||
|
|
|
|||
|
|
\end{enumerate}
|
|||
|
|
|
|||
|
|
|
|||
|
|
\section*{Aufgabe 4}
|
|||
|
|
|
|||
|
|
Die multiplikativen Inversen wurden aus der Tabelle 4.2 im Buch abgelesen.
|
|||
|
|
|
|||
|
|
$P(x) = x^8 + x^4 + x^3 + x + 1$
|
|||
|
|
|
|||
|
|
\begin{enumerate}[a)]
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{align*}
|
|||
|
|
A(x) &= x^5 + x^4 + x^3 + x^2 \\
|
|||
|
|
B(x) &= x^5 + x^2 + 1 \widehat{=} 0x25 \\
|
|||
|
|
B^{-1}(x) &= 0x4D \widehat{=} x^6 + x^3 + x^2 + 1 \\
|
|||
|
|
A(x) \div B(x) &= A(x) * B^{-1}(x) = \\
|
|||
|
|
&= x^{11} + x^{10} + x^9 + x^7 + x^5 + x^3 \equiv x^5 + x^3 + x &\mod P(x) \\
|
|||
|
|
&\widehat{=} 0x2A
|
|||
|
|
\end{align*}
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{align*}
|
|||
|
|
A(x) &= x^7 + x^2 + x + 1 \\
|
|||
|
|
B(x) &= x^6 + x^4 + x^3 + x^2 \widehat{=} 0x5C \\
|
|||
|
|
B^{-1}(x) &= 0x51 \widehat{=} x^6 + x^4 + 1 \\
|
|||
|
|
A(x) \div B(x) &= A(x) * B^{-1}(x) = \\
|
|||
|
|
&= x^{13} + x^{11} + x^8 + x^5 + x^4 + x^2 + x + 1 \equiv x^7 + x^5 + x^4 + x^3 + 1 &\mod
|
|||
|
|
P(x) \\
|
|||
|
|
&\widehat{=} 0xB9
|
|||
|
|
\end{align*}
|
|||
|
|
|
|||
|
|
\end{enumerate}
|
|||
|
|
|
|||
|
|
|
|||
|
|
\section*{Aufgabe 5}
|
|||
|
|
|
|||
|
|
$P(x) = x^4 + x^2 + 3x + 5$
|
|||
|
|
|
|||
|
|
\begin{enumerate}[a)]
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{tabular}{|c||c|c|c|c|c|c|c|}
|
|||
|
|
\hline
|
|||
|
|
$+$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\hline\hline
|
|||
|
|
0 & 0 & 1 & 2 & 3 & 4 & 4 & 6 \\\hline
|
|||
|
|
1 & 1 & 2 & 3 & 4 & 5 & 6 & 0 \\\hline
|
|||
|
|
2 & 2 & 3 & 4 & 5 & 6 & 0 & 1 \\\hline
|
|||
|
|
3 & 3 & 4 & 5 & 6 & 0 & 1 & 2 \\\hline
|
|||
|
|
4 & 4 & 5 & 6 & 0 & 1 & 2 & 3 \\\hline
|
|||
|
|
5 & 5 & 6 & 0 & 1 & 2 & 3 & 4 \\\hline
|
|||
|
|
6 & 6 & 0 & 1 & 2 & 3 & 4 & 5 \\\hline
|
|||
|
|
\end{tabular}
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{tabular}{|c||c|c|c|c|c|c|c|}
|
|||
|
|
\hline
|
|||
|
|
$\cdot$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\hline
|
|||
|
|
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\hline
|
|||
|
|
1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\\hline
|
|||
|
|
2 & 0 & 2 & 4 & 6 & 1 & 3 & 5 \\\hline
|
|||
|
|
3 & 0 & 3 & 6 & 2 & 5 & 1 & 4 \\\hline
|
|||
|
|
4 & 0 & 4 & 1 & 5 & 2 & 6 & 3 \\\hline
|
|||
|
|
5 & 0 & 5 & 3 & 1 & 6 & 4 & 2 \\\hline
|
|||
|
|
6 & 0 & 6 & 5 & 4 & 3 & 2 & 1 \\\hline
|
|||
|
|
\end{tabular}
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{enumerate}[i)]
|
|||
|
|
\item
|
|||
|
|
\begin{align*}
|
|||
|
|
A(x) &= 3x^2 + x + 2 \\
|
|||
|
|
B(x) &= 6x^4 + 4x^2 + 3x + 5 \\
|
|||
|
|
A(x) + B(x) &= 6x^4 + 4x &\mod P(x)
|
|||
|
|
\end{align*}
|
|||
|
|
|
|||
|
|
\item Nein, da $A(x) + B(x)$ auch ohne die Moduloreduktion keinen Koeffizienten mit einem Grad $> 6$ hat
|
|||
|
|
\end{enumerate}
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{enumerate}[i)]
|
|||
|
|
\item
|
|||
|
|
\begin{align*}
|
|||
|
|
A(x) &= 3x^3 + x + 2 \\
|
|||
|
|
B(x) &= 6x^3 + 4x^2 + 3x + 5 \\
|
|||
|
|
A(x) - B(x) &= 4x^3 + 3x^2 + 5x + 4 &\mod P(x)
|
|||
|
|
\end{align*}
|
|||
|
|
|
|||
|
|
\item Nein, da $A(x) - B(x)$ auch ohne die Moduloreduktion keinen Koeffizienten mit einem Grad $> 6$ hat
|
|||
|
|
\end{enumerate}
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{enumerate}[i)]
|
|||
|
|
\item
|
|||
|
|
\begin{align*}
|
|||
|
|
A(x) &= 5x^4 + x + 2 \\
|
|||
|
|
B(x) &= 3x^3 + 5x^2 + 4 \\
|
|||
|
|
A(x) * B(x) &= x^7 + 4x^6 + 2x^4 + 4x^3 + 3x^2 + 4x + 1 &\mod P(x) \\
|
|||
|
|
&\equiv 2x^3 + 5x^2 + 3x + 5 &\mod P(x)
|
|||
|
|
\end{align*}
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{align*}
|
|||
|
|
A(x) &= x^3 + 2x + 4 \\
|
|||
|
|
B(x) &= 4x^4 + 6x^3 + 3 \\
|
|||
|
|
A(x) * B(x) &= 4x^7 + 6x^6 + 6x^5 + 6x^3 + 6x + 5 &\mod P(x) \\
|
|||
|
|
&\equiv x^3 + 3x^2 + x + 4 &\mod P(x)
|
|||
|
|
\end{align*}
|
|||
|
|
\end{enumerate}
|
|||
|
|
|
|||
|
|
\item
|
|||
|
|
\begin{align*}
|
|||
|
|
x^4 &\equiv 6x^2 + 4x + 2 &\mod P(x) \\
|
|||
|
|
x^5 &\equiv 6x^3 + 4x^2 + 2x &\mod P(x) \\
|
|||
|
|
x^6 &\equiv 4x^3 + x^2 + 5x + 5 &\mod P(x) \\
|
|||
|
|
x^7 &\equiv 3x^3 + 6x^2 + 1 &\mod P(x) \\
|
|||
|
|
\end{align*}
|
|||
|
|
|
|||
|
|
\end{enumerate}
|
|||
|
|
|
|||
|
|
|
|||
|
|
\end{document}
|
|||
|
|
|