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10000000) diff --git a/Zusammenfassung_DC.pdf b/Zusammenfassung_DC.pdf index dafeee6..8f0678b 100644 Binary files a/Zusammenfassung_DC.pdf and b/Zusammenfassung_DC.pdf differ diff --git a/Zusammenfassung_DC.synctex.gz b/Zusammenfassung_DC.synctex.gz index ea02904..309d1d8 100644 Binary files a/Zusammenfassung_DC.synctex.gz and b/Zusammenfassung_DC.synctex.gz differ diff --git a/Zusammenfassung_DC.tex b/Zusammenfassung_DC.tex index b671138..3fb949b 100644 --- a/Zusammenfassung_DC.tex +++ b/Zusammenfassung_DC.tex @@ -1,4 +1,6 @@ -\documentclass[a4paper,11pt,oneside]{scrartcl} +\documentclass[a4paper,11pt,oneside]{report} +%numbering depth +\setcounter{secnumdepth}{5} %page setup \usepackage[a4paper, margin=2cm]{geometry} %encoding @@ -11,6 +13,8 @@ \graphicspath{ {./images/} } %mathsymbols \usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{mathrsfs} %for code highlighting \usepackage{xcolor} \usepackage{listings} @@ -30,6 +34,6 @@ \tableofcontents{} \pagebreak -\input{sections/Substitutionsverfahren.tex} +\input{chapters/Substitutionsverfahren.tex} \end{document} diff --git a/Zusammenfassung_DC.toc b/Zusammenfassung_DC.toc index debd67d..137b5c5 100644 --- a/Zusammenfassung_DC.toc +++ b/Zusammenfassung_DC.toc @@ -1,3 +1,16 @@ \babel@toc {ngerman}{} -\contentsline {section}{\numberline {1}Substitionsverfahren}{3}% -\contentsline {subsection}{\numberline {1.1}Skytale}{3}% +\contentsline {chapter}{\numberline {1}Substitionsverfahren}{2}% +\contentsline {section}{\numberline {1.1}Skytale}{2}% +\contentsline {section}{\numberline {1.2}Monoalphabetische Substitionsverfahren}{2}% +\contentsline {subsection}{\numberline {1.2.1}Caesar-Verschlüsselung}{3}% +\contentsline {subsection}{\numberline {1.2.2}Häufigkeitsanalyse}{3}% +\contentsline {section}{\numberline {1.3}Polyalphabetische Substitutionsverfahren}{3}% +\contentsline {subsection}{\numberline {1.3.1}Vignère-Verfahren}{3}% +\contentsline {subsubsection}{\numberline {1.3.1.1}Verschlüsselung}{3}% +\contentsline {subsection}{\numberline {1.3.2}One-Time-Pad}{4}% +\contentsline {subsubsection}{\numberline {1.3.2.1}Verschlüsselung}{4}% +\contentsline {subsubsection}{\numberline {1.3.2.2}Perfekte Sicherheit}{4}% +\contentsline {section}{\numberline {1.4}algebraische Substitionsverfahren}{4}% +\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}% diff --git a/chapters/Substitutionsverfahren.tex b/chapters/Substitutionsverfahren.tex new file mode 100644 index 0000000..96ca5ed --- /dev/null +++ b/chapters/Substitutionsverfahren.tex @@ -0,0 +1,87 @@ +\chapter{Substitionsverfahren} + \section{Skytale} + Ein Streifen wird um einen Stock gewickelt und dann beschrieben. + Nach dem abwickeln erhält man den entsprechenden Code. + Durch aufwickeln auf einen Stock mit dem gleichen Umfang lässt sich die Nachricht wieder entschlüsseln.\\ + \includegraphics[width=1.0\textwidth]{images/Skytale.png}\\ + \noindent\textbf{Tipp:}\\ + Zum entschlüsseln der Nachricht am PC ist der Editor mit automatischen Zeilen-Wrap gut geeignet + + \section{Monoalphabetische Substitionsverfahren} \label{monoalphabet} + Jeder Buchstabe wird bijektiv durch einen anderen Buchstaben des gleichen Alphabets ersetzt.\\ + \begin{tabbing} + Alphabet: \hspace{35mm}\=$\mathscr{L} := \{A,B,...,Z\}$\\ + Schlüsselraum: \>$\mathscr{K} := \{k:\mathscr{L}\rightarrow\mathscr{L} | k \text{ ist bijektiv}\}$\\ + Verschlüsselung von $z\in \mathscr{L}$: \>$k(z)$\\ + Entschlüsselung: \> $E(z) := k(z)$ + \end{tabbing} + + \subsection{Caesar-Verschlüsselung} + Alle Buchstaben des Alphabets werden um einen konstanten Wert verschoben. + \newline + \noindent\textbf{Beispiel:}\\ + \noindent\textbf{Code:} $n=2$ \\ + \noindent\textbf{Verschlüsselung:} jeder Buchstabe wird durch den übernächsten Buchstaben im Alphabet ersetzt ($E(m)=(m+3) \mod 26$).\\ + \noindent\textbf{Entschlüsselung:} jeder Buchstabe wird durch den vor-vorherigen Buchstaben ersetzt. + + \subsection{Häufigkeitsanalyse} \label{häufigkeit} + Monoalphabetische Substitionsverfahren lassen sich mit moderner Technik sehr einfach durch die Verwendung von Häufigkeitsanalysen entschlüsseln. + In jeder Sprache gibt es Buchstaben die deutlich häufiger vorkommen als andere. + Durch einer Analyse der Häufigkeit der einzelnen Buchstaben im Geheimtext lassen sich diese den Ausgangsbuchstaben zuordnen. + + \section{Polyalphabetische Substitutionsverfahren} + Damit sich ein Geheimtext nicht durch eine Häufigkeitsanalyse (\ref{häufigkeit}) entschlüsseln lässt wird bei polyalphabetischen Verschlüsselungsverfahren der Schlüssel regelmäßig gewechselt. + Hierbei wird der Schlüsselwechsel meist selbst durch ein Schlüsselwort kodiert. + + \subsection{Vignère-Verfahren} + \begin{tabbing} + Alphabet: \hspace{35mm}\=$\mathscr{L} := \{A,B,...,Z\}$\\ + Menge aller Wörter: \>$\mathscr{W} = \mathscr{L}^{>0} := \bigcup_{n\in\mathbb{N}} \mathscr{L}^n = \{(m_1,m_2,...,m_n)|m_i\in\mathscr{L},n\in\mathbb{N}\}$\\ + Schlüsselraum: \>$\mathscr{K} := \Sigma_\mathscr{L} \times \mathscr{W}$ + \end{tabbing} + Ein Schlüssel $k = (f,w)\in \mathscr{K}$ besteht aus einer Permutation $f\in \Sigma_\mathscr{L}$ (siehe \ref{monoalphabet}) und einem Schlüsselwort $w$.\\ + Das Vignère-Verfahren ist ein Blockverschlüsselungsverfahren, bei dem jeweils ein Block von Zeichen (im Beispiel 4) nach dem gleichen Verfahren verschlüsselt wird. + Wenn die Blockgröße klein genug oder die Textlänge groß genug sind lässt sich ein solches Verfahren ebenfalls durch eine Häufigkeitsanalyse (siehe \ref{häufigkeit}) knacken. + + \subsubsection{Verschlüsselung} + Der Schlüssel $f$ wird zyklisch mithilfe des Schlüsselwortes $w$ verschoben.\\ + \noindent\textbf{Beispiel:}\\ + $f=\text{VEYZNTCWOMFQPRGDXIABHULKJS}$ und $w=\text{SKAT}$\\ + \includegraphics[width=0.75\textwidth]{images/Vignère-Verfahren.png}\\ + + \subsection{One-Time-Pad}\label{otp} + Bei dem One-Time-Pad handelt es sich um ein absolut sicheres Substitionsverfahren + (Nachrichten, bei denen sich Informationen von der Länge der Nachricht ableiten lassen müssen auf eine konstante Länge gebracht werden). + + \subsubsection{Verschlüsselung} + \begin{tabbing} + Annahme: \hspace{30mm}\= Klar- und Geheimtext sind eine Folge von Zeichen der Menge $\mathscr{L} = \mathbb{Z}$ ($n\in\mathbb{N}$)\\ + Klartextnachricht: \>$m=(m_1,m_2,...m_l)\in {\mathbb{Z}_n}^l$\\ + One-Time-Pad (zufällig): \>$k=(k_1,k_2,...k_l)\in {\mathbb{Z}_n}^l$\\ + Geheimtext: \>$E(m) := m+k := (m_1+k_1,m_2+k_2,...,m_l+k_l)$\\ + \end{tabbing} + + \subsubsection{Perfekte Sicherheit} + Wenn die Zeichen $k_i$ des Schlüssels mit einem perfekten Zufallsgenerator erzeugt lassen sich \textbf{keine} Rückschlüsse auf die Klartextnachricht ziehen (außer Länge).\\ + Allerdings darf ein One-Time-Pad nur ein einziges Mal für die Verschlüsselung verwendet werden. + Andernfalls könnte durch eine Berechnung der Differenz der mit dem gleichen OTP verschlüsselten Nachrichten Informationen gewonnen werden. + So kann eine Häufigkeitsanalyse (siehe \ref{häufigkeit}) oder sogar eine komplette Entschlüsselung (falls Klartext einer Nachricht bekannt ist) möglich werden. + + \section{algebraische Substitionsverfahren} + \subsection{Hill-Verfahren} + Das Hill-Verfahren ver- und entschlüsselt die Nachrichten mithilfe einer \textbf{invertierbaren} Matrix. + + \subsubsection{Verschlüsselung} + \begin{tabbing} + Annahme: \hspace{35mm}\=$n,k\in\mathbb{N}$ sind vorgegeben\\ + Klartextnachricht: \> Menge von Blöcken $\mathscr{B}={\mathbb{Z}_n}^k$\\ + Schlüssel: \> \textbf{invertierbare} ($k,k$)-Matrix $K$\\ + Geheimtext: \> $E_K(b) := K\cdot b$ \hspace{5mm} ($b\in\mathscr{B}$)\\ + \end{tabbing} + + \subsubsection{Entschlüsselung} + \begin{tabbing} + Annahme: \hspace{35mm}\= gleiche Bedingungen wie bei der Verschlüsselung \\ + Geheimtextnachricht: \> Menge von Blöcken $\mathscr{C}={\mathbb{Z}_n}^k$\\ + Klartext: \>$E_K(c) := K^{-1}\cdot c$ \hspace{5mm} ($c\in\mathscr{C}$)\\ + \end{tabbing} \ No newline at end of file diff --git a/images/Skytale.png b/images/Skytale.png new file mode 100644 index 0000000..a97db0d Binary files /dev/null and b/images/Skytale.png differ diff --git a/images/Vignère-Verfahren.png b/images/Vignère-Verfahren.png new file mode 100644 index 0000000..20b365c Binary files /dev/null and b/images/Vignère-Verfahren.png differ diff --git a/sections/Substitutionsverfahren.tex b/sections/Substitutionsverfahren.tex deleted file mode 100644 index ef1c833..0000000 --- a/sections/Substitutionsverfahren.tex +++ /dev/null @@ -1,5 +0,0 @@ -\section{Substitionsverfahren} -\subsection{Skytale} -Ein Streifen wird um einen Stock gewickelt und dann beschrieben. -Nach dem abwickeln erhält man den entsprechenden Code. -Durch aufwickeln auf einen Stock mit dem gleichen Umfang lässt sich die Nachricht wieder entschlüsseln. \ No newline at end of file