added complete chapter for substitutionsverfahren
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@ -1,9 +1,24 @@
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\relax
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\providecommand*\new@tpo@label[2]{}
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\@nameuse{bbl@beforestart}
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\catcode `"\active
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\@writefile{toc}{\contentsline {section}{\numberline {1}Substitionsverfahren}{3}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Skytale}{3}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {1.2}Monoalphabetische Substitionsverfahren}{2}\protected@file@percent }
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\newlabel{monoalphabet}{{1.2}{2}}
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\newlabel{häufigkeit}{{1.2.2}{3}}
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\@writefile{toc}{\contentsline {section}{\numberline {1.3}Polyalphabetische Substitutionsverfahren}{3}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.1}Vignère-Verfahren}{3}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.3.1.1}Verschlüsselung}{3}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.2}One-Time-Pad}{4}\protected@file@percent }
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\newlabel{otp}{{1.3.2}{4}}
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.3.2.1}Verschlüsselung}{4}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.3.2.2}Perfekte Sicherheit}{4}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\numberline {1.4}algebraische Substitionsverfahren}{4}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsection}{\numberline {1.4.1}Hill-Verfahren}{4}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.4.1.1}Verschlüsselung}{4}\protected@file@percent }
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\@writefile{toc}{\contentsline {subsubsection}{\numberline {1.4.1.2}Entschlüsselung}{4}\protected@file@percent }
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@ -1,18 +1,32 @@
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\contentsline {subsubsection}{\numberline {1.4.1.1}Verschlüsselung}{4}%
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\contentsline {subsubsection}{\numberline {1.4.1.2}Entschlüsselung}{4}%
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87
chapters/Substitutionsverfahren.tex
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87
chapters/Substitutionsverfahren.tex
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\chapter{Substitionsverfahren}
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\section{Skytale}
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Ein Streifen wird um einen Stock gewickelt und dann beschrieben.
|
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Nach dem abwickeln erhält man den entsprechenden Code.
|
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Durch aufwickeln auf einen Stock mit dem gleichen Umfang lässt sich die Nachricht wieder entschlüsseln.\\
|
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\includegraphics[width=1.0\textwidth]{images/Skytale.png}\\
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\noindent\textbf{Tipp:}\\
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Zum entschlüsseln der Nachricht am PC ist der Editor mit automatischen Zeilen-Wrap gut geeignet
|
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|
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\section{Monoalphabetische Substitionsverfahren} \label{monoalphabet}
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Jeder Buchstabe wird bijektiv durch einen anderen Buchstaben des gleichen Alphabets ersetzt.\\
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\begin{tabbing}
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Alphabet: \hspace{35mm}\=$\mathscr{L} := \{A,B,...,Z\}$\\
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Schlüsselraum: \>$\mathscr{K} := \{k:\mathscr{L}\rightarrow\mathscr{L} | k \text{ ist bijektiv}\}$\\
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Verschlüsselung von $z\in \mathscr{L}$: \>$k(z)$\\
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Entschlüsselung: \> $E(z) := k(z)$
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\end{tabbing}
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\subsection{Caesar-Verschlüsselung}
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Alle Buchstaben des Alphabets werden um einen konstanten Wert verschoben.
|
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\newline
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\noindent\textbf{Beispiel:}\\
|
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\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}
|
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Monoalphabetische Substitionsverfahren lassen sich mit moderner Technik sehr einfach durch die Verwendung von Häufigkeitsanalysen entschlüsseln.
|
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In jeder Sprache gibt es Buchstaben die deutlich häufiger vorkommen als andere.
|
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Durch einer Analyse der Häufigkeit der einzelnen Buchstaben im Geheimtext lassen sich diese den Ausgangsbuchstaben zuordnen.
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|
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\section{Polyalphabetische Substitutionsverfahren}
|
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Damit sich ein Geheimtext nicht durch eine Häufigkeitsanalyse (\ref{häufigkeit}) entschlüsseln lässt wird bei polyalphabetischen Verschlüsselungsverfahren der Schlüssel regelmäßig gewechselt.
|
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Hierbei wird der Schlüsselwechsel meist selbst durch ein Schlüsselwort kodiert.
|
||||
|
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\subsection{Vignère-Verfahren}
|
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\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}
|
||||
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images/Vignère-Verfahren.png
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@ -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.
|
||||
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Reference in New Issue
Block a user