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|
||||
Output written on "/home/paul/Documents/TH/Datensicherheit (DC)/Zusammenfassung/Zusammenfassung_DC.pdf" (12 pages, 2271338 bytes).
|
||||
Output written on "/home/paul/Documents/TH/Datensicherheit (DC)/Zusammenfassung/Zusammenfassung_DC.pdf" (16 pages, 2289929 bytes).
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@ -18,6 +18,14 @@
|
||||
%for code highlighting
|
||||
\usepackage{xcolor}
|
||||
\usepackage{listings}
|
||||
%quotation
|
||||
\usepackage[%
|
||||
left = \glqq,%
|
||||
right = \grqq,%
|
||||
leftsub = \glq,%
|
||||
rightsub = \grq%
|
||||
]{dirtytalk}
|
||||
|
||||
|
||||
%opening
|
||||
\title{Zusammenfassung DC}
|
||||
@ -40,4 +48,6 @@
|
||||
|
||||
\input{chapters/IT-Sicherheit.tex}
|
||||
|
||||
\input{chapters/Verschlüsselungsverfahren.tex}
|
||||
|
||||
\end{document}
|
||||
|
||||
@ -1,52 +1,67 @@
|
||||
\babel@toc {ngerman}{}
|
||||
\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}%
|
||||
\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 {subsubsection}{\numberline {2.2.1.1}Inverse bezüglich der Addition}{5}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.1.2}Inverse bezüglich der Multiplikation}{5}%
|
||||
\contentsline {subsection}{\numberline {2.2.2}Subtraktion}{6}%
|
||||
\contentsline {subsection}{\numberline {2.2.3}Teiler, Vielfache}{6}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.3.1}Teilerregeln}{6}%
|
||||
\contentsline {subsection}{\numberline {2.2.4}Kongruenz}{6}%
|
||||
\contentsline {subsection}{\numberline {2.2.5}Matrizen}{6}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.5.1}Determinantenberechnung}{6}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.5.2}Inverse Matrix}{6}%
|
||||
\contentsline {section}{\numberline {2.3}Der erweiterte Euklid'sche Algorithmus}{6}%
|
||||
\contentsline {subsection}{\numberline {2.3.1}Euklid'scher Algorithmus}{7}%
|
||||
\contentsline {subsection}{\numberline {2.3.2}erweiterter Euklid'scher Algorithmus}{7}%
|
||||
\contentsline {subsubsection}{\numberline {2.3.2.1}Beispiel}{7}%
|
||||
\contentsline {section}{\numberline {2.4}Euler'sche $\varphi $-Funktion}{8}%
|
||||
\contentsline {subsection}{\numberline {2.4.1}$\varphi $-Funktion und Primzahlen}{8}%
|
||||
\contentsline {chapter}{\numberline {3}IT-Sicherheit: Gefährdungen und Maßnahmen}{9}%
|
||||
\contentsline {section}{\numberline {3.1}Vertraulichkeit}{9}%
|
||||
\contentsline {subsection}{\numberline {3.1.1}Schutzmaßnahmen: Verschlüsselungsverfahren}{9}%
|
||||
\contentsline {section}{\numberline {3.2}Integrität}{9}%
|
||||
\contentsline {subsection}{\numberline {3.2.1}Schutzmaßnahme: Hashfunktionen, Whitelists}{10}%
|
||||
\contentsline {section}{\numberline {3.3}Authenzität der Daten}{10}%
|
||||
\contentsline {subsection}{\numberline {3.3.1}Schutzmaßnahme: Signaturen}{10}%
|
||||
\contentsline {subsection}{\numberline {3.3.2}Schutz vor Replay-Angriffen}{10}%
|
||||
\contentsline {section}{\numberline {3.4}Authenzität von Nutzern}{10}%
|
||||
\contentsline {subsection}{\numberline {3.4.1}Schutzmaßnahmen}{10}%
|
||||
\contentsline {section}{\numberline {3.5}Zugriffskontrolle}{10}%
|
||||
\contentsline {subsection}{\numberline {3.5.1}Schutzmaßname: Zugriffskontrollsystem}{11}%
|
||||
\contentsline {section}{\numberline {3.6}Nichtabstreitbarkeit, Verbindlichkeit}{11}%
|
||||
\contentsline {subsection}{\numberline {3.6.1}Schutzmaßname: Signaturen und PKI}{11}%
|
||||
\contentsline {section}{\numberline {3.7}Verfügbarkeit}{11}%
|
||||
\contentsline {subsection}{\numberline {3.7.1}Schutzmaßnahmen}{11}%
|
||||
\contentsline {section}{\numberline {3.8}Anonymität}{11}%
|
||||
\contentsline {chapter}{\numberline {1}Substitionsverfahren}{3}%
|
||||
\contentsline {section}{\numberline {1.1}Skytale}{3}%
|
||||
\contentsline {section}{\numberline {1.2}Monoalphabetische Substitionsverfahren}{3}%
|
||||
\contentsline {subsection}{\numberline {1.2.1}Caesar-Verschlüsselung}{4}%
|
||||
\contentsline {subsection}{\numberline {1.2.2}Häufigkeitsanalyse}{4}%
|
||||
\contentsline {section}{\numberline {1.3}Polyalphabetische Substitutionsverfahren}{4}%
|
||||
\contentsline {subsection}{\numberline {1.3.1}Vignère-Verfahren}{4}%
|
||||
\contentsline {subsubsection}{\numberline {1.3.1.1}Verschlüsselung}{4}%
|
||||
\contentsline {subsection}{\numberline {1.3.2}One-Time-Pad}{5}%
|
||||
\contentsline {subsubsection}{\numberline {1.3.2.1}Verschlüsselung}{5}%
|
||||
\contentsline {subsubsection}{\numberline {1.3.2.2}Perfekte Sicherheit}{5}%
|
||||
\contentsline {section}{\numberline {1.4}algebraische Substitionsverfahren}{5}%
|
||||
\contentsline {subsection}{\numberline {1.4.1}Hill-Verfahren}{5}%
|
||||
\contentsline {subsubsection}{\numberline {1.4.1.1}Verschlüsselung}{5}%
|
||||
\contentsline {subsubsection}{\numberline {1.4.1.2}Entschlüsselung}{5}%
|
||||
\contentsline {chapter}{\numberline {2}Modulare Arithmetik}{6}%
|
||||
\contentsline {section}{\numberline {2.1}Exkurs: Division mit Rest}{6}%
|
||||
\contentsline {section}{\numberline {2.2}Der Ring $\mathbb {Z}_n$}{6}%
|
||||
\contentsline {subsection}{\numberline {2.2.1}Addition und Multiplikation}{6}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.1.1}Inverse bezüglich der Addition}{6}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.1.2}Inverse bezüglich der Multiplikation}{6}%
|
||||
\contentsline {subsection}{\numberline {2.2.2}Subtraktion}{7}%
|
||||
\contentsline {subsection}{\numberline {2.2.3}Teiler, Vielfache}{7}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.3.1}Teilerregeln}{7}%
|
||||
\contentsline {subsection}{\numberline {2.2.4}Kongruenz}{7}%
|
||||
\contentsline {subsection}{\numberline {2.2.5}Matrizen}{7}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.5.1}Determinantenberechnung}{7}%
|
||||
\contentsline {subsubsection}{\numberline {2.2.5.2}Inverse Matrix}{7}%
|
||||
\contentsline {section}{\numberline {2.3}Der erweiterte Euklid'sche Algorithmus}{7}%
|
||||
\contentsline {subsection}{\numberline {2.3.1}Euklid'scher Algorithmus}{8}%
|
||||
\contentsline {subsection}{\numberline {2.3.2}erweiterter Euklid'scher Algorithmus}{8}%
|
||||
\contentsline {subsubsection}{\numberline {2.3.2.1}Beispiel}{8}%
|
||||
\contentsline {section}{\numberline {2.4}Euler'sche $\varphi $-Funktion}{9}%
|
||||
\contentsline {subsection}{\numberline {2.4.1}$\varphi $-Funktion und Primzahlen}{9}%
|
||||
\contentsline {chapter}{\numberline {3}IT-Sicherheit: Gefährdungen und Maßnahmen}{10}%
|
||||
\contentsline {section}{\numberline {3.1}Vertraulichkeit}{10}%
|
||||
\contentsline {subsection}{\numberline {3.1.1}Schutzmaßnahmen: Verschlüsselungsverfahren}{10}%
|
||||
\contentsline {section}{\numberline {3.2}Integrität}{10}%
|
||||
\contentsline {subsection}{\numberline {3.2.1}Schutzmaßnahme: Hashfunktionen, Whitelists}{11}%
|
||||
\contentsline {section}{\numberline {3.3}Authenzität der Daten}{11}%
|
||||
\contentsline {subsection}{\numberline {3.3.1}Schutzmaßnahme: Signaturen}{11}%
|
||||
\contentsline {subsection}{\numberline {3.3.2}Schutz vor Replay-Angriffen}{11}%
|
||||
\contentsline {section}{\numberline {3.4}Authenzität von Nutzern}{11}%
|
||||
\contentsline {subsection}{\numberline {3.4.1}Schutzmaßnahmen}{11}%
|
||||
\contentsline {section}{\numberline {3.5}Zugriffskontrolle}{11}%
|
||||
\contentsline {subsection}{\numberline {3.5.1}Schutzmaßname: Zugriffskontrollsystem}{12}%
|
||||
\contentsline {section}{\numberline {3.6}Nichtabstreitbarkeit, Verbindlichkeit}{12}%
|
||||
\contentsline {subsection}{\numberline {3.6.1}Schutzmaßname: Signaturen und PKI}{12}%
|
||||
\contentsline {section}{\numberline {3.7}Verfügbarkeit}{12}%
|
||||
\contentsline {subsection}{\numberline {3.7.1}Schutzmaßnahmen}{12}%
|
||||
\contentsline {section}{\numberline {3.8}Anonymität}{12}%
|
||||
\contentsline {chapter}{\numberline {4}Verschlüsselungsverfahren}{13}%
|
||||
\contentsline {section}{\numberline {4.1}Das Kerckhoffs'sche Prinzip}{13}%
|
||||
\contentsline {section}{\numberline {4.2}Mathematische Modelierung von Verschlüsselungsverfahren}{13}%
|
||||
\contentsline {section}{\numberline {4.3}Schlüsselaustausch}{13}%
|
||||
\contentsline {section}{\numberline {4.4}Angriffsszenarien}{14}%
|
||||
\contentsline {subsection}{\numberline {4.4.1}Ciphertext-only Angriffe}{14}%
|
||||
\contentsline {subsection}{\numberline {4.4.2}Known-plaintext Angriffe}{14}%
|
||||
\contentsline {subsection}{\numberline {4.4.3}Chosen-plaintext Angriffe}{14}%
|
||||
\contentsline {section}{\numberline {4.5}Brute-Force Angriffe}{14}%
|
||||
\contentsline {subsection}{\numberline {4.5.1}Beispiel: Brute-Force Angriff auf $k$}{14}%
|
||||
\contentsline {subsection}{\numberline {4.5.2}Beispiel: Brute-Force Angriff auf $m$}{14}%
|
||||
\contentsline {subsection}{\numberline {4.5.3}Anforderungen zum Schutz vor Brute-Force}{14}%
|
||||
\contentsline {section}{\numberline {4.6}Wörterbuchangriffe}{15}%
|
||||
\contentsline {subsection}{\numberline {4.6.1}Schutz vor Wörterbuchangriffen}{15}%
|
||||
\contentsline {subsubsection}{\numberline {4.6.1.1}Nonce-Verschlüsselung}{15}%
|
||||
|
||||
92
chapters/Verschlüsselungsverfahren.tex
Normal file
92
chapters/Verschlüsselungsverfahren.tex
Normal file
@ -0,0 +1,92 @@
|
||||
\chapter{Verschlüsselungsverfahren}
|
||||
\section{Das Kerckhoffs'sche Prinzip}\label{kerckhoff}
|
||||
\say{Ein Verschlüsselungssystem darf nicht der Geheimhaltung bedürfen und soll ohne Schaden in Feindeshand fallen können.}\\
|
||||
|
||||
\noindent Folglich ist für die Entschlüsselung der Nachricht nicht die Kenntnis über das Verfahren, sondern der Schlüssel die relevante Information.\\
|
||||
|
||||
\noindent\say{Es soll nicht möglich sein, einen Geheimtext ohne Kenntnis des hierfür vorgesehenen Schlüssels \textbf{effizient} zu entschlüsseln.}
|
||||
|
||||
\section{Mathematische Modelierung von Verschlüsselungsverfahren}
|
||||
Klartextnachrichten und Geheimtextnachrichten sind Elemente einer Menge $\mathscr{M}$.\\
|
||||
Schlüssel sind Elemente eines Menge $\mathscr{K}$, die als Schlüsselraum bezeichnet wird.\\
|
||||
Die Verschlüsselungsverfahren bestehen aus einem Paar von Funktionen zur Verschlüsselung ($E$) und Entschlüsselung ($D$):
|
||||
$$E:\mathscr{K}\times\mathscr{M}\rightarrow\mathscr{M}$$
|
||||
$$D:\mathscr{K}\times\mathscr{M}\rightarrow\mathscr{M}$$
|
||||
Hierbei sind die definierten Abbildungen bijektiv ($D_k:=E_k^{-1}$):
|
||||
$$E_k:\mathscr{M}\rightarrow\mathscr{M},\space E_k(m):=E(k,m)$$
|
||||
$$D_k:\mathscr{M}\rightarrow\mathscr{M},\space D_k(m):=D(k,m)$$
|
||||
Die Nachrichtenmenge $\mathscr{M}$ besteht in Realität aus einer endlichen Folgen (Tupeln) von Bit- oder Bytewerten.
|
||||
Diese können entweder die gleiche Länge besitzen (Blockverschlüsselung) oder von beliebiger Länge sein (Stromverschlüsselung).
|
||||
$$\mathscr{M} = \mathscr{L}^n = \{(z_1,z_2,...,z_n)\mid z_i\in\mathscr{L}\} \text{ für ein festes } n\in \mathbb{N} \text{ (Blockverschlüsselung)}$$
|
||||
$$\mathscr{M} = \mathscr{L}^{>0} = \bigcup_{n\in\mathbb{N}} \mathscr{L}^n = \{(z_1,z_2,...,z_n)\mid z_i\in\mathscr{L}, n\in\mathbb{N}\} \text{ (Stromverschlüsselung)}$$
|
||||
|
||||
\section{Schlüsselaustausch}
|
||||
Um mithilfe eines symmetrischen Schlüssels Daten austauschen zu können muss der Schlüssel auf eine Sichere Art und Weise ausgetauscht werden.
|
||||
Dies ist zwar offline möglich, stellt allerdings kein praktikables Verfahren für die Kommunikation im Internet dar.
|
||||
Daher werden für einen sicheren Schlüsselaustausch über eine unsichere Infrastruktur asymmetrische Verschlüsselungsverfahren benötigt.
|
||||
|
||||
\section{Angriffsszenarien}
|
||||
Da für eine effiziente Entschlüsselung einer Geheimtextnachrichten die Kenntnis des Schlüssels essentiell ist versuchen die meisten Angriffe diesen herauszufinden.
|
||||
|
||||
\subsection{Ciphertext-only Angriffe}
|
||||
\begin{tabbing}
|
||||
Vorraussetzung: \= ein oder mehrere Geheimtexte $c_i=E_k(m_i)$ bekannt\\
|
||||
Angriffsziel: \> Bestimmung von $m$ oder von $k$
|
||||
\end{tabbing}
|
||||
|
||||
\subsection{Known-plaintext Angriffe}\label{known-plaintext}
|
||||
\begin{tabbing}
|
||||
Vorraussetzung: \= Geheimtext $c=E_k(m)$ und eine Reihe von bekannten Paaren $(m_i,E_k(m_i))$ sind bekannt\\
|
||||
Angriffsziel: \> Bestimmung von $m$ oder von $k$
|
||||
\end{tabbing}
|
||||
|
||||
\subsection{Chosen-plaintext Angriffe}\label{chosen-plaintext}
|
||||
\begin{tabbing}
|
||||
Vorraussetzung: \= Geheimtext $c=E_k(m)$ und für eine Reihe von beliebig vorgegebenen Klartextnachrichten $m_i$ kann die zugehörige Geheimtextnachricht $c_i=E_k(m_i)$ errechnet werden\\
|
||||
Angriffsziel: \> Bestimmung von $m$ oder von $k$
|
||||
\end{tabbing}
|
||||
|
||||
\section{Brute-Force Angriffe}
|
||||
Da die Vorraussetzung für ein gutes Verschlüsselungsverfahren ist, dass es nicht \textbf{effizient} entschlüsselt werden kann (siehe \ref{kerckhoff}) muss die \textbf{Effizienz} definiert sein.
|
||||
An dieser Stelle setzen Brute-Force Angriffe an.
|
||||
Sie versuchen wie der Name schon sagt mit roher Rechenleistung den Schlüssel zu ermitteln.\\
|
||||
\subsection{Beispiel: Brute-Force Angriff auf $k$}
|
||||
Unter der Annahme, dass ein known-plaintext Angriff (siehe \ref{known-plaintext}) vorliegt lässt sich der Schlüssel bestimmen,
|
||||
indem alle möglichen Schlüssel $k$ des Schlüsselraums $\mathscr{K}$ \say{ausprobiert} werden.
|
||||
Hierbei können für die einzelnen bekannten Klartextnachrichten mehrere Schlüssel passen:
|
||||
$$\mathscr{K}_i := \{\tilde{k}\in \mathscr{K} \mid E_{\tilde{k}}(m_i)=c_i\}$$
|
||||
Bei einer ausreichenden Menge bekannter Klartextnachrichten lässt sich der Schlüssel aus der Schnittmenge der möglichen Schlüssel bestimmen:
|
||||
$$\bigcap_{i=1}^N \mathscr{K}_i=\{k\}$$
|
||||
|
||||
\subsection{Beispiel: Brute-Force Angriff auf $m$}
|
||||
Liegen die Vorraussetzungen für einen chosen-plaintext Angriff vor, so kann die Klartextnachrichten herausgefunden werden,
|
||||
indem alle möglichen Klartextnachrichten $\tilde{m}\in\mathscr{M}$ \say{ausprobiert} werden:
|
||||
$$E_k(\tilde{m}) = c = E_k(m) \Longrightarrow \tilde{m}=m$$
|
||||
|
||||
\subsection{Anforderungen zum Schutz vor Brute-Force}
|
||||
\begin{enumerate}
|
||||
\item Es soll keinen Angriff auf den Schlüssel $k$ geben, der durchschnittlich weniger als $\frac{|\mathscr{K}|}{2}$ Ver- oder Entschlüsselungsoperationen braucht.
|
||||
\item Es soll keinen Angriff auf die Klartextnachricht $m$ geben, der durchschnittlich weniger als $\min \big\{\frac{|\mathscr{K}|}{2}, \frac{|\mathscr{M}|}{2} \big\}$ Ver- oder Entschlüsselungsoperationen braucht.
|
||||
\end{enumerate}
|
||||
|
||||
\section{Wörterbuchangriffe}
|
||||
Nachrichtenpaare (Geheim- und Klarttext), die mithilfe eines Schlüssels $k$ verschlüsselt wurden können in einem Wörterbuch abgespeichert werden.
|
||||
Mithilfe des Wörterbuchs können diese Paare jederzeit entschlüsselt werden.
|
||||
Zudem kann für einen chosen-plaintext Angriff (siehe \ref{chosen-plaintext}) ein Wörterbuch mit häufig vorkommenden Wörtern erstellt werden,
|
||||
sodass eine Entschlüsselung in wesentlich kürzerer Zeit möglich wird.
|
||||
|
||||
\subsection{Schutz vor Wörterbuchangriffen}
|
||||
Um einem Wörterbuchangriff vorzubeugen ist es wichtig, dass sich der Geheimtext bei jeder Verschlüsselung des gleichen Klartextes unterscheidet.
|
||||
Dies wird meist dadurch erreicht, dass die Klartextnachricht vor Anwendung des Verschlüsselungsverfahrens abgeändert wird.
|
||||
Meist wird hierfür eine Nonce (Number used Once) verwendet.
|
||||
Die Nonce kann ein Zähler, ein Zeitstempel oder eine Zufallszahl sein.
|
||||
Der Nonce-Wert muss beiden Seiten bekannt sein.
|
||||
Hierbei kann er entweder mitverschlüsselt übertragen werden oder besteht aus einem Wert (z.B. Zähler), der beiden Seiten bekannt ist.
|
||||
|
||||
\subsubsection{Nonce-Verschlüsselung}
|
||||
Ein Nonce-Wert $v\in\mathscr{M}$ wird dazu genutzt die Klartextnachricht $m\in\mathscr{M}$ abzuändern:
|
||||
$$\tilde{m}=m+v$$
|
||||
Anschließend wird die veränderte Nachricht verschlüsselt:
|
||||
$$c = E_k(\tilde{m})$$
|
||||
Für die Entschlüsselung wird der Schlüssel $k$ und der Nonce-Wert $v$ benötigt:
|
||||
$$m=D_k(c)+v$$
|
||||
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Reference in New Issue
Block a user