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ze^yha1Z0Gx;bjE%PCIhExLbCP-w(qwd^mnp{!N4mHj{Mb*BbwPlap|47a_vqjY(&e zF>%%!1n#=&2?KAvf2_}Vu2xXNYq8RHe`fRH@pO5$?deFj?fzo*@A;zqhlhvXtgDZa zPj|~1ZBKs!Grr#b*7bT=UwwWm+Wt=Net6nHYJR$1WLFM?& znt=)u+qRr@PMkf)9^J^%-BSe1RVx=@~F#kCUulZ|k3{Gt91f SxjkKeMdH7_rBet*eepku3^Awx diff --git a/Zusammenfassung_DC.tex b/Zusammenfassung_DC.tex index dc185c8..10f3705 100644 --- a/Zusammenfassung_DC.tex +++ b/Zusammenfassung_DC.tex @@ -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} diff --git a/Zusammenfassung_DC.toc b/Zusammenfassung_DC.toc index 3b39c1c..074ee9d 100644 --- a/Zusammenfassung_DC.toc +++ b/Zusammenfassung_DC.toc @@ -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}% diff --git a/chapters/Verschlüsselungsverfahren.tex b/chapters/Verschlüsselungsverfahren.tex new file mode 100644 index 0000000..c223b96 --- /dev/null +++ b/chapters/Verschlüsselungsverfahren.tex @@ -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: + $$m=D_k(c)+v$$ \ No newline at end of file