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+++ b/Zusammenfassung_DC.toc @@ -65,3 +65,12 @@ \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}% +\contentsline {chapter}{\numberline {5}Stromverschlüsselungsverfahren}{16}% +\contentsline {section}{\numberline {5.1}Synchrone Stromverschlüsselungsverfahren}{16}% +\contentsline {section}{\numberline {5.2}Zustandsabhängige Stromverschlüsselungsverfahren}{17}% +\contentsline {subsection}{\numberline {5.2.1}Additive zustandsabhängige Stromverschlüsselungsverfahren}{18}% +\contentsline {subsubsection}{\numberline {5.2.1.1}Synchrone additive Stromverschlüsselungsverfahren}{18}% +\contentsline {section}{\numberline {5.3}Schlüsselstrom vs. One-Time-Pad}{19}% +\contentsline {section}{\numberline {5.4}Nonces zur Initialisierung eines Schlüsselstromgenerators}{19}% +\contentsline {section}{\numberline {5.5}ChaCha20}{20}% +\contentsline {section}{\numberline {5.6}Cipher-Instanzen: Verschlüsselungsalgorithmen in Java-Laufzeitumgebungen}{21}% diff --git a/chapters/Modulare Arithmetik.tex b/chapters/Modulare Arithmetik.tex index d673a2d..6f2d0bc 100644 --- a/chapters/Modulare Arithmetik.tex +++ b/chapters/Modulare Arithmetik.tex @@ -10,7 +10,7 @@ \section{Der Ring $\mathbb{Z}_n$} Ein Ring $\mathbb{Z}_n$ ist definiert durch: $$\mathbb{Z}_n := {0,1,...,n-1}$$ - \subsection{Addition und Multiplikation} + \subsection{Addition und Multiplikation}\label{modulare_addition} \begin{equation} \begin{aligned} a +_{\mathbb{Z}_n} b :&= (a+b) \mod n\\ diff --git a/chapters/Stromverschlüsselungsverfahren.tex b/chapters/Stromverschlüsselungsverfahren.tex new file mode 100644 index 0000000..06723a4 --- /dev/null +++ b/chapters/Stromverschlüsselungsverfahren.tex @@ -0,0 +1,139 @@ +\chapter{Stromverschlüsselungsverfahren} + Bei einem Stromverschlüsselungsverfahren wird eine Nachricht zeichenweise verschlüsselt. + Die Menge der verschlüsselten Zeichen $\mathscr{M}$ ist als endliche Folge von Elementen einer Zeichenmenge $\mathscr{Z}$ definiert. + $$\mathscr{M} = \mathscr{Z}^{>0} := \bigcup_{l\in \mathbb{N}}\mathscr{Z}^l=\{(m_1,m_2,...,m_l)\mid m_i\in\mathscr{Z},l\in\mathbb{N}\}$$ + Die Verschlüsselungsfunktion $E_k$ lässt sich zeichenweise beschreiben: + $$E^{(i)}:\mathscr{K}\times\mathscr{Z}^i\rightarrow\mathscr{Z}\space\text{für alle }i\in\mathbb{N}$$ + mit: + $$E_k((m_1,m_2,...,m_l))=(E^{(1)}(k,(m_1)),E^{(2)}(k,(m_1,m_2)),...,E^{(l)}(k,(m_1,m_2,...,m_l)))$$ + Bei manchen Verfahren können bei der Berechnung des $i$-ten Geheimtextzeichens $c_i=E^{(i)}(k,(m_1,m_2,...,m_i))$ auch die vorherigen Zeichen miteinfließen. + + \includegraphics{Stromverschlüsselung.png} + + \section{Synchrone Stromverschlüsselungsverfahren} + Bei synchronen Stromverschlüsselungsverfahren hängen die Zeichen einer Geheimtextnachricht nicht von den vorherigen ab. + So ist es möglich die einzelnen Zeichen des Klartextes gleichzeitig (synchron) zu verschlüsseln. + Ein Beispiel hierfür sind monoalphabetische Verschlüsselungsverfahren (siehe \ref{monoalphabet}). + $$E^{(i)}:\mathscr{K}\times\mathscr{Z}\rightarrow\mathscr{Z}$$ + wobei gelten muss: + $$E_k((m_1,m_2,...,m_l))=(E^{(1)}(k,m_1),E^{(2)}(k,m_2),...,E^{(l)}(k,m_l))$$ + + \includegraphics{synchrone Stromverschlüsselung.png} + + \section{Zustandsabhängige Stromverschlüsselungsverfahren} + Bei einer zustandsabhängigen Stromverschlüsselung werden die Zeichen mithilfe eines Zustands verschlüsselt, der sich abhängig vom vorherigen Zeichen ändert. + Hierzu werden benötigt: + $$\begin{aligned} + \mathscr{S}&:\text{Menge der Zustände}\\ + s_0&:\mathscr{K}\rightarrow\mathscr{S}\\ + s&:\mathscr{S}\times\mathscr{Z}\rightarrow\mathscr{S}\\ + E&:\mathscr{S}\times\mathscr{Z}\rightarrow\mathscr{Z} \space \text{(bijektiv)} + \end{aligned}$$ + Mithilfe von $s_0$ und einem Schlüssel $k$ wird ein Startzustand $\sigma_1 := s_0(k)$ festgelegt. + Die folgenden Zustände $\sigma_i$ werden mithilfe der Abbildung $s$ errechnet: + $$\sigma_i := s(\sigma_{i-1},m_{i-1})$$ + Die einzelnen Zeichen lassen sich dann mithilfe von $E$ in Abhängigkeit von dem jeweiligen Zustand bestimmen: + $$c_l := E(\sigma_l,m_l)$$ + + \includegraphics{zustandsabhängige Stromverschlüsselung.png}\label{zustandsabhängige Stromverschlüsselung} + + \subsection{Additive zustandsabhängige Stromverschlüsselungsverfahren} + Additive zustandsabhängige Stromverschlüsselungsverfahren sind zustandsabhängige Stromverschlüsselungsverfahren, mit einer Funktion $S:\mathscr{S}\rightarrow\mathscr{Z}$. + Die Verschlüsselungsfunktion ist hierbei durch $E(\sigma,m) = S(\sigma) + m$ definiert, wobei hierbei eine modulare Addition (siehe \ref{modulare_addition}) über $\mathscr{Z}$ stattfindet. + $s_0$,$s$ und $S$ bilden den Schlüsselstromgenerator, der den folgenden Schlüsselstrom erzeugt: + $$z_i := S(\sigma_i) = S(s(\sigma_{i-1},m_{i-1}))$$ + + \includegraphics{additive zustandsabhängige Stromverschlüsselung.png} + + \subsubsection{Synchrone additive Stromverschlüsselungsverfahren} + eine additive Stromverschlüsselung lässt sich synchron durchführen, falls die Übergangsfunktion $s$ nicht von den Zeichen des Klartextes abhängt. + $$\sigma_{i+1}=s(\sigma_i) \text{ mit } s:\mathscr{S}\rightarrow\mathscr{S}$$ + Hierdurch definiert sich der Schlüsselstrom: + $$z_i=S(s^{i-1}(s_0(k))) \hspace{5mm} (i\in \mathbb{N})$$ + + \includegraphics{synchrone additive Stromverschlüsselung.png} + + \section{Schlüsselstrom vs. One-Time-Pad} + Ein One-Time-Pad (siehe \ref{otp}) ist eine spezielle Form einer additiven Stromverschlüsselung, wobei die Schlüsselstormzeichen $S{\sigma_i}$ direkt aus dem Schlüssel $k$ entnommen werden. + Aufgrund dieser Tatsache kann eine Schlüsselstrom als Ersatz für ein One-Time-Pad betrachtet werden. + Hierbei ist allerdings darauf zu achten, dass folgendende Bedingungen erfüllt sind: + \begin{itemize} + \item Schlüsselstromzeichen dürfen keinen Aufschluss über die nachfolgenden Zeichen geben + \begin{itemize} + \item Schlüsselstromzeichen müssen statistisch gleichverteilt und unkorreliert sein + \item ein Schlüssel $k\in \mathscr{K}$ dient gut zur Initialisierung eines Pseudozufallsgenerators + \end{itemize} + \item die Bestimmung des Schlüssels $k$ darf nicht effizienter als mit einer Brute-Force-Suche (siehe \ref{brute-force}) im Schlüsselraum $\mathscr K$ möglich sein. + \end{itemize} + + \section{Nonces zur Initialisierung eines Schlüsselstromgenerators} + Wie bei einem One-Time-Pad (siehe \ref{otp}) sollte ein Schlüsselstrom nur einmalig verwendet werden. + Da der Austausch von Schlüsseln bei synchronen Verschlüsselungsverfahren allerdings sehr aufwendig ist, wird versucht den Schlüssel weiter zu verwenden. + Hierfür gibt es mehrere Möglichkeiten: + \begin{itemize} + \item Der Schlüsselstrom wird nicht neu initialisiert und durchgehend weiterverwendet (Schlüsselstrom muss zwischen Sender und Empfänger synchron bleiben). + \item Der verwendete Schlüssel wird durch einen Nonce-Wert $\eta_m$ verändert. + \begin{itemize} + \item $\eta_m$ muss mit übertragen werden und zwischen Sender und Empfänger synchron gehalten werden + \item Der Schlüssleraum $\mathscr k$ wird kleiner, da er sich in einen Raum $\mathscr K_{eff}$, in dem $k$ liegt und einen Bereich $\mathscr N$ für den Nonce-Wert aufteilt + \end{itemize} + \item Die Funktion $s_0$ (siehe \ref{zustandsabhängige Stromverschlüsselung}) wird so erweitert, dass sowohl Werte aus $\mathscr K$, als auch aus $\mathscr N$ einfließen: + $$s_0:\mathscr K \times \mathscr N \rightarrow \mathscr S$$ + \includegraphics{Schlüsselstromgenerator mit Nonce.png} + \end{itemize} + + \section{ChaCha20} + Der ChaCha20-Schlüsselstromgenerator nutzt für die Initialisierung des Zustands $s_0$ einen Schlüsselwert $k\in\mathscr K$ und einen Nonce-Wert $\eta\in\mathscr N$. + Für die Beschreibung des Algorithmus werden die folgenden Mengen benötigt: + $$\begin{aligned} + (\text{byte-basiert})\hspace{10mm} \mathscr Z &= \mathbb{Z}_{2^8}\\ + (\text{int-basiert})\hspace{10mm} \mathscr K &= (\mathbb{Z}_{2^{32}})^8\\ + (\text{int-basiert})\hspace{10mm} \mathscr S &= (\mathbb{Z}_{2^{32}})^{16}\times(\mathbb{Z}_{2^{32}})^{16}\times\mathbb{Z}_{2^{38}+1}\\ + (\text{int-basiert})\hspace{10mm} \mathscr N &= (\mathbb{Z}_{2^{32}})^3\\ + \end{aligned}$$ + Elemente im Zustandsraum $\mathscr S$ bestehen aus 3-Tupeln: + $$(s^0,s,c)= (\mathbb{Z}_{2^{32}})^{16}\times(\mathbb{Z}_{2^{32}})^{16}\times\mathbb{Z}_{2^{38}+1}$$ + In $s^0$ werden der Schlüssel $k$ und der Nonce-Wert $\eta$ mit fest definierten Konstanten nach der Initialisierung abgespeichert. + Bei der Initialisierung und nach Ausgabe von jeweils 64 Bytes wird der Wert von $s^0$ nach $s$ kopiert und in einen Block von 64 Schlüsselstrombytes überführt. + $c$ speichert die Anzahl der bereits verwendeten Schlüsselstrombytes (nach Spezifikation max. $2^{38}$ Bytes)\\ + \textbf{Initalisierung:} + $$s_0:\mathscr K \times \mathscr N \rightarrow \mathscr S$$ + daraus ergibt sich: + $$\begin{aligned} + s_0((k,\eta)) =& s_0(((k_1,...,k_8),(n_1,n_2,n_3)))\\ + :=& s(((c_1,c_2,c_3,c_4,k_1,...,k_8,0,n_1,n_2,n_3),(0,...,0),0))\\ + =& s((s^0,0,0)) + \end{aligned}$$ + wobei $c_i$ die folgende Konstanten sind: + $$\begin{aligned} + c_1 =&0x61707865\\ + c_2 =&0x3320646e\\ + c_3 =&0x79622d32\\ + c_4 =&0x6b206574\\ + \end{aligned}$$ + und + $$s^0 := (c_1,c_2,c_3,c_4,k_1,...,k_8,0,n_1,n_2,n_3)$$ + \textbf{Update-Funktion:} $s: \mathscr S \rightarrow\mathscr S$\\ + Solange $c<2^{38}$ gilt: + $$s((s^i,s,c)):=\begin{cases} + (s^i,s,c+1) &\text{falls }c \mod 64 \ne 0\\ + (s^{i+1},s^i\oplus {f_{ib}}^{10}(s^i),c+1) &\text{falls }c\mod 64 = 0 + \end{cases}$$ + $f_{ib}$ wird 10 mal wiederholt und bildet die folgende Abbildung: + $$f_{ib}:(\mathbb{Z}_{2^{32}})^{16}\rightarrow(\mathbb{Z}_{2^{32}})^{16}$$ + Die Funktion ist in RFC 8439 definiert.\\ + \textbf{Extraktion der Schlüsselstromzeichen:} $S;\mathscr S \rightarrow \mathscr S$\\ + Mit der Funktion $S$ werden die Bytewerte, aus denen die Zahlen $s_0,s_1,...,s_{15}$ der zweiten Komponente eines Elements aus $\mathscr S$ zusammengesetzt sind, extrahiert:\\ + Für + $$(s^0,s,c) = (s^0,(s_0,s_1,...,s_{15}),c)\in\mathscr S$$ + werden Hierzu + $$\begin{aligned} + i&=\lfloor c/4 \rfloor \mod 16\\ + j&=c \mod 4 + \end{aligned}$$ + berechnet und + $$S((s^0,(s_0,s_1,...,s_{15}),c)) := \lfloor \frac{s_i}{2^{8j}} \rfloor \mod 2^8$$ + gesetzt. + + \section{Cipher-Instanzen: Verschlüsselungsalgorithmen in Java-Laufzeitumgebungen} + siehe Skript2 1.5 (Seite 19). diff --git a/chapters/Substitutionsverfahren.tex b/chapters/Substitutionsverfahren.tex index 35cada2..830a0e7 100644 --- a/chapters/Substitutionsverfahren.tex +++ b/chapters/Substitutionsverfahren.tex @@ -10,9 +10,9 @@ \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 $z\in \mathscr{L}$: \>$k(z)$\\ + Alphabet: \hspace{35mm}\=$\mathscr{Z} := \{A,B,...,Z\}$\\ + Schlüsselraum: \>$\mathscr{K} := \{k:\mathscr{Z}\rightarrow\mathscr{Z} | k \text{ ist bijektiv}\}$\\ + Verschlüsselung von $z\in \mathscr{Z}$: \>$k(z)$\\ Entschlüsselung: \> $E(z) := k(z)$ \end{tabbing} @@ -35,11 +35,11 @@ \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: \>$\mathscr{K} := \Sigma_\mathscr{L} \times \mathscr{W}$ + Alphabet: \hspace{35mm}\=$\mathscr{Z} := \{A,B,...,Z\}$\\ + Menge aller Wörter: \>$\mathscr{W} = \mathscr{Z}^{>0} := \bigcup_{n\in\mathbb{N}} \mathscr{Z}^n = \{(m_1,m_2,...,m_n)|m_i\in\mathscr{Z},n\in\mathbb{N}\}$\\ + Schlüsselraum: \>$\mathscr{K} := \Sigma_\mathscr{Z} \times \mathscr{W}$ \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 $w$.\\ + Ein Schlüssel $k = (f,w)\in \mathscr{K}$ besteht aus einer Permutation $f\in \Sigma_\mathscr{Z}$ (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. @@ -55,7 +55,7 @@ \subsubsection{Verschlüsselung} \begin{tabbing} - Annahme: \hspace{30mm}\= Klar- und Geheimtext sind eine Folge von Zeichen der Menge $\mathscr{L} = \mathbb{Z}$ ($n\in\mathbb{N}$)\\ + Annahme: \hspace{30mm}\= Klar- und Geheimtext sind eine Folge von Zeichen der Menge $\mathscr{Z} = \mathbb{Z}$ ($n\in\mathbb{N}$)\\ 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: \>$E(m) := m+k := (m_1+k_1,m_2+k_2,...,m_l+k_l)$\\ diff --git a/chapters/Verschlüsselungsverfahren.tex b/chapters/Verschlüsselungsverfahren.tex index c223b96..1906f55 100644 --- a/chapters/Verschlüsselungsverfahren.tex +++ b/chapters/Verschlüsselungsverfahren.tex @@ -17,8 +17,8 @@ $$D_k:\mathscr{M}\rightarrow\mathscr{M},\space D_k(m):=D(k,m)$$ 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). - $$\mathscr{M} = \mathscr{L}^n = \{(z_1,z_2,...,z_n)\mid z_i\in\mathscr{L}\} \text{ für ein festes } n\in \mathbb{N} \text{ (Blockverschlüsselung)}$$ - $$\mathscr{M} = \mathscr{L}^{>0} = \bigcup_{n\in\mathbb{N}} \mathscr{L}^n = \{(z_1,z_2,...,z_n)\mid z_i\in\mathscr{L}, n\in\mathbb{N}\} \text{ (Stromverschlüsselung)}$$ + $$\mathscr{M} = \mathscr{Z}^n = \{(z_1,z_2,...,z_n)\mid z_i\in\mathscr{Z}\} \text{ für ein festes } n\in \mathbb{N} \text{ (Blockverschlüsselung)}$$ + $$\mathscr{M} = \mathscr{Z}^{>0} = \bigcup_{n\in\mathbb{N}} \mathscr{Z}^n = \{(z_1,z_2,...,z_n)\mid z_i\in\mathscr{Z}, n\in\mathbb{N}\} \text{ (Stromverschlüsselung)}$$ \section{Schlüsselaustausch} Um mithilfe eines symmetrischen Schlüssels Daten austauschen zu können muss der Schlüssel auf eine Sichere Art und Weise ausgetauscht werden. @@ -46,7 +46,7 @@ Angriffsziel: \> Bestimmung von $m$ oder von $k$ \end{tabbing} - \section{Brute-Force Angriffe} + \section{Brute-Force Angriffe}\label{brute-force} 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. 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