From 23f91641edd6565624855dcfc8ea69d51f43fc5b Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Paul=20L=C3=B6dige?= Date: Wed, 16 Sep 2020 18:11:00 +0200 Subject: [PATCH] =?UTF-8?q?finished=20chapter=20Blockverschl=C3=BCsselungs?= =?UTF-8?q?verfahren?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Zusammenfassung_DC.aux | 9 +- Zusammenfassung_DC.fdb_latexmk | 34 ++++--- Zusammenfassung_DC.fls | 43 +++++++- Zusammenfassung_DC.log | 51 +++++++--- Zusammenfassung_DC.pdf | Bin 2512320 -> 2599251 bytes Zusammenfassung_DC.synctex.gz | Bin 162900 -> 181403 bytes Zusammenfassung_DC.tex | 2 + Zusammenfassung_DC.toc | 6 +- chapters/Blockverschlüsselungsverfahren.tex | 104 +++++++++++++++++++- chapters/Endliche Körper.tex | 1 + images/AES S-Box.png | Bin 0 -> 23090 bytes images/meet-in-the-middle.png | Bin 0 -> 13863 bytes 12 files changed, 220 insertions(+), 30 deletions(-) create mode 100644 chapters/Endliche Körper.tex create mode 100644 images/AES 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    diff --git a/Zusammenfassung_DC.tex b/Zusammenfassung_DC.tex
    index 39fbf58..42af0a8 100644
    --- a/Zusammenfassung_DC.tex
    +++ b/Zusammenfassung_DC.tex
    @@ -55,4 +55,6 @@ rightsub = \grq%
     
     \input{chapters/Blockverschlüsselungsverfahren.tex}
     
    +\input{chapters/Endliche Körper.tex}
    +
     \end{document}
    diff --git a/Zusammenfassung_DC.toc b/Zusammenfassung_DC.toc
    index f167a28..e7fe1a3 100644
    --- a/Zusammenfassung_DC.toc
    +++ b/Zusammenfassung_DC.toc
    @@ -89,5 +89,9 @@
     \contentsline {subsection}{\numberline {6.2.6}CFB (Cipher-Feedback)}{26}%
     \contentsline {subsubsection}{\numberline {6.2.6.1}CFB-8$n$}{26}%
     \contentsline {section}{\numberline {6.3}Konstruktionsprinzipien von Blockverschlüsselungsverfahren}{27}%
    -\contentsline {section}{\numberline {6.4}DES und Tripple-DES}{27}%
    +\contentsline {section}{\numberline {6.4}DES}{27}%
     \contentsline {subsection}{\numberline {6.4.1}Triple-DES (3DES)}{27}%
    +\contentsline {section}{\numberline {6.5}Meet-in-the-Middle-Angriff}{28}%
    +\contentsline {section}{\numberline {6.6}AES (Advanced Encryption Standard)}{29}%
    +\contentsline {subsection}{\numberline {6.6.1}AES-128}{29}%
    +\contentsline {chapter}{\numberline {7}Endliche Körper $\mathbb {F}_{p^n}$}{31}%
    diff --git a/chapters/Blockverschlüsselungsverfahren.tex b/chapters/Blockverschlüsselungsverfahren.tex
    index 2a3d0c4..705e5db 100644
    --- a/chapters/Blockverschlüsselungsverfahren.tex
    +++ b/chapters/Blockverschlüsselungsverfahren.tex
    @@ -116,7 +116,7 @@
             Aus ihm werden Rundenschlüssel $k_i$ mit $i\in\{1,...,r\}$ abgeleitet, welche bei jeder Runde mit in die Berechnung einfließen.
             Bei einer Entschlüsselung wird das durch die Runden beschriebene Verfahren einfach rückwärts angewandt.
     
    -    \section{DES und Tripple-DES}
    +    \section{DES}
             Das einfach DES wurde 1977 veröffentlicht und 2005 zurückgezogen.
             Sowohl die Blöcke als auch die Schlüssel setzten sich jeweils aus 8 Bytes zusammen:
             $$\begin{aligned}
    @@ -141,4 +141,104 @@
                 \end{aligned}$$ 
                 Die 3-DES Entschlüsselungsfunktion definiert sich durch:
                 $$E((k^{(1)},k^{(2)},m)) := E_{DES}\left(k^{(1)},D_{DES}\left(k^{(2)},E_{DES}\left(k^{(1)},m \right)\right)\right)$$
    -            \includegraphics[scale=0.8]{3DES.png}
    \ No newline at end of file
    +            \includegraphics[scale=0.8]{3DES.png}
    +
    +    \section{Meet-in-the-Middle-Angriff}
    +        Aufgrund des Meet-in-the-Middle-Angriffs führt eine \textbf{zweifache} Hintereinanderreihung von Verschlüsselungsfunktionen nur zu einer kleinen Vergrößerung des Schlüsselraums.
    +        Bei diesem Angriff handelt es sich um einen known-plaintext-Angriff (siehe \ref{known-plaintext}).
    +        Es wird angenommen, dass es ein $2E_{k_1,k_2}$ verfahren gibt, dass sich wie folgt definiert:\\
    +        \includegraphics{meet-in-the-middle.png}\\
    +        Es wird zudem angenommen, dass genügend Speicher zur Verfügung steht um alle $\tilde{k}_1\in\mathscr K$ zusammen mit den jeweiligen $c_{\tilde{k}_1}(m_1):=E(\tilde{k}_1,m_1)$ in eine Tabelle zu schreiben:
    +        \begin{center}
    +            \begin{tabular}{c|c}
    +                $\tilde{k}_1$ & $c_{\tilde{k}_1}(m_1):=E(\tilde{k}_1,m_1)$\\
    +                \hline
    +                0101010101010101 & 8A549EC56733AB66\\
    +                0101010101010102 & 653148AE6B688132\\
    +                \vdots & \vdots \\
    +                FEFEFEFEFEFEFEFE & CE55464B6485684E
    +            \end{tabular}
    +        \end{center}
    +        Anschließend wird die Tabelle nach der zweiten Spalte sortiert.
    +        Beim Ausprobieren alle Möglichkeiten von $\tilde{k}_2 \in\mathscr K$ kann für $D_{\tilde{k}_2}(c_1)$ nachgeschlagen werden, ob sich dieser in der Tabelle befindet.
    +        Hierdurch ergeben sich dann $(\tilde{k}_1,\tilde{k}_2)$ Paare, für die gilt:
    +        $$2E_{\tilde{k}_1,\tilde{k}_2}(m_{\color{red}{\mathbf{1}}})=c_{\color{red}{\mathbf{1}}}$$
    +        Durchschnittlich ist mit $1+\frac{|\mathscr K|}{|\mathscr M|}$ solcher Paare zu rechnen.
    +        Die Paare können nun durch Ausprobieren der anderen bekannten Nachrichten $\{m_i\setminus m_1\}$ schnell ausgeschlossen werden.
    +
    +    \section{AES (Advanced Encryption Standard)}
    +        Der AES ist für Schlüssel mit den Bitlängen 128, 192 und 256 definiert.
    +        Im folgenden wird AES-128 als Synonym für alle möglichen Schlüssellängen verwendet.
    +
    +        \subsection{AES-128}
    +            Sowohl die Blöcke als auch die Schlüssel setzen sich aus jeweils 16 Bytes zusammen:
    +            $$\begin{aligned}
    +                \mathscr M =& ({\mathbb{Z}_2}^8)^{16}=\{(m_1,...m_{16})\mid m_i\in{\mathbb{Z}_2}^8\}\\
    +                \mathscr K =& ({\mathbb{Z}_2}^8)^{16}=\{(k_1,...k_{16})\mid k_i\in{\mathbb{Z}_2}^8\}\\
    +            \end{aligned}$$
    +            Blöcke werden in Form von Matrizen dargestellt, die als State-Array bezeichnet werden:
    +            $$S=\begin{pmatrix}
    +                s_{0,0} & s_{0,1} &s_{0,2} &s_{0,3}\\
    +                s_{1,0} & s_{1,1} &s_{1,2} &s_{1,3}\\
    +                s_{2,0} & s_{2,1} &s_{2,2} &s_{2,3}\\
    +                s_{3,0} & s_{3,1} &s_{3,2} &s_{3,3}\\
    +            \end{pmatrix}:=\begin{pmatrix}
    +                m_1 &m_5 & m_9 & m_{13}\\
    +                m_2 &m_6 & m_{10} & m_{14}\\
    +                m_3 &m_7 & m_{11} & m_{15}\\
    +                m_4 & m_8 & m{12} & m_{16}
    +            \end{pmatrix}$$
    +            \vspace{5mm}
    +            Im AES gibt es 4 elementare Operationen auf den State-Arrays:
    +            \begin{enumerate}
    +                \item SubBytes()\\
    +                    Es wird auf Basis der folgenden Tabelle eine Substitution von jedem Element des State-Arrays durchgeführt.\\
    +                    \includegraphics[scale=0.8]{AES S-Box.png}
    +                \item ShiftRows()\\
    +                    auf dem S-Array  wird die folgende Transposition durchgeführt:
    +                    $$\begin{pmatrix}
    +                        s_{0,0} & s_{0,1} &s_{0,2} &s_{0,3}\\
    +                        s_{1,0} & s_{1,1} &s_{1,2} &s_{1,3}\\
    +                        s_{2,0} & s_{2,1} &s_{2,2} &s_{2,3}\\
    +                        s_{3,0} & s_{3,1} &s_{3,2} &s_{3,3}\\
    +                    \end{pmatrix}
    +                    \mapsto
    +                    \begin{pmatrix}
    +                        s_{0,0} & s_{0,1} &s_{0,2} &s_{0,3}\\
    +                        s_{1,1} & s_{1,2} &s_{1,3} &s_{1,0}\\
    +                        s_{2,2} & s_{2,3} &s_{2,0} &s_{2,1}\\
    +                        s_{3,3} & s_{3,0} &s_{3,1} &s_{3,2}\\
    +                    \end{pmatrix}$$
    +                \item MixColumns()\\
    +                    auf dem S-Array wird eine Substitution ausgeführt, die durch die folgende Multiplikation definiert ist:
    +                    $$S\mapsto M\cdot S\hspace{5mm} \text{mit }M=
    +                    \begin{pmatrix}
    +                        02 & 03& 01& 01\\
    +                        01 & 02 & 03 & 01 \\
    +                        01 & 01 & 02 & 03 \\
    +                        03 & 02 & 01 & 02 \\
    +                    \end{pmatrix}$$
    +                    Hierbei ist wichtig anzumerken, dass die Multiplikation nicht in $\mathbf{Z}_{256}$ berechnet wird.
    +                    Stattdessen wird sind die Elemente als Elemente des Körpers $\mathbf{F}_{256}$ aufzufassen (siehe \ref{endliche Körper})
    +                \item AddRoundKey()\\
    +                    Es wird eine XOR-Operation der Einträge des S-Arrays mit den entsprechenden Einträgen aus \textbf{K} durchgeführt.
    +                    Dies entspricht einer Matrixaddition in $\mathbb{F}_{256}$:
    +                    $$ \textbf{S}\mapsto\textbf{K}+\textbf{S}$$
    +            \end{enumerate}
    +            \vspace{2mm}
    +            AES-128 führt die zuvor beschriebenen Schritte nach dem folgenden Algorithmus aus:
    +            \texttt{
    +            \begin{tabbing}
    +                AES\_ENCRYPT(\textbf{S},$\textbf{K}_0,...,\textbf{K}_10$)\{\\
    +                \hspace{5mm}\=AddRoundKey(\textbf{S},$\textbf{K}_0$)\\
    +                \>for( int i = 1; i < 10; ++i)\{\\
    +                \>\hspace{5mm}\=SubBytes(\textbf{S});\\
    +                \>\>ShiftRows(\textbf{S});\\
    +                \>\>MixColumns(\textbf{S});\\
    +                \>\>AddRoundKey(\textbf{S},$\textbf{K}_0$)\\
    +                \>\}\\
    +                \>SubBytes(\textbf{S});\\
    +                \>ShiftRows(\textbf{S});\\
    +                \>AddRoundKey(\textbf{S},$\textbf{K}_0$)\\
    +                \}
    +            \end{tabbing}}
    \ No newline at end of file
    diff --git a/chapters/Endliche Körper.tex b/chapters/Endliche Körper.tex
    new file mode 100644
    index 0000000..787ae9e
    --- /dev/null
    +++ b/chapters/Endliche Körper.tex	
    @@ -0,0 +1 @@
    +\chapter{Endliche Körper $\mathbb{F}_{p^n}$} \label{endliche Körper}
    \ No newline at end of file
    diff --git a/images/AES S-Box.png b/images/AES S-Box.png
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