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provisional version of Modulare Arithmetik - Teil 2 finished

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Inhalt.tex Normal file
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\newmdenv[roundcorner=2mm, linecolor=blue, backgroundcolor=blue!5]{mybox}
\input{chapters/Substitutionsverfahren.tex}
\input{chapters/Modulare Arithmetik.tex}
\input{chapters/IT-Sicherheit.tex}
\input{chapters/Verschlüsselungsverfahren.tex}
\input{chapters/Stromverschlüsselungsverfahren.tex}
\input{chapters/Blockverschlüsselungsverfahren.tex}
\input{chapters/Hashfunktionen.tex}
\input{chapters/MAC-Verfahren.tex}
\input{chapters/Modulare Arithmetik - Teil 2.tex}

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Packages.tex Normal file
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%numbering depth
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%automatic links
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%graphics
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%boxes
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%enumeration
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Zusammenfassung_DC.aux
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\BOOKMARK [0][-]{chapter.1}{Substitionsverfahren}{}% 1
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\BOOKMARK [0][-]{chapter.2}{Modulare Arithmetik}{}% 11
\BOOKMARK [1][-]{section.2.1}{Exkurs: Division mit Rest}{chapter.2}% 12
\BOOKMARK [1][-]{section.2.2}{Der Ring Zn}{chapter.2}% 13
\BOOKMARK [2][-]{subsection.2.2.1}{Addition und Multiplikation}{section.2.2}% 14
\BOOKMARK [2][-]{subsection.2.2.2}{Subtraktion}{section.2.2}% 15
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\BOOKMARK [2][-]{subsection.2.2.5}{Matrizen}{section.2.2}% 18
\BOOKMARK [2][-]{subsection.2.2.6}{Kartesisches Produkt von Ringen}{section.2.2}% 19
\BOOKMARK [2][-]{subsection.2.2.7}{Isomorphie von Ringen}{section.2.2}% 20
\BOOKMARK [1][-]{section.2.3}{Der erweiterte Euklid'sche Algorithmus}{chapter.2}% 21
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\BOOKMARK [2][-]{subsection.9.2.1}{Z2e*}{section.9.2}% 102
\BOOKMARK [1][-]{section.9.3}{Der chinesische Restsatz}{chapter.9}% 103
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\documentclass[a4paper,11pt,oneside]{report}
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%load packages
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%opening
\title{Zusammenfassung DC}
\author{Paul Lödige \\ Matrikel: 15405036}
@ -42,20 +18,7 @@ rightsub = \grq%
\tableofcontents
\pagebreak
\input{chapters/Substitutionsverfahren.tex}
\input{chapters/Modulare Arithmetik.tex}
\input{chapters/IT-Sicherheit.tex}
\input{chapters/Verschlüsselungsverfahren.tex}
\input{chapters/Stromverschlüsselungsverfahren.tex}
\input{chapters/Blockverschlüsselungsverfahren.tex}
\input{chapters/Hashfunktionen.tex}
\input{chapters/MAC-Verfahren.tex}
%load content
\input{Inhalt.tex}
\end{document}

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\babel@toc {ngerman}{}
\contentsline {chapter}{\numberline {1}Substitionsverfahren}{4}%
\contentsline {section}{\numberline {1.1}Skytale}{4}%
\contentsline {section}{\numberline {1.2}Monoalphabetische Substitionsverfahren}{4}%
\contentsline {subsection}{\numberline {1.2.1}Caesar-Verschlüsselung}{5}%
\contentsline {subsection}{\numberline {1.2.2}Häufigkeitsanalyse}{5}%
\contentsline {section}{\numberline {1.3}Polyalphabetische Substitutionsverfahren}{5}%
\contentsline {subsection}{\numberline {1.3.1}Vignère-Verfahren}{5}%
\contentsline {subsubsection}{\numberline {1.3.1.1}Verschlüsselung}{5}%
\contentsline {subsection}{\numberline {1.3.2}One-Time-Pad}{6}%
\contentsline {subsubsection}{\numberline {1.3.2.1}Verschlüsselung}{6}%
\contentsline {subsubsection}{\numberline {1.3.2.2}Perfekte Sicherheit}{6}%
\contentsline {section}{\numberline {1.4}algebraische Substitionsverfahren}{6}%
\contentsline {subsection}{\numberline {1.4.1}Hill-Verfahren}{6}%
\contentsline {subsubsection}{\numberline {1.4.1.1}Verschlüsselung}{6}%
\contentsline {subsubsection}{\numberline {1.4.1.2}Entschlüsselung}{6}%
\contentsline {chapter}{\numberline {2}Modulare Arithmetik}{7}%
\contentsline {section}{\numberline {2.1}Exkurs: Division mit Rest}{7}%
\contentsline {section}{\numberline {2.2}Der Ring $\mathbb {Z}_n$}{7}%
\contentsline {subsection}{\numberline {2.2.1}Addition und Multiplikation}{7}%
\contentsline {subsubsection}{\numberline {2.2.1.1}Inverse bezüglich der Addition}{7}%
\contentsline {subsubsection}{\numberline {2.2.1.2}Inverse bezüglich der Multiplikation}{7}%
\contentsline {subsection}{\numberline {2.2.2}Subtraktion}{8}%
\contentsline {subsection}{\numberline {2.2.3}Teiler, Vielfache}{8}%
\contentsline {subsubsection}{\numberline {2.2.3.1}Teilerregeln}{8}%
\contentsline {subsection}{\numberline {2.2.4}Kongruenz}{8}%
\contentsline {subsection}{\numberline {2.2.5}Matrizen}{8}%
\contentsline {subsubsection}{\numberline {2.2.5.1}Determinantenberechnung}{8}%
\contentsline {subsubsection}{\numberline {2.2.5.2}Inverse Matrix}{8}%
\contentsline {section}{\numberline {2.3}Der erweiterte Euklid'sche Algorithmus}{8}%
\contentsline {subsection}{\numberline {2.3.1}Euklid'scher Algorithmus}{9}%
\contentsline {subsection}{\numberline {2.3.2}erweiterter Euklid'scher Algorithmus}{9}%
\contentsline {subsubsection}{\numberline {2.3.2.1}Beispiel}{9}%
\contentsline {section}{\numberline {2.4}Euler'sche $\varphi $-Funktion}{10}%
\contentsline {subsection}{\numberline {2.4.1}$\varphi $-Funktion und Primzahlen}{10}%
\contentsline {chapter}{\numberline {3}IT-Sicherheit: Gefährdungen und Maßnahmen}{11}%
\contentsline {section}{\numberline {3.1}Vertraulichkeit}{11}%
\contentsline {subsection}{\numberline {3.1.1}Schutzmaßnahmen: Verschlüsselungsverfahren}{11}%
\contentsline {section}{\numberline {3.2}Integrität}{11}%
\contentsline {subsection}{\numberline {3.2.1}Schutzmaßnahme: Hashfunktionen, Whitelists}{12}%
\contentsline {section}{\numberline {3.3}Authenzität der Daten}{12}%
\contentsline {subsection}{\numberline {3.3.1}Schutzmaßnahme: Signaturen}{12}%
\contentsline {subsection}{\numberline {3.3.2}Schutz vor Replay-Angriffen}{12}%
\contentsline {section}{\numberline {3.4}Authenzität von Nutzern}{12}%
\contentsline {subsection}{\numberline {3.4.1}Schutzmaßnahmen}{12}%
\contentsline {section}{\numberline {3.5}Zugriffskontrolle}{12}%
\contentsline {subsection}{\numberline {3.5.1}Schutzmaßname: Zugriffskontrollsystem}{13}%
\contentsline {section}{\numberline {3.6}Nichtabstreitbarkeit, Verbindlichkeit}{13}%
\contentsline {subsection}{\numberline {3.6.1}Schutzmaßname: Signaturen und PKI}{13}%
\contentsline {section}{\numberline {3.7}Verfügbarkeit}{13}%
\contentsline {subsection}{\numberline {3.7.1}Schutzmaßnahmen}{13}%
\contentsline {section}{\numberline {3.8}Anonymität}{13}%
\contentsline {chapter}{\numberline {4}Verschlüsselungsverfahren}{14}%
\contentsline {section}{\numberline {4.1}Das Kerckhoffs'sche Prinzip}{14}%
\contentsline {section}{\numberline {4.2}Mathematische Modelierung von Verschlüsselungsverfahren}{14}%
\contentsline {section}{\numberline {4.3}Schlüsselaustausch}{14}%
\contentsline {section}{\numberline {4.4}Angriffsszenarien}{15}%
\contentsline {subsection}{\numberline {4.4.1}Ciphertext-only Angriffe}{15}%
\contentsline {subsection}{\numberline {4.4.2}Known-plaintext Angriffe}{15}%
\contentsline {subsection}{\numberline {4.4.3}Chosen-plaintext Angriffe}{15}%
\contentsline {section}{\numberline {4.5}Brute-Force Angriffe}{15}%
\contentsline {subsection}{\numberline {4.5.1}Beispiel: Brute-Force Angriff auf $k$}{15}%
\contentsline {subsection}{\numberline {4.5.2}Beispiel: Brute-Force Angriff auf $m$}{15}%
\contentsline {subsection}{\numberline {4.5.3}Anforderungen zum Schutz vor Brute-Force}{15}%
\contentsline {section}{\numberline {4.6}Wörterbuchangriffe}{16}%
\contentsline {subsection}{\numberline {4.6.1}Schutz vor Wörterbuchangriffen}{16}%
\contentsline {subsubsection}{\numberline {4.6.1.1}Nonce-Verschlüsselung}{16}%
\contentsline {chapter}{\numberline {5}Stromverschlüsselungsverfahren}{17}%
\contentsline {section}{\numberline {5.1}Synchrone Stromverschlüsselungsverfahren}{17}%
\contentsline {section}{\numberline {5.2}Zustandsabhängige Stromverschlüsselungsverfahren}{18}%
\contentsline {subsection}{\numberline {5.2.1}Additive zustandsabhängige Stromverschlüsselungsverfahren}{19}%
\contentsline {subsubsection}{\numberline {5.2.1.1}Synchrone additive Stromverschlüsselungsverfahren}{19}%
\contentsline {section}{\numberline {5.3}Schlüsselstrom vs. One-Time-Pad}{20}%
\contentsline {section}{\numberline {5.4}Nonces zur Initialisierung eines Schlüsselstromgenerators}{20}%
\contentsline {section}{\numberline {5.5}ChaCha20}{21}%
\contentsline {section}{\numberline {5.6}Cipher-Instanzen: Verschlüsselungsalgorithmen in Java-Laufzeitumgebungen}{22}%
\contentsline {chapter}{\numberline {6}Blockverschlüsselungsverfahren}{23}%
\contentsline {section}{\numberline {6.1}Padding-Verfahren}{23}%
\contentsline {section}{\numberline {6.2}Betriebsmodi}{23}%
\contentsline {subsection}{\numberline {6.2.1}ECB (Electronic Code Book)}{23}%
\contentsline {subsection}{\numberline {6.2.2}CBC (Cipher Block Chaining)}{24}%
\contentsline {subsection}{\numberline {6.2.3}CBC-CS (Chiphertext Stealing for CBC Mode)}{25}%
\contentsline {subsubsection}{\numberline {6.2.3.1}CBC-CS1}{25}%
\contentsline {subsubsection}{\numberline {6.2.3.2}CBC-CS2}{26}%
\contentsline {subsubsection}{\numberline {6.2.3.3}CBC-CS3}{26}%
\contentsline {subsection}{\numberline {6.2.4}CTR (Counter)}{26}%
\contentsline {subsection}{\numberline {6.2.5}OFB (Output Feedback)}{26}%
\contentsline {subsubsection}{\numberline {6.2.5.1}OFB-8$n$}{27}%
\contentsline {subsection}{\numberline {6.2.6}CFB (Cipher-Feedback)}{27}%
\contentsline {subsubsection}{\numberline {6.2.6.1}CFB-8$n$}{27}%
\contentsline {section}{\numberline {6.3}Konstruktionsprinzipien von Blockverschlüsselungsverfahren}{28}%
\contentsline {section}{\numberline {6.4}DES}{28}%
\contentsline {subsection}{\numberline {6.4.1}Triple-DES (3DES)}{28}%
\contentsline {section}{\numberline {6.5}Meet-in-the-Middle-Angriff}{29}%
\contentsline {section}{\numberline {6.6}AES (Advanced Encryption Standard)}{30}%
\contentsline {subsection}{\numberline {6.6.1}AES-128}{30}%
\contentsline {chapter}{\numberline {7}Hashfunktionen}{32}%
\contentsline {section}{\numberline {7.1}schwache Kollisionsfreiheit}{32}%
\contentsline {section}{\numberline {7.2}\texttt {MessageDigest}-Instanzen: Hashfunktionen in Java}{32}%
\contentsline {section}{\numberline {7.3}Anwendungsbeispiele}{33}%
\contentsline {subsection}{\numberline {7.3.1}Anwendungsbeispiel: Passwortdatei}{33}%
\contentsline {subsubsection}{\numberline {7.3.1.1}Anwendungsbeispiel: Passwortdatei mit Salt und Iteration Count}{33}%
\contentsline {subsection}{\numberline {7.3.2}Anwendungsbeispiel: Integritätsschutz von Dateien}{33}%
\contentsline {subsection}{\numberline {7.3.3}Anwendungsbeispiel: Integritätsschutz bei einem Dateidownload}{33}%
\contentsline {section}{\numberline {7.4}Brute-Force-Angriffe auf Hashfunktionen}{33}%
\contentsline {subsection}{\numberline {7.4.1}Brute-Force-Urbildsuche}{33}%
\contentsline {subsection}{\numberline {7.4.2}Brute-Force-Kollisionssuche}{34}%
\contentsline {section}{\numberline {7.5}Konstruktionsverfahren von Hashfunktionen}{34}%
\contentsline {chapter}{\numberline {8}MAC-Verfahren}{36}%
\contentsline {section}{\numberline {8.1}HMAC}{36}%
\contentsline {section}{\numberline {8.2}CMAC, CBC-MAC}{36}%
\contentsline {subsection}{\numberline {8.2.1}CMAC}{36}%
\contentsline {subsection}{\numberline {8.2.2}CBC-MAC}{37}%
\contentsline {chapter}{\numberline {1}Substitionsverfahren}{4}{chapter.1}%
\contentsline {section}{\numberline {1.1}Skytale}{4}{section.1.1}%
\contentsline {section}{\numberline {1.2}Monoalphabetische Substitionsverfahren}{4}{section.1.2}%
\contentsline {subsection}{\numberline {1.2.1}Caesar-Verschlüsselung}{5}{subsection.1.2.1}%
\contentsline {subsection}{\numberline {1.2.2}Häufigkeitsanalyse}{5}{subsection.1.2.2}%
\contentsline {section}{\numberline {1.3}Polyalphabetische Substitutionsverfahren}{5}{section.1.3}%
\contentsline {subsection}{\numberline {1.3.1}Vignère-Verfahren}{5}{subsection.1.3.1}%
\contentsline {subsubsection}{\numberline {1.3.1.1}Verschlüsselung}{5}{subsubsection.1.3.1.1}%
\contentsline {subsection}{\numberline {1.3.2}One-Time-Pad}{6}{subsection.1.3.2}%
\contentsline {subsubsection}{\numberline {1.3.2.1}Verschlüsselung}{6}{subsubsection.1.3.2.1}%
\contentsline {subsubsection}{\numberline {1.3.2.2}Perfekte Sicherheit}{6}{subsubsection.1.3.2.2}%
\contentsline {section}{\numberline {1.4}algebraische Substitionsverfahren}{6}{section.1.4}%
\contentsline {subsection}{\numberline {1.4.1}Hill-Verfahren}{6}{subsection.1.4.1}%
\contentsline {subsubsection}{\numberline {1.4.1.1}Verschlüsselung}{6}{subsubsection.1.4.1.1}%
\contentsline {subsubsection}{\numberline {1.4.1.2}Entschlüsselung}{6}{subsubsection.1.4.1.2}%
\contentsline {chapter}{\numberline {2}Modulare Arithmetik}{7}{chapter.2}%
\contentsline {section}{\numberline {2.1}Exkurs: Division mit Rest}{7}{section.2.1}%
\contentsline {section}{\numberline {2.2}Der Ring $\mathbb {Z}_n$}{7}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}Addition und Multiplikation}{7}{subsection.2.2.1}%
\contentsline {subsubsection}{\numberline {2.2.1.1}Inverse bezüglich der Addition}{7}{subsubsection.2.2.1.1}%
\contentsline {subsubsection}{\numberline {2.2.1.2}Inverse bezüglich der Multiplikation}{7}{subsubsection.2.2.1.2}%
\contentsline {subsection}{\numberline {2.2.2}Subtraktion}{8}{subsection.2.2.2}%
\contentsline {subsection}{\numberline {2.2.3}Teiler, Vielfache}{8}{subsection.2.2.3}%
\contentsline {subsubsection}{\numberline {2.2.3.1}Teilerregeln}{8}{subsubsection.2.2.3.1}%
\contentsline {subsection}{\numberline {2.2.4}Kongruenz}{8}{subsection.2.2.4}%
\contentsline {subsection}{\numberline {2.2.5}Matrizen}{8}{subsection.2.2.5}%
\contentsline {subsubsection}{\numberline {2.2.5.1}Determinantenberechnung}{8}{subsubsection.2.2.5.1}%
\contentsline {subsubsection}{\numberline {2.2.5.2}Inverse Matrix}{8}{subsubsection.2.2.5.2}%
\contentsline {subsection}{\numberline {2.2.6}Kartesisches Produkt von Ringen}{8}{subsection.2.2.6}%
\contentsline {subsection}{\numberline {2.2.7}Isomorphie von Ringen}{9}{subsection.2.2.7}%
\contentsline {section}{\numberline {2.3}Der erweiterte Euklid'sche Algorithmus}{9}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}Euklid'scher Algorithmus}{9}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}erweiterter Euklid'scher Algorithmus}{9}{subsection.2.3.2}%
\contentsline {subsubsection}{\numberline {2.3.2.1}Beispiel}{9}{subsubsection.2.3.2.1}%
\contentsline {section}{\numberline {2.4}Euler'sche $\varphi $-Funktion}{10}{section.2.4}%
\contentsline {subsection}{\numberline {2.4.1}$\varphi $-Funktion und Primzahlen}{10}{subsection.2.4.1}%
\contentsline {chapter}{\numberline {3}IT-Sicherheit: Gefährdungen und Maßnahmen}{11}{chapter.3}%
\contentsline {section}{\numberline {3.1}Vertraulichkeit}{11}{section.3.1}%
\contentsline {subsection}{\numberline {3.1.1}Schutzmaßnahmen: Verschlüsselungsverfahren}{11}{subsection.3.1.1}%
\contentsline {section}{\numberline {3.2}Integrität}{11}{section.3.2}%
\contentsline {subsection}{\numberline {3.2.1}Schutzmaßnahme: Hashfunktionen, Whitelists}{12}{subsection.3.2.1}%
\contentsline {section}{\numberline {3.3}Authenzität der Daten}{12}{section.3.3}%
\contentsline {subsection}{\numberline {3.3.1}Schutzmaßnahme: Signaturen}{12}{subsection.3.3.1}%
\contentsline {subsection}{\numberline {3.3.2}Schutz vor Replay-Angriffen}{12}{subsection.3.3.2}%
\contentsline {section}{\numberline {3.4}Authenzität von Nutzern}{12}{section.3.4}%
\contentsline {subsection}{\numberline {3.4.1}Schutzmaßnahmen}{12}{subsection.3.4.1}%
\contentsline {section}{\numberline {3.5}Zugriffskontrolle}{12}{section.3.5}%
\contentsline {subsection}{\numberline {3.5.1}Schutzmaßname: Zugriffskontrollsystem}{13}{subsection.3.5.1}%
\contentsline {section}{\numberline {3.6}Nichtabstreitbarkeit, Verbindlichkeit}{13}{section.3.6}%
\contentsline {subsection}{\numberline {3.6.1}Schutzmaßname: Signaturen und PKI}{13}{subsection.3.6.1}%
\contentsline {section}{\numberline {3.7}Verfügbarkeit}{13}{section.3.7}%
\contentsline {subsection}{\numberline {3.7.1}Schutzmaßnahmen}{13}{subsection.3.7.1}%
\contentsline {section}{\numberline {3.8}Anonymität}{13}{section.3.8}%
\contentsline {chapter}{\numberline {4}Verschlüsselungsverfahren}{14}{chapter.4}%
\contentsline {section}{\numberline {4.1}Das Kerckhoffs'sche Prinzip}{14}{section.4.1}%
\contentsline {section}{\numberline {4.2}Mathematische Modelierung von Verschlüsselungsverfahren}{14}{section.4.2}%
\contentsline {section}{\numberline {4.3}Schlüsselaustausch}{14}{section.4.3}%
\contentsline {section}{\numberline {4.4}Angriffsszenarien}{15}{section.4.4}%
\contentsline {subsection}{\numberline {4.4.1}Ciphertext-only Angriffe}{15}{subsection.4.4.1}%
\contentsline {subsection}{\numberline {4.4.2}Known-plaintext Angriffe}{15}{subsection.4.4.2}%
\contentsline {subsection}{\numberline {4.4.3}Chosen-plaintext Angriffe}{15}{subsection.4.4.3}%
\contentsline {section}{\numberline {4.5}Brute-Force Angriffe}{15}{section.4.5}%
\contentsline {subsection}{\numberline {4.5.1}Beispiel: Brute-Force Angriff auf $k$}{15}{subsection.4.5.1}%
\contentsline {subsection}{\numberline {4.5.2}Beispiel: Brute-Force Angriff auf $m$}{15}{subsection.4.5.2}%
\contentsline {subsection}{\numberline {4.5.3}Anforderungen zum Schutz vor Brute-Force}{15}{subsection.4.5.3}%
\contentsline {section}{\numberline {4.6}Wörterbuchangriffe}{16}{section.4.6}%
\contentsline {subsection}{\numberline {4.6.1}Schutz vor Wörterbuchangriffen}{16}{subsection.4.6.1}%
\contentsline {subsubsection}{\numberline {4.6.1.1}Nonce-Verschlüsselung}{16}{subsubsection.4.6.1.1}%
\contentsline {chapter}{\numberline {5}Stromverschlüsselungsverfahren}{17}{chapter.5}%
\contentsline {section}{\numberline {5.1}Synchrone Stromverschlüsselungsverfahren}{17}{section.5.1}%
\contentsline {section}{\numberline {5.2}Zustandsabhängige Stromverschlüsselungsverfahren}{18}{section.5.2}%
\contentsline {subsection}{\numberline {5.2.1}Additive zustandsabhängige Stromverschlüsselungsverfahren}{19}{subsection.5.2.1}%
\contentsline {subsubsection}{\numberline {5.2.1.1}Synchrone additive Stromverschlüsselungsverfahren}{19}{subsubsection.5.2.1.1}%
\contentsline {section}{\numberline {5.3}Schlüsselstrom vs. One-Time-Pad}{20}{section.5.3}%
\contentsline {section}{\numberline {5.4}Nonces zur Initialisierung eines Schlüsselstromgenerators}{20}{section.5.4}%
\contentsline {section}{\numberline {5.5}ChaCha20}{21}{section.5.5}%
\contentsline {section}{\numberline {5.6}Cipher-Instanzen: Verschlüsselungsalgorithmen in Java-Laufzeitumgebungen}{22}{section.5.6}%
\contentsline {chapter}{\numberline {6}Blockverschlüsselungsverfahren}{23}{chapter.6}%
\contentsline {section}{\numberline {6.1}Padding-Verfahren}{23}{section.6.1}%
\contentsline {section}{\numberline {6.2}Betriebsmodi}{23}{section.6.2}%
\contentsline {subsection}{\numberline {6.2.1}ECB (Electronic Code Book)}{23}{subsection.6.2.1}%
\contentsline {subsection}{\numberline {6.2.2}CBC (Cipher Block Chaining)}{24}{subsection.6.2.2}%
\contentsline {subsection}{\numberline {6.2.3}CBC-CS (Chiphertext Stealing for CBC Mode)}{25}{subsection.6.2.3}%
\contentsline {subsubsection}{\numberline {6.2.3.1}CBC-CS1}{25}{subsubsection.6.2.3.1}%
\contentsline {subsubsection}{\numberline {6.2.3.2}CBC-CS2}{26}{subsubsection.6.2.3.2}%
\contentsline {subsubsection}{\numberline {6.2.3.3}CBC-CS3}{26}{subsubsection.6.2.3.3}%
\contentsline {subsection}{\numberline {6.2.4}CTR (Counter)}{26}{subsection.6.2.4}%
\contentsline {subsection}{\numberline {6.2.5}OFB (Output Feedback)}{26}{subsection.6.2.5}%
\contentsline {subsubsection}{\numberline {6.2.5.1}OFB-8$n$}{27}{subsubsection.6.2.5.1}%
\contentsline {subsection}{\numberline {6.2.6}CFB (Cipher-Feedback)}{27}{subsection.6.2.6}%
\contentsline {subsubsection}{\numberline {6.2.6.1}CFB-8$n$}{27}{subsubsection.6.2.6.1}%
\contentsline {section}{\numberline {6.3}Konstruktionsprinzipien von Blockverschlüsselungsverfahren}{28}{section.6.3}%
\contentsline {section}{\numberline {6.4}DES}{28}{section.6.4}%
\contentsline {subsection}{\numberline {6.4.1}Triple-DES (3DES)}{28}{subsection.6.4.1}%
\contentsline {section}{\numberline {6.5}Meet-in-the-Middle-Angriff}{29}{section.6.5}%
\contentsline {section}{\numberline {6.6}AES (Advanced Encryption Standard)}{30}{section.6.6}%
\contentsline {subsection}{\numberline {6.6.1}AES-128}{30}{subsection.6.6.1}%
\contentsline {chapter}{\numberline {7}Hashfunktionen}{32}{chapter.7}%
\contentsline {section}{\numberline {7.1}schwache Kollisionsfreiheit}{32}{section.7.1}%
\contentsline {section}{\numberline {7.2}\texttt {MessageDigest}-Instanzen: Hashfunktionen in Java}{32}{section.7.2}%
\contentsline {section}{\numberline {7.3}Anwendungsbeispiele}{33}{section.7.3}%
\contentsline {subsection}{\numberline {7.3.1}Anwendungsbeispiel: Passwortdatei}{33}{subsection.7.3.1}%
\contentsline {subsubsection}{\numberline {7.3.1.1}Anwendungsbeispiel: Passwortdatei mit Salt und Iteration Count}{33}{subsubsection.7.3.1.1}%
\contentsline {subsection}{\numberline {7.3.2}Anwendungsbeispiel: Integritätsschutz von Dateien}{33}{subsection.7.3.2}%
\contentsline {subsection}{\numberline {7.3.3}Anwendungsbeispiel: Integritätsschutz bei einem Dateidownload}{33}{subsection.7.3.3}%
\contentsline {section}{\numberline {7.4}Brute-Force-Angriffe auf Hashfunktionen}{33}{section.7.4}%
\contentsline {subsection}{\numberline {7.4.1}Brute-Force-Urbildsuche}{33}{subsection.7.4.1}%
\contentsline {subsection}{\numberline {7.4.2}Brute-Force-Kollisionssuche}{34}{subsection.7.4.2}%
\contentsline {section}{\numberline {7.5}Konstruktionsverfahren von Hashfunktionen}{34}{section.7.5}%
\contentsline {chapter}{\numberline {8}MAC-Verfahren}{36}{chapter.8}%
\contentsline {section}{\numberline {8.1}HMAC}{36}{section.8.1}%
\contentsline {section}{\numberline {8.2}CMAC, CBC-MAC}{36}{section.8.2}%
\contentsline {subsection}{\numberline {8.2.1}CMAC}{36}{subsection.8.2.1}%
\contentsline {subsection}{\numberline {8.2.2}CBC-MAC}{37}{subsection.8.2.2}%
\contentsline {chapter}{\numberline {9}Modulare Arithmetik - Teil 2}{38}{chapter.9}%
\contentsline {section}{\numberline {9.1}Potenzen}{38}{section.9.1}%
\contentsline {subsection}{\numberline {9.1.1}Erzeugnis und Ordnung eines invertierbaren Elements}{38}{subsection.9.1.1}%
\contentsline {subsection}{\numberline {9.1.2}Faktorenzerlegung}{39}{subsection.9.1.2}%
\contentsline {subsection}{\numberline {9.1.3}Kleiner Satz von Fermat}{39}{subsection.9.1.3}%
\contentsline {section}{\numberline {9.2}Exkurs: Einheitengruppe $\mathbb {Z}_{p^e}^*$}{39}{section.9.2}%
\contentsline {subsection}{\numberline {9.2.1}$\mathbb {Z}_{2^e}^*$}{39}{subsection.9.2.1}%
\contentsline {section}{\numberline {9.3}Der chinesische Restsatz}{39}{section.9.3}%
\contentsline {subsection}{\numberline {9.3.1}Beispiel}{40}{subsection.9.3.1}%
\contentsline {section}{\numberline {9.4}Elemente gerader und ungerader Ordnung in $\mathbb {Z}_n$**}{40}{section.9.4}%

2
chapters/Blockverschlüsselungsverfahren.tex
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@ -219,7 +219,7 @@
03 & 02 & 01 & 02 \\
\end{pmatrix}$$
Hierbei ist wichtig anzumerken, dass die Multiplikation nicht in $\mathbf{Z}_{256}$ berechnet wird.
Stattdessen sind die Elemente als Elemente des Körpers $\mathbf{F}_{256}$ aufzufassen (siehe \ref{endliche Körper}),
Stattdessen sind die Elemente als Elemente des Körpers $\mathbf{F}_{256}$ aufzufassen,
der bezüglich des irreduziblen Polynoms $m(x)=x^8+x^4+x^3+x+1\in\mathbb{F}_2[x]$ definiert ist.
\item AddRoundKey()\\
Es wird eine XOR-Operation der Einträge des S-Arrays mit den entsprechenden Einträgen aus \textbf{K} durchgeführt.

2
chapters/Endliche Körper.tex
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@ -1,4 +1,4 @@
\chapter{Endliche Körper $\mathbb{F}_{p^n}$} \label{endliche Körper}
\chapter{Endliche Körper $\mathbb{F}_{p^n}$}\label{endliche Körper}
$\mathbb{F}_p:=\mathbb{Z}_p$ ($p$ ist eine Primzahl) ist ein Körper.
Zudem können Körper auch für beliebige Primzahlpotenzen $p^n$ mit $n\in\mathbb{N},n>0$ mithilfe der Polynome aus $\mathbb{F}_p$ vom Grad $<n$ konstruiert werden:
$$\begin{aligned}

128
chapters/Modulare Arithmetik - Teil 2.tex Normal file
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@ -0,0 +1,128 @@
\chapter{Modulare Arithmetik - Teil 2}
\section{Potenzen}
Potenzen sind für beliebige Zahlen $a\in\mathbb{Z}_n$ für natürliche Exponenten $i\in\mathbb{N}$ wie üblich definiert:
$$a^i:=\underbrace{a\cdot a\cdots a}_i$$
Ist $a$ invertierbar ($a\in\mathbb{Z}_n^*$) so sind die Potenzen für alle ganzzahligen Exponenten $i\in\mathbb{Z}$ definiert,
indem $a^0 = 1$ und $a^{-i}:=(a^{-1})^i$ definiert werden.
Folglich gelten die Folgenden Regeln:
\begin{mybox}
für $a,b\in\mathbb{Z}_n$ und $i,j\in \mathbb{N}$ oder für $a,b\in\mathbb{Z}_n^*$ und $i,j\in\mathbb{Z}$ gilt:
$$\begin{aligned}
a^i\cdot a^j &= a^{i+j}\\
(a^i)^j &= a^{ij}\\
a^i\cdot b^i &= (ab)^i\\
\end{aligned}$$
\end{mybox}
\subsection{Erzeugnis und Ordnung eines invertierbaren Elements}
Die zyklische Untergruppe, die aus den Potenzen eines Elementes $a\in\mathbb{Z}_n^*$ entsteht wird als Erzeugnis von $a$ in $\mathbb{Z}_n^*$ bezeichnet.
$$<a>:=\{a^i\mid i\in\mathbb{Z}\}$$
Die Anzahl der verschiedenen Potenzen von $a$ wird als Ordnung von $a$ in $\mathbb{Z}_n^*$ bezeichnet:
$$o_n(a):=|<a>|$$
Hierbei ergibt sich $a^{o(a)}=1$ und $<a>=\{1,a,a^2,a^3,...,a^{o(a)-1}\}$.\\
\begin{mybox}
für $n\in\mathbb{N}(n\ge 2);a,b\in \mathbb{Z}_n^*$ und $i\in\mathbb{Z}$ gilt:
\begin{enumerate}[(i)]
\item $a^i=1 \Longrightarrow o(a)|i$
\item $o(a^i)|o(a)$
\item $i|o(a) \Longrightarrow o(a^i)=\frac{o(a)}{|i|}$
\item $\text{ggT}(o(a),i)=1 \Longrightarrow o(a^i)=o(a)$
\item $o(ab)|o(a)\cdot o(b)$
\item $\text{ggT}(o(a),o(b))=1\Longrightarrow o(a\cdot b)=o(a)\cdot o(b)$
\end{enumerate}
\end{mybox}
\begin{mybox}
Wenn $\mu_n$ für $n\in\mathbb{N},n\ge2$ die maximale Ordnung eines Elementes aus $\mathbb{Z}_n^*$ ($\mu_n:=\max\{o(a)\mid a\in\mathbb{Z}_n^*\}$),
ist jede Ordnung von $a$ ein Teiler von $\mu_n$
\end{mybox}
\subsection{Faktorenzerlegung}
Falls es für ein Polynom $p(x) \in \mathbb{Z}_n[x]$ ($n\in\mathbb{N}$) vom Grad $k$ eine Nullstelle $a\in\mathbb{Z}_n$ gibt,
muss es ein Polynom $q(x) \in \mathbb{Z}_n[x]$ vom Grad $k-1$ geben, für das gilt:
$$p(x)=(x-a)\cdot q(x)$$
\subsection{Kleiner Satz von Fermat}
Für eine Primzahl $p$ gilt für alle $a\in\mathbb{Z}$ mit $\text{ggT}(a,p)=1$:
$$a^{p-1}\equiv 1 \hspace{10mm}(\mod p)$$
\section{Exkurs: Einheitengruppe $\mathbb{Z}_{p^e}^*$}
Für eine Primzahl $p$ gibt es ein Element $g\in\mathbb{Z}_p^*$ mit $o(g)=p-1$ ($\mathbb{Z}_p^*=<g>=\{1,g,g^2,...,g^{p-2}\}$).
Diese Elemente $g$ werden als \textbf{primitive Elemente} bezeichnet.\\
Mithilfe der Potenzen der primitiven Elemente lassen sich alle Elemente von $\mathbb{Z}_p^*$ generieren:
$$<g>=\mathbb{Z}_p^*$$
Diese Elemente werden daher auch als \textbf{Generatoren} von $\mathbb{Z}_p^*$ bezeichnet.
Auch für Einheitengruppen $\mathbb{Z}_{p^e}$ für Primzahlpotenzen mit $e>1$ gibt es solche zyklische Generatoren:
\begin{mybox}
für eine Primzahl $p\ne2$ und $e\in\mathbb{N}$ gibt es ein $g\in\mathbb{Z}_{p^e}^*$ für das gilt:
$$\begin{aligned}
o(g)=p^{e-1}(p-1) &= \varphi(p^e)=|\mathbb{Z}_{p^e}^*\\
\Leftrightarrow\hspace{10mm}<g>&=\mathbb{Z}_{p^e}^*
\end{aligned}$$
\end{mybox}
Hierbei gilt:
\begin{mybox}
Für ein primitives Element $g\in\mathbb{Z}_p^*$ gilt mit $a:=g^{\frac{o_{p^e}(g)}{p-1}}$:
$$\begin{aligned}
\mathbb{Z}_{p^e}^*&=<a>\cdot <1+p>\\
:&=\{a^i\cdot (1+p)^j\mid i=0,1,...,p-2; j=0,1,...p^{e-1}-1\}
\end{aligned}$$
\end{mybox}
\subsection{$\mathbb{Z}_{2^e}^*$}
Die Primzahl 2 stellt eine besonderheit dar, da $\mathbb{Z}_{2^e}^*$ für $e\ge3$ nicht zyklisch ist.
So gilt $o(5)=2^{e-2}$ und allgemein:
$$\begin{aligned}
\mathbb{Z}_{2^e}^*&=<2^e-1>\cdot <5>\\
&= \{(2^e-1)^i\cdot 5^j\mid i=0,1;j=0,1,...,2^{e-2}-1\}
\end{aligned}$$
\section{Der chinesische Restsatz}
Für eine Menge teilerfremder Zahlen $\{(n_1,...,n_r)\mid n_i\in\mathbb{N}\}$ bildet einen Menge $\{(a_1,...,a_r)\mid a_i\in\mathbb{Z}\}$ ein System aus Kongruenzen:
$$\begin{aligned}
x\equiv&a_1\hspace{10mm} &(\textbf{mod }n_1)\\
x\equiv&a_2 &(\textbf{mod } n_2)\\
\vdots\\
x\equiv&a_r &(\textbf{mod } n_r)\\
\end{aligned}$$
Die Menge der ganzzahligen Lösungen ist gegeben durch:
$$\mathbb{L}=\{a+v\cdot n_1n_2\cdots n_r\mid v\in\mathbb{Z}\}$$
Hierbei ist $0\le a < n_1n_2\cdots n_r$ eindeutig.
$a$ wird im folgenden Beispiel mithilfe des erweiterten Euklid'schen Algorithmus (siehe \ref{erweiterter Euklid}) berechnet.
Aus dem Restsatz leitet sich folgender Satz ab:
\begin{mybox}
Sind $n_1,n_2,...,n_r$ paarweise teilerfremde Zahlen gibt es die folgende Isomorphie (\ref{Isomorphie}):
$$\mathbb{Z}_{n_1n_2\cdots n_r}\cong \mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times\cdots\times\mathbb{Z}_{n_r}$$
\end{mybox}
\subsection{Beispiel}
folgende Kongruenzen sind gegeben:
$$\begin{aligned}
x\equiv&2\hspace{10mm}&(\textbf{mod }5)\\
x\equiv&7 &(\textbf{mod }2)\\
x\equiv&2 &(\textbf{mod }5)\\
\end{aligned}$$
\begin{itemize}
\item \textbf{Bestimmung von} $e_1$\\
für $n_1=5$ und $m_1=21\cdot 11=231$ liefert der erweiterte Euklid'sche Algorithmus (\ref{erweiterter Euklid}):
$$(-46)\cdot 5+1\cdot 231 = 1$$
Daher ist $e_1 = 1\cdot 231$
\item \textbf{Bestimmung von} $e_2$\\
für $n_2=21$ und $m_2=5\cdot 11=55$ liefert der erweiterte Euklid'sche Algorithmus (\ref{erweiterter Euklid}):
$$(-21)\cdot 21+(-8)\cdot 55 = 1$$
Daher ist $e_2 = (-8)\cdot 55=-440$
\item \textbf{Bestimmung von} $e_3$\\
für $n_3=11$ und $m_3=5\cdot 21=105$ liefert der erweiterte Euklid'sche Algorithmus (\ref{erweiterter Euklid}):
$$(-19)\cdot 11+2\cdot 105 = 1$$
Daher ist $e_3 = 2\cdot 105=210$
\item \textbf{Berechnung der Lösung}\\
$n=5\cdot 21\cdot 11=1155$:
$$\begin{aligned}
a=&(2\cdot e_1+7\cdot e_2+6\cdot e_3)\mod 1155\\
=&(2\cdot 231-7\cdot 440+6\cdot 210)\mod 1155\\
=&-1358\mod 1155\\
=952
\end{aligned}$$
\end{itemize}
\section{Elemente gerader und ungerader Ordnung in $\mathbb{Z}_n$**}
siehe Skript 3 Kapitel 1.4 auf Seite 17(23).

17
chapters/Modulare Arithmetik.tex
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@ -72,6 +72,23 @@
Die Inverse einer quadratischen Matrix $A$ über $\mathbb{Z}_n$ lässt sich mithilfe der Adjunkten berechnen:
$$A^{-1} = (\det(A))^{-1} \cdot adj(A)$$
Die Adjunkte lässt sich über $\mathbb{Z}_n$ berechnen, da lediglich Summen und Differenzen von Produkten berechnet werden müssen.
\subsection{Kartesisches Produkt von Ringen}
Sind $R_1,R_2,...,R_r$ Ringe, so bildet das kartesische Produkt ($R_1\times R_2\times\cdots\times R_r$)ebenfalls einen Ring mit den folgenden Verknüpfungen:
$$\begin{aligned}
(a_1,...,a_r)+(b_1,...,b_r)&=(a_1+b_1,...,a_r+b_r)\\
(a_1,...,a_r)\cdot(b_1,...,b_r)&=(a_1\cdot b_1,...,a_r\cdot b_r)\\
\end{aligned}$$
Auch das kartesische Produkt der Einheitengruppen bildet einen Ring:
$$R^*=R_1^*\times\cdots\times R_r^*$$
\subsection{Isomorphie von Ringen}\label{Isomorphie}
Eine bijektive Abbildung $f:R_1\rightarrow R_2$ wird Isomorphismus genannt, falls für alle $a,b\in R_1$ gilt:
$$\begin{aligned}
f(a+b)&=f(a)+f(b)\\
f(a\cdot b)&=f(a)\cdot f(b)\\
\end{aligned}$$
Zwei isomorphe Ringe werden als $R_1\cong R_2$ geschrieben.
\section{Der erweiterte Euklid'sche Algorithmus}
Der Euklid'sche Algorithmus ist ein sehr effizienter Weg den ggT zweier Zahlen zu ermitteln.