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+\usepackage{color,graphicx,overpic} +\usepackage{amsmath,amsthm,amsfonts,amssymb} +\usepackage{tabularx} +\usepackage{listings} +\usepackage[many]{tcolorbox} +\usepackage{multicol} +\usepackage{hyperref} +\usepackage{pgfplots} +\usepackage{bussproofs} + +\pdfinfo{ + /Title (Kryptographie - Übung) + /Creator (TeX) + /Producer (pdfTeX 1.40.0) + /Author (Robert Jeutter) + /Subject () +} +\title{Kryptographie - Übung} +\author{} +\date{} + +% Don't print section numbers +\setcounter{secnumdepth}{0} + +\newtcolorbox{myboxii}[1][]{ + breakable, + freelance, + title=#1, + colback=white, + colbacktitle=white, + coltitle=black, + fonttitle=\bfseries, + bottomrule=0pt, + boxrule=0pt, + colframe=white, + overlay unbroken and first={ + \draw[red!75!black,line width=3pt] + ([xshift=5pt]frame.north west) -- + (frame.north west) -- + (frame.south west); + \draw[red!75!black,line width=3pt] + ([xshift=-5pt]frame.north east) -- + (frame.north east) -- + (frame.south east); + }, + overlay unbroken app={ + \draw[red!75!black,line width=3pt,line cap=rect] + (frame.south west) -- + ([xshift=5pt]frame.south west); + \draw[red!75!black,line width=3pt,line cap=rect] + (frame.south east) -- + ([xshift=-5pt]frame.south east); + }, + overlay middle and last={ + \draw[red!75!black,line width=3pt] + (frame.north west) -- + (frame.south west); + \draw[red!75!black,line width=3pt] + (frame.north east) -- + (frame.south east); + }, + overlay last app={ + \draw[red!75!black,line width=3pt,line cap=rect] + (frame.south west) -- + ([xshift=5pt]frame.south west); + \draw[red!75!black,line width=3pt,line cap=rect] + (frame.south east) -- + ([xshift=-5pt]frame.south east); + }, +} + +\begin{document} +\begin{myboxii}[Disclaimer] + Die Übungen die hier gezeigt werden stammen aus der Vorlesung \textit{Kryptographie}! Für die Korrektheit der Lösungen wird keine Gewähr gegeben. +\end{myboxii} + +%########################################## +\begin{questions} + \question Possibilistisch sichere Kryptosysteme + + Bestimmen Sie alle possibilistisch sicheren Kryptosysteme $S=(X,K,Y,e,d)$ mit $X=\{a,b\}$ und $K=\{1,2\}$ (bis auf das Umbenennen von Chiffretexten). + \begin{solution} + \end{solution} + + \question Possibilistische Sicherheit: Eine alternative Definition? + Beweisen oder widerlegen Sie: Ein Kryptosystem $S=(X,K,Y,e,d)$ ist possibilistisch sicher genau dann, wenn Folgendes gilt: $\forall x\in X\forall y\in Y\exists k\in K:d(y,k)=x$. + \begin{solution} + \end{solution} + + Bemerkung: Im Gegensatz zur Definition der possibilistischen Sicherheit wird hier eine Aussage über die Entschlüsselungsfunktion gemacht. + + \question Possibilistische Sicherheit bei komponentenweiser Verschlüsselung + + Gegeben seien ein Kryptosystem $S=(X,K,Y,e,d)$ und $l\in\mathbb{N}^+$. Wir können $S$ benutzen, um längere Klartexte (Elemente aus $X^l$) zu verschlüsseln. + + Das Kryptosystem $S'=(X^l,K,Y^l,e',d')$ mit $e'((x_1,...,x^l),k)=(e(x_1,k),...,e(x_l,k))$ verschlüsselt komponentenweise unter Verwendung eines einzigen Schlüssels $k$. + \begin{parts} + \part Definieren Sie $d'$ so, dass $S'$ tatsächlich ein Kryptosystem ist. + \begin{solution} + \end{solution} + \part Zeigen Sie, dass $S'$ für $|X|,l\geq 2$ nicht possibilistisch sicher ist. (Dies gilt auch dann, wenn S selber possibilistisch sicher ist!) + \begin{solution} + \end{solution} + \end{parts} + + Das Kryptosystem $S^*=(X^l,K^l,Y^l,e^*,d^*)$ mit $e^*((x_1,...,x_l),(k_1,...,k_l))=(e(x_1,k_1),..., e(x_l,k_l))$ verschlüsselt komponentenweise unter Verwendung mehrerer Schlüssel $k_1,...,k_l$. + \begin{parts} + \part Definieren Sie $d^*$ so, dass $S^*$ tatsächlich ein Kryptosystem ist. + \begin{solution} + \end{solution} + \part Zeigen Sie, dass $S^*$ genau dann possibilistisch sicher ist, wenn $S$ possibilistisch sicher ist. + \begin{solution} + \end{solution} + \end{parts} + + Notation: Für eine natürliche Zahl $n\geq 2$ sei $Z_n$ die Menge der Zahlen $\{0,1,...,n-1\}$. Die Addition $+_n$ und Multiplikation $*_n$ auf $Z_n$ sind wie folgt definiert: $a +_n b =(a+b)\ mod\ n$ und $a *_n b =(a*b)\ mod\ n$, wobei $x\ mod\ n$ der Rest von $x$ bei Division durch $n$ ist. + + \question Verschiebe- und affines Kryptosystem + + Für $n\in N^+$ betrachten wir zwei Kryptosysteme, um Elemente aus $Z_n$ zu verschlüsseln. + Das Verschiebekryptosystem (Cäsar-Chiffre) mit Parameter $n$ ist gegeben durch $C_n=(Z_n,Z_n,Z_n,e_n,d_n)$ mit $e_n(x,k)=x +_n k$. + \begin{parts} + \part Wie muss $d_n$ definiert werden, damit $C_n$ tatsächlich ein Kryptosystem ist? + \begin{solution} + \end{solution} + \part Zeigen Sie, dass $C_n$ possibilistisch sicher ist. + \begin{solution} + \end{solution} + \end{parts} + + Das affine Kryptosystem mit Parameter $n\geq 2$ ist gegeben durch $A_n=(Z_n,A_n\times Z_n,Z_n,e_n',d_n')$ mit $A_n=\{a\in Z_n| ggT(a, n) = 1\}$ und $e_n'(x,(a,b)=a *_n x +_n b$. + Hinweis: Falls $ggT(a,n)=1$, d.h., $a$ und $n$ teilerfremd sind, dann gilt: Es existert genau ein $b\in A_n\subseteq Z_n\backslash\{0\}$, so dass $a*_n b=b*_n a=1$. Dieses Element $b$ heißt ,,multiplikatives Inverses von a modulo n''. + \begin{parts} + \part Definieren Sie $d_n'$ so, dass $A_n$ tatsächlich ein Kryptosystem ist. + \begin{solution} + \end{solution} + \part Zeigen Sie, dass $A_n$ possibilistisch sicher ist. + \begin{solution} + \end{solution} + \end{parts} + +\end{questions} +\end{document} diff --git a/README.md b/README.md index 63bba5d..dac98e2 100644 --- a/README.md +++ b/README.md @@ -26,6 +26,7 @@ Mitschriften und selbst geschriebene Unterlagen und Hilfen zu Informatik Vorlesu - [GudS - Cheatsheet](Grundlagen%20und%20Diskrete%20Strukturen%20-%20Cheatsheet.pdf) - [GudS - Short Paper](Grundlagen%20und%20Diskrete%20Strukturen%20-%20short.pdf) - [Kryptographie](Kryptographie.md) + - [Kryptographie - Übung](Kryptographie%20-%20Übung.pdf) - [Logik und Logikprogrammierung](Logik%20und%20Logikprogrammierung.md) - [Cheatsheet](Logik%20und%20Logikprogrammierung%20-%20Cheatsheet.pdf) - [Flashcards - Karteikarten für Definitionen](Logik%20und%20Logikprogrammierung%20-%20Flashcards.pdf)