diff --git a/Inhalt.tex b/Inhalt.tex index b84aae0..de1cd13 100644 --- a/Inhalt.tex +++ b/Inhalt.tex @@ -16,4 +16,6 @@ \input{chapters/MAC-Verfahren.tex} -\input{chapters/Modulare Arithmetik - Teil 2.tex} \ No newline at end of file +\input{chapters/Modulare Arithmetik - Teil 2.tex} + +\input{chapters/Diffie-Hellman-Schlüsselaustausch.tex} \ No newline at end of file diff --git a/Packages.tex b/Packages.tex index c8c3a9b..307025b 100644 --- a/Packages.tex +++ b/Packages.tex @@ -27,8 +27,6 @@ rightsub = \grq% ]{dirtytalk} %automatic links \usepackage{hyperref} -%graphics -\usepackage{tikz} %boxes \usepackage[framemethod=tikz]{mdframed} %enumeration diff --git a/Zusammenfassung_DC.aux b/Zusammenfassung_DC.aux index 74fb9ad..c134601 100644 --- a/Zusammenfassung_DC.aux +++ b/Zusammenfassung_DC.aux @@ -146,6 +146,7 @@ \@writefile{toc}{\contentsline {section}{\numberline {6.4}DES}{28}{section.6.4}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline 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PDF statistics: - 1001 PDF objects out of 1200 (max. 8388607) - 897 compressed objects within 9 object streams - 211 named destinations out of 1000 (max. 500000) - 968 words of extra memory for PDF output out of 10000 (max. 10000000) + 1107 PDF objects out of 1200 (max. 8388607) + 994 compressed objects within 10 object streams + 239 named destinations out of 1000 (max. 500000) + 1061 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/Zusammenfassung_DC.out b/Zusammenfassung_DC.out index 751ae14..87819d4 100644 --- a/Zusammenfassung_DC.out +++ b/Zusammenfassung_DC.out @@ -98,8 +98,19 @@ \BOOKMARK [2][-]{subsection.9.1.1}{Erzeugnis und Ordnung eines invertierbaren Elements}{section.9.1}% 98 \BOOKMARK [2][-]{subsection.9.1.2}{Faktorenzerlegung}{section.9.1}% 99 \BOOKMARK [2][-]{subsection.9.1.3}{Kleiner Satz von Fermat}{section.9.1}% 100 -\BOOKMARK [1][-]{section.9.2}{Exkurs: Einheitengruppe Zpe*}{chapter.9}% 101 -\BOOKMARK [2][-]{subsection.9.2.1}{Z2e*}{section.9.2}% 102 -\BOOKMARK [1][-]{section.9.3}{Der chinesische Restsatz}{chapter.9}% 103 -\BOOKMARK [2][-]{subsection.9.3.1}{Beispiel}{section.9.3}% 104 -\BOOKMARK [1][-]{section.9.4}{Elemente gerader und ungerader Ordnung in Zn**}{chapter.9}% 105 +\BOOKMARK [2][-]{subsection.9.1.4}{Berechnung modularer Potenzen}{section.9.1}% 101 +\BOOKMARK [1][-]{section.9.2}{Exkurs: Einheitengruppe Zpe*}{chapter.9}% 102 +\BOOKMARK [2][-]{subsection.9.2.1}{Z2e*}{section.9.2}% 103 +\BOOKMARK [1][-]{section.9.3}{Der chinesische Restsatz}{chapter.9}% 104 +\BOOKMARK [2][-]{subsection.9.3.1}{Beispiel}{section.9.3}% 105 +\BOOKMARK [1][-]{section.9.4}{Elemente gerader und ungerader Ordnung in Zn**}{chapter.9}% 106 +\BOOKMARK [0][-]{chapter.10}{Das Diffie-Hellman-Schl\374sselaustauschverfahren}{}% 107 +\BOOKMARK [1][-]{section.10.1}{Das DH-Verfahren in Einzelschritten}{chapter.10}% 108 +\BOOKMARK [1][-]{section.10.2}{Das Diskrete-Logarithmus-Problem \(DL-Problem\)}{chapter.10}% 109 +\BOOKMARK [2][-]{subsection.10.2.1}{Das Diffie-Hellman-Problem \(DH-Problem\)}{section.10.2}% 110 +\BOOKMARK [1][-]{section.10.3}{Beispielanwendung des DH-Verfahrens}{chapter.10}% 111 +\BOOKMARK [1][-]{section.10.4}{Angriffe auf das DH-Verfahren}{chapter.10}% 112 +\BOOKMARK [2][-]{subsection.10.4.1}{Pohlig-Hellman-Reduktion}{section.10.4}% 113 +\BOOKMARK [2][-]{subsection.10.4.2}{Shanks' \214Baby Steps Giant Steps\215-Verfahren \(BSGS-Verfahren\)}{section.10.4}% 114 +\BOOKMARK [2][-]{subsection.10.4.3}{Pollard's Rho-Methode}{section.10.4}% 115 +\BOOKMARK [2][-]{subsection.10.4.4}{Zahlk\366rpersieb}{section.10.4}% 116 diff --git a/Zusammenfassung_DC.pdf b/Zusammenfassung_DC.pdf index f56543f..6685a93 100644 Binary files a/Zusammenfassung_DC.pdf and b/Zusammenfassung_DC.pdf differ diff --git a/Zusammenfassung_DC.synctex.gz b/Zusammenfassung_DC.synctex.gz index ca6e4da..bc8acb2 100644 Binary files a/Zusammenfassung_DC.synctex.gz and b/Zusammenfassung_DC.synctex.gz differ diff --git a/Zusammenfassung_DC.toc b/Zusammenfassung_DC.toc index c85161b..5050e3b 100644 --- a/Zusammenfassung_DC.toc +++ b/Zusammenfassung_DC.toc @@ -118,8 +118,19 @@ \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}% +\contentsline {subsection}{\numberline {9.1.4}Berechnung modularer Potenzen}{39}{subsection.9.1.4}% +\contentsline {section}{\numberline {9.2}Exkurs: Einheitengruppe $\mathbb {Z}_{p^e}^*$}{40}{section.9.2}% +\contentsline {subsection}{\numberline {9.2.1}$\mathbb {Z}_{2^e}^*$}{40}{subsection.9.2.1}% +\contentsline {section}{\numberline {9.3}Der chinesische Restsatz}{40}{section.9.3}% +\contentsline {subsection}{\numberline {9.3.1}Beispiel}{41}{subsection.9.3.1}% +\contentsline {section}{\numberline {9.4}Elemente gerader und ungerader Ordnung in $\mathbb {Z}_n$**}{41}{section.9.4}% +\contentsline {chapter}{\numberline {10}Das Diffie-Hellman-Schlüsselaustauschverfahren}{42}{chapter.10}% +\contentsline {section}{\numberline {10.1}Das DH-Verfahren in Einzelschritten}{42}{section.10.1}% +\contentsline {section}{\numberline {10.2}Das Diskrete-Logarithmus-Problem (DL-Problem)}{43}{section.10.2}% +\contentsline {subsection}{\numberline {10.2.1}Das Diffie-Hellman-Problem (DH-Problem)}{43}{subsection.10.2.1}% +\contentsline {section}{\numberline {10.3}Beispielanwendung des DH-Verfahrens}{44}{section.10.3}% +\contentsline {section}{\numberline {10.4}Angriffe auf das DH-Verfahren}{44}{section.10.4}% +\contentsline {subsection}{\numberline {10.4.1}Pohlig-Hellman-Reduktion}{44}{subsection.10.4.1}% +\contentsline {subsection}{\numberline {10.4.2}Shanks' \glqq Baby Steps Giant Steps\grqq -Verfahren (BSGS-Verfahren)}{44}{subsection.10.4.2}% +\contentsline {subsection}{\numberline {10.4.3}Pollard's Rho-Methode}{45}{subsection.10.4.3}% +\contentsline {subsection}{\numberline {10.4.4}Zahlkörpersieb}{45}{subsection.10.4.4}% diff --git a/chapters/Blockverschlüsselungsverfahren.tex b/chapters/Blockverschlüsselungsverfahren.tex index 45e3157..16e4032 100644 --- a/chapters/Blockverschlüsselungsverfahren.tex +++ b/chapters/Blockverschlüsselungsverfahren.tex @@ -143,7 +143,7 @@ $$E((k^{(1)},k^{(2)},m)) := E_{DES}\left(k^{(1)},D_{DES}\left(k^{(2)},E_{DES}\left(k^{(1)},m \right)\right)\right)$$ \includegraphics[scale=0.8]{3DES.png} - \section{Meet-in-the-Middle-Angriff} + \section{Meet-in-the-Middle-Angriff}\label{meet-in-the-middle} 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:\\ diff --git a/chapters/Diffie-Hellman-Schlüsselaustausch.tex b/chapters/Diffie-Hellman-Schlüsselaustausch.tex new file mode 100644 index 0000000..7f95d45 --- /dev/null +++ b/chapters/Diffie-Hellman-Schlüsselaustausch.tex @@ -0,0 +1,75 @@ +\chapter{Das Diffie-Hellman-Schlüsselaustauschverfahren} + In allen Fällen, in denen ein symmetrisches Schlüsselverfahren (inkl. MAC-Verfahren) verwendet wird ist es nötig den Schlüssel sicher auszutauschen. + Eine Möglichkeit ist es diese Ãœbertragung über einen sicheren Kanal (z.B. persönliche Ãœbergabe) zu organisieren. + Dies ist allerdings mit einem hohen Aufwand verbunden und erlaubt nicht einen sicheren Kanal in Echtzeit (z.B. für HTTPS) aufzubauen. + Hierfür wurde von W.Diffie udn M.E.Hellman auf Basis der Arbeit von R.C.Merkle ein asymmetrisches Schlüsselaustauschverfahren entwickelt, welches als DH-Verfahren bekannt ist. + Asymmetrische Verschlüsselungsverfahren zeichnen sich durch die Verwendung von \textbf{Schlüsselpaaren} ($k=(k_{\text{priv}},k_{\text{pub}})$) aus. + Diese Schlüsselpaare sollten folgende Eigenschaften haben: + \begin{itemize} + \item Ein Schlüsselpaar ist genau einem Besitzer zugeordnet + \item Niemals hat jemand anderes als der Besitzer zugriff auf den privaten Schlüssel $k_\text{priv}$ + \item Der private Schlüssel wird vom Besitzer selber generiert + \item Der öffentliche Schlüssel $k_\text{pub}$ lässt sich effizient aus $k_\text{priv}$ errechnen + \item Es gibt keinen effizienten Algorithmus um mithilfe von $k_\text{pub}$ $k_\text{priv}$ zu bestimmen + \end{itemize} + Vor dem Austausch eines symmetrischen Schlüssels besitzen beide Seiten jeweils ein unabhängiges asymmetrisches Schlüsselpaar. + + \section{Das DH-Verfahren in Einzelschritten} + Im Folgenden werden Alice und Bob als Synonyme für zwei beliebige Sender/Empfänger genutzt: + \begin{itemize} + \item Alice und Bob vereinbaren zwei Parameter (meist durch einen Standard spezifiziert): + \begin{itemize} + \item eine große Zahl $n\in\mathbb{N}$ + \item eine Zahl $g\in\mathbb{Z}_n^*$ deren Ordnung $o(g)$ mindestens durch eine sehr große Primzahl geteilt wird. + \end{itemize} + \item Alice bestimmt eine zufällig Zahl $e_A\in\mathbb{Z}_n^*$ als privaten Schlüssel: + $$k_{A,\text{priv}}=e_A$$ + und errechnet den öffentlichen Schlüssel $k_{A,\text{pub}}=g^{e_A}\in\mathbb{Z}_n$ + \item Bob bestimmt ebenfalls eine Zufallszahl $e_B\in\mathbb{Z}_n^*$ und damit ein Schlüsselpaar: + $$(k_{B,\text{priv}},k_{B,\text{pub}})=(e_B,g^{e_B})$$ + \item Alice und Bob tauschen ihre öffentlichen Schlüssel aus. + \item Alice berechnet in $\mathbb{Z}n$: + $$s_A:=(g^{e_B})^{e_A}$$ + Bob berechnet in $\mathbb{Z}n$: + $$s_B:=(g^{e_A})^{e_B}$$ + \item Der gemeinsame Schlüssel (shared secret) ergibt sich durch: + $$s_A:=(g^{e_B})^{e_A}=g^{e_B\cdot e_A}=g^{e_A\cdot e_B}=(g^{e_A})^{e_B}=s_B$$ + \end{itemize} + + \section{Das Diskrete-Logarithmus-Problem (DL-Problem)} + Das Diskrete-Logarithmus-Problem bildet die mathematische Grundlage dafür, dass Schlüssel mittels der DH-Verfahrens über einen potentiell unsicheren Kanal übertragen werden können. + Das Problem definiert sich wie folgt: + \begin{mybox} + \textbf{DL-Problem}\\ + Gegeben seien $n\in\mathbb{N},g\in\mathbb{Z}_n^*$ und eine Potenz von $g$ in $\mathbb{Z}_n$: + $$a\in\langle g\rangle:=\{g^e\mid e\in\mathbb{N}\}$$ + Als DL-Problem wird die Aufgabe bezeichnet nur mittels $a$ und $g$ einen passenden Exponenten $e$ für $a=g^{e}$ zu finden.\\ + Es wird vermutet, dass es keinen effizienten Algorithmus gibt. + \end{mybox} + + \subsection{Das Diffie-Hellman-Problem (DH-Problem)} + Das DH-Problem ist eine Variante des DL-Problems, bei der zwei Potenzen $g^{e_A},g^{e_B}\in\langle g\rangle$ gegeben sind. + Hierbei wird die Zahl $g^{e_A\cdot e_B}$ gesucht. + Auch hier wird vermutet, dass es keinen effizienten Algorithmus gibt. + + \section{Beispielanwendung des DH-Verfahrens} + \includegraphics[scale=0.75]{DH-Verfahren.png} + + \section{Angriffe auf das DH-Verfahren} + \subsection{Pohlig-Hellman-Reduktion} + Eine Brute-Force-Suche für das DL-Problem wird dadurch vereinfacht, dass sich die Brute-Force-Suche auf eine Brute-Force-Suche auf die Untergruppen von $\langle g \rangle$ mit maximaler Primzahlpotenz reduzieren lässt. + Ein optimaler Schutz bietet die Wahl einer Primzahl $n$ bei der $n-1$ die Primfaktorzerlegung $n-1=2\cdot q$ hat ($q$ ist eine Primzahl). + + \subsection{Shanks' \glqq Baby Steps Giant Steps\grqq-Verfahren (BSGS-Verfahren)} + Falls die Möglichkeit besteht $\tilde{m}:=\lceil \sqrt{m}\rceil$ viele Zahlen von $\mathbb{Z}n$ in einer Tabelle zu speichern, + lässt sich ein gesuchter Exponent $x$ mit dem BSGS-Verfahren bereits mit der Berechnung von höchsten $2\tilde{m}$ vielen Potenzen finden. + Das Verfahren ähnelt dabei dem Meet-in-the-middle-Angriff (siehe \ref{meet-in-the-middle}). + + \subsection{Pollard's Rho-Methode} + siehe Skript 3 Kapitel 2.2.3 auf Seite 26(42) + + \subsection{Zahlkörpersieb} + Zu Beginn der 1990er Jahre wurde das Zahlkörpersieb (Number Field Sieve (NFS)) als Methode zur Faktorisierung großer Zahlen $n$ und zur Bestimmung diskreter Logarithmen in $\mathbb{Z}_n^*$ entwickelt. + Die Laufzeit zur Bestimmung diskreter Logarithmen in $\mathbb{Z}_n^*$ mithilfe des Zahlkörpersiebs kann durch + $$O\left(e^{\left(\left(\frac{64}{9}\right)^{1/3}+o(1)\right)(\ln(n))^{1/3}(\ln(\ln(n)))^{2/3}}\right)$$ + von oben abgeschätzt werden. diff --git a/chapters/Modulare Arithmetik - Teil 2.tex b/chapters/Modulare Arithmetik - Teil 2.tex index 7057068..e7475e5 100644 --- a/chapters/Modulare Arithmetik - Teil 2.tex +++ b/chapters/Modulare Arithmetik - Teil 2.tex @@ -45,6 +45,41 @@ Für eine Primzahl $p$ gilt für alle $a\in\mathbb{Z}$ mit $\text{ggT}(a,p)=1$: $$a^{p-1}\equiv 1 \hspace{10mm}(\mod p)$$ + \subsection{Berechnung modularer Potenzen} + Es gibt zwei verschiedene Varianten zur effizienten Berechnung modularer Potenzen, die sich beide an der Darstellung des Exponenten $e$ im Stellenwertsystem zur Basis 2 (binär) orientieren: + Es lässt sich ein Algorithmus aufstellen, bei dem sukzessive die Quadrate $a^2,a^4=(a^2)^2, a^8=(a^4)^2,...,a^{2^l-1}=\left(a^{2^{l-2}}\right)^2$ in $\mathbb{Z}_n$ berechnet werden. + Diese fließen als Faktor in die Berechnung von $p=a^e$ ein, falls das entsprechende Bit $e_i=1$ ist. + Es folgen die beiden Algorithmen: + \begin{mybox} + \textbf{Algorithmus 1}\\ + Sei $n\in\mathbb{N},a\in\mathbb{Z}_n$ und $e\in\mathbb{N}$ (in binärer Schreibweise): + $$e=(e_{l-1}...e_2 e_1 e_0)_2=\sum_{i=0}^{l-1}e_i\cdot 2^i$$ + Für die Berechnung von $p=a^e\mod n$ wird der folgende Algorithmus angewandt: + \begin{enumerate}[(1)] + \item $i:=0,p:=1$ und $s:=a$ + \item if($e_i=1$){$p:=p\cdot s\mod n$} + \item if($i=l-1$){beende den Algorithmus und gebe $p$ aus} + \item $i:=i+1$ + \item $s:=s^2\mod n$ + \item goto (2) + \end{enumerate} + \end{mybox} + \begin{mybox} + \textbf{Algorithmus 2}\\ + Sei $n\in\mathbb{N},a\in\mathbb{Z}_n$ und $e\in\mathbb{N}$ (in binärer Schreibweise): + $$e=(e_{l-1}...e_2 e_1 e_0)_2=\sum_{i=0}^{l-1}e_i\cdot 2^i$$ + Für die Berechnung von $p=a^e\mod n$ wird der folgende Algorithmus angewandt: + \begin{enumerate}[(1)] + \item $i:=l-1,p:=1$ und $s:=a$ + \item if($e_i=1$){$p:=p\cdot a\mod n$} + \item if($i=0$){beende den Algorithmus und gebe $p$ aus} + \item $i:=i-1$ + \item $p:=p^2\mod n$ + \item goto (2) + \end{enumerate} + \end{mybox} + + \section{Exkurs: Einheitengruppe $\mathbb{Z}_{p^e}^*$} Für eine Primzahl $p$ gibt es ein Element $g\in\mathbb{Z}_p^*$ mit $o(g)=p-1$ ($\mathbb{Z}_p^*==\{1,g,g^2,...,g^{p-2}\}$). Diese Elemente $g$ werden als \textbf{primitive Elemente} bezeichnet.\\ diff --git a/images/DH-Verfahren.png b/images/DH-Verfahren.png new file mode 100644 index 0000000..d73e183 Binary files /dev/null and b/images/DH-Verfahren.png differ