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z1N{puswiwSwqUz|9RL5}Bq;TSxWy#-K>S=1>{4P9QXqEDFFd^LU&OfhCB!%-c{q8* zL@58iDi|XF4E^u%9!g$5&NwVM27o4B&?!;{_vfO(w@#cEe#e-QPmW!i>vkU{0;Mtk z_+oM7U8BChMbd)lEfb9`xw%nH2c{Ut*ij;-?6^_TJvL^A>h8SmmMzXZGJVFz-pt2L zu=Qco^rXj6RpL~}&BkWIjL1aQ5(^RuesI-N3(^Tcn44u9gG({3O+Bc;8v)t~VZ$gx zM?{H!sW+)$B1hWDY`zdOp~ zd}MxKwzeUp^)vKi3uJvmM1z)&hCo*$Z#7uP$ItjqENNsG!F6Sy!3{i#wRR3^L7iPhPBhWkw4e zm@C -war das oben nicht 60..? HI 20.1.17 -\end{equation*} - -\begin{equation*} -7^{-1} \bmod \varphi(60) -\end{equation*} -\begin{equation*} -\textit{e}^{-1} \bmod \varphi\textit{(N)} \equiv 7^{-1} \bmod -\siebenmineinsxelfmineins -\end{equation*} +Formen diese nach \textit{d} um :\\[1em] +\fbox{% + \begin{minipage}[t][40mm][t]{0.3\textwidth} +Theorie: + \begin{align*} + \textit{d} = \frac{\bmod \varphi\textit{(N)}}{\textit{e}} + \end{align*} +\end{minipage}% +} +\fbox{% + \begin{minipage}[t][40mm][t]{0.3\textwidth} +Beispiel: + \begin{align*} + \frac{\bmod \varphi\textit{(77)}}{\textit{7}} + \end{align*} + \end{minipage}% +}\\ +Danach können wir die Zahlen in die Formel setzen :\\ +\fbox{% + \begin{minipage}[t][20mm][t]{0.3\textwidth} +Theorie: + \begin{align*} + \textit{e}^{-1} \bmod \varphi\textit{(N)} + \end{align*} + \end{minipage}% +} +\fbox{% + \begin{minipage}[t][20mm][t]{0.3\textwidth} +Beispiel: + \begin{align*} + \frac{\bmod \varphi(77)}{\textit{7}}\\ + \frac{\bmod \siebenmineinsxelfmineins}{\textit{7}} + \end{align*} + \end{minipage}% +}\\[2em] Aus der letzten Gleichung berechnen wir die Inverse, indem wir erstmal aus 60 und 7 den euklidischen Algorithmus formen, als wollten wir den ggT dieser Zahlen ermitteln. Wir wissen schon dass der ggT(60,7) 1 ist, brauchen jedoch wie man @@ -264,7 +273,7 @@ Theorie: \begin{minipage}[t][40mm][t]{0.3\textwidth} Beispiel: \begin{align*} - 60 & = 8 \cdot 7 + 4\\ % IH | 60 und 7 kommen aus e und m und wir rechnen hier wieviel mal 7 + der rest 60 gibt + 60 & = 8 \cdot 7 + 4\\ 7 & = 1 \cdot 4 + 3\\ 4 & = 1 \cdot 3 + 1 \label{eq:bspd1}\\ % IH | Labels können mit \ref{} dynamisch wieder aufgerufen werden 3 & = 3 \cdot 1 + 0 @@ -279,7 +288,7 @@ diese nacheinander eingesetzt werden:\\ \\ \fbox{% \begin{minipage}[t][40mm][t]{0.3\textwidth} -Theorie: +%Theorie: \begin{align*} \\ \\ % hier könnte ein Theorieteil eingefügt werden @@ -302,9 +311,9 @@ Beispiel: } \\ Zu beachten ist, dass wir alle Klammern, jedoch nicht alle Produkte %das verstehe ich nicht HI | 10.1.17 -ausmultiplizieren\\ +ausmultiplizieren.\\ \\ -Da unsere Inverse positiv und kleiner 60 sein soll, addieren wir \(60*7\) auf +Da unsere Inverse von \textit{e} positiv und kleiner 60 sein soll, addieren wir \(60 \cdot 7\) auf der linken Seite. Die rechte Seite verändert sich nicht, da wir mod 60 rechnen.\\ \\ \fbox{% @@ -314,23 +323,25 @@ Theorie: \\ \\ % hier könnte ein Theorieteil eingefügt werden \\ - \\ + \\ + \textit{e} ^\textit{-1} \, \bmod \textit{N} \end{align*} - \end{minipage}% + \end{minipage}% isi empfielt: https://www.youtube.com/watch?v=Grd-sxx5dEQ } \fbox{% \begin{minipage}[t][40mm][t]{0.3\textwidth} Beispiel: \begin{align*} 1 & = 2 \cdot 60 -17 \cdot 7 \,\mid \bmod\\ + 1 & = -17 \cdot 7 \bmod 60 \mid + 60 \cdot 7\\ 1 & = 43 \cdot 7 \, \bmod 60 \,\mid (60 - 17 = 43)\\ - 1 & = \textit{d} ^\textit{e} \, \bmod \textit{n} \, \, \,\mid Umformen \, nach + 1 & = \textit{d} ^\textit{e} \, \bmod \textit{N} \, \, \,\mid Umformen \, nach \, \textit{d} \, \\ \textit{d} & = 7^{-1} \, \bmod 60 \\ - \textit{d} & = 43 + \textit{d} & = 43 \\ \end{align*} - \end{minipage}% -} \\ + \end{minipage}% +} \\[1em] % IH | in Formeln werden leerzeichen ignoriert darum kann mit " \, " , " \: " , %" \; " erstellt einen "schmalen", "mittleren" und "breiten" Leerschlag. Es geht %auch mit \thinspace, \medspace oder \thickspace... @@ -340,57 +351,77 @@ Danach lösen wir die Gleichung nach d auf. \section{Verschlüsselung} -Im Beispiel der Schlüsselkonstruktion werden die Variablen e und N als +Im Beispiel der Schlüsselkonstruktion werden die Variablen \textit{e} und \textit{N} als öffentlicher Schlüssel festgelegt. Dieser wird benötigt um eine Nachricht für den dafür entsprechenden Empfänger zu verschlüsseln. Mit der daraus -resultierenden Zahl sowie dem privaten Schlüssel, welcher aus den Variablen d -und N besteht, kann die Nachricht wieder entschlüsselt werden. - -In unserem Beispiel lautet der private Schlüssel also: \(43+77\) +resultierenden Zahl sowie dem privaten Schlüssel, welcher aus den Variablen \textit{d} +und \textit{N} besteht, kann die Nachricht wieder entschlüsselt werden. \\ +\\ +In unserem Beispiel lautet der private Schlüssel also: \(43+77\)\\ und der öffentliche Schlüssel: \(7+77\) \subsection {Der eigentliche Akt der Verschlüsselung} Wollen wir nun eine Nachricht mit dem öffentlichen Schlüssel verschlüsseln, so -das sie also nur noch für den Empfänger mit dem entsprechenden privaten +das sie nur noch für den Empfänger mit dem entsprechenden privaten Schlüssel zu entschlüsseln ist, gehen wir folgendermassen vor: Wir kennen die beiden Zahlen des öffentlichen Schlüssels: \(7+77\) -Unsere zu verschlüsselnde Nachricht x: 45 (muss kleiner sein als N) +Unsere zu verschlüsselnde Nachricht x: 47 (muss kleiner sein als \textit{N}) (Wie bereits in einem früheren Kapitel erwähnt sind solche öffentliche Schlüssel Primzahlen mit mehreren hundert Stellen, somit ist diese Regel im Normalfall irrelevant. Da wir aber in unserem Beispiel keine so grossen Primzahlen verwenden müssen wir diesen Punkt beachten um sicherzustellen das wir auch ein korrektes Ergebnis erhalten.) -Die Nachricht wird nun mit folgender Formel verschlüsselt: - -\begin{align*} -\textit{y} & = \textit{x}^\textit{a} \bmod \textit{n} \\ %Sollte n nicht N sein? -\textit{y} & = 45^{43} \bmod 77 \\ -\textit{y} & = 45 \\ -\end{align*} - -\subsection{Die Übermittlung} -TODO: Muss ergänzt werden. - -45 (y) ist nun also unsere Verschlüsselte Nachricht, welche nun an den Empfänger +Die Nachricht wird nun mit folgender Formel verschlüsselt: \\ +\fbox{% + \begin{minipage}[t][40mm][t]{0.3\textwidth} +Theorie: + \begin{align*} + \textit{y} & = \textit{x}^\textit{e} \bmod \textit{N} \\ + \\ + \end{align*} + \end{minipage}% +} +\fbox{% + \begin{minipage}[t][40mm][t]{0.3\textwidth} +Beispiel: + \begin{align*} + \textit{y} & = 47^{7} \bmod 77 \\ + \textit{y} & = 75 \\ + \end{align*} + \end{minipage}% +}\\ +47 (y) ist unsere Verschlüsselte Nachricht, welche an den Empfänger übermittelt wird. - \section{Entschlüsselung} -Um die Nachricht zu entschlüsseln muss zuerst d errechnet werden, dies geschieht +Um die Nachricht zu entschlüsseln muss zuerst \textit{d} errechnet werden, dies geschieht mithilfe des erweiterten euklidischen Algorithmus. Diese Berechnung wurde bereits im Kapitel Schlüsselerzeugung erledigt. -Unsere gesuchte Zahl lautet demnach 43 (d) +Unsere gesuchte Zahl lautet demnach 47 (\textit{d}) Da nun alle benötigten Variablen bekannt sind kann die Nachricht mit folgender -Formel entschlüsselt werden. +Formel entschlüsselt werden.\\ +\fbox{% + \begin{minipage}[t][40mm][t]{0.3\textwidth} +Theorie: + \begin{align*} + \textit{x} & = \textit{y}^\textit{d} \bmod \textit{N} \\ + \\ + \end{align*} + \end{minipage}% +} +\fbox{% + \begin{minipage}[t][40mm][t]{0.3\textwidth} +Beispiel: + \begin{align*} + \textit{x} & = 75^{43} \bmod 77 \\ + \textit{x} & = 47 \\ + \end{align*} + \end{minipage}% +}\\ -\begin{align*} -\textit{x} & = \textit{y}^\textit{d} \bmod \textit{n} \\ -\textit{x} & = 45^{43} \bmod 77 \\ -\textit{x} & = 45 \\ -\end{align*} \section{Schwachstellen} Obwohl schon einige verkündet haben die RSA Verschlüsselung geknackt zu haben