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Wiesenbauer. +Das ist die "ge {\TeX} te" Mitschrift zur Vorlesung $118 \dot 186$ \emph{AKDIS Zahlentheorie und Anwendungen} vom Sommersemester 2012, gehalten von Prof. J. Wiesenbauer. \newpage @@ -581,17 +581,17 @@ Das Euler'sche Kriterium kann nicht verallgemeinert werden. \\ @(4) stark multiplikativ im Zähler. \\ \begin{proof}[Beweis zu Satz $3.2$, Seite $10$] - \begin{itemize} + \begin{enumerate} \item Folgt unmittelbar aus der Definition des Legendre-Symbols, wegen \begin{equation} a \equiv b \mod p \implies \left[ x^{2} \equiv a \mod p \textsl{ ist lösbar } \Leftrightarrow x^{b} \equiv b \mod p \textsl{ ist lösbar} \right] \end{equation} \item Ist $g$ eine Primitivwurzel $\mod p$, so ist dann $\lbrace g, g^{2}, \ldots, g^{p-1} \rbrace$ ein volles primes Restsystem $\mod p$ und daher sind \begin{equation} - \underbrace{g^{2}, g^{4}, \ldots, g^{\left(\frac{p-1}{2}\right)^{2}}=g^{p-1}\equiv 1 \mod p}_{\textsl{versch. quadr. Reste}}, \underbrace{g^{\lbrace \frac{p+1}{2} \right)^{2}}}_{\textsl{Wiederholung}} \equiv g^{2} \mod p, \ldots + \underbrace{g^{2}, g^{4}, \ldots, g^{\left( \frac{p-1}{2} \right)^{2}} = g^{p-1}\equiv 1 \mod p}_{\textsl{versch. quadr. Reste}}, \underbrace{g^{ \left( \frac{p+1}{2} \right)^{2}}}_{\textsl{Wiederholung}} \equiv g^{2} \mod p, \ldots \end{equation} daher sind $g^{2}, g^{4}, g^{p-1}$ alle quadratischen Reste, und die ungeraden Potenzen sind quadratische Nichtreste. -\item \hfill +\item Sei $g$ eine Primitivwurzel. \begin{itemize} \item[1. Fall] $a$ ist quadratischer Rest $\implies \exists k \in \N: a \equiv g^{2k} \mod p$, woraus folgt dass \begin{equation} @@ -599,15 +599,60 @@ daher sind $g^{2}, g^{4}, g^{p-1}$ alle quadratischen Reste, und die ungeraden P \end{equation} \item[2. Fall] $a$ ist quadratischer Nichtrest, d.h. $\exists k \in \N: a \equiv g^{2k+1} \mod p$, dann gilt \begin{equation} - a^{\left( \frac{p-1}{2} \right)} \equiv \left( g^{2k+1} \right)^{\left( \frac{p-1}{2} \right) } \equiv \underbrace{g^{(p-1)k}}_{\equiv 1 \mod p} \cdot \underbrace{g^{\left( \frac{p-1}{2} \right)}}_{\equiv -1 \mod p} \equiv 1 \equiv \lbrace \frac{a}{p} \rbrace \mod p + a^{\left( \frac{p-1}{2} \right)} \equiv \left( g^{2k+1} \right)^{\left( \frac{p-1}{2} \right) } \equiv \underbrace{g^{(p-1)k}}_{\equiv 1 \mod p} \cdot \underbrace{g^{\left( \frac{p-1}{2} \right)}}_{\equiv -1 \mod p} \equiv -1 \mod p = \left( \frac{a}{p} \right) \end{equation} -Warum $\equiv -1 \mod p$? da $g^{\frac{p-1}{2}}$ Lösung von $x^{2} \equiv 1 \mod p$ ist $\implies$ - \end{itemize} +Warum $\equiv -1 \mod p$? Da $g^{\frac{p-1}{2}}$ Lösung von $x^{2} \equiv 1 \mod p$ ist folgt, dass +\begin{equation} +g^{\frac{p-1}{2}} \equiv \pm 1 \mod p +\end{equation} +da ein Polynom über einem Körper höchstens so viele Nullstellen hat, wie sein Grad angiebt. Weiters ist gilt $ord_{Z_{p}}(g)p-1$, daher $g^{\frac{p-1}{2}} \nequiv 1 \mod p$. \end{itemize} +\item Diese Eigenschaft nennt man auch ``starke Multiplikativität im Zähler''. Hier ist eine Fallunterscheidung wie folgt durchzuführen: +\begin{itemize} +\item +\begin{equation} +\left( \frac{a}{p} \right) = \left( \frac{b}{p} \right) = 1 +\end{equation} +\item oBdA: +\begin{equation} +\left( \frac{a}{p} \right) = 1, \left( \frac{b}{p} \right) = -1 +\end{equation} +\item +\begin{equation} +\left( \frac{a}{p} \right) = \left( \frac{b}{p} \right) = -1 +\end{equation} +\end{itemize} +Beispielsweise werde der erste Fall bewiesen (Rest analog): aus der Voraussetzung erhält man aus Punkt (2), dass +\begin{subequations} +\begin{align} +\exists k,\ell \in \N: a \equiv g^{2k} \mod p, b \equiv g^{2 \ell} \mod p \\ +\left( \frac{ab}{p} \right) = \left( \frac{g^{2k} \cdot g^{2 \ell}}{p} \right) = \left( \frac{g^{2(k+\ell)}}{p} \right) \stackrel{\textsl{Pkt (2)}} = 1 = \left( \frac{a}{p} \right) \cdot \left( \frac{b}{p} \right) +\end{align} +\end{subequations} + \end{enumerate} \end{proof} +\begin{bem}[Bemerkung zu Satz $3.4$, Seite $10$] +Der erste Ergänzungssatz beantwortet daher die Frage, wann es eine imaginäre Einheit modulo $p$ gibt. Im Falle $p \equiv 3 \mod 4$ muss man für eine imaginäre Einheit zu einem Erweiterungskörper $\mathbb{F}_{p^{2}}$ mit $p^{2}$ Elementen übergehen. +\end{bem} + +\begin{bem} +Mögliche Prüfungsfrage: Nennen Sie eine Primzahl zwischen $50$ und $100$. Antwort z.B.: $97$. +\end{bem} + +\begin{bem} +Für die Berechnung des Legendre-Symbols kann eine Faktorisierung des ``Zählers'' notwendig sein. +\end{bem} + +\begin{bem} +Definition $3.7$ ist eine beliebte Prüfungsfrage. +\end{bem} + +\begin{bem}[Bemerkung $3.8$, Seite $11$] +Gilt $\left( \frac{x}{y} \right)_{J} = -1$, so ist $x$ sicher \emph{kein} quadratischer Rest $\mod y$. Aus $\left( \frac{x}{y} \right)_{J}=1$ lässt sich allerdings keine Aussage mehr darüber treffen. +\end{bem} %Vorlesung 2.5.2012 -Zu Lucas-Folgen: sind Verallgemeinerung von Fibonacci-Folgen, sind wichtig für Primzahltests. \\ +Zu Lucas-Folgen kann man als Verallgemeinerung von Fibonacci-Folgen interpretieren, sind beispielsweise wichtig für Primzahltests. \\ \begin{proof}[Beweis zu Satz $4.2$, Seite $13$] Wegen $(x-\alpha)(x-\beta)=x^{2} - Px + Q = 0$ erhält man durch einen Koeffizientenvergleich \begin{equation} @@ -616,7 +661,7 @@ Zu Lucas-Folgen: sind Verallgemeinerung von Fibonacci-Folgen, sind wichtig für Durch unmittelbares Einsetzen erhält man: \begin{subequations} \begin{align} - U_{m}V_{n} - Q^{n}U_{m-n} = \frac{\alpha^{m}-\beta^{m}}{\alpha-\beta} \left( \alpha^{n} - \beta^{n} \right) - \left( \alpha \beta \right)^{n} \cdot \left( \frac{\alpha^{m-n} - \beta^{m-n}}{\alpha-\beta} \right)= \\ + U_{m}V_{n} - Q^{n}U_{m-n} = \frac{\alpha^{m}-\beta^{m}}{\alpha-\beta} \left( \alpha^{n} + \beta^{n} \right) - \left( \alpha \beta \right)^{n} \cdot \left( \frac{\alpha^{m-n} - \beta^{m-n}}{\alpha-\beta} \right)= \\ = \left( \frac{1}{\alpha - \beta} \right) \cdot \left( \alpha^{m+n} + \alpha^{m}\beta^{n} - \alpha^{n} \beta^{m} - \beta^{m+n} - \alpha^{m} \beta^{n} + \alpha^{n} \beta^{m} \right) = \\ = \frac{\alpha^{m+n} - \beta^{m+n}}{\alpha-\beta} \stackrel{\textsl{nach Def.}} = U_{m+n} \end{align} @@ -655,7 +700,7 @@ also dass, was zu zeigen war. \begin{align} U_{2n} = U_{n+n} = U_{n}V_{n} - Q^{n}U_{0} = U_{n} V_{n} \\ U_{2n+1} = U_{(n+1)+n}=U_{n+1}V_{n} - Q^{n} U_{1} = U_{n+1}V_{n} - Q^{n} \\ -V_{2n} = V_{n+n} = V_{n} V_{n} - Q^{n} V_{0} = V_{2}^{2} - 2Q^{n} \\ +V_{2n} = V_{n+n} = V_{n} V_{n} - Q^{n} V_{0} = V_{n}^{2} - 2Q^{n} \\ V_{2n+1} = V_{(n+1)+n} = V_{n+1}V_{n} - Q^{n} V_{1} = V_{n+1} V_{n} - PQ^{n} \end{align} \end{subequations} @@ -675,9 +720,9 @@ Man berechne: \begin{subequations} \begin{align} \left( 2 \alpha \right)^{r} = \left( R+\sqrt{D} \right)^{r} = \\ -= P^{r} \underbrace{\sum \limits_{k=1}^{r-1} \binom{r}{k} P^{k} \left( \sqrt{D} \right)^{r-k}}_{=:*} + \left( \sqrt(D) \right)^{r} = \\ -\equiv P^{r} + \left( \sqrt{D} \right)^{r} \equiv P^{r} + D^{\frac{r-1}{2}} \sqrt{D} \\ -\stackrel{\textsl{(**)}} \equiv P + \left( \frac{D}{r} \right) \equiv \begin{cases}P + \sqrt{D} \equiv 2 \alpha \mod r, \left( \frac{D}{r} \right) = 1 \\ P - \sqrt{D} \equiv 2 \beta \mod r, \left( \frac{D}{r} \right) = -1 \end{cases} += P^{r} + \underbrace{\sum \limits_{k=1}^{r-1} \binom{r}{k} P^{k} \left( \sqrt{D} \right)^{r-k}}_{=:*} + \left( \sqrt(D) \right)^{r} \equiv \\ +\equiv P^{r} + \left( \sqrt{D} \right)^{r} \equiv P + D^{\frac{r-1}{2}} \sqrt{D} \equiv \\ +\stackrel{\textsl{(**)}} \equiv P + \left( \frac{D}{r} \right)\cdot \sqrt{D} \equiv \begin{cases}P + \sqrt{D} \equiv 2 \alpha \mod r, \left( \frac{D}{r} \right) = 1 \\ P - \sqrt{D} \equiv 2 \beta \mod r, \left( \frac{D}{r} \right) = -1 \end{cases} \end{align} \end{subequations} @(*): es gilt @@ -687,7 +732,7 @@ Man berechne: denn angenommen $r \mid 1 \cdot 2 \cdots k \Rightarrow r \mid i$ für ein $i \in \lbrace 1,2, \ldots, r-1 \rbrace$, denn wenn eine Primzahl ein Produkt teilt, teilt sie einen Faktor, daher WS zu r ist Primzahl. \\ @(**): Es gilt nach dem ``kleinen Fermat'': $P^{r} \equiv P \mod r$. Weiters gilt nach dem Euler'schen Kriterium: \begin{equation} - D^{\frac{r-1}{2}} \equiv \left( \frac{D}{r} \right) \mod r + D^{\frac{r-1}{2}} \equiv \left( \frac{D}{r} \right) \mod p \end{equation} Nun gilt aber \begin{equation} @@ -702,7 +747,7 @@ Durch Kürzen durch $2$ ($2^{-1} \mod r$ existiert, da ja $\gcd(2,r)=1$ da $r$ u \item $\alpha, \beta \in \Z_{r} \implies$ ``kleiner Fermat'' $\implies$ fertig. \item $\alpha \vee \beta \notin \Z_{r} \implies \alpha, \beta \in \Z_{r^{2}} \geq \Z_{r}$. $\Z_{r^{2}}$ hat genau einen nichtrivialen Automorphismus und es gilt \begin{equation} - \mathcal{F} = \lbrace a + b \sqrt{D} \mid a,b \in \Z_{r} \rbrace + \mathbb{F} = \lbrace a + b \sqrt{D} \mid a,b \in \Z_{r} \rbrace \end{equation} Es folgt daher, dass die Funktionen $x \mapsto x^{r}$ und $a+b\sqrt{D} \mapsto a - b \sqrt{D}$ der gleiche Automorphismus sind. \end{itemize} @@ -711,7 +756,7 @@ Es folgt daher, dass die Funktionen $x \mapsto x^{r}$ und $a+b\sqrt{D} \mapsto a \begin{proof}[Beweis zu Satz $4.7$, Seite $14$] Es gilt $\left( \frac{D}{r} \right) \in \lbrace \pm 1 \rbrace$ wegen $\gcd(r,QD)=1$. \begin{itemize} - \item[(1)] Sei zunächst $\lbrace \frac{D}{r} \rbrace = 1$. Dann gilt nach Lemma $4.6$ + \item[(1)] Sei zunächst $\left( \frac{D}{r} \right) = 1$. Dann gilt nach Lemma $4.6$ \begin{equation} \alpha^{r} \equiv \alpha \mod r, \beta^{r} \equiv \beta \mod r \end{equation} @@ -720,9 +765,9 @@ Es gilt daher \begin{equation} \left( \alpha - \beta \right) U_{r-1} = \alpha^{r-1} - \beta^{r-1} \equiv 1 -1 \equiv 0 \mod r \end{equation} -Nun ist aber $\alpha - \beta = \sqrt{D} \nequiv 0 \mod r$ (weil $\sqrt{D} \mod r \Rightarrow D = (\sqrt{D})^{2} \equiv 0 \mod r \Rightarrow \gcd(r,QD) \neq 1 \blitza$), woraus durch Kürzen tatsächlich $U_{r-1} \equiv 0 \mod r$ folgt. \\ +Nun ist aber $\alpha - \beta = \sqrt{D} \nequiv 0 \mod r$ (weil wäre $\sqrt{D} \equiv 0 \mod r \Rightarrow D = (\sqrt{D})^{2} \equiv 0 \mod r \Rightarrow \gcd(r,QD) \neq 1$ WS), woraus durch Kürzen tatsächlich $U_{r-1} \equiv 0 \mod r$ folgt. \\ Sei nun $\left( \frac{D}{r} \right) = -1$. Daher gilt nach Lemma $4.6$ -\begin{subequations} +\begin{subequations}\label{eq1} \begin{align} \alpha^{r} \equiv \beta \mod r \\ \beta^{r} \equiv \alpha \mod r @@ -730,14 +775,14 @@ Sei nun $\left( \frac{D}{r} \right) = -1$. Daher gilt nach Lemma $4.6$ \end{subequations} Daher gilt \begin{equation} -(\alpha - \beta)U_{r+1} = (\alpha^{r+1} - \beta^{r+1} )=alpha \alpha^{r} - \beta \beta^{r} \equiv \alpha \beta - \alpha \beta \equiv 0 \mod r, +(\alpha - \beta)U_{r+1} = \alpha^{r+1} - \beta^{r+1} = \alpha \alpha^{r} - \beta \beta^{r} \stackrel{\eqref{eq1}} \equiv \alpha \beta - \alpha \beta \equiv 0 \mod r, \end{equation} woraus wie vorhin durch Kürzen $U_{r+1} \equiv 0 \mod r$ folgt. \item[(2)] Wegen $U_{2n} = U_{n}V_{n}$ nach $4.4$ gilt \begin{subequations} \begin{align} U_{r-\left( \frac{D}{r} \right)} = U_{s \cdot 2^{t}} = U_{s2^{t-1}}V_{s2^{t-1}} = \cdots = \\ -= U_{s}V_{s}V_{2s}V_{4s} \cdots V_{s2^{t-1}} \equiv 0 \mod r \textsl{ nach (1)} += U_{s}V_{s}V_{2s}V_{4s} \cdots V_{s2^{t-1}} \equiv 0 \mod r \textsl{ nach Punkt (1)} \end{align} \end{subequations} $r$ ist Primzahl, daher teilt r einen Faktor, woraus die Behauptung direkt folgt. @@ -746,6 +791,62 @@ $r$ ist Primzahl, daher teilt r einen Faktor, woraus die Behauptung direkt folgt (\alpha - \beta)U_{r} \equiv \alpha^{r} - \beta^{r} \equiv \begin{cases}\alpha - \beta \mod r, \left( \frac{D}{r} \right)=1 \\ \beta - \alpha \mod r, \left( \frac{D}{r} \right) = -1 \end{cases} \end{equation} Woraus durch Kürzen folgt: $U_{r} \equiv \left( \frac{D}{r} \right) \mod r$. - \end{itemize} +%VO 9.5.2012 +\item[(4)] + \begin{equation} + V_{r} = \alpha^{r} + \beta^{r} \equiv \begin{cases} \alpha + \beta \mod r, \textsl{ falls } \left( \frac{D}{r} \right) = 1 \\ \beta + \alpha \mod r, \textsl{ falls } \left( \frac{D}{r} \right) = -1 \end{cases} \equiv P \mod r + \end{equation} +Die letze Kongruenz gilt wegen $\alpha + \beta = P$. +\item[(5)] +\begin{equation} +V_{r-\left( \frac{D}{r} \right)} = \alpha^{r-\left( \frac{D}{r} \right)} + \beta^{r - \left( \frac{D}{r} \right) } \equiv \begin{cases} \alpha^{r-1} + \beta^{r-1} \equiv 1+1 \equiv 2 \mod 4, \textsl{ falls } \left( \frac{D}{r} \right) = 1 \\ +\alpha^{r+1} + \beta^{r+1} = \beta \alpha + \alpha \beta = 2 \alpha \beta = 2Q, \textsl{ falls } \left( \frac{D}{r} \right) = -1 \end{cases} = 2 \cdot Q^{\frac{1-\left( \frac{D}{r} \right)}{2}} +\end{equation} +Man beachte beim Fall $\left( \frac{D}{r} \right) = -1$ das Folgende: $\alpha^{r} \equiv \beta \mod r \land \beta^{r} \equiv \alpha \mod r$. Die letzte Gleichung erhält man durch einfaches Einsetzen. +\end{itemize} \end{proof} + +\begin{bem}[Bemerkung $4.9$, Seite $14$] +\begin{itemize} +\item[(2)] Sei $P:=a+1, Q:=a$ und $a \geq 1$. +\begin{subequations} +\begin{align} +D = P^{2} - 4Q = (a+1)^{2} - 4a = (a-1)^{2} \Rightarrow \left( \frac{D}{r} \right) = 1 \land \sqrt{D}=a-1 \\ +\alpha = \frac{P + \sqrt{D}}{2} = \frac{a+1+(a-1)}{2} = a, \beta = \frac{P-\sqrt{D}}{2} = \frac{a+1-(a-1)}{2} = 1 +\end{align} +\end{subequations} +Teste: $n \mid U_{n-1} = \frac{a^{n-1}-1}{a-1}$, d.h. falls $\gcd(n,a-1)=1$ ist dies äquivalent mit $n \mid a^{n-1} - 1 (\Leftrightarrow a^{n-1} \equiv 1 \mod n)$. +\item[(3)] Falls $\left( \frac{D}{n} \right) = 1$ vermeide, besser ist $=-1$: setzte $P:=\lceil \sqrt{D} \rceil$ +\begin{equation} +D=P^{2} - 4Q \Rightarrow Q = \frac{P^{2}-D}{4} +\end{equation} +Dadurch erhält man +\begin{subequations} +\begin{align} +\begin{cases}P \equiv \pm 1 \mod 4 \Rightarrow P^{2} \equiv 1 \mod 4 \\ D \equiv \mod 4 \end{cases} \\ +\Rightarrow Q = \frac{P^{2}-D}{4} \in \Z +\end{align} +\end{subequations} +Der Baillie-Wagstaff-Test ist ein starker Lucas-Test mit den oben genannten Parametern, basierend auf Punkt (2) von Satz $4.7$. +\item Gilt $p \in \P \land p+2 \in \P$, so betrachte alle Paare $(P,Q)$ mit gleichem $D \equiv P^{2}-4Q \mod N$. Teste ob $N$ die quadratische Gleichung (in x) $x^{2} + 2 x = 0$ löst, was leicht zu realisieren ist. +\end{itemize} +\end{bem} + +\begin{bem}[Signaturen, Seite $15$] +Angenommen Alice und Bob kommunizieren, wählen eine Hashfunktion $h$. Bob schickt nun an Alice eine Nachricht m und eine Signatur s: +\begin{equation} +(c,s)=\left( E_{A}(m), D_{B}(h(m)) \right) +\end{equation} +Die erste Komponente ist die mit dem öffentlichen Schlüssel von Alice verschlüsselte Nachricht $c$. Die zweite Komponente besteht aus der Unterschrift/Signatur $s$. \\ +Beim Empfang der Nachricht geht Alice wie folgt vor: +\begin{enumerate} +\item Entschlüsseln: $D_{A}(E_{A}(m))=m$, damit hat Alice die Klartextnachricht. +\item Alice berechnet $h(m)$. +\item Mit dem öffentlichen Schlüssel von Bob berechnet Alice nun $E_{B}(s)$ und testet $E_{B}(s) == h(m)$? Falls "$=$", dann stammt die Nachricht von Bob. Falls "$\neq$", dann stammt die Nachricht nicht von Bob. +\end{enumerate} +\end{bem} + +\begin{bem}[Bemerkung zu Knapsack-Problem] +Sind die Gewichte beispielsweise (alle) Potenzen von $2$ und kommt jedes Item höchstens einmal vor, so erhält man aus der Binärdarstellung die (eindeutige) Lösung. +\end{bem} \end{document} -- 2.47.3