From 036b7c4dca85ede42a6dacbc80527673f5e8c1c5 Mon Sep 17 00:00:00 2001 From: Felix-Dev Date: Sat, 11 Apr 2015 19:36:44 +0200 Subject: [PATCH] 05-asymmenc.tex: ElGamal: Erster Entwurf MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit einer Erläuterung zum Verschlüsseln beliebiger Nachrichten mit ElGamal verfasst. --- 05-asymmenc.tex | 18 ++++++++++-------- skript.pdf | Bin 1163495 -> 1168475 bytes 2 files changed, 10 insertions(+), 8 deletions(-) diff --git a/05-asymmenc.tex b/05-asymmenc.tex index afd9a82..c48280e 100644 --- a/05-asymmenc.tex +++ b/05-asymmenc.tex @@ -276,13 +276,11 @@ \subsection{Bedeutung von RSA} \section{ElGamal} \label{ch:asymenc:elgamal} -Das ElGamal-Verfahren (1985) macht sich die Schwierigkeit zunutze, das Diffie-Hellman-Problem, also die Berechnung von diskreten Logarithmen in zyklischen Gruppen zu -lösen. Unter einer zyklischen Gruppe versteht man eine Gruppe $\mathbbm{G}$, bei der ein sogenanntes Erzeugerelement $g$ existiert, so dass $\mathbbm{G} = \langle g \rangle := \{g^k \mid k \in \mathbbm{Z}\}$. +Das ElGamal-Verfahren (1985) macht sich die Schwierigkeit zunutze, das Diffie-Hellman-Problem, also die Berechnung von diskreten Logarithmen in zyklischen Gruppen zu lösen. Unter einer zyklischen Gruppe versteht man eine Gruppe $\mathbbm{G}$, bei der ein sogenanntes Erzeugerelement $g$ existiert, so dass $\mathbbm{G} = \langle g \rangle := \{g^k \mid k \in \mathbbm{Z}\}$. \subsection{Vorgehen} Für die Erzeugung der Schlüssel benötigen wir eine angemessen große, zyklische Gruppe $\mathbbm{G}$ mit dem Erzeuger $g$. -Geeignete Kandidaten für eine solche Gruppe sind (echte) Untergruppen von $\mathbbm{Z}^*_P$ mit $P$ prim oder allgemeiner Untergruppen von $\mathbbm{F}^*_q$ -mit $q$ Primpotenz mit einer Gruppengröße von $|\mathbbm{G}| \approx 2^{2048}$. Effizienter sind Untergruppen von elliptischen Kurven +Geeignete Kandidaten für eine solche Gruppe sind (echte) Untergruppen von $\mathbbm{Z}^*_p$ mit $p$ prim oder allgemeiner Untergruppen von $\mathbbm{F}^*_q$ mit $q$ Primpotenz mit einer Gruppengröße von $|\mathbbm{G}| \approx 2^{2048}$. Effizienter sind Untergruppen von elliptischen Kurven $\boldsymbol{\mathsf{E}}(\mathbbm{F}^*_q)$ mit einer Gruppengröße von $|\mathbbm{G}| \approx 2^{200}$. Wir wählen außerdem eine Zufallszahl $x \in \{0, \dots, \vert \mathbbm{G} \vert - 1\}$, so dass $\text{ggT}(x, \vert \mathbbm{G} \vert) = 1$ und berechnen $h = g^x \in \mathbbm{G}$. Bemerke, dass in Gruppen primer Ordnung jedes Element der Menge gewählt werden kann. In einer Gruppe, die zusätzlich auf Äquivalenzklassen arbeitet, kann aufgrund der Modulorechnung $x \in \mathbbm{Z}$ sogar zufällig gewählt werden. Für unser Schlüsselpaar gilt damit: @@ -303,14 +301,18 @@ \subsection{Vorgehen} \item $\enc(pk, M) = (g^y, h^y \cdot M)$ \item $\dec(sk, (Y, C)) = \frac{C}{Y^x}$ \end{itemize} - -Durch die zufällige Wahl von $y$ ist das Chiffrat $\enc({pk, M})$ randomisiert. ElGamal ist somit semantisch sicher. Allerdings ist ElGamal wie RSA -homomorph: +Durch die zufällige Wahl von $y$ ist das Chiffrat $\enc({pk, M})$ randomisiert. ElGamal ist somit semantisch sicher. Allerdings ist ElGamal wie RSA homomorph: \begin{align*} \enc({pk,M}) \cdot \enc({pk,M'}) &= (g^y, g^{xy} \cdot M) \cdot (g^{y'}, g^{xy'} \cdot M')\\ &= (g^{y+y'}, g^{x(y + y')} \cdot M \cdot M')\\ &= \enc({pk, M \cdot M'}) \end{align*} +Es existieren allerdings bereits nicht-homomorphe Varianten von ElGamal. %TODO Verweis auf so ein Verfahren? + + +Ein Problem des obigen Verfahrens ist, dass wir nur Nachrichten $M \in \mathbbm{G}$ verschlüsseln können. Wollen wir beliebige Nachrichten $M$ (mit Einschränkung bei der Länge) verschlüsseln, so müssen wir vor der Verschlüsselung $M$ in eine Nachricht $\tilde{M} \in \mathbbm{G}$ konvertieren. Im Folgenden fassen wir $M$ als Bit-String auf. Betrachte hierfür folgendes Beispiel: + +Wir wählen $p > 2 $ prim und setzen $\mathbbm{G} \subseteq \mathbbm{Z}^*_p$ als Untergruppe der Quadrate von $\mathbbm{Z}^*_p$, wobei $\mathbbm{G}$ Ordnung $q = \frac{(p - 1)}{2}$ hat.\footnote{Die Untergruppe der Quadrate von $\mathbbm{Z}^*_p$ besteht aus den Elementen $\{y \equiv x^2 \text{ mod } p\ \vert\ x \in \mathbbm{Z}^*_p\}$. Falls $p > 2$ prim ist, besteht diese Untergruppe aus $\frac{p - 1}{2}$ Elementen.} Wir können $M \in \{0, 1\}^{n - 1}$ beliebig wählen und interpretieren $M$ in weiteren Verlauf als ganze Zahl äquivalent zu ihrer gewählten Binärdarstellung. Da $M$ auch die Null darstellen kann und die Null in multiplikativen Gruppen nicht vorhanden ist, setzen wir $\tilde{M} = M + 1$. Folglich ist $ 1 \leq \tilde{M} \leq q$ und daher $\tilde{M} \in \mathbbm{Z}^*_p$. Nach der Eigenschaft einer quadratischen Untergruppe ist somit $\hat{M} = \tilde{M}^2 \text{ mod } p \in \mathbbm{G}$. -Es existieren allerdings bereits nicht-homomorphe Varianten von ElGamal. +Damit kann $\hat{M}$ analog zum obigen Verfahren verschlüsselt werden. Zum Entschlüsseln berechnet der Empfänger aus $\hat{M}$ als Zwischenschritt $\tilde{M} = \sqrt{\hat{M}} \in [1, q]$ und erhält mit $M = \tilde{M} - 1$ die ursprüngliche Nachricht $M$ in der Binärdarstellung. $\hat{M}$ ist durch normales Entschlüsseln mit ElGamal zu berechnen. \ No newline at end of file diff --git a/skript.pdf b/skript.pdf index 348f92dfd3db05ae9aeb25dad0942c2ea25a44ca..a4b15c8ada6039c626c668d72689fc0a934cdefb 100644 GIT binary patch delta 168407 zcmZ@=cOaGj_Ye2p=i1}io0J*ubZ?1-2GK-TR;dVu_P8od4N~4{(NNN&aS0(&R%s6- z4Me1=6!kmLbMO6rTE9O&dXMLMKkswiXS~jNpZn=tKe6fL#Pg zdDTKYzG35)i5I#z2cM1GHYY{nJI$(-IlGOeCg(mu)49_li8DUdYE#ajan!g>X{!;~ ze2w8%`O}PNZr!-)Vd#jEEdDXqw4)lfW8G)Es^z#%>J7}9e_UtaMf8ZP$1*MUgpDkk z5gC%B7CxTVCAIFT_2Tb|oS@Fb`^$b(7G4R?JUecvu&Jc%WrCec_Kd6+iRlft!zJfw z-N>}PXn7~5>BN<(Gb{xCtdZt32l}JBD?1iiJ?+1pSQDi&OPe|6tlPS0LtOsqaLK%Y z>3zN`2POxZRc?57Cx0z3^T&c7%~9MVl5r{Yn3Qn2xT-a8)FESgj>sykQfaFNop zcSCIo(x!0_rmgYcak>9$MN;*KqkH)a>u1IH8))f!JlcOpVOZtQuWG7X$!`gk^0p1T 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