\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Arial;}{\f4\fmodern\fprq1 Courier New;}{\f5\fswiss\fprq2\fcharset1 Arial;}{\f6\fswiss\fprq2 Helvetica;}{\f7\froman\fcharset1 Times New Roman;}} {\colortbl\red0\green0\blue0;\red255\green0\blue0;\red0\green0\blue255;\red0\green128\blue0;\red255\green255\blue255;} \deflang1031\pard\ri4\plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \par \plain\f4\fs20\cf0 Inhalt....: ElGamal-Algorithmus \par Kategorie.: Arbeitsblatt \par Mathematik: Kryptographie, Zahlentheorie \par MuPAD.....: 3.0.0 \par Datum.....: 2003-06-23 \par Autoren...: Julia Faflek \par Funktionen: ithprime, nextprime, igcd, random, powermod, \par Funktionen: numlib::toAscii, numlib::fromAscii \par \plain\f4\fs20\cf0\b ________________________________________________________________________________ \par \plain\f3\fs36\cf0\b \par \plain\f3\fs40\cf0\b Der ElGamal-Algorithmus zur Nachrichten- \par verschl\'fcsselung und zur Erstellung digitaler \par Unterschriften\plain\f3\fs36\cf0\b \par \plain\f3\fs22\cf0 \par \plain\f3\fs24\cf3 Der ElGamal-Algorithmus zur Verschl\'fcsselung ist eine Variation des Diffie-Hellman- \par Algorithmus. Dieser wird eingehend in dem Notebook "DiffieHellman.mnb", welches \par unter \plain\f3\fs24\cf3\b\i www.schule.mupad.de/material\plain\f3\fs24\cf3 zum Download zur Verf\'fcgung steht, untersucht. \par Die Mathematik, die dem Verfahren unterliegt, ist die im wesentlichen die gleiche wie \par beim Diffie-Hellman-Algorithmus. Daher wird auf weitere zahlentheoretische Hinter- \par gr\'fcnde im Rahmen dieses Arbeitsblattes nicht eingegangen. \par \plain\f4\fs22\cf2 \par \par \plain\f3\fs28\cf0 Wie bei dem Diffie-Hellman-Algorithmus ben\'f6tigen wir zun\'e4chst wieder \par eine m\'f6glichst gro\'dfe Primzahl \plain\f3\fs28\cf0\i p\plain\f3\fs28\cf0 und eine Basis \plain\f3\fs28\cf0\i g\plain\f3\fs28\cf0 aller Elemente aus \plain\f3\fs28\cf0\i Z_p\plain\f3\fs28\cf0 , \par die ungleich 0 sind. Zur Erzeugung dieser Basis benutzen wir die Prozedur \par aus dem Notebook "Diffie-Hellman.mnb". Zun\'e4chst die Prozedur, die das \par Produkt der ersten \plain\f4\fs28\cf1 n\plain\f3\fs28\cf0 Primzahlen berechnet. \par \plain\f3\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}kleinePrim := proc(n) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 begin \par _mult(ithprime(i) $ i = 1..n) \par end_proc:\plain\f3\fs28\cf1 \par \pard\ri4\plain\f7\fs22\cf0 \par \plain\f5\fs28\cf0 Dann die Berechnung des Produkts der entfernten Primfaktoren und des \par \'fcbrig gebliebenen gro\'dfen Teilers: \par \plain\f7\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}entferneFaktoren := proc(p, n) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 local grFaktor, gemFaktor, ProdFaktoren; \par begin \par grFaktor := p-1: \par gemFaktor := igcd(kleinePrim(n), grFaktor): \par while (gemFaktor > 1) do \par grFaktor := grFaktor/gemFaktor: \par gemFaktor := igcd(kleinePrim(n), grFaktor): \par end_while: \par ProdFaktoren := (p-1)/grFaktor: \par return(grFaktor, ProdFaktoren): \par end_proc: \par \par \pard\ri4\plain\f5\fs28\cf0 Abschlie\'dfend die Erzeugung einer Basis \plain\f5\fs28\cf0\i g\plain\f5\fs28\cf0 : \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Basis := proc(p, n) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 local Ergebnis, x, g, test; \par begin \par Ergebnis := entferneFaktoren(p, n): \par x := random(1..p)(): \par g := powermod(x, Ergebnis[2], p): \par test := powermod(g, Ergebnis[1], p): \par while (test > 1) do \par Ergebnis := entferneFaktoren(p,n): \par x := random(1..p)(): \par g := powermod(x, Ergebnis[2], p): \par test := powermod(g, Ergebnis[1], p): \par end_while: \par return(g) \par end_proc: \par \pard\ri4\plain\f5\fs28\cf0 \par Wir w\'e4hlen also eine zuf\'e4llige gro\'dfe Primzahl: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}p := nextprime(2^300) \par \pard\ri4\plain\f5\fs28\cf0 \par und erzeugen die Basis: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}g := Basis(p, 300) \par \pard\ri4\plain\f5\fs28\cf0 \par Nun k\'f6nnen wir mit der Verschl\'fcsselung beginnen. Dazu muss jeder Teil- \par nehmer seinen \'f6ffentlichen Schl\'fcssel ver\'f6ffentlichen. Als erstes w\'e4hlt jede \par Partei eine zuf\'e4llige Zahl zwischen 1 und \plain\f5\fs28\cf0\i p\plain\f5\fs28\cf0 -1. Dies ist der geheime \par Schl\'fcssel: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}geheim1 := random(1..p-1)() \par \pard\ri4\plain\f5\fs28\cf0 \par Mittels der diskreten Exponentialfunktion wird nun der \'f6ffentliche Schl\'fcssel \par berechnet: \par \plain\f4\fs22\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}oeff1 := powermod(g, geheim1, p) \par \pard\ri4\plain\f3\fs28\cf0 \par Wir ver\'f6ffentlichen unseren \'f6ffentlichen Schl\'fcssel, damit jeder, der uns \par eine Nachricht schicken m\'f6chte, wei\'df, mit welchen Parametern er ver- \par schl\'fcsseln muss. \par \par \plain\f5\fs28\cf0 Die Partei mit der wir kommunizieren wollen erzeugt ihre Schl\'fcssel auf \par gleiche Weise: \par \plain\f7\fs22\cf0 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}geheim2 := random(1..p-1)() \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}oeff2 := powermod(g, geheim2, p) \par \pard\ri4\plain\f3\fs28\cf0 \par Nun k\'f6nnen wir mit dem Ver- und Entschl\'fcsseln beginnen. Stellen wir uns \par vor, die andere Partei m\'f6chte uns folgende Nachricht zu kommen lassen: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Nachricht := "Hallo! Bist du auch ein MuPAD-Nutzer?". \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 "Kryptografie macht wirklich Spass.". \par "Bin ja mal gespannt, ob du das entschluesseln kannst." \par \pard\ri4\plain\f3\fs28\cf0 \par Nun muss der andere erst einmal aus dem Text Zahlen machen. Dazu wird \par der Textstring einfach in das ASCII-Format umgewandelt: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Umwandlung := numlib::toAscii(Nachricht) \par \pard\ri4\plain\f5\fs28\cf0 \par Er sucht sich unseren \'f6ffentlichen Schl\'fcssel heraus und benutzt zus\'e4tzlich \par seinen geheimen Schl\'fcssel, um die nun vorliegende Liste von Zahlen zu \par verschl\'fcsseln. Dabei wird jede Zahl der Liste einzeln in folgender Weise \par verschl\'fcsselt: \par \pard\li2500\ri4\plain\f4\fs22\cf2 {\pict\wmetafile8\picw7201\pich1138\picscalex98\picscaley98\picwgoal4124\pichgoal651 0100090000036E0400000A001C0000000000050000000B0200000000050000000C027204211C03 0000001E00050000000C027E046B1C050000000B0200000000030000001E00050000000C028B04 B61C050000000B0200000000050000000B0200000000030000001E00050000000C029804011D05 0000000B0200000000050000000B0200000000050000000B0200000000030000001E0005000000 0C02A4044D1D050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000030000001E00050000000C02B2049B1D050000000B0200000000050000000B02 00000000050000000B0200000000050000000B0200000000050000000B0200000000030000001E 00050000000C02BF04E91D050000000B0200000000050000000B0200000000050000000B020000 0000050000000B0200000000050000000B0200000000050000000B0200000000030000001E0005 0000000C02CC04381E050000000B0200000000050000000B0200000000050000000B0200000000 050000000B0200000000050000000B0200000000050000000B0200000000050000000B02000000 00030000001E00050000000C02D904871E050000000B0200000000050000000B02000000000500 00000B0200000000050000000B0200000000050000000B0200000000050000000B020000000005 0000000B0200000000050000000B0200000000030000001E00050000000C02DA04881E05000000 0B0200000000050000000B0200000000050000000B0200000000050000000B0200000000050000 000B0200000000050000000B0200000000050000000B0200000000050000000B02000000000500 00000B0200000000030000001E00030000001E00050000000C02C0024F11050000000B02000000 00050000000B0200000000050000000B0200000000050000000B0200000000050000000B020000 0000050000000B0200000000050000000B0200000000050000000B0200000000050000000B0200 000000050000000B020000000008000000FA0200000000000000000000040000002D0100001C00 0000FB0238FF00000000000090010000000107000000417269616C000000810E0A1C38E91200D8 9FF177E19FF1772020F3778F0D66CD040000002D01010005000000020101000000050000000102 FFFFFF00050000002E01180000000500000009020000000004000000080100001C000000FB02E8 FE00000000000090010000000107000000417269616C0000007D0E0A4238E91200D89FF177E19F F1772020F3778F0D66CD040000002D0102000B00000026060F000C004D6174685479706500009D 001C000000FB023AFF00000000000090010100000107000000417269616C000000DD0D0A9A38E9 1200D89FF177E19FF1772020F3778F0D66CD040000002D0103001C000000FB023AFF0000000000 0090010000000107000000417269616C000000740E0A1538E91200D89FF177E19FF1772020F377 8F0D66CD040000002D010400040000002D010200040000002D0104001C000000FB02E8FE000000 00000090010100000107000000417269616C000000DB0D0A6D38E91200D89FF177E19FF1772020 F3778F0D66CD040000002D010500040000002D0102001C000000FB02E8FE000000000000900100 000002070000005346204D617468204578740038E91200D89FF177E19FF1772020F3778F0D66CD 040000002D010600040000002D010200040000002D010600040000002D010200040000002D0106 001C000000FB02E8FE0000000000009001000000020700000053796D626F6C0000B8040A5D38E9 1200D89FF177E19FF1772020F3778F0D66CD040000002D010700040000002D010500040000002D 010200040000002D010600040000002D010200040000002D010600040000002D01020004000000 2D010600040000002D010700040000002D010200040000002D010400040000002D010200070000 00210501004C00FB01640007000000210501006900FB01000107000000210501007300FB013E01 07000000210501007400FB01CA0107000000210501006500FB011802040000002D010300070000 002105010069003302B402040000002D0107000700000021050100D700FB012203040000002D01 0600040000002D010200040000002D010600040000002D010200040000002D010600040000002D 010200040000002D010600040000002D010200040000002D0106000700000021050100B3006400 A003040000002D01020007000000210501006F00FB01470407000000210501006500FB01E30407 000000210501006600FB017F0507000000210501006600FB01CD0507000000210501003100FB01 1B06040000002D010400070000002105010067003001B706070000002105010065003001250707 000000210501006800300193070700000021050100650030010108070000002105010069003001 6F0807000000210501006D0030019B080700000021050100320030014009040000002D01020007 000000210501006D00FB01F40907000000210501006F00FB01DD0A07000000210501006400FB01 790B040000002D01050007000000210501007000FB015D0C040000002D010600040000002D0102 00040000002D010600040000002D010200040000002D0106000700000021050100B4006400F90C 040000002D01020007000000210501006D00FB01E60D07000000210501006F00FB01CF0E070000 00210501006400FB016B0F040000002D01050007000000210501007000FB014F1008000000FA02 00000000000000000000040000002D0108001C000000FB021000070000000000BC020000000001 02022253797374656D0000D1080AEE38E91200D89FF177E19FF1772020F3778F0D66CD04000000 2D010900040000002701FFFF04000000F001000004000000F001010004000000F0010200040000 00F001030004000000F001040004000000F001050004000000F001060004000000F00107000400 00002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF04 0000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF 030000000000 }\plain\f4\fs22\cf2 \par \pard\ri4\plain\f4\fs22\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Verschluesselung := [(Umwandlung[i] * \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 powermod(oeff1, geheim2, p) mod p) \par $ i = 1..nops(Umwandlung)]: \par \pard\ri4\plain\f5\fs28\cf0 \par Diese Verschl\'fcsselung bekommen wir nun zugeschickt. Aber wie kommen \par wir nun an den Klartext heran? Dazu m\'fcssen wir die Zahlenliste erst einmal \par entschl\'fcsseln. Wir benutzen unseren geheimen und den \'f6ffentlichen Schl\'fcs- \par sel der anderen Partei und machen die Verschl\'fcsselung folgenderweise r\'fcck- \par g\'e4ngig: \par \pard\li1000\ri4\plain\f4\fs22\cf2 {\pict\wmetafile8\picw11076\pich1138\picscalex99\picscaley98\picwgoal6281\pichgoal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}\plain\f4\fs22\cf2 \par \plain\f4\fs28\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Entschluesselung := [(Verschluesselung[i] * \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 powermod(oeff2, p-1-geheim1, p) mod p) \par $ i = 1..nops(Verschluesselung)] \par \par \pard\ri4\plain\f5\fs28\cf0 Das sieht doch schon aus wie eine Liste im ASCII-Format! Wir m\'fcssen sie \par lediglich noch in einen Textstring zur\'fcckwandeln: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}numlib::fromAscii(Entschluesselung) \par \pard\ri4\plain\f4\fs22\cf2 \par \plain\f5\fs28\cf0 Toll! Es hat alles geklappt! Aber warum? Sei \plain\f5\fs28\cf0\i N\plain\f5\fs28\cf0 die Klartext Nachricht, \plain\f5\fs28\cf0\i V\plain\f5\fs28\cf0 die \par Verschl\'fcsselung und \plain\f5\fs28\cf0\i E\plain\f5\fs28\cf0 die Entschl\'fcsselung. Wir m\'fcssen zeigen, dass \par \plain\f5\fs28\cf0\i E\plain\f5\fs28\cf0 = \plain\f5\fs28\cf0\i N\plain\f5\fs28\cf0 gilt. Im Weiteren gelten alle Gleichungen modulo \plain\f5\fs28\cf0\i p\plain\f5\fs28\cf0 . Erinnern wir uns \par an die Entschl\'fcsselung: \par \pard\li2500\ri4\plain\f4\fs22\cf2 {\pict\wmetafile8\picw5015\pich1010\picscalex98\picscaley98\picwgoal2872\pichgoal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002701FFFF040000002701FFFF040000002701FFFF040000002701FFFF030000000000 }\plain\f5\fs28\cf0 \par \pard\ri4\plain\f5\fs28\cf0 Damit ist bewiesen, dass das Protokoll korrekt arbeitet. \par \par Wir haben nun schon gesehen, wie der ElGamal-Algorithmus zum Ver- \par schl\'fcsseln benutzt wird. Als n\'e4chstes setzen wir die Methode von ElGamal \par als Signaturverfahren ein. \par \par Zun\'e4chst einiges \'fcber digitale Unterschriften: \par \par Ziel der Signaturverfahren ist es, sowohl den Absender einer Nachricht ein- \par deutig zu identifizieren als auch Ver\'e4nderungen an einer Nachricht belegen \par zu k\'f6nnen. Denken wir nur einmal an die \'fcblichen E-Mails. Dort kann z.B. \par vor Weiterleiten einer Mail der Inhalt unbemerkt ver\'e4ndert werden. Ebenso \par k\'f6nnen Absenderadressen gef\'e4lscht werden. Digitale Unterschriften verhin- \par dern diesen Missbrauch. \par \par Die Funktionsweise einer digitalen Signatur l\'e4sst sich mit einem durchsichtigem \par Safe vergleichen. Durch das Signieren wird das Dokument in den Tresor gelegt, \par der dann verschlossen wird. Nur der Besitzer kann den Safe \'f6ffnen. Aber jeder \par Teilnehmer kann durch den Glastresor das Dokument einsehen und verifizieren, \par dass es dem Besitzer geh\'f6rt. \par \par Wir \'fcberlegen nun, wie wir dies mit der ElGamal-Methode umsetzen k\'f6nnen. \par Stellen wir uns vor, wir wollen die Nachricht von oben signieren. Wir behalten \par alle Parameter und Schl\'fcssel bei. Zun\'e4chst m\'fcssen wir eine zu \plain\f5\fs28\cf0\i p\plain\f5\fs28\cf0 -1 teiler- \par fremde Zahl \plain\f5\fs28\cf0\i r\plain\f5\fs28\cf0 w\'e4hlen, damit \plain\f5\fs28\cf0\i r\plain\f5\fs28\cf0 invertierbar modulo \plain\f5\fs28\cf0\i p\plain\f5\fs28\cf0 -1 ist: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}r := random(1..p-1)(); \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 while (igcd(r, p-1) > 1) do \par r := random(1..p-1)() \par end_while: \par r\plain\f4\fs22\cf1 \par \pard\ri4\plain\f3\fs28\cf0 Wir berechnen eine Zahl \plain\f3\fs28\cf0\i k\plain\f3\fs28\cf0 mittels der diskreten Exponentialfunktion: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}k := powermod(g, r, p) \par \pard\ri4\plain\f5\fs28\cf0 \par Jetzt m\'fcssen wir folgende Kongruenz l\'f6sen: \par \pard\li1500\ri4\plain\f4\fs22\cf2 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}\plain\f4\fs22\cf2 \par \pard\ri4\plain\f5\fs28\cf0 Nun wird klar, warum \plain\f5\fs28\cf0\i r\plain\f5\fs28\cf0 invertierbar sein musste. Wir berechnen also zun\'e4chst \par das Inverse von \plain\f5\fs28\cf0\i r\plain\f5\fs28\cf0 : \par \plain\f4\fs22\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}invr := powermod(r, -1, p-1) \par \pard\ri4\plain\f3\fs28\cf0 \par und berechnen \plain\f3\fs28\cf0\i s\plain\f3\fs28\cf0 f\'fcr jede Zahl in der Liste der umgewandelten Nachricht: \par \plain\f4\fs22\cf1 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}s := [0 $ nops(Umwandlung)]: \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 for i from 1 to nops(Umwandlung) do \par s[i] := (Umwandlung[i] - geheim1*k) * invr mod p-1 \par end_for: \par s: \par \pard\ri4\plain\f5\fs28\cf0 \par Die digitale Unterschrift ist das Paar (\plain\f5\fs28\cf0\i k\plain\f5\fs28\cf0 , \plain\f5\fs28\cf0\i s\plain\f5\fs28\cf0 ). Wir schicken nun die umge- \par wandelte Nachricht und die Unterschrift an die andere Partei. \par Diese muss nun durch einige Rechnungen die Echtheit der Unterschrift \par verifizieren. Dazu muss \'fcberpr\'fcft werden, ob folgende Gleichheit gilt: \par \pard\li3000\ri4\plain\f4\fs22\cf2 {\pict\wmetafile8\picw5734\pich974\picscalex98\picscaley99\picwgoal3285\pichgoal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}\plain\f4\fs22\cf2 \par \pard\ri4\plain\f5\fs28\cf0 Wenn dies zutrifft, kann der Andere sicher sein, dass die Unterschrift von \par uns stammt. Also berechnet er: \par \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Erg1 := [powermod(g, Umwandlung[i], p) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 $ i = 1..nops(Umwandlung)]: \par \pard\ri4\plain\f4\fs22\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}Erg2 := [(powermod(oeff1, k, p) * powermod(k, s[i], p) \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 mod p) $ i = 1..nops(s)]: \par \pard\ri4\plain\f4\fs22\cf2 \par \plain\f5\fs28\cf0 Nun \'fcberpr\'fcft er jeden Eintrag der Liste folgenderweise: \par \plain\f4\fs22\cf2 \par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs28\cf1 {\pntext\f1\'b7\tab}if(nops(Erg1) - nops(Erg2)=0) then \par \pard\li600\ri1\fi-300\plain\f4\fs28\cf1 Pruefe := [0 $ nops(Erg1)]: \par for i from 1 to nops(Erg1) do \par Pruefe[i] := is(Erg1[i] = Erg2[i]) \par end_for: \par else \par error("Listen nicht gleich lang") \par end_if: \par Pruefe \par \pard\ri4\plain\f3\fs28\cf0 \par Da alle Eintr\'e4ge \plain\f4\fs28\cf2 TRUE\plain\f3\fs28\cf0 lauten, wei\'df die andere Partei genau, dass auch \par wirklich wir es waren, die diese Nachricht geschickt haben. Wir \'fcberlegen \par noch kurz, warum das ganze funktioniert. Auch hier gelten alle Gleichungen \par modulo \plain\f3\fs28\cf0\i p\plain\f3\fs28\cf0 . F\'fcr oeff1 und \plain\f3\fs28\cf0\i k\plain\f3\fs28\cf0 setzen wir deren Erzeugungen ein: \par \pard\li2000\ri4\plain\f4\fs22\cf2 {\pict\wmetafile8\picw6291\pich1138\picscalex99\picscaley98\picwgoal3569\pichgoal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}\plain\f4\fs22\cf2 \par \pard\ri4\plain\f3\fs28\cf0 Nun folgt: \par \pard\li2000\ri4\plain\f4\fs22\cf2 {\pict\wmetafile8\picw6442\pich1010\picscalex98\picscaley98\picwgoal3690\pichgoal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Funktionen \plain\f4\fs20\cf1 numlib::toAscii\plain\f3\fs20\cf2 und \plain\f4\fs20\cf1 numlib::fromAscii\plain\f3\fs20\cf2 vertraut. Besuchen \par \plain\f3\fs20\cf4 ss\plain\f3\fs20\cf2 Sie dazu die Hilfeseiten \plain\f4\fs20\cf1 ?numlib::toAscii\plain\f3\fs20\cf2 und \plain\f4\fs20\cf1 ?numlib::fromAscii\plain\f3\fs20\cf2 . \par \par \plain\f3\fs20\cf2\b 2.\plain\f3\fs20\cf2 Erzeugen Sie eine gr\'f6\'dfere Primzahl, berechnen Sie die Basis und verschl\'fcsseln Sie beliebige Textnachrichten. \par \plain\f4\fs20\cf0\b _____________________________________________________________________________________ \par \plain\f3\fs22\cf0 \par \plain\f3\fs22\cf3\b Anmerkungen:\plain\f3\fs22\cf3 \par \plain\f3\fs20\cf3\b 1. \plain\f3\fs20\cf3 Unter \plain\f3\fs20\cf3\b\i www.schule.mupad.de/material/\plain\f3\fs20\cf3 befinden sich Notebooks, die sich ebenfalls mit \par \plain\f3\fs20\cf4 ss\plain\f3\fs20\cf3 Kryptographie besch\'e4ftigen. \par \par \plain\f3\fs20\cf3\b 2. \plain\f3\fs20\cf3 Weitere Anregungen finden Sie in der Buchreihe \plain\f3\fs20\cf1 Mathematik 1 x anders\plain\f3\fs20\cf3 . In dieser Reihe wird eine Vielzahl \par \plain\f3\fs20\cf4 ss\plain\f3\fs20\cf3 unterschiedlichster mathematischer Probleme mit MuPAD gel\'f6st. Die B\'fccher k\'f6nnen unter \par \plain\f3\fs20\cf4 ss\plain\f6\fs20\cf2 www.schule.mupad.de/literatur\plain\f3\fs20\cf3 kostenfrei kopiert werden. \par \plain\f3\fs20\cf1 \par \plain\f4\fs20\cf0\b _____________________________________________________________________________________ \par \par \par }