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0 0 0 0 1 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 415 1 {CSTYLE "" -1 -1 "" 1 10 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 19 416 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 19 417 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 314 "" 0 "" {TEXT -1 66 "\" Bill Clinton, Bertie Ahern und\ndigitale Unterschriften \" . . . . " }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 416 "" 0 "" {TEXT 256 66 ". . . eine nicht ganz wortgetreue \334 bertragung einer Vorlesung von " }{TEXT 605 13 "John Cosgrave" }{TEXT 607 2 " (" }{URLLINK 17 "John Cosgrave" 4 "http://www.spd.dcu.ie/johnb cos/" "" }{TEXT 606 1 ")" }}{PARA 417 "" 0 "" {TEXT 604 27 "f\374r jed ermann und jedefrau." }}{PARA 19 "" 0 "" {TEXT 257 13 "7. Juli 1999 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " . . . " } {TEXT 556 19 "mit einigen online-" }{TEXT 557 5 "MAPLE" }{TEXT 558 6 " -Demos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 411 "" 0 "" {TEXT -1 0 "" }{TEXT 602 0 "" }{TEXT -1 17 "pr\344sentiert von: " }}{PARA 412 " " 0 "" {TEXT -1 14 "Wiland Schmale" }}{PARA 413 "" 0 "" {TEXT -1 22 "F achbereich Mathematik" }}{PARA 414 "" 0 "" {TEXT -1 40 "Carl von Ossie tzky Universit\344t Oldenburg" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 415 "" 0 "" {TEXT 0 263 "Wichtig er Hinweis: Weiter unten werden ein paar Prozeduren benutzt die dieses Arbeitsblatt einlesen will. Dazu m\374ssen Sie die Prozeduren selbst \+ laden. Sie liegen im gleichen Verzeichnis, wie dieses Arbeitsblatt. Be i Problemen wenden Sie sich ruhig direkt an mich." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 384 "" 0 "" {TEXT -1 3 "( " }{HYPERLNK 17 "zur " 1 "" "Inhalts\374bersicht" } {TEXT -1 32 " Inhalts\374bersicht weiter unten )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT 258 105 "\304usserer Anlass f\374r den Titel dieser \366ffentlic hen Vorlesung ist ein Ereignis im September letzten Jahres:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 " " 0 "" {TEXT 259 41 "Dublin, Irland --- 4-ter September 1998. " } {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 6 " " }}{PARA 264 "" 0 "" {TEXT -1 302 "Die erste internationale digitale Unterzeichnung ei nes Dokuments wurde in Dublin in einer feierlichen Zeremonie mit dem P r\344sidenten der USA Bill Clinton und dem Premierminister von Irland \+ Bertie Ahern durchgef\374hrt. Digital unterzeichnet wurde ein US-Irisc hes Kommuniqu\351 \374ber den elektronischen Handel. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 315 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 " " {TEXT 260 52 "\"Ahern to seal historic digital accord with Clinton\" " }}{PARA 271 "" 0 "" {TEXT 263 28 " (Irish Times, Sept. 1st.) " }} {PARA 290 "" 0 "" {TEXT -1 0 "" }}{PARA 287 "" 0 "" {TEXT 392 34 " \"D igital history made in Dublin\" " }}{PARA 270 "" 0 "" {TEXT 262 148 "O n Sept. 4th. Bill Clinton and Bertie Ahern 'digitally signed' a US-Iri sh government communiqu\351 on e-commerce, using cryptographic softwar e produced" }{TEXT -1 2 " " }{TEXT 261 25 "by Baltimore Technologies " }{TEXT -1 2 " ." }}{PARA 289 "" 0 "" {TEXT -1 2 " " }{TEXT 393 25 " (Irish Times, Sept. 5th.)" }}{PARA 288 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 506 15 "Cebit M\344rz '99:" }{TEXT -1 15 " Die deutsc he " }{TEXT 570 7 "Telekom" }{TEXT -1 47 ", Europa's gr\366\337ter Int ernet-Anbieter, wird mit " }{TEXT 571 22 "Baltimore Technologies" } {TEXT -1 79 " zusammenarbeiten, um sichere E-mail anzubieten f\374r de n elektronischen Handel ." }}{PARA 350 "" 0 "" {TEXT 504 1 "(" }{TEXT 572 22 "Baltimore Technologies" }{TEXT 573 77 " mit Sitz in Dublin ist ein weltweit f\374hrendes Unternehmen insbesondere beim " }{TEXT 505 26 "\"Verkauf von Primzahlen\".)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "..... da\337 " }{TEXT 507 10 "Primzahlen " }{TEXT -1 6 " und " }{TEXT 508 34 "geschickte mathematische Verfahr en" }{TEXT -1 136 " die Hauptrolle spielen bei den am h\344ufigsten be nutzten Sicherungsmethoden, wird im Laufe dieser Vorlesung - so hoffe \+ ich - klar werden." }}{PARA 361 "" 0 "" {TEXT -1 0 "" }{TEXT 516 0 "" }}{PARA 353 "" 0 "Inhalts\374bersicht" {TEXT 515 0 "" }{TEXT 514 17 "I nhalts\374bersicht:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Vorbemerkungen" 1 "" "Vorbemerkungen" }{TEXT -1 0 "" }}{PARA 399 "" 0 "" {TEXT -1 0 "" }}{PARA 398 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "Allgemeines zur Kryptologie" 1 "" "Allgem eines" }{TEXT -1 0 "" }}{PARA 400 "" 0 "" {TEXT -1 1 "*" }}{PARA 354 " " 0 "" {HYPERLNK 17 "Eine einfache Idee " 1 "" "Eine einfache Idee " } {TEXT -1 0 "" }}{PARA 401 "" 0 "" {TEXT -1 0 "" }}{PARA 355 "" 0 "" {HYPERLNK 17 "Fallt\374rfunktionen f\374r RSA" 1 "" "Fallt\374rfunktio nen f\374r RSA" }{TEXT -1 0 "" }}{PARA 402 "" 0 "" {TEXT -1 0 "" }} {PARA 356 "" 0 "" {HYPERLNK 17 "Beispiele zur Verschl\374sselung" 1 " " "Beispiele zur Verschl\374sselung" }{TEXT -1 0 "" }}{PARA 403 "" 0 " " {TEXT -1 0 "" }}{PARA 357 "" 0 "" {HYPERLNK 17 "Beispiele zum elektr onischen Signieren" 1 "" "Beispiele zum elektronischen Signieren" } {TEXT -1 0 "" }}{PARA 404 "" 0 "" {TEXT -1 0 "" }}{PARA 358 "" 0 "" {HYPERLNK 17 "Ein ber\374hmtes Beispiel von und zu RSA und zum Zerlege n einer Zahl in ihre Primfaktoren" 1 "" "Zum Zerlegen in Primzahlen" }{TEXT -1 0 "" }}{PARA 405 "" 0 "" {TEXT -1 1 "*" }}{PARA 378 "" 0 "" {HYPERLNK 17 "Viel mehr - als es nach diesem Vortrag scheinen mag - und ganz moderne und tiefliegende Mathematik ist n\366tig, damit RSA sicher ist und bleibt" 1 "" "Viel mehr - als es nach diesem Vortrag scheinen mag - und ganz moderne und tiefliegende Mathematik ist n \366tig, damit RSA sicher ist und bleibt:" }{TEXT -1 0 "" }}{PARA 406 "" 0 "" {TEXT -1 0 "" }}{PARA 359 "" 0 "" {HYPERLNK 17 "Was bei echten Anwendungen sonst noch zu beachten ist " 1 "" "Was sonst noch zu beac hten ist :" }{TEXT -1 0 "" }}{PARA 407 "" 0 "" {TEXT -1 0 "" }}{PARA 360 "" 0 "" {HYPERLNK 17 "Literatur" 1 "" "Literatur" }{TEXT -1 0 "" } }{PARA 408 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 1 " " }} {SECT 1 {PARA 3 "" 0 "Vorbemerkungen" {TEXT -1 0 "" }{TEXT 396 0 "" } {TEXT 397 16 " Vorbemerkungen:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 395 0 "" }{TEXT -1 118 "In den letzten Jahren ist der sogenannte \"elektronische Handel\" immer h\344ufiger Gegenstand \366ffentliche r Diskussionen. " }}{PARA 292 "" 0 "" {TEXT -1 86 "Ein wichtiges Thema ist dabei die Sicherheit bei der \334bertragung vertraulicher Daten. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 297 "" 0 "" {TEXT -1 13 "Es \+ wird also " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 301 "" 0 "" {TEXT 394 5 "nicht" }{TEXT -1 52 " um die technische Zuverl\344ssigkeit oder Korrektheit " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 302 "" 0 "" {TEXT -1 173 "von Daten\374bertragung gehen, obwohl auch dabei \" ohne Mathematik nichts l\344uft \". Aber das w\344re ein ganz anderes Them a ... vielleicht f\374r einen n\344chsten \366ffentlichen Vortrag." } }{PARA 299 "" 0 "" {TEXT -1 0 "" }}{PARA 298 "" 0 "" {TEXT -1 74 "Sond ern es wird uns um folgende Aspekte der \334bertragungssicherheit gehe n: " }}{PARA 300 "" 0 "" {TEXT -1 0 "" }}{PARA 351 "" 0 "" {TEXT 20 47 "Schutz der Vertraulichkeit und Identifizierung." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 293 "" 0 "" {TEXT -1 66 "Wie kann ich ein Doku ment so \374ber das Internet verschicken, dass " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 303 "" 0 "" {TEXT -1 62 " (1) nur der berechti gte Empf\344nger mein Dokument lesen kann:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 305 "" 0 "" {TEXT 20 35 "Verschl\374sselung und Entsch l\374sselung" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 304 "" 0 "" {TEXT -1 110 " (2) der Empf\344nger an Hand des Dokuments zweifelsfrei feststellen kann, ob es auch tats\344chlich von mir komm t:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 306 "" 0 "" {TEXT 20 21 "d igitale Unterschrift" }}{PARA 295 "" 0 "" {TEXT -1 0 "" }}{PARA 294 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "Wir werden sehen, wie die weltweit am h\344ufigsten benutzten Sicherungsverfahren im Pr inzip funktionieren." }}{PARA 0 "" 0 "" {TEXT -1 33 "Dabei wird deutli ch werden, dass " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 307 "" 0 "" {TEXT 0 58 "solche Verfahren nur m\366glich sind auf Grund von Mathema tik" }}{PARA 309 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "und dass sie " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 308 "" 0 "" {TEXT 0 57 "nur durch mehr Mathematik zu Fall gebracht werden k\366nne n." }{TEXT -1 2 " " }}{PARA 296 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 26 30 "Ein interessanter Nebe naspekt:" }{TEXT 20 1 " " }}{PARA 313 "" 0 "" {TEXT 20 0 "" }}{PARA 310 "" 0 "" {TEXT 0 71 "Ein wichtiger Teil der dabei benutzten Mathema tik wurde mit Sicherheit " }}{PARA 311 "" 0 "" {TEXT 0 19 "ganz und ga r nicht " }}{PARA 312 "" 0 "" {TEXT 0 28 "zu diesem Zweck entwickelt. \+ " }}{PARA 0 "" 0 "" {TEXT 0 0 "" }}{PARA 0 "" 0 "" {TEXT 0 132 "Die \+ \374bertriebene Orientierung der Forschung an kurzfristiger Verwertun g kann durchaus zuk\374nftige Anwendungsm\366glichkeiten verbauen. " } }{PARA 0 "" 0 "" {TEXT 0 77 "Auf jedenfall kann sie aber schon jetzt d ie Freude an der Mathematik tr\374ben. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "zur\374ck" 1 "" "In halts\374bersicht" }{TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "Allgemeines " {TEXT 277 62 " Allgemeines zu Verschl\374sselung, Entschl\374sselung , Kryptologie:" }{TEXT -1 0 "" }{TEXT 278 2 " " }}{PARA 0 "" 0 "" {TEXT 283 15 "Kryptologie : " }{TEXT -1 33 " Lehre von den Geheimsch riften, " }}{PARA 0 "" 0 "" {TEXT -1 121 "heute haupts\344chlich: Ve rschl\374sselung (Kryptografie) und Entschl\374sselung (Kryptoanalyse) von digitalisierten Nachrichten" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 16 "" 0 "" {TEXT -1 356 "Ein ganz altes Beispiel ist der Verschl \374sselungsstock. Ein dicker gleichm\344\337ig runder Stock wird mit \+ einem schmalen Papierband so umwickelt, da\337 er l\374ckenlos bedeckt ist vom Papierband. Dann wird er beschrieben und der Streifen wieder \+ abgewickelt. Dies f\374hrt zu einer Vermischung des Textes. Wer den Ra dius des Stockes kennt, kann auch wieder entschl\374sseln." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 16 "" 0 "" {TEXT -1 154 "Eine bei Kinder n beliebte Geheimschrift entsteht durch Vermischung des Alphabets. Ei n ganz einfaches Beispiel daf\374r wurde schon von Caesar benutzt: \+ " }{XPPEDIT 18 0 "proc (A) options operator, arrow; D end;" "6#f*6#% \"AG7\"6$%)operatorG%&arrowG6\"%\"DGF*F*F*" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "proc (B) options operator, arrow; E end;" "6#f*6#%\"BG7 \"6$%)operatorG%&arrowG6\"%\"EGF*F*F*" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "proc (C) options operator, arrow; F end;" "6#f*6#%\"CG7\"6$%)operat orG%&arrowG6\"%\"FGF*F*F*" }{TEXT -1 25 ", . . . . . . . . . . , " } {XPPEDIT 18 0 "proc (X) options operator, arrow; A end;" "6#f*6#%\"XG7 \"6$%)operatorG%&arrowG6\"%\"AGF*F*F*" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "proc (Y) options operator, arrow; B end" "6#f*6#%\"YG7\"6$%)operato rG%&arrowG6\"%\"BGF*F*F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "proc (Z) op tions operator, arrow; C end;" "6#f*6#%\"ZG7\"6$%)operatorG%&arrowG6\" %\"CGF*F*F*" }}{PARA 0 "" 0 "" {TEXT -1 16 " Also z.B.: " } {XPPEDIT 18 0 "proc (LEO) options operator, arrow; OHR end" "6#f*6#%$L EOG7\"6$%)operatorG%&arrowG6\"%$OHRGF*F*F*" }{TEXT -1 5 " " }} {PARA 0 "" 0 "" {TEXT -1 69 " Wer den Vermischungsvorgang kennt, kan n auch wieder entschl\374sseln." }}{PARA 0 "" 0 "" {TEXT -1 68 " \+ " }} {PARA 16 "" 0 "" {TEXT 554 46 "typisch von ca. 2000 vor Christus bis c a 1976:" }{TEXT -1 105 " wenn man weiss, wie verschl\374sselt wurde, \+ dann kann man auch entschl\374sseln ; solche Verfahren nennt man " } {TEXT 536 11 "symmetrisch" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 126 "Die spannende 4000-j\344hrige Vorgeschichte der m odernen Kryptologie wird auf \374ber 1000 Seiten erz\344hlt in dem ein zigartigen Buch " }{TEXT 560 18 " The Codebreakers " }{TEXT -1 43 "von David Kahn, siehe Literaturverzeichnis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 16 "" 0 "" {TEXT 20 23 "v\366llig neu ab ca. 1977:" } {TEXT -1 2 " " }}{PARA 352 "" 0 "" {TEXT 561 34 "Verschl\374sselungsm ethode \366ffentlich" }{TEXT -1 13 ", sogenannte " }{TEXT 555 22 "\366 ffentliche Schl\374ssel," }}{PARA 352 "" 0 "" {TEXT -1 0 "" }}{PARA 352 "" 0 "" {TEXT 562 30 "Entschl\374sselungsmethode geheim" }{TEXT -1 31 " ; solche Verfahren nennt man " }{TEXT 537 13 "unsymmetrisch" }}{PARA 0 "" 0 "" {TEXT -1 32 " " }} {PARA 385 "" 0 "" {TEXT 384 23 "Wie ist so etwas \374berha" }{TEXT 563 0 "" }{TEXT 564 14 "upt m\366glich ??" }}{PARA 286 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 354 38 "Zus\344tzliches Problem und un ser Thema: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Wie kann ich nach erfolgreicher Entschl\374sselung sicher sein , dass die Nachricht auch wirklich vom genannten Absender stammt ?" }} {PARA 0 "" 0 "" {TEXT -1 55 "Wie kann ein elektronisches Dokument sign iert werden ? " }}{PARA 0 "" 0 "" {TEXT -1 26 "Wie ist so etwas wie ei ne " }{TEXT 309 21 "digitale Unterschrift" }{TEXT -1 10 " m\366glich ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT 32 89 "Man sollte sich hier zun\344chst klarmachen, dass das Versenden digitalis ierter Schriftz\374ge " }{TEXT 387 5 "keine" }{TEXT 32 19 " L\366sung \+ darstellt !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 310 97 "Die z.Zt. weltweit am meisten benutzten \"digitalen Unterschri ften\" wurden erst m\366glich durch die " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 376 "" 0 "" {TEXT 538 47 "geschickte Benutzung \"\366ffe ntlicher Schl\374ssel\"." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Das ist unser Thema im n\344chsten A bschnitt mit der \334berschrift: " }}{PARA 388 "" 0 "" {TEXT -1 15 "\" Eine einfache" }{TEXT 574 0 "" }{TEXT -1 7 " Idee \"" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 385 54 "\"\366ff entliche Schl\374ssel\" wurden erst m\366glich durch die " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 377 "" 0 "" {TEXT 539 57 "geschickte Benu tzung von Mathematik, hier: Zahlentheorie." }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "Damit befassen wir uns im Abschnitt:" }}{PARA 389 "" 0 "" {TEXT -1 23 "\" Fallt\374rfunk tionen f\374r" }{TEXT 388 1 " " }{TEXT 390 4 "RSA " }{TEXT -1 1 "\"" } {TEXT 389 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 386 1 " " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "zur\374ck" 1 "" "Inhalts \374bersicht" }{TEXT -1 2 " " }}{SECT 1 {PARA 3 "" 0 "Eine einfache I dee " {TEXT -1 0 "" }{TEXT 311 20 " Eine einfache Idee " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 355 58 "So einfach ist die \+ Grundidee \366ffentlicher Verschl\374sselung:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Nehmen wir an wir h\344tt en zwei verschiedene " }{TEXT 314 16 "undurchsichtige " }{TEXT -1 11 " Lackfarben " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 289 15 " \366ffentlich-1" }{TEXT -1 5 " ( " }{TEXT 312 5 "al len" }{TEXT -1 29 " Interessierten zug\344nglich) " }}{PARA 15 "" 0 " " {TEXT 303 0 "" }{TEXT 304 11 " privat-1" }{TEXT 305 0 "" }{TEXT -1 12 " (" }{TEXT 313 7 "nur mir" }{TEXT -1 13 " zug\344ngl ich)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "... mit folgenden Eigenschaften:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 22 "1. Ein e zuerst mit " }{TEXT 290 12 "\366ffentlich-1" }{TEXT -1 24 " und da nn dar\374ber mit " }{TEXT 306 0 "" }{TEXT 307 8 "privat-1" }{TEXT -1 60 " lackierte Oberfl\344che macht diese wieder sichtbar, kurz: \+ " }{TEXT 315 0 "" }{TEXT 316 8 "privat-1" }{TEXT -1 8 " \374ber " } {TEXT 317 12 "\366ffentlich-1" }{TEXT -1 30 " wird zusammen durchsicht ig ! " }}{PARA 15 "" 0 "" {TEXT -1 21 "2. Eine zuerst mit " }{TEXT 320 0 "" }{TEXT 322 8 "privat-1" }{TEXT -1 25 " und dann dar\374ber \+ mit " }{TEXT 321 12 "\366ffentlich-1" }{TEXT -1 70 " lackierte Oberfl \344che macht diese ebenfalss wieder sichtbar, kurz: " }{TEXT 319 12 "\366ffentlich-1" }{TEXT -1 8 " \374ber " }{TEXT 318 8 "privat-1 " }{TEXT -1 40 " wird zusammen ebenfalls durchsichtig ! " }}{PARA 15 " " 0 "" {TEXT -1 38 "3. Niemand kann in absehbarer Zeit " }{TEXT 291 8 "privat-1" }{TEXT -1 16 " ausgehend von " }{TEXT 292 12 "\366ff entlich-1" }{TEXT -1 14 " herstellen.\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 356 65 "Jetzt k\366nnen Sie mir wie folgt eine geheime Botschaft zuschicken:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 73 " Schreibe die Botschaft z.B. auf ein Brett, lackiere mit der Farbe " }{TEXT 323 12 "\366ffentlich-1" } {TEXT -1 30 ", und schicke mir das Brett. " }}{PARA 0 "" 0 "" {TEXT -1 77 " Nachdem ich das Brett erhalten habe, lackiere ich dar\374b er mit der Farbe " }{TEXT 324 9 " privat-1" }{TEXT -1 35 " und kann da nn die Botschaft lesen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Kann ich jetzt sicher sein dass die Botschaft von Ih nen kommt ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Nat\374rlich nicht !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 357 66 "Die gleiche Idee zweimal geschickt kombiniert \+ l\366st dieses Problem:" }{TEXT -1 2 "\n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Nehmen wir an wir h\344tten " } {TEXT 333 24 "zwei weitere Lackfarben " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT 325 12 "\366ffentlich-2" }{TEXT -1 5 " ( " }{TEXT 329 5 "allen" }{TEXT -1 30 " Interessierten zug\344nglich) " }}{PARA 15 "" 0 "" {TEXT 326 0 "" }{TEXT 327 8 "privat-2" }{TEXT 328 0 "" }{TEXT -1 12 " (" }{TEXT 330 9 "nur Ihnen" }{TEXT -1 14 " zug\344nglich)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 37 "mit den gleichen Eigenschaften wie " }{TEXT 331 12 "\366ffentlich-1" }{TEXT -1 6 " und " }{TEXT 332 8 "privat-1" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Jetzt kann wie folgt vers chl\374sselt und signiert werden:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 63 "Schreibe die Botschaft z.B. auf ein Brett und lackiere zweimal:" }}{PARA 15 "" 0 "" {TEXT -1 12 "zuerst mit " }{TEXT 334 0 "" }{TEXT 335 6 "Ihrer " }{TEXT -1 7 " Farbe " }{TEXT 338 1 " " }{TEXT 336 0 "" }{TEXT 293 8 "privat-2" }{TEXT 337 0 "" } {TEXT -1 2 ", " }}{PARA 15 "" 0 "" {TEXT -1 48 "dann mit \"meiner\" al lgemein zug\344nglichen Farbe " }{TEXT 294 12 "\366ffentlich-1" } {TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT -1 142 "Nach Erhalt des Bret ts mit der nun doppelt verdeckten Botschaft lackiere ich ebenfalls noc h zweimal, um durchsichtige Schichten zu erhalten: " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 11 "zuerst mit " }{TEXT 339 0 "" }{TEXT 340 6 "meiner" }{TEXT 341 1 " " }{TEXT -1 16 "geheimen Farbe " }{TEXT 295 8 "privat-1" }{TEXT -1 1 " " }}{PARA 15 "" 0 "" {TEXT -1 53 "und danach mit \"Ihrer\" allgemein zug\344nglichen Farbe \+ " }{TEXT 296 12 "\366ffentlich-2" }}{PARA 15 "" 0 "" {TEXT -1 41 "Jet zt wird die Botschaft wieder sichtbar " }{TEXT 342 0 "" }{TEXT 343 0 " " }{TEXT 344 3 "und" }{TEXT 345 1 " " }{TEXT 308 0 "" }{TEXT -1 35 "ic h weiss, dass Sie von Ihnen ist " }{TEXT 346 1 "!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 297 0 "" }{TEXT -1 0 "" }{TEXT 298 0 "" }}{PARA 0 "" 0 "" {TEXT 352 14 "Beachten Sie: " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "Es wurde nicht wirklich von Ihnen unt erschrieben. " }}{PARA 0 "" 0 "" {TEXT -1 68 "Vielmehr verschl\374ssel ten Sie mit der allein Ihnen verf\374gbaren Farbe " }{TEXT 348 0 "" } {TEXT -1 2 " " }{TEXT 347 8 "privat-2" }{TEXT 349 2 " ." }}{PARA 0 " " 0 "" {TEXT 353 26 "Das ist die Unterschrift !" }}{PARA 0 "" 0 "" {TEXT -1 18 "Alle Besitzer von " }{TEXT 350 12 "\366ffentlich-2" } {TEXT -1 119 " k\366nnen feststellen, dass die Botschaft von Ihnen is t und sie lesen, wenn Sie nur diese eine Farbe angewendet h\344tten. \+ " }}{PARA 0 "" 0 "" {TEXT -1 27 "Da Sie aber ausserdem noch " }{TEXT 351 12 "\366ffentlich-1" }{TEXT -1 88 " angewendet haben, kann nur ich , die von Ihnen signierte Botschaft \374berhaupt freilegen. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 272 "" 0 "" {TEXT -1 113 "Diese recht \+ unrealistisch klingenden Ideen lassen sich auf einem Computer mit Hilf e von Mathematik verwirklichen." }}{PARA 0 "" 0 "" {TEXT 288 1 "\n" } {TEXT -1 42 "Die bahnbrechenden Ver\366ffentlichung waren:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 285 54 " 'The fundame ntal idea of public-key cryptography' " }}{PARA 0 "" 0 "" {TEXT 284 1 "(" }{TEXT 299 24 "Diffie and Hellman, 1976" }{TEXT 300 3 "), " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT 286 48 " 'The realization of public-key crypt ography'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 287 1 "(" }{TEXT 301 1 "R" }{TEXT 549 7 "ivest, " }{TEXT 550 1 "S" }{TEXT 551 10 "hamir and " }{TEXT 552 1 "A" }{TEXT 553 12 "dleman, 1977" }{TEXT 302 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Die An fangsbuchstaben der Autoren der zweiten Arbeit ergeben den Abk\374rzun gsnamen f\374r das Verfahren das wir heute besprechen: " }{TEXT 575 3 "RSA" }{TEXT -1 2 " " }{TEXT 576 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Man k\366nnte sagen: " }}{PARA 0 "" 0 "" {TEXT -1 60 "In der ersten Arbeit wurde die Idee der \"Farben\" entwickelt:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 273 " " 0 "" {TEXT 358 56 "\"Fallt\374rfunktionen\" mit geheimhaltbarem \"Sc hnellausgang\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "und in der zweiten wurde gezeigt, wie man sie mit Hilfe \+ von vergleichsweise einfacher Mathematik \"herstellen\" kann." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 222 "Die Gew \344hrleistung der Sicherheit erfordert allerdings recht anspruchsvoll e Mathematik und macht es f\374r die Anbieter von RSA-Software notwend ig, st\344ndig Kontakt zu halten zu den neuesten Entwicklungen in der \+ Mathematik. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 274 "" 0 "" {TEXT 359 47 "Die speziellen Fallt\374rfunktionen sehen so aus :" }} {PARA 363 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "proc (x) options operator, arrow; `mod`(x^s,M) end;" "6#f*6#%\"xG7\"6$%)operatorG%&arr owG6\"-%$modG6$)F%%\"sG%\"MGF*F*F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 285 "" 0 "" {TEXT 521 51 "Wir werden das gleic h noch ausf\374hrlich besprechen !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "zur\374ck" 1 "" "Inhalts \374bersicht" }{TEXT -1 2 " " }}{SECT 1 {PARA 3 "" 0 "Fallt\374rfunkt ionen f\374r RSA" {TEXT -1 1 " " }{TEXT 360 22 "Fallt\374rfunktionen f \374r " }{TEXT -1 3 "RSA" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 73 "Schon bei einfachen Funktionen ist die Berechnung \+ eines Funktionswertes " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" } {TEXT -1 16 " zu gegebenem " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 42 " meist leichter, als die Berechnung von " }{XPPEDIT 18 0 "x; " "6#%\"xG" }{TEXT -1 5 " aus " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"x G" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT 361 9 "Beispiel:" }{TEXT -1 7 " " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"# " }{TEXT -1 9 " f\374r " }{XPPEDIT 18 0 "0 < x;" "6#2\"\"!%\"xG" } {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 35 " g egeben " }{XPPEDIT 18 0 "x = 1234;" "6#/%\"xG\"%M7" }{TEXT -1 13 ", berechne " }{XPPEDIT 18 0 "f(1234) = 1234^2;" "6#/-%\"fG6#\"%M 7*$F'\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 " \+ gegeben " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1522756;" "6#\"(cF_\"" }{TEXT -1 3 ", " }} {PARA 0 "" 0 "" {TEXT -1 46 " berech ne " }{XPPEDIT 18 0 "x = sqrt(1522756);" "6#/%\"xG-%%sqrtG6#\"(cF_\" " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Funktionen, bei denen die Berechnung von " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 6 " aus " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG 6#%\"xG" }{TEXT -1 63 " noch \"viel, viel schwieriger\" ist, hei\337en in der Kryptographie" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 275 "" 0 "" {TEXT 364 18 "Fallt\374rfunktionen " }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "Interessanterweise gibt es ganz einfach gebaute Fallt\374rfunktionen, bei denen die Bere chnung von " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 57 " ( Bruchteile von) Sekunden dauert, die R\374ckgewinnung von " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 79 " hingegen Monate oder gar Jahrhunder te dauern kann, je nach Wahl der Parameter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 362 0 "" }{TEXT 363 46 "Beispiel (f\374r einen \+ PC noch keine Fallt\374r !) :" }{TEXT -1 2 " " }}{PARA 371 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x) = `mod`(x^23,77)" "6#/-%\"fG6#%\"x G-%$modG6$*$F'\"#B\"#x" }{TEXT -1 7 " " }{TEXT 526 6 "f\374r \+ " }{TEXT -1 3 " " }{XPPEDIT 18 0 "0 <= x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 76;" "6#1%!G\"#w" }{TEXT -1 3 " " }}{PARA 370 "" 0 "" {TEXT -1 1 "\"" }{TEXT 527 3 "mod" }{TEXT -1 30 "\" ist dabei die Abk \374rzung f\374r " }{TEXT 403 9 "\"modulo\":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Was hei\337t hier " } {XPPEDIT 18 0 "`mod`(x^23,77)" "6#-%$modG6$*$%\"xG\"#B\"#x" }{TEXT -1 8 " ???" }}{PARA 0 "" 0 "" {TEXT -1 35 " . . . . \+ berechne " }{XPPEDIT 18 0 "x^23" "6#*$%\"xG\"#B" }{TEXT -1 5 " = " } }{PARA 369 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*x*x*x*x*x*x*x*x*x* x*x*x*x*x*x*x*x*x*x*x*x*x" "6#*P%\"xG\"\"\"F$F%F$F%F$F%F$F%F$F%F$F%F$F %F$F%F$F%F$F%F$F%F$F%F$F%F$F%F$F%F$F%F$F%F$F%F$F%F$F%F$F%F$F%" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 " dividiere das Ergebnis durch 77 " }}{PARA 0 "" 0 "" {TEXT -1 50 " und behalte d en " }{TEXT 522 4 "Rest" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Diese " }{TEXT 523 4 "Rest" }{TEXT -1 82 "bildung k\366nnen wir uns mit Hilfe von ganz verschiedenen Mode llen veranschaulichen:" }}{PARA 16 "" 0 "" {TEXT -1 43 "Wickeln eines \+ Zentimeterbandes der L\344nge " }{XPPEDIT 18 0 "x^23" "6#*$%\"xG\"#B " }{TEXT -1 66 " um ein Rohr mit Umfang 77 cm. (Wird in der Vorlesung vorgef\374hrt)" }}{PARA 16 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2 3" "6#*$%\"xG\"#B" }{TEXT -1 124 " Bierflaschen in Bierkisten packen . Jede volle Kiste fa\337t genau 77 Flaschen. Wieviele Flaschen sind i n der letzten Kiste. " }}{PARA 0 "" 0 "" {TEXT -1 24 "Ein Rechenbeispi el: " }{XPPEDIT 18 0 "x = 5" "6#/%\"xG\"\"&" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 8 "Was ist " }{XPPEDIT 18 0 "`5`^23" "6#*$%\" 5G\"#B" }{TEXT -1 4 " ? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 39 "Mit MAPLE k\366nnen wir das so ausrechnen:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "`5`^23=5^23;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "length(5^23)*` - stellig`;" }}}{PARA 0 "" 0 " " {TEXT -1 31 "\334berlesen Sie getrost die mit " }{TEXT 565 1 " " } {TEXT 0 1 ">" }{TEXT 566 2 " " }{TEXT -1 67 "gekennzeichneten Zeilen \+ und schauen Sie sich nur die Ergebnisse an." }}{PARA 0 "" 0 "" {TEXT -1 11 "Was ist " }{XPPEDIT 18 0 "f(5)" "6#-%\"fG6#\"\"&" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "`mod`(`5`^23,77);" "6#-%$modG6$*$%\"5G\"#B\"#x " }{TEXT -1 3 " ?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f(5)=5 ^23 mod 77;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Wievielmal mu\337te daf\374r das Band um das Rohr gewickelt wer den ? " }}{PARA 0 "" 0 "" {TEXT -1 6 "oder: " }}{PARA 0 "" 0 "" {TEXT -1 50 "Wieviele 77-er Bierkisten wurden dabei ganz voll ?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "iquo(5^23,77)*Bierkisten;" }}} {PARA 0 "" 0 "" {TEXT -1 39 "In der letzten Bierkiste sind dann noch" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "(5^23- 154817259156858 \+ * 77 ) * Flaschen;" }}}{PARA 0 "" 0 "" {TEXT -1 11 "Nur dieser " } {TEXT 524 9 "Rest: 59" }{TEXT -1 20 " interessiert uns !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Wir machen eine W ertetabelle z.B. von " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" } {TEXT -1 7 " bis " }{XPPEDIT 18 0 "x = 5;" "6#/%\"xG\"\"&" }{TEXT -1 3 " : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a:=1:b:=7:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "M:=7 7: s:=23:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "read `Tabelle1.txt`:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Tabelle1(M,a,b,s);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 281 "" 0 "" {TEXT -1 55 " Wenn ich Ihnen einen Funktionswert (Rest) vorgebe z.B. " }{TEXT 20 6 " f(x)=8" }{TEXT -1 68 " , dann k\366nnen Sie mit einem PC noch relativ \+ leicht und schnell ein " }{TEXT 20 1 "x" }{TEXT -1 39 " finden, mit de m sich f(x)=8 ergibt ???" }}{PARA 0 "" 0 "" {TEXT -1 51 "Man kann das \+ aber beliebig schwieriger machen. Z.B." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 365 "" 0 "" {XPPEDIT 18 0 "x^6789*`mod 77665544332211`;" "6# *&%\"xG\"%*y'%3mod~77665544332211G\"\"\"" }}{PARA 367 "" 0 "" {TEXT -1 0 "" }}{PARA 366 "" 0 "" {TEXT -1 94 "Praktisch angewendet werden n och viel gr\366\337ere Zahlen. Nach oben sind ja keine Grenzen gesetzt ." }}{PARA 364 "" 0 "" {TEXT -1 8 "Die bei " }{TEXT 383 3 "RSA" } {TEXT -1 60 " benutzten Fallt\374rfunktionen sind jedenfalls vom selbe n Typ:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 276 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x) = `mod`(x^s,M);" "6#/-%\"fG6#%\"xG-%$modG 6$)F'%\"sG%\"MG" }{TEXT -1 4 " " }}{PARA 372 "" 0 "" {TEXT -1 6 "f \374r " }{XPPEDIT 18 0 "0 <= x" "6#1\"\"!%\"xG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "`` <= M-1;" "6#1%!G,&%\"MG\"\"\"F'!\"\"" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "... \+ allerdings " }{TEXT 414 40 "mit sehr speziell gew\344hlten Parametern \+ " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT 391 5 " und " }{XPPEDIT 18 0 " M;" "6#%\"MG" }{TEXT 540 12 ", bei denen " }{XPPEDIT 18 0 "M;" "6#%\"M G" }{TEXT 541 51 " bis zu mehreren hundert Dezimalstellen haben kann. " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "In unseren Modellen entspricht das " }{TEXT 528 1 "M" } {TEXT -1 281 " der Anzahl der Bierflaschen, die in eine einzelne Kiste passen oder dem Rohrumfang. Bei einer hundertstelligen Dezimalzahl be tr\374ge der Rohrdurchmesser mehr als 100000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 Kilo meter. (94 Nullen)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Wie will man da noch " }}{PARA 373 "" 0 "" {TEXT -1 14 "z u gegebenem " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 19 " das urspr\374ngliche " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 11 " f inden ???" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 525 80 "Es gibt aber einen leicht geheimhaltbaren \"Schnellausgang\" a us der \"Fallgrube\": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 279 " " 0 "" {TEXT -1 7 "wenn " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 51 " geschickt gew\344hlt wird, dann gibt es eine Zahl " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 2 " " }}{PARA 284 "" 0 "" {TEXT -1 48 " mit der folgenden extrem n\374tzlichen Eigenschaft:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 280 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`mo d`(f(x)^t,M) = x;" "6#/-%$modG6$)-%\"fG6#%\"xG%\"tG%\"MGF+" }{TEXT -1 2 " " }}{PARA 374 "" 0 "" {TEXT -1 6 "f\374r " }{XPPEDIT 18 0 "0 <= x" "6#1\"\"!%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= M-1;" "6#1%! G,&%\"MG\"\"\"F'!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 171 "Diese mathematische Erkenntnis stam mt i.W. bereits aus dem 18. Jahrhundert (Leonhard Euler, 1707-1783) od er aus dem 2. oder h\366heren Semester in unserem Mathematikstudium." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 376 23 "Im obigen Beispiel ist:" }{TEXT -1 3 " " }{XPPEDIT 18 0 "s = 23;" "6#/%\"sG\"#B" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "M = 77;" "6#/%\"MG\"#x" }{TEXT -1 2 " ;" }}{PARA 0 "" 0 "" {TEXT -1 47 " \+ und mit " }{XPPEDIT 18 0 "t = 47;" "6# /%\"tG\"#Z" }{TEXT -1 22 " erh\344lt man z.B. f\374r " }{XPPEDIT 18 0 "x = a,` . . . `,x = b;" "6%/%\"xG%\"aG%(~.~.~.~G/F$%\"bG" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a:=1:b:=7:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M:=7 7:s:=23:t:=47:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "read`Tabelle2.txt `;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Tabelle2(M,a,b,s,t);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "(5^23 mod 77);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "59^47 mod 77;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "`59`^`47`=59^47;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "length(59^47)*`-stellig`;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 415 22 "Auffallen sollte dabei" }{TEXT -1 103 ", dass die Erhebung zur 23-ten Potenz durch weiteres Potenzieren \"hoc h 47\" wieder aufgehoben wird " }{TEXT 418 87 "\"modulo M \" , \+ in unseren Modellen also modulo Wickeln oder modulo voller Bierkisten. " }}{PARA 0 "" 0 "" {TEXT 416 21 "Eine Fallt\374r entsteht" }{TEXT -1 17 ", wenn z.B. zu " }{XPPEDIT 18 0 "s = 23;" "6#/%\"sG\"#B" }{TEXT -1 22 " das erforderliche " }{XPPEDIT 18 0 "t = 47;" "6#/%\"tG\"#Z " }{TEXT -1 29 " sehr schwer zu finden ist. " }}{PARA 0 "" 0 "" {TEXT -1 55 "Die Mathematik kann viele solcher Fallt\374ren liefern \+ !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Bevo r wir dazu kommen, zuerst noch die " }{TEXT 509 36 "wichtige mathemati sche Beobachtung :" }}{PARA 383 "" 0 "" {TEXT 559 40 " es kommt auf di e Reihenfolge nicht an !" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 19 "Ob man zuerst mit " }{XPPEDIT 18 0 "s" "6#%\"sG" }{TEXT -1 27 " \+ potenziert und dann mit " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 21 " , oder zuerst mit " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 16 " \+ und dann mit " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 80 " ist egal , auch wenn man sich am Ende nur f\374r die (Wickel-) Reste interessie rt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "In der Mathematik dr\374ckt man das formelm\344\337ig so aus:" }}{PARA 317 "" 0 "" {TEXT -1 2 " " }{TEXT 19 0 "" }{XPPEDIT 18 0 "`mod`(`mod` (x^` s`,M)^` t `,M);" "6#-%$modG6$)-F$6$)%\"xG%#~sG%\"MG%(~~t~~~~G F," }{TEXT -1 11 " = " }{XPPEDIT 18 0 "`mod`(`mod`(x^` t`,M)^` s `,M);" "6#-%$modG6$)-F$6$)%\"xG%#~tG%\"MG%(~~s~~~~GF," }{TEXT -1 6 " = " }}{PARA 368 "" 0 "" {TEXT -1 0 "" }}{PARA 375 "" 0 "" {XPPEDIT 18 0 "`` = x;" "6#/%!G%\"xG" }{TEXT 529 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 390 "" 0 "" {TEXT -1 6 "f\374r " }{XPPEDIT 18 0 "0 <= x" "6#1\"\"!%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= M- 1;" "6#1%!G,&%\"MG\"\"\"F'!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "In unserem Beispiel sieht das so aus:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a:=1:b:=7:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M:=77:s:=23:t:=47:" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "Tabelle2(M, a, b ,t,s);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "Tabelle2(M, a, b, s,t);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 596 64 "Das entspricht genau den Eigenschaften eines Lack farbenpaares : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 386 "" 0 "" {TEXT 511 7 "(M,s) " }{TEXT 591 5 "<===>" }{TEXT 592 14 " \366ffent lich " }}{PARA 387 "" 0 "" {TEXT 512 7 "(M,t) " }{TEXT 594 0 "" } {TEXT 595 6 "<===> " }{TEXT 593 11 " privat ." }}{PARA 362 "" 0 "" {TEXT 517 11 "___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 320 "" 0 "" {TEXT 375 43 "Die ganz speziellen Fallt\374rfunktionen bei m " }{TEXT 428 1 " " }{TEXT -1 0 "" }{TEXT 378 0 "" }{TEXT 426 0 "" } {TEXT 425 5 "RSA -" }{TEXT 427 1 " " }{TEXT 379 0 "" }{TEXT 380 0 "" } {TEXT 381 10 "Verfahren " }{TEXT 377 41 "machen es besonders schwer, z u gegebenem " }{TEXT 534 3 " " }{XPPEDIT 18 0 "M,s;" "6$%\"MG%\"sG" }{TEXT 417 3 " " }{TEXT 535 14 " das geeignete" }{TEXT 532 3 " " } {XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT 531 2 " " }{TEXT 533 11 " zu find en." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 577 38 "I m Prinzip wird wie folgt vorgegangen:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 35 "W\344hle zwei sehr grosse Primzahlen \+ " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT -1 7 " und " }{XPPEDIT 18 0 "q;" "6#%\"qG" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 19 "Bilde i hr Produkt " }{XPPEDIT 18 0 "M = p*q" "6#/%\"MG*&%\"pG\"\"\"%\"qGF'" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 18 "finde eine Zahl " } {XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 23 " mit der Eigenschaft:" }} {PARA 409 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ggT(s,(p-1)*(q-1)) = \+ 1" "6#/-%$ggTG6$%\"sG*&,&%\"pG\"\"\"F+!\"\"F+,&%\"qGF+F+F,F+F+" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Mathematik " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 410 "" 0 "" {TEXT -1 79 "- Leonhard Euler 18. Jhdt, 2. oder h\366heres \+ Semester im Mathematikstudium -" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "liefert dann eine Zahl " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 29 " mit der Umkehreigenschaft:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {XPPEDIT 18 0 "`mod`(`mod`(x^ s,M)^t,M) = x;" "6#/-%$modG6$)-F%6$)%\"xG%\"sG%\"MG%\"tGF-F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 20 33 "Falls alles \"gut\" gemacht wurde " }}{PARA 391 "" 0 "" {TEXT 20 36 "- die Mathematik sagt, wie - ," }{TEXT -1 3 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 55 " dann ist die uns inzwischen schon \+ vertraute Funktion " }}{PARA 291 "" 0 "" {XPPEDIT 18 0 "f(x) = `mod`( x^s,M)" "6#/-%\"fG6#%\"xG-%$modG6$)F'%\"sG%\"MG" }{TEXT -1 2 " " }} {PARA 283 "" 0 "" {TEXT -1 8 " f\374r " }{XPPEDIT 18 0 "0 <= x" "6# 1\"\"!%\"xG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= M-1;" "6#1%!G,&%\"M G\"\"\"F'!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 278 "" 0 "" {TEXT 374 21 "eine Fallt\374rfunktion," }{TEXT -1 1 " " }{TEXT 367 4 "wenn" }{TEXT 530 1 " " }{TEXT 365 3 "nur" }{TEXT 368 1 " " }{TEXT 371 20 " M (=pq) und s , " }{TEXT 372 1 " " } {TEXT 366 5 "aber " }{TEXT 373 5 "nicht" }{TEXT 369 1 " " }{TEXT 370 39 " p , q und t bekannt gemacht werden." }}{PARA 282 "" 0 "" {TEXT 382 0 "" }{TEXT -1 0 "" }}{PARA 352 "" 0 "" {TEXT -1 0 "" }} {PARA 352 "" 0 "" {TEXT -1 4 "Die " }{TEXT 510 18 "Fallt\374reigenscha ft" }{TEXT -1 47 " kommt vor allem daher, dass bei unbekannten " } {XPPEDIT 18 0 "p,q,t;" "6%%\"pG%\"qG%\"tG" }{TEXT -1 13 " die Zahl \+ " }{XPPEDIT 18 0 "M*`(=pq)`;" "6#*&%\"MG\"\"\"%&(=pq)GF%" }{TEXT -1 25 " in ihre Primfaktoren " }{XPPEDIT 18 0 "p,q;" "6$%\"pG%\"qG" } {TEXT -1 96 " zerlegt werden mu\337 (bzw. gleichwertig komplizierte \+ Rechnungen notwendig sind) , um die Zahl " }{XPPEDIT 18 0 "t;" "6#%\" tG" }{TEXT -1 35 " zu berechnen, mit der man dann " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 8 " aus " }{XPPEDIT 18 0 "`mod`(x^s,M);" " 6#-%$modG6$)%\"xG%\"sG%\"MG" }{TEXT -1 172 " berechnen k\366nnte. Die Berechnung von Primzerlegungen gro\337er Zahlen dauert aber z.Zt. und wohl auch in Zukunft wesentlich l\344nger, als das recht schnelle Pot enzieren mod " }{XPPEDIT 18 0 "M" "6#%\"MG" }{TEXT -1 33 " selbst b ei riesigen Exponenten " }}{PARA 352 "" 0 "" {TEXT -1 0 "" }}{PARA 352 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 75 "MAPLE-Neuli nge \374berspringen einfach den nachfolgenden eingeklammerten Teil." } }{PARA 0 "" 0 "" {TEXT 567 1 "\{" }{TEXT 568 2 " " }{TEXT 513 69 "Ein einfaches Rechenbeispiel, bei dem wir die Rechenzeit vergleichen:" } {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Z:=10^25;" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p:=nextpr ime(Z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "q:=nextprime(p); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "M:=p*q;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " s:=time():" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "Faktorzerlegung=ifact or(M);Rechenzeit:=time()-s;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "R:=rand(ceil(M/2)..M-1):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "a:=R();b:=R();" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 10 "s:=time():" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "a&^b mod M;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Rechenzeit:=ti me()-s;" }{TEXT -1 0 "" }{TEXT 569 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 578 36 " \}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 69 " \+ " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Je gr\366 \337er der Fallt\374refekt ist, desto l\344nger ist der Sicherheitszei traum." }}{PARA 0 "" 0 "" {TEXT -1 90 "F\374r extreme Sicherheitsanspr \374che werden zur Zeit 300- bis 500-stellige Primzahlen benutzt." }} {PARA 0 "" 0 "" {TEXT -1 249 "Extreme Sicherheit kann z.B. bedeuten, d a\337 selbst weltweit vernetzte parallel rechnende superschnelle Compu ter der neuesten Generation mit den jeweils modernsten mathematischen \+ Verfahren mehr als ein Jahr brauchen, um die Verschl\374sselung zu kna cken." }}{PARA 0 "" 0 "" {TEXT -1 64 "Ein Beispiel, das Geschichte mac hte lernen wir demn\344chst kennen." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "zur\374ck" 1 "" "Inhalts\374bersicht" }{TEXT -1 2 " " }}{SECT 1 {PARA 3 "" 0 "Beispiele zur Verschl\374sselung" {TEXT 267 0 "" }{TEXT 281 0 "" }{TEXT 282 30 " Beispiele zur Verschl\374sselung" } }{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Zuerst vereinbaren wir unser Alph abet und dessen Anordnung:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re start:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "read `alphabet.tx t`;" }}}{PARA 0 "" 0 "" {TEXT -1 65 "Das sind 98 Zeichen inklusive der unsichtbaren Leestelle am Ende." }}{PARA 0 "" 0 "" {TEXT -1 140 "F \374r einen Computer und f\374r die Mathematik, die wir benutzen wolle n, sind Zahlen besser geeignet, als so seltsame Buchstaben, wie etwa: \+ @ . " }}{PARA 0 "" 0 "" {TEXT -1 149 "Deswegen bauen wir uns einen Te xt-Zahl-Wandler, der jedem Zeichen des obigen Alphabets in aufsteigend er Reihenfolge eine Zahl von 01 bis 98 zuordnet." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 20 "read `textzahl.txt`:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# op(Text_Zahl);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Ein Beispiel:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "Text_Zahl(` E \+ s f u n k t i o n i e r t !!! `); " }}}{PARA 0 "" 0 "" {TEXT -1 42 "Achten Sie auf die Zahl 98 (=Leerstelle) !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Ein Zahl-Text-Wandler gib t uns den urspr\374nglichen Text wieder zur\374ck:" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "read`zahltext.txt`:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "#op(Zahl_Text);" }}}{PARA 0 "" 0 "" {TEXT -1 23 "...i n unserem Beispiel:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "Zahl_ Text(98369821989898079824981598129823981098169815981098069820982398989 87373739898);" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Noch ein Beispiel:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Zahl:=Text_Zahl(`digitale Un terschrift ?`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Zahl_Tex t( Zahl );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Auch Quatsch l\344\337t sich in Zahlen ausdr\374cken:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Qu atsch:=Text_Zahl(`aha ! @uni-oldenburg.de`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Zahl_Text( Quatsch );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 234 "Texte, Briefe Dokumente k\366nnen recht lang sein. Die s w\374rde zu beliebig langen Zahlen als Ergebnis der Text-Zahl-Wandlu ng f\374hren. Unsere Verschl\374sselungsmethode funktioniert aber nur \+ f\374r die begrenzte Zahlenmenge \{0 , . . . , (M-1)\} ." }}{PARA 0 " " 0 "" {TEXT -1 33 "Daher teilt man die Text-Zahl in " }{TEXT 443 25 " lauter gleichlange Bl\366cke" }{TEXT -1 5 " ein:" }}{PARA 0 "" 0 "" {TEXT -1 4 "Die " }{TEXT 429 14 "Blockl\344nge B " }{TEXT -1 10 " sei hier " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B:=6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "read `Zahl_Blockliste.txt`;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "'Zahl'=Zahl;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "`Blockliste f\374r Zahl`=Zahl_Blockliste(Zahl,B);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 597 44 "Jetzt verschl\374sseln wir uns ere Texte/Zahlen:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "Dazu brauchen wir zun\344chst zwei " }{TEXT 399 7 "geheim e" }{TEXT -1 70 " gro\337e Primzahlen p1 und q1 , die wir miteinander \+ multiplizieren zur " }{TEXT 400 12 "\366ffentlichen" }{TEXT -1 12 " Z ahl M1 . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Ganz so einfach ist es in Wirklichkeit nicht !!! Dazu sp \344ter ein paar Bemerkungen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 36 "Die Mathematik fordert dann f\374r den " }{TEXT 401 12 "\366ffentlichen" }{TEXT -1 36 " verschl\374sselnden Exp onenten s1 :" }}{PARA 337 "" 0 "" {TEXT 398 37 "s1 darf keine geme insamen Teiler mit" }}{PARA 336 "" 0 "" {TEXT 454 44 "dem Produkt ( p1 - 1 )( q1 - 1 ) haben" }}{PARA 0 "" 0 "" {TEXT -1 21 "Den entsc hl\374sselnden " }{TEXT 402 8 "geheimen" }{TEXT -1 121 " Exponenten t 1 kann man dann bei bekannten p1, q1, s1 leicht berechnen aber eben \+ nicht, wenn nur (M1,s1) bekannt ist." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 85 "Ein einfaches Beispiel (in den allt \344glichen Anwendungen sind die Zahlen viel gr\366\337er):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p1 :=nextprime(1234);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "q1:=nextprime (p1+20);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "dann ist " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "M1:=p1*q1;" }}}{PARA 0 "" 0 "" {TEXT -1 21 "und wir k\366nnten z.B. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "s1:=11111;" }}}{PARA 0 "" 0 "" {TEXT -1 80 "w\344hlen, denn s1 und (p1 - 1)(q1 - 1) haben keinen \+ echten gemeinsamen Teiler:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "igcd(s1,(p1-1)*(q1-1));" }}}{PARA 0 "" 0 "" {TEXT -1 45 "Ein kurze Rechnung liefert den Exponenten t1:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "t1:=s1^(-1) mod (p1-1)*(q1-1);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 26 "Versuchen Sie mal dieses " }{TEXT 20 2 "t1" }{TEXT -1 31 " herauszufinden, wenn S ie nur " }}{PARA 392 "" 0 "" {TEXT 579 31 "M1 = 1557383 und s1 = 1 1111" }}{PARA 393 "" 1 "" {TEXT -1 14 "kennen ??? " }}{PARA 316 "" 0 "" {TEXT -1 3 "\" " }{TEXT 20 2 "t1" }{TEXT -1 36 " ist wie eine N adel im Heuhaufen \"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 33 "Jetzt zur RSA - Verschl\374sselung:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 0 "" {TEXT 275 12 "Das P aar ( " }{TEXT 20 2 "M1" }{TEXT 410 3 " , " }{TEXT 20 2 "s1" }{TEXT 411 12 " ) ist der " }{TEXT 404 21 "\366ffentliche Schl\374ssel" } {TEXT 272 29 " (die \366ffentliche Lackfarbe " }{TEXT 406 12 "\366ffe ntlich-1" }{TEXT 407 4 ") \n" }}{PARA 11 "" 0 "" {TEXT 273 7 "( M1, \+ " }{TEXT 20 3 "t1 " }{TEXT 455 11 ") ist der " }{TEXT 405 17 "private Schl\374ssel" }{TEXT 274 1 " " }}{PARA 11 "" 0 "" {TEXT 444 23 "(die \+ private Lackfarbe " }{TEXT 408 8 "privat-1" }{TEXT 409 1 ")" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "In unserem kleinen Beispiel ist also" }}{PARA 394 "" 0 "" {TEXT -1 0 "" }{XPPMATH 20 "6#/6$%#M1G%#s1G6$\"($Qd:\"&66\"" }}{PARA 0 "" 0 "" {TEXT -1 58 "der \366ffentlichen Schl\374ssel und der geheime Schl\374 ssel ist :" }}{PARA 395 "" 0 "" {TEXT -1 0 "" }{XPPMATH 20 "6#/6$%#M1G %#t1G6$\"($Qd:\"'JIS" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 268 68 "Den \366ffentlichen Schl\374ssel (M1,s1) mache ich jet zt weltweit bekannt." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 86 "I ch kann dann von jedem, der meinen \366ffentlichen Schl\374ssel kennt, geheime Nachrichten " }{TEXT 412 9 "empfangen" }{TEXT -1 55 " (bei de nen ich aber nicht wei\337, von wem sie stammen) " }{TEXT 580 3 "und " }{TEXT -1 64 " jedem , der meinen \366ffentlichen Schl\374ssel kenn t, Nachrichten " }{TEXT 413 8 "zusenden" }{TEXT -1 65 " (die dann nu r von mir stammen k\366nnen, also signiert sind !) . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "John Cosgrave benutz t hier folgendes Beispiel:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 381 "" 0 "" {TEXT -1 43 "Nehmen Sie an, Sie sind Mary und wollen mir" }}{PARA 382 "" 0 "" {TEXT -1 47 "die folgende Nachricht verschl\374sse lt schicken :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 35 "Meet at Cinema, 9.30 P.M. -- Mary" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 33 "Sie - Mary - w\374rden f olgendes tun" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 419 19 "Text-Zahl-Wandlung:" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 47 "MaryText:=`Meet at Cinema, 9.30 P.M. -- Mary`;" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 31 "MaryZahl:= Tex t_Zahl(MaryText);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT 420 46 "Zerlegen der Text-Zahl in Bl\366cke der \+ L\344nge B: " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "B:=6:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "MaryBL:=Zahl_Blockliste(MaryZahl,B) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 318 "" 0 "" {TEXT -1 27 "Verschl\374sselung mit meinem " }}{PARA 338 "" 0 "" {TEXT -1 32 "\366ffentlichen Schl\374ssel (M1,s1) :" }}{PARA 319 "" 0 "" {TEXT 456 27 "(Anwendung der Lackfarbe " }{TEXT 421 12 "\366ffent lich-1" }{TEXT 422 4 " ) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "read `RSA.txt`;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "#op(RSA);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "MaryBL_s1:=RSA(MaryBL, M1,s1);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 321 "" 0 "" {TEXT -1 33 "Aus dieser Zahlenfolge l\344\337t sich " }}{PARA 339 "" 0 "" {TEXT -1 19 "ohne Kenntnis von " }{TEXT 20 2 "t1" }{TEXT -1 1 " \+ " }}{PARA 340 "" 0 "" {TEXT -1 31 "die Nachricht nicht entnehmen !" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 423 53 "Entschl\374sselung mit meinem geheimen Schl \374ssel (M1,t1)" }{TEXT -1 2 " :" }}{PARA 0 "" 0 "" {TEXT -1 37 "(Anw endung meiner geheimen Lackfarbe " }{TEXT 424 8 "privat-1" }{TEXT -1 2 " )" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "MaryBL_s1_t1:=RSA(M aryBL_s1, M1,t1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 56 "Zum Vergleich die urspr\374ngliche Blockl iste von von Mary:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "MaryBL; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Dies e Blockliste verketten wir wieder zu einer Zahl:" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "read `Blockliste_Zahl.txt`;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`MaryZahl ? `:= Blockliste_Zahl(MaryBL_s1_t1,B);" }}}{PARA 0 "" 0 "" {TEXT -1 56 "Da ja schon die Bl\366cke \374bereinst immen ist nat\374rlich auch" }}{PARA 18 "" 0 "" {TEXT 20 24 "MaryZahl \+ ? = MaryZahl " }}{PARA 0 "" 0 "" {TEXT -1 116 "Dies sehen wir auch s p\344testens dann, wenn wir diese Zahl mit Hilfe unseres Alphabets wie der als Text interpretieren:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Zahl_Text(`MaryZahl ? `);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 445 38 "Geschafft, es klappt wirklich \+ ...... " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 329 "" 0 "" {TEXT -1 59 ".....doch die Sache hat noch einen gro\337en Sch\366nheitsfehle r:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 330 "" 0 "" {TEXT 270 90 " Jeder, der meinen \366ffentlichen Schl\374ssel kennt, k\366nnte mir di e Nachricht geschickt haben ." }}{PARA 331 "" 0 "" {TEXT -1 0 "" }} {PARA 332 "" 0 "" {TEXT 430 41 "Woher wei\337 ich, da\337 Sie \"Mary\" sind ??? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 396 "" 0 "" {TEXT -1 117 "Um dieses Problem auch noch zu l\366sen, folgen wir im n \344chsten Abschnitt weiter der einfachen Idee mit den Lackfarben ." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "Damit a ber kein falscher Eindruck entsteht, ist es wichtig, auch hier schon d arauf hinzuweisen, da\337 im allt\344glichen Datenverkehr " }}{PARA 0 "" 0 "" {TEXT -1 47 "die Primzahlen p und q viel viel gr\366\337er si nd :" }}{PARA 322 "" 0 "" {TEXT -1 29 " je nach Sicherheitsanspruch " }}{PARA 333 "" 0 "" {TEXT -1 26 "ungef\344hr 100-300-stellig ," }} {PARA 0 "" 0 "" {TEXT -1 103 "und da\337 ihre \"sichere\" Auswahl zus \344tzlichen (hier nicht besprochenen) mathematischen Aufwand erforder t." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "zur\374ck" 1 "" "I nhalts\374bersicht" }{TEXT -1 2 " " }}{SECT 1 {PARA 3 "" 0 "Beispiele zum elektronischen Signieren" {TEXT 265 38 " Beispiel zum elektronisc hen Signieren" }}{PARA 0 "" 0 "" {TEXT -1 17 " [John Cosgrave: " }} {PARA 0 "" 0 "" {TEXT -1 148 "That is what President Clinton and Mr. A hern were doing in a very public way, but it is exactly what everyone \+ else is doing all the time ... .] (?)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 77 "So k\366nnen Sie bzw. Mary ihre Nachr icht digital signieren bzw. unterschreiben:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Sie brauchen ein eigenes Paar v on \"geeigneten\" Primzahlen " }}{PARA 323 "" 0 "" {TEXT -1 12 " p2 un d q2 " }}{PARA 0 "" 0 "" {TEXT -1 19 "Damit bestimmen Sie" }}{PARA 324 "" 0 "" {TEXT -1 14 "M2, s2 und t2 " }}{PARA 0 "" 0 "" {TEXT -1 40 "f\374r Ihren eigenen \366ffentlichen Schl\374ssel" }}{PARA 325 "" 0 "" {TEXT -1 2 " " }{TEXT 441 12 "( M2 , s2 ) " }}{PARA 328 "" 0 "" {TEXT -1 29 "entsprechend der Lackfarbe " }{TEXT 433 12 "\366ffentli ch-2" }{TEXT 432 0 "" }{TEXT 431 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "und den dazugeh\366rigen " }}{PARA 326 "" 0 "" {TEXT -1 20 "privaten Schl\374ssel " }{TEXT 442 11 "( M2 , t2 )" }}{PARA 327 "" 0 "" {TEXT -1 28 "entsprechend der Lackfarbe " } {TEXT 436 8 "privat-2" }{TEXT 435 0 "" }{TEXT 434 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "als einfaches Beispiel w\344hlen wir:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p2:=nextp rime(1111);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "q2:=nextprime(p2+100 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "M2:=p2*q2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "s2:=nextprime(ceil(M2/2));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 87 "Sie m\374ssen sicher sein, da\337 s2 richtig gew\344hlt wurde. Die Mathematik for dert (s.o.) :" }}{PARA 334 "" 0 "" {TEXT -1 24 "s2 und (p2-1)(q2-1) " }}{PARA 335 "" 0 "" {TEXT -1 47 "d\374rfen keinen echten gemeinsam en Teiler haben. " }}{PARA 0 "" 0 "" {TEXT -1 15 "Dies trifft zu:" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "igcd(s2,(p2-1)*(q2-1));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Jetzt ist t2 wieder leicht zu berechnen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "t2:=s2&^(-1) mod (p2-1)*(q2-1);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 50 "Sie bzw. Mary benutzen nun zuerst Ihre Lackfarbe " }{TEXT 437 9 "privat-2 " }}{PARA 0 "" 0 "" {TEXT -1 30 "bzw. Ihren privaten Schl \374ssel " }{TEXT 439 7 "(M2,t2)" }{TEXT -1 65 " und dann gleich hinte rher meine allgemein zug\344ngliche Lackfarbe " }{TEXT 438 12 "\366ffe ntlich-1" }{TEXT -1 48 " bzw. meinen \366ffentlich zug\344nglichen Sc hl\374ssel " }{TEXT 440 7 "(M1,s1)" }{TEXT -1 2 " :" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Wir schauen uns dies in unserem Beispiel an. " }}{PARA 0 "" 0 "" {TEXT -1 23 "Zur Erinnerung: es war" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "MaryText;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "MaryZahl;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "MaryBL;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 17 "Jetzt geht's los:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Anwendung von " }{TEXT 446 11 " privat-2 " }{TEXT 447 0 "" }{TEXT -1 6 "bzw. " }{TEXT 449 7 "(M2,t2)" }{TEXT 448 3 " " }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "MaryBL_t2:=RSA(MaryBL, M2,t2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Anwendung von " } {TEXT 450 15 " \366ffentlich-1 " }{TEXT 451 0 "" }{TEXT -1 6 "bzw. \+ " }{TEXT 453 7 "(M1,s1)" }{TEXT 452 3 " " }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "MaryBL_t2_s1 := " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " RSA(MaryBL_t2, M1,s1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Jetzt k\366nnen Sie die B otschaft an mich verschicken !" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 63 "Nachdem ich sie erhalten habe wende ich m eine private Lackfarbe" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 457 12 " privat-1 " }{TEXT 458 0 "" }{TEXT -1 11 "an, bzw. " }{TEXT 460 7 "(M1,t1)" }{TEXT 459 3 " " }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "MaryBL_t2_s1_t1 :=" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 " RSA(MaryBL_t2_s1, M1,t1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Danach wende ich Ihre \+ \366ffentliche Lackfarbe " }{TEXT 461 15 " \366ffentlich-2 " }{TEXT 462 0 "" }{TEXT -1 11 " an, bzw. " }{TEXT 464 7 "(M2,s2)" }{TEXT 463 3 " " }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Ma ryBL_t2_s1_t1_s2 :=" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " RSA(Mary BL_t2_s1_t1, M2,s2);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Das ist tats\344chlich wieder die alte Blockliste wi e ein Vergleich zeigt:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Mar yBL;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 66 "Wir schenken uns die Verwandlung in Text, den wir ja sc hon kennen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Damit es nicht zu un\374bersich tlich wird, habe ich dabei verschwiegen, da\337 dabei " }{TEXT 465 17 "M2 kleiner als M1" }{TEXT -1 6 " sein " }{TEXT 466 3 "mu\337" } {TEXT -1 2 " !" }}{PARA 0 "" 0 "" {TEXT -1 40 "Wenn dies nicht der Fal l ist, wenn also " }{TEXT 467 16 "M2 gr\366\337er als M1" }{TEXT -1 87 " ist, dann mu\337 lediglich die Reihenfolge ge\344ndert werden. Es ist dann so zu verfahren: " }}{PARA 341 "" 0 "" {TEXT -1 3 " " } {TEXT 468 64 "(M1,s1) ---> (M2,t2) ---> (M2,s2) ---> \+ (M1,t1)" }}{PARA 0 "" 0 "" {TEXT -1 4 "bzw." }}{PARA 342 "" 0 "" {TEXT -1 1 " " }{TEXT 469 19 "\366ffentlich-1 --->" }{TEXT -1 4 " \+ " }{TEXT 471 14 "privat-2 --->" }{TEXT -1 3 " " }{TEXT 470 19 " \366ffentlich-2 --->" }{TEXT -1 5 " " }{TEXT 472 8 "privat-1" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "Diese Reihenfolgen\344nderung kann jeder Sender und jede r Empf\344nger selbst vornehmen, denn sowohl M1, als auch M2 sind ja b ekannt." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "Jetzt noch ein Beispiel - mit denselben M1,s1,t1,M2,s2,t2 - \+ in dem die einzelnen Schritte zusammengefa\337t sind:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "CC_Text:=`Es wurden mir zuverl\344ssig v on ...... die folgenden geheimen Absprachen zwischen ....... aus Rambo uillet gemeldet : ....... `;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "CC_senden:= " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "RSA( RSA( Zah l_Blockliste( Text_Zahl( CC_Text),B),M2,t2),M1,s1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "CC_entschl||`\374`||sselt:= " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 75 "Zahl_Text( Blockliste_Zahl( RSA( RSA( CC_ senden ,M1,t1), M2,s2),B));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "zur\374ck" 1 "" "Inhalts\374bersicht" }{TEXT -1 2 " " }}{SECT 1 {PARA 3 "" 0 "Zu m Zerlegen in Primzahlen" {TEXT -1 0 "" }{TEXT 266 35 " Ein ber\374hmt es Beispiel von und zu " }{TEXT -1 3 "RSA" }{TEXT 519 51 " und zum Zer legen einer Zahl in ihre Primfaktoren " }}{PARA 343 "" 0 "" {TEXT 474 161 "Die beim RSA-Verfahren benutzten Fallt\374rfunktionen sind nu r solange wirkliche Fallt\374ren, als es nicht schnell genug gelingt, \+ ausgehend vom \366ffentlichen Schl\374ssel" }{TEXT 477 1 " " }{TEXT 480 5 "(M,s)" }{TEXT 481 1 " " }{TEXT 478 50 "den zur Entschl\374sselu ng n\366tigen geheimen Schl\374ssel" }{TEXT 484 1 " " }{TEXT 475 1 " \+ " }{TEXT 482 3 "(M," }{TEXT 473 1 "t" }{TEXT 476 1 ")" }{TEXT 483 1 " \+ " }{TEXT 479 89 "zu berechnen. Diebstahl oder \344hnliche u.U. \"schne llere\" Methoden lassen wir mal beiseite." }}{PARA 344 "" 0 "" {TEXT -1 98 "Sobald es gelingt M in seine beiden Primfaktoren zu zerlegen, k ann sekundenschnell das unbekannte " }{TEXT 485 1 "t" }{TEXT -1 21 " \+ berechnet werden. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 27 "Hinter der Fallt\374rfunktion " }}{PARA 345 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "proc (x) options operator, arrow; `mo d`(x^s,M) end;" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"-%$modG6$)F%%\" sG%\"MGF*F*F*" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 90 "die \+ man beim RSA-Verfahren st\344ndig benutzt, steckt die eigentliche giga ntische Fallt\374re : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "W\344hrend ein" }}{PARA 346 "" 0 "" {TEXT -1 17 " Pr odukt zweier " }{TEXT 486 8 "riesiger" }{TEXT -1 8 " Zahlen " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 488 12 "sehr sch nell" }{TEXT -1 31 " berechnet werden kann, ist die" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 347 "" 0 "" {TEXT -1 36 " Wiedergewinnung der \+ beiden Faktoren" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "i.A. " }{TEXT 489 17 "extrem langwierig" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 487 38 "Folgend es Beispiel machte Geschichte:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 70 "1977 erschien in der Mathematik-Spiele-Ko lumne von Martin Gardener in " }{TEXT 518 20 "Scientific American " } {TEXT -1 149 " u.A. folgende Aufgabe von Rivest, Shamir und Adleman (R SA) (die Aufgabe ist unwesentlich abge\344ndert da, Rivest e.a. ein an deres Alphabet benutzten):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Angegeben sind die " }{TEXT 581 27 "vier Zahlen \+ C1, M, s, C2 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 177 "Die Zahl C1 wurde mit dem \366ffentlichen Schl\374ssel \+ ( M , s ) verschl\374sselt und die Zahl C2 wurde mit dem nicht angege ben zugeh\366rigen privaten Schl\374ssel ( M , t ) verschl\374sselt. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Die Z ahlen C1, M, s, C2 gebe ich gleich noch an." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "C2 ist mit dem \366ffent lichen Schl\374ssel ( M , s ) sofort lesbar und soll beweisen, dass di e Aufgabe von den Autoren stammt." }}{PARA 0 "" 0 "" {TEXT -1 20 "Die \+ Aufgabe lautet: " }}{PARA 348 "" 0 "" {TEXT 502 19 "Entschl\374ssele C 1 !!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "J etzt die Zahlen C1 , (M, s) , C2 :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "C1:=196512823380697612509938304734234102608862525343 6887425763012567669836904600894577175082733830812111742894261300461484 6229600376:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "`Anzahl der Ziffern \+ von C1`:=length(C1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "M:=1143816257578888676692357799761466120 1021829672124236256256184293570693524573389783059712356395870505898907 5147599290026879543541:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "`Anzahl \+ der Ziffern von M`:=length(M);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Diese Zahl M tr\344gt auch den Namen " }{TEXT 497 7 "RSA-129" } {TEXT -1 74 " . Sie ist durch die besondere Geschichte dieser Aufgabe \+ ber\374hmt geworden." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "s:=9007:" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "C2:=6968148142569390426 8252495235552568891070737366406610747314084258977590964276669530597769 089495503402436944011336153053886139929:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "`Anzahl der Ziffern von C2`:=length(C2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "Da wir ja ( M , s ) kennen, ist f\374r uns die Entschl\374ss elung der Zahl C2 kein Problem, sofern sie tats\344chlich von den Besi tzern des geheimen Schl\374ssels ( M , s ) stammt:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Zahl_Text(Blockliste_Zahl(RSA([C2],M,s),1 28));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Es wird berichtet, da\337 Rivest " }{TEXT 499 4 "1977" }{TEXT -1 183 " gesch\344tzt hatte, da\337 mit der damaligen \+ Technologie und den besten damals bekannten mathematischen Verfahren d ie Zerlegung von M in seine beiden Faktoren und damit die Gewinnung vo n " }{TEXT 498 1 "t" }{TEXT -1 53 " ungef\344hr 40.000.000.000.000.0 00 Jahre dauern w\374rde." }}{PARA 0 "" 0 "" {TEXT -1 24 "Bereits am a m 27. April " }{TEXT 500 4 "1994" }{TEXT -1 255 " wurde die Zerlegung \+ von A.Lenstra und anderen bekanntgegeben. Die Zerlegung wurde innerhal b von 8 Monaten von ca 600 weltweit \374ber das Internet vernetzten mo dernsten und parallel rechnenden Computern und mit ganz neuen mathemat ischen Verfahren gewonnen. " }}{PARA 0 "" 0 "" {TEXT -1 136 "Mehr Einz elheiten dieser bisher gr\366\337ten gobalen Zerlegungsberechnung find en Sie in dem Artikel von David Wright im Literaturverzeichnis." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Da wir nur einen einzigen nicht b esonders schnellen PC und auch nicht die schnellste Software zur Verf \374gung haben, ist von der Aktivierung der Anweisung " }}{PARA 349 " " 0 "" {TEXT 501 11 "ifactor(M);" }}{PARA 0 "" 0 "" {TEXT -1 19 "in MA PLE abzuraten." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Die beiden inzwischen ja bekannten Faktoren von M sind:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "p:= 349052951084765094914 7849619903898133417764638493387843990820577:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "q:= 32769132993266709549961988190834461413177642 967992942539798288533:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "i sprime(p),isprime(q);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Da\337 M zerlegbar ist l\344\337t sich erstaunlicherweise sekundenschnell test en:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "isprime(M);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Da\337 M tats\344chlich Produkt de r angegebenen Primzahlen p und q ist, sieht man so:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "M-p*q;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Mit Hilfe von " }{TEXT 583 1 "p" }{TEXT -1 5 " und " }{TEXT 584 1 "q" }{TEXT -1 19 " k\366nnen wir jetzt " }{TEXT 582 1 "t" } {TEXT -1 37 " ebenfalls sekundenschnell berechnen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "t:=s&^(-1) mod (p-1)*(q-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "length(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " } {TEXT 586 1 "t" }{TEXT -1 88 " ist zwar immer kleiner als M, hat hier aber doch immerhin genausoviele Stellen wie M ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Mit diesem " }{TEXT 585 1 "t" }{TEXT -1 61 " m\374\337ten wir die Nachricht C1 problemlos ents chl\374sseln k\366nnen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " Zahl_Text(op(RSA([ C1 ],M,t)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Diese se kundenschnellen Berechnungen mit doch immerhin recht gro\337en Zahlen \+ verglichen mit den M\374hen, die Primzahlen " }{TEXT 598 1 "p" }{TEXT -1 5 " und " }{TEXT 599 1 "q" }{TEXT -1 69 " zu finden, machen drastis ch den gigantischen Fallt\374reffekt deutlich." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "Mit diesen Beispielen wi rd auch die eingangs gemachte Bemerkung verst\344ndlich, da\337 es bei m Verkauf von RSA-Verfahren letztendlich auch um den Verkauf von Primz ahlen geht." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "zur\374ck" 1 "" "Inhalts\374bersicht" }{TEXT -1 2 " " }}{SECT 1 {PARA 3 "" 0 "Viel mehr - als es nach diesem Vor trag scheinen mag - und ganz moderne und tiefliegende Mathematik ist n\366tig, damit RSA sicher ist und bleibt:" {TEXT 542 149 " Viel mehr - als es nach diesem Vortrag scheinen mag - und ganz moderne und \+ tiefliegende Mathematik ist n\366tig, damit RSA sicher ist und bleibt: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Die s sei durch ein paar Fragen verdeutlicht:" }}{PARA 15 "" 0 "" {TEXT -1 58 "Wann ist eine Primzahl schwer zu entdecken als Faktor in " } {TEXT 587 1 "M" }{TEXT -1 40 " und damit nutzbar (und verk\344uflich) ? " }}{PARA 397 "" 0 "" {TEXT 543 73 "Primzahlen werden zwar verkauft , aber trotzdem geh\366ren sie uns allen !!!" }}{PARA 15 "" 0 "" {TEXT -1 59 "Wie funktionieren die schnellsten Faktor-Such-Algorithmen ?" }}{PARA 15 "" 0 "" {TEXT -1 99 "Wie k\366nnte jemand noch schnelle re Algorithmen erfinden, und ist das prinzipiell \374berhaupt m\366gli ch ?" }}{PARA 15 "" 0 "" {TEXT -1 53 "Gibt es auch in Zukunft gen\374g end sichere Primzahlen ?" }}{PARA 15 "" 0 "" {TEXT -1 43 "Wie funktion ieren schnelle Primzahltests ? " }}{PARA 15 "" 0 "" {TEXT -1 1 "?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "Hier m \366chte ich auf den Fachartikel von Dan Boneh: " }{TEXT 588 44 "20 ye ars of attacks on the RSA-cryptosystem " }{TEXT -1 74 " und auf die B \374cher von Paulo Ribenboim im Literaturverzeichnis verweisen." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "zur\374ck" 1 "" "Inhalts \374bersicht" }{TEXT -1 2 " " }}{SECT 1 {PARA 3 "" 0 "Was sonst noch \+ zu beachten ist :" {TEXT 503 57 " Was bei echten Anwendungen sonst noc h zu beachten ist :" }}{PARA 0 "" 0 "" {TEXT -1 137 "Es gab bereits e in paar Andeutungen, da\337 bei realistischen Anwendungen des RSA-Verf ahrens noch verschiedene Feinheiten zu beachten seien." }}{PARA 0 "" 0 "" {TEXT -1 54 "Hierzu jetzt ein paar Stichworte. Viel detaillierter e " }}{PARA 0 "" 0 "" {TEXT -1 32 "Informationen gibt es z.B. dort " } {URLLINK 17 "http://www.rsa.com/rsalabs/" 4 "http://www.rsa.com/rsalab s/" "" }}{PARA 0 "" 0 "" {TEXT -1 28 "und \374ber die Literaturliste. " }}{PARA 15 "" 0 "" {TEXT -1 276 "RSA gilt z.Zt. als sicherstes Verfa hren und wird weltweit am meisten benutzt, insbesondere bei digitalen \+ Unterschriften, beim Versenden kurzer vertraulicher Dokumente wie etwa : Kontonummern, Kreditkartennummern, Verschl\374sselungsdaten f\374r a ndere Verschl\374sselungsverfahren, etc." }}{PARA 15 "" 0 "" {TEXT -1 173 "F\374r l\344ngere Dokumente ist das RSA-Verfahren aufwendiger als andere Verschl\374sselungsverfahren. Daher wird in solchen F\344llen \+ oft kombiniert und RSA nur zum signieren benutzt. " }}{PARA 15 "" 0 " " {TEXT -1 582 "Auch f\374r die sorgf\344ltige Auswahl der Primzahlen \+ p1,q1, p2,q2 gibt es automatisierte Verfahren. Es kommt dabei darauf a n, da\337 man den Produkten M1=p1*p2 und M2=p2*q2 nicht sofort (d.h. \+ je nach Sicherheitsanforderung m\366glichst erst nach mehrt\344giger, \+ mehrw\366chiger, mehrmonatiger oder gar mehrj\344hriger bester Compute rleistung mit schlauesten Verfahren) ansieht, wie sie zustande kamen. \+ Dabei ist die Gr\366\337e der Primzahlen ein wichtiger Gesichtspunkt \+ aber bei weitem nicht der einzige ! Hierauf geht John Cosgrave in sei ner \366ffentlichen Vorlesung n\344her ein. Siehe Nr.9 auf der Seite: \+ " }{URLLINK 17 "http://web.usna.navy.mil/~wdj/crypto.htm" 4 "http://w eb.usna.navy.mil/~wdj/crypto.htm" "" }}{PARA 15 "" 0 "" {TEXT -1 143 " Ob M1 gr\366\337er ist als M2 oder nicht, beeinflu\337t zwar - wie wi r gesehen haben - den Ablauf des Verfahrens, ist aber ansonsten unpr oblematisch." }}{PARA 15 "" 0 "" {TEXT -1 171 "Nicht so deutlich hervo rgehoben wurde au\337erdem die Notwendigkeit, da\337 die Blockl\344nge B bei der Zerlegung der Text-Zahl in gleichlange Bl\366cke nicht beli ebig gro\337 sein darf." }}{PARA 352 "" 0 "" {TEXT -1 23 "Es mu\337 au f jeden Fall " }{XPPEDIT 18 0 "10^B < M1" "6#2)\"#5%\"BG%#M1G" } {TEXT -1 2 " " }{TEXT 544 3 "und" }{TEXT -1 2 " " }{XPPEDIT 18 0 "10 ^B < M2;" "6#2)\"#5%\"BG%#M2G" }{TEXT -1 9 " sein." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "zur \374ck" 1 "" "Inhalts\374bersicht" }{TEXT -1 2 " " }}{SECT 1 {PARA 3 "" 0 "Literatur" {TEXT 279 0 "" }{TEXT 280 11 " Literatur " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "Die Grundidee zu \+ dieser \"\366ffentlichen Vorlesung f\374r jedermann und jedefrau\" sta mmt von" }}{PARA 16 "" 0 "" {TEXT -1 15 "John Cosgrave: " }{TEXT 495 82 "Bill Clinton, Bertie Ahern and digital signatures - A public lectu re using MAPLE, " }{TEXT -1 86 "Dublin, 25.Nov. 1998, das Original MAP LE-Arbeitsblatt ist zu erhalten als Nr. 9 bei : " }{URLLINK 17 "http:/ /web.usna.navy.mil/~wdj/crypto.htm" 4 "http://web.usna.navy.mil/~wdj/c rypto.htm" "" }{TEXT -1 83 " dort finden Sie auch weitere \366ffentli che Vorlesungen insbesondere zu Primzahlen. " }{TEXT 603 26 "Hinzugef \374gt am 23.7 1999: " }{TEXT -1 64 "Das College von Cosgrave hat inzw ischen eine eigene Heimseite: " }{URLLINK 17 "http://www.spd.dcu.ie/j ohnbcos/" 4 "http://www.spd.dcu.ie/johnbcos/" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 16 "" 0 "" {TEXT -1 77 "Diese \366ffentliche Vor lesung wird ab 7.Juli 1999 im Internet zug\344nglich sein. " }}{PARA 352 "" 0 "" {TEXT -1 209 "Als Maple- Arbeitsblatt - Sie k\366nnen dan n selbst\344ndig Beispiele bearbeiten, allerdings nur, wenn Sie MAP LE auf Ihrem PC haben und benutzen k\366nnen. Alle notwendigen Prozedu ren liegen im selben Verzeichnis: " }{URLLINK 17 "http://www.uni-olden burg.de/~schmale/dateien/Oeffentlich/RSA_Oeffentlich.mws" 4 "http://ww w.uni-oldenburg.de/~schmale/dateien/Oeffentlich/RSA_Oeffentlich.mws" " " }{TEXT -1 0 "" }}{PARA 352 "" 0 "" {TEXT -1 173 "Als automatisch gen erierte Hypertextdatei - darin k\366nnen Sie dann zwar herumh\374pfen , aber alle MAPLE-Befehle sind inaktiv. Au\337erdem ist das Layout dab ei nicht immer optimal:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {URLLINK 17 "http://www.uni-oldenburg.de/~schmale/dateien/Oeffentlich/ RSA_Oeffentlich.html" 4 "http://www.uni-oldenburg.de/~schmale/dateien/ Oeffentlich/RSA_Oeffentlich.html" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 73 "Viele praktische und technische Fragen w erden beantwortet in dem Report: " }{TEXT 600 64 "Answers to frequentl y Asked Questions About Today\222s Cryptography" }{TEXT -1 22 ", den S ie als Datei " }{TEXT 601 11 "LABSFAQ.PDF" }{TEXT -1 11 " bei \+ " }{URLLINK 17 "http://www.rsa.com/rsalabs/ " 4 "http://www.rsa.com/ rsalabs/" "" }{TEXT -1 18 "beziehen k\366nnen. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "Ein mit Abitur-Kenntniss en in Mathematik lesbares Buch, in dem unter vielem anderen auch RSA, \+ Primzerlegung, sichere Primzahlen behandelt werden, ist z.B.:" }} {PARA 16 "" 0 "" {TEXT -1 16 "Lindsay Childs: " }{TEXT 546 37 "A concr ete approach to higher algebra" }{TEXT -1 27 ", 2.Auflage, Springer, 1 995" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We itere Literatur:" }}{PARA 16 "" 0 "" {TEXT -1 24 "Albrecht Beutelspach er: " }{TEXT 545 228 "Kryptologie : eine Einf\374hrung in die Wissensc haft vom Verschl\374sseln, Verbergen und Verheimlichen ; ohne alle Geh eimniskr\344merei, aber nicht ohne hinterlistigen Schalk, dargestellt \+ zum Nutzen und Erg\366tzen des allgemeinen Publikums" }{TEXT -1 27 ", \+ 3. Auflage, Vieweg, 1993" }}{PARA 16 "" 0 "" {TEXT -1 11 "Dan Boneh: \+ " }{TEXT 490 47 "Twenty Years of Attacks on the RSA Cryptosystem" } {TEXT -1 82 ", Notices of the American Mathematical Society, Band 46 , Nummer 2, Februar 1999 " }}{PARA 16 "" 0 "" {TEXT -1 17 "Martin Gard ener: " }{TEXT 520 19 "Mathematical games " }{TEXT -1 27 ", Scientific American, 1977" }}{PARA 16 "" 0 "" {TEXT -1 13 "Mark Janeba: " } {TEXT 493 69 "Factoring Challenge conquered - with a little help from \+ Williamette, " }{TEXT -1 16 "anzusehen bei : " }{URLLINK 17 "http://ww w.willamette.edu/~mjaneba/rsa129.html" 4 "http://www.willamette.edu/~m janeba/rsa129.html" "" }{TEXT -1 2 " " }}{PARA 16 "" 0 "" {TEXT -1 12 "David Kahn: " }{TEXT 491 103 "The Codebreakers -The Comprehensive \+ History of Secret Communication from Ancient Times to the Internet " } {TEXT -1 38 " Scribner, 1996 (2nd. new edition) " }}{PARA 16 "" 0 " " {TEXT -1 17 "Paulo Ribenboim: " }{TEXT 590 31 "The little book of bi g primes, " }{TEXT -1 14 "Springer, 1991" }}{PARA 16 "" 0 "" {TEXT -1 17 "Paulo Ribenboim: " }{TEXT 589 32 "The book of prime number records " }{TEXT -1 29 ", 2.Auflage, Springer, 1989. " }}{PARA 16 "" 0 "" {TEXT 494 42 "RSA Factoring-by-web: General Information " }{TEXT -1 9 "mit einem" }{TEXT 496 1 " " }{TEXT -1 29 "Vorwort von Arjen Lenstra : " }{URLLINK 17 "http://www.npac.syr.edu/factoring/info.html " 4 "htt p://www.npac.syr.edu/factoring/info.html " "" }{TEXT -1 47 " \+ " }}{PARA 16 "" 0 "David Wright" {TEXT -1 13 "David Wright:" }{TEXT 492 24 " The Cracking of RSA-129" } {TEXT -1 42 " , 1994, zu erhalten als dvi-Datei bei : " }{URLLINK 17 "http://www.math.okstate.edu/~wrightd/numthry/rsa129.html" 4 "http://w ww.math.okstate.edu/~wrightd/numthry/rsa129.html " "" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 547 5 "MAPLE" }{TEXT -1 170 " hei\337t die in Ol denburg an der Universit\344t durch eine entsprechende Linzenz zug\344 ngliche \"Computer-Algebra\"-Software aus Waterloo,Canada. Es gibt auc h andere Systeme. Wie " }{TEXT 548 5 "MAPLE" }{TEXT -1 62 " bereits in Gymnasien vordringt sehen Sie am besten unter : " }{URLLINK 17 "htt p://www.ikg.rt.bw.schule.de/maple.html" 4 "http://www.ikg.rt.bw.schule .de/maple.html" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 379 "" 0 "" {TEXT -1 54 "Bei der Literaturbescha ffung bin ich gerne behilflich." }}{PARA 380 "" 0 "" {TEXT -1 156 "Ers te Fragen wegen MAPLE k\366nnen Sie auch gerne an mich richten. Dies i st z.B. m\366glich \374ber meine Heimseite an der Carl von Ossietzky U niversit\344t Oldenburg: " }{URLLINK 17 "http://www.uni-oldenburg.de/m ath/personen/schmale/schmale.html" 4 "http://www.uni-oldenburg.de/math /personen/schmale/schmale.html" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{HYPERLNK 17 "zur\374ck" 1 "" "Inhalts \374bersicht" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT 264 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }