(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 72421, 1926] NotebookOptionsPosition[ 23489, 871] NotebookOutlinePosition[ 69418, 1838] CellTagsIndexPosition[ 69375, 1835] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Grote getallen en geheimschrift\n", StyleBox["workshop over RSA en PGP", FontSize->24, FontSlant->"Italic"] }], "Title", TextAlignment->Center, FontFamily->"Garamond", FontColor->RGBColor[0.699229, 0.347662, 0], Background->GrayLevel[0.900008]], Cell[CellGroupData[{ Cell["Initialisatie", "Section"], Cell["De programmacode die hieronder staat kun je overslaan.", "Text"], Cell[BoxData[ RowBox[{ RowBox[{"rep", "[", "n_", "]"}], ":=", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"10", "^", "n"}], "-", "1"}], ")"}], "/", "9"}]}]], "Input", InitializationCell->True], Cell[BoxData[ RowBox[{ RowBox[{"nextprime", "[", "n_", "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"x", "=", "n"}], "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"While", "[", RowBox[{ RowBox[{"!", RowBox[{"PrimeQ", "[", "x", "]"}]}], ",", RowBox[{"x", "++"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", "x", "]"}]}]}], "\[IndentingNewLine]", "]"}]}]], "Input", InitializationCell->True], Cell[BoxData[ RowBox[{ RowBox[{"makekey", "[", RowBox[{"e_", ",", "p_", ",", "q_"}], "]"}], ":=", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "}"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"GCD", "[", RowBox[{"e", ",", RowBox[{ RowBox[{"(", RowBox[{"p", "-", "1"}], ")"}], "*", RowBox[{"(", RowBox[{"q", "-", "1"}], ")"}]}]}], "]"}], ">", "1"}], ",", "\[IndentingNewLine]", RowBox[{ RowBox[{ "Print", "[", "\"\\"", "]"}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", "0", "]"}]}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Return", "[", RowBox[{"PowerMod", "[", RowBox[{"e", ",", RowBox[{"-", "1"}], ",", RowBox[{ RowBox[{"(", RowBox[{"p", "-", "1"}], ")"}], "*", RowBox[{"(", RowBox[{"q", "-", "1"}], ")"}]}]}], "]"}], "]"}]}]}], "\[IndentingNewLine]", "]"}]}]], "Input", InitializationCell->True], Cell[BoxData[ RowBox[{ RowBox[{"randominteger", "[", "n_", "]"}], ":=", RowBox[{"RandomInteger", "[", RowBox[{"{", RowBox[{ SuperscriptBox["10", RowBox[{"n", "-", "1"}]], ",", SuperscriptBox["10", "n"]}], "}"}], "]"}]}]], "Input", InitializationCell->True], Cell[BoxData[ RowBox[{ RowBox[{"randomprime", "[", "n_", "]"}], ":=", RowBox[{"nextprime", "[", RowBox[{"randominteger", "[", "n", "]"}], "]"}]}]], "Input", InitializationCell->True], Cell[BoxData[ RowBox[{ RowBox[{"string2number", "[", "s_", "]"}], ":=", RowBox[{"FromDigits", "[", RowBox[{ RowBox[{"ToCharacterCode", "[", "s", "]"}], ",", "128"}], "]"}]}]], "Input", InitializationCell->True], Cell[BoxData[ RowBox[{ RowBox[{"number2string", "[", "n_", "]"}], ":=", RowBox[{"FromCharacterCode", "[", RowBox[{"IntegerDigits", "[", RowBox[{"n", ",", "128"}], "]"}], "]"}]}]], "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell["Iets over Mathematica", "Section"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " is een programma waarmee je wiskunde kunt doen. Hieronder staan een paar \ voorbeelden. In de blauwe vakken staat steeds tekst en uitleg, de roze vakken \ kun je ", StyleBox["evalueren", FontSlant->"Italic"], " als je er met de cursor op gaat staan en dan op CTRL+ENTER drukt. \ Evalueren wil zeggen dat je de computer opdracht geeft om de wiskundige \ opdrachten die er staan, uit te voeren." }], "Text"], Cell[TextData[{ "Algebra, bijvoorbeeld machten van ", Cell[BoxData[ FormBox[ RowBox[{"1", "+", "x"}], TraditionalForm]]], " uitwerken: " }], "Text"], Cell[BoxData[ RowBox[{"Expand", "[", RowBox[{ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "^", "2"}], "]"}]], "Input"], Cell["Differentieren:", "Text"], Cell[BoxData[ RowBox[{"D", "[", RowBox[{ SuperscriptBox["x", "n"], ",", "x"}], "]"}]], "Input"], Cell["Integreren:", "Text"], Cell[BoxData[ RowBox[{"\[Integral]", RowBox[{ SuperscriptBox["x", "n"], RowBox[{"\[DifferentialD]", "x"}]}]}]], "Input"], Cell["Grafieken tekenen:", "Text"], Cell[BoxData[ RowBox[{"Plot", "[", RowBox[{ RowBox[{"Sin", "[", "x", "]"}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "10"}], ",", "10"}], "}"}]}], "]"}]], "Input"], Cell["Met hoge precisie rekenen:", "Text"], Cell[BoxData[ RowBox[{"N", "[", RowBox[{"Pi", ",", "99"}], "]"}]], "Input"], Cell["Met grote getallen rekenen:", "Text"], Cell[BoxData[ RowBox[{"a", "=", RowBox[{"2", "^", "1000"}]}]], "Input"], Cell[TextData[{ "Tot zover de voorbeelden. Zoals je hebt kunnen zien zijn er een groot \ aantal opdrachten die ", StyleBox["Mathematica", FontSlant->"Italic"], " kan uitvoeren. Die hoef je niet te kennen; aan hun namen kun je wel \ ongeveer zien wat ze doen." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["(Grote) priemgetallen", "Section"], Cell["\<\ We bekijken het getal dat je schrijft met 71 enen achterelkaar, \ kortheidshalve zullen we het E71 noemen. \ \>", "Text"], Cell[BoxData[ RowBox[{"E71", "=", RowBox[{"rep", "[", "71", "]"}]}]], "Input"], Cell[TextData[{ "Misschien is E71 een priemgetal of misschien toch niet. en zo nee, kun je \ het ontbinden?\nEerst even warm draaien met getallen van 40 cijfers. Als je \ de volgende formule een aantal keer achter elkaar evalueert, krijg je steeds \ een ander getal en zijn ontbinding in priemfactoren. Wat betekenen de \ getallen die ", StyleBox["Mathematica", FontSlant->"Italic"], " uitbraakt?" }], "Text"], Cell[BoxData[{ RowBox[{"n", "=", RowBox[{"randominteger", "[", "40", "]"}]}], "\[IndentingNewLine]", RowBox[{"FactorInteger", "[", "n", "]"}]}], "Input"], Cell["\<\ Nu proberen we E71 te ontbinden. Als je het te lang vindt duren, kun je de \ computer laten stoppen door ALT+. (alt+punt) in te drukken.\ \>", "Text"], Cell[BoxData[ RowBox[{"FactorInteger", "[", "E71", "]"}]], "Input"], Cell[TextData[{ "Getallen van 40 cijfers ontbinden kon de computer best snel. Maar je ziet \ dat het zelfs voor een computer moeilijk kan zijn om nog veel grote getallen \ in factoren te ontbinden. Sommige cryptografische technieken maken daar \ gebruik van. Hieronder leer je hoe dat werkt. ", "\n", "Even de stelling checken met het commando ", StyleBox["PowerMod[", FontWeight->"Bold"], StyleBox["a, b, m", FontWeight->"Bold", FontSlant->"Italic"], StyleBox["]", FontWeight->"Bold"], " dat ", Cell[BoxData[ FormBox[ RowBox[{" ", StyleBox[ RowBox[{ SuperscriptBox["a", "b"], " ", "modulo", " ", "m"}], FontWeight->"Bold"]}], TraditionalForm]]], " uitrekent. Probeer maar voor verschillende waarden van ", Cell[BoxData[ FormBox[ RowBox[{"a", Cell[""]}], TraditionalForm]]], "en ", Cell[BoxData[ FormBox["p", TraditionalForm]]], "." }], "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"a", "=", "71"}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"p", "=", "71"}], ";"}], "\[IndentingNewLine]", RowBox[{"PowerMod", "[", RowBox[{"a", ",", "p", ",", "p"}], "]"}], "\[IndentingNewLine]", RowBox[{"Mod", "[", RowBox[{"a", ",", "p"}], "]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"PowerMod", "[", RowBox[{"a", ",", "p", ",", "p"}], "]"}], "\[Equal]", RowBox[{"Mod", "[", RowBox[{"a", ",", "p"}], "]"}]}]}], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Cryptografie", "Section"], Cell["\<\ Stel Bob wil een code maken om emailberichten te ontvangen. Hij wil er zeker \ van zijn dat een afluisteraar zijn email niet kan lezen. Hiertoe maakt hij \ een geheimschrift met publieke sleutel. Eerst maakt hij twee grote \ priemgetallen aan en bepaalt hun product.\ \>", "Text"], Cell[BoxData[{ RowBox[{"p", "=", RowBox[{"randomprime", "[", "50", "]"}]}], "\[IndentingNewLine]", RowBox[{"q", "=", RowBox[{"randomprime", "[", "50", "]"}]}], "\[IndentingNewLine]", RowBox[{ StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], "=", RowBox[{"p", "*", "q"}]}]}], "Input"], Cell[TextData[{ "Bob kiest als publieke sleutel 101 (het mag ook een ander getal zijn) en \ noemt deze ", StyleBox["public", FontWeight->"Bold"], ". Vervolgens berekenent Bob de geheime sleutel die hij ", StyleBox["private", FontWeight->"Bold"], " noemt." }], "Text"], Cell[BoxData[{ RowBox[{ StyleBox["public", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], "=", "101"}], "\[IndentingNewLine]", RowBox[{ StyleBox["private", FontSize->24, FontColor->RGBColor[1, 0, 1]], "=", RowBox[{"makekey", "[", RowBox[{ StyleBox["public", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], ",", "p", ",", "q"}], "]"}]}]}], "Input"], Cell[TextData[{ "Vervolgens vernietigt Bob de priemgetallen p en q. Zelfs een hacker kan ze \ niet meer vinden op zijn computer. Voor (bijna) alle getallen a geldt nu:", "\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["a", RowBox[{"public", "\[CenterDot]", "private"}]], "=", RowBox[{"a", " ", "modulo", " ", "n"}]}], TraditionalForm]], FontSize->24], "\n", "We checken de uitspraak weer even voor een \"willekeurig\" getal:" }], "Text", TextAlignment->Left], Cell[BoxData[ RowBox[{"PowerMod", "[", RowBox[{"12345678", ",", StyleBox[ RowBox[{ StyleBox["public", FontColor->RGBColor[0, 0.500008, 0]], "*", StyleBox["private", FontColor->RGBColor[1, 0, 1]]}], FontSize->24], ",", StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], "]"}]], "Input"], Cell[TextData[{ "Nu wil vriendin Alice een mededeling via email versturen naar Bob. Bob \ heeft de publieke code ", StyleBox["public", FontWeight->"Bold"], " op zijn website staan, samen met het getal ", StyleBox["n. ", FontWeight->"Bold"], StyleBox["Bob bewaart echter zorgvuldig de ", FontVariations->{"CompatibilityType"->0}], "geheime sleutel ", StyleBox["private", FontWeight->"Bold"], ". \nHier is het bericht dat Alice wil versturen: \n", StyleBox["Vandaag eten we spruitjes\n", FontSize->24], "Eerst kent Alice een getal aan deze tekst toe:" }], "Text"], Cell[BoxData[ RowBox[{"bericht", "=", RowBox[{ "string2number", "[", "\"\\"", "]"}]}]], "Input"], Cell["\<\ Hoewel je aan zo'n getal niet direct de boodschap herkent, kan het bericht \ toch ook weer heel snel door iedereen ontcijferd worden:\ \>", "Text"], Cell[BoxData[ RowBox[{"number2string", "[", "bericht", "]"}]], "Input"], Cell["\<\ Zo kan Alice het bericht dus niet op internet versturen. Eerst versleutelt \ Alice het bericht met Bob's publieke code. Dat gaat als volgt:\ \>", "Text"], Cell[BoxData[ RowBox[{"codebericht", "=", RowBox[{"PowerMod", "[", RowBox[{"bericht", ",", StyleBox["public", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], StyleBox[",", FontSize->24], StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], "]"}]}]], "Input"], Cell["\<\ Dit getal kan Alice rustig het internet op sturen omdat het onleesbaar is \ geworden. Immers:\ \>", "Text"], Cell[BoxData[ RowBox[{"number2string", "[", "codebericht", "]"}]], "Input"], Cell["\<\ Bob ontvangt het gecodeerde bericht via de email en gebruikt nu zijn geheime \ sleutel om te decoderen.\ \>", "Text"], Cell[BoxData[ RowBox[{"decoded", "=", RowBox[{"PowerMod", "[", RowBox[{"codebericht", ",", StyleBox["private", FontSize->24, FontColor->RGBColor[1, 0, 1]], ",", StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], "]"}]}]], "Input"], Cell["Dit getal geeft het bericht van Alice weer terug.", "Text"], Cell[BoxData[ RowBox[{"number2string", "[", "decoded", "]"}]], "Input"], Cell[TextData[{ "Hoe kan een afluisteraar het bericht ontcijferen? Hij kent alleen maar de \ getallen ", StyleBox["public", FontWeight->"Bold"], " en ", StyleBox["n. ", FontWeight->"Bold"], StyleBox["De ", FontVariations->{"CompatibilityType"->0}], StyleBox["enige", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" manier om hieruit ", FontVariations->{"CompatibilityType"->0}], StyleBox["private", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" te berekenen is als de afluisteraar de factoren ", FontVariations->{"CompatibilityType"->0}], StyleBox["p", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" en ", FontVariations->{"CompatibilityType"->0}], StyleBox["q", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" van ", FontVariations->{"CompatibilityType"->0}], StyleBox["n", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" kent. En daar zit nou juist het probleem, grote getallen zijn \ practisch niet te ontbinden!! Probeer maar ", FontVariations->{"CompatibilityType"->0}], StyleBox["n ", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox["te ontbinden in", FontVariations->{"CompatibilityType"->0}], StyleBox[" p", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[" en ", FontVariations->{"CompatibilityType"->0}], StyleBox["q", FontWeight->"Bold", FontVariations->{"CompatibilityType"->0}], StyleBox[":", FontVariations->{"CompatibilityType"->0}] }], "Text"], Cell[BoxData[ RowBox[{"FactorInteger", "[", "n", "]"}]], "Input"], Cell["\<\ We noemen bovenstaande procedure RSA-cryptografie, naar de uitvinders \ Rivest-Shamir-Adleman.\ \>", "Text"], Cell[TextData[{ "Natuurlijk hoeven Bob en Alice zich geen zorgen te maken over het \ uitrekenen van priemgetallen en machtverheffingen. Dit gebeurt allemaal met \ standaard software. Het bekendste programma is ", StyleBox["PGP", FontSlant->"Italic"], " (Pretty Good Privacy)." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Hoe werkt het precies?", "Section"], Cell["\<\ We laten nu zien waarom het decoderen in RSA werkt. Dat doen we door \ eenvoudige getallen te kiezen, waaraan je wel ziet hoe het werkt maar \ waardoor de code in het echt wel heel makkelijk te kraken is. We kiezen \ namelijk:\ \>", "Text"], Cell[BoxData[{ RowBox[{"p", "=", "2"}], "\[IndentingNewLine]", RowBox[{"q", "=", "5"}]}], "Input"], Cell["Je krijgt dan:", "Text"], Cell[BoxData[ RowBox[{ StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], "=", RowBox[{"p", " ", "q"}]}]], "Input"], Cell[TextData[{ "Ga nu zelf even na welke getallen ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"1.", ".."}], StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], TraditionalForm]]], " geen priemfactoren hebben die ook een priemfactor van ", Cell[BoxData[ FormBox[ StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], TraditionalForm]]], " zijn. Hoeveel zijn het er? " }], "Text"], Cell[TextData[{ "In het algemeen, ook voor andere priemgetallen ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " en ", Cell[BoxData[ FormBox["q", TraditionalForm]]], ", geldt dat het er ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"p", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"q", "-", "1"}], ")"}]}], TraditionalForm]]], " zijn. Ga na dat dat in dit geval klopt. ", "\n", "Vanaf nu noemen we:" }], "Text"], Cell[BoxData[ RowBox[{"d", "=", RowBox[{ RowBox[{"(", RowBox[{"p", "-", "1"}], ")"}], RowBox[{"(", RowBox[{"q", "-", "1"}], ")"}]}]}]], "Input"], Cell[TextData[{ " Er is een stelling van Euler die beweert dat:\n", StyleBox["als ", FontSize->18], Cell[BoxData[ FormBox["a", TraditionalForm]], FontSize->16], StyleBox[" een getal is dat met ", FontSize->16], Cell[BoxData[ FormBox[ StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], TraditionalForm]], FontSize->16], StyleBox[" geen delers gemeenschappelijk heeft, dan is ", FontSize->16], Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["a", "d"], "=", RowBox[{"1", " ", RowBox[{"(", RowBox[{"mod", " ", StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], ")"}], Cell[""]}]}], TraditionalForm]], FontSize->16], StyleBox[".\n", FontSize->16], "Ga zelf weer even na met een paar voorbeelden:" }], "Text"], Cell[BoxData[{ RowBox[{"a", "=", "3"}], "\[IndentingNewLine]", RowBox[{"PowerMod", "[", RowBox[{"a", ",", "d", ",", StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], "]"}]}], "Input"], Cell[TextData[{ "Nu kiezen we ", Cell[BoxData[ FormBox[ StyleBox["public", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], TraditionalForm]]], " en ", Cell[BoxData[ FormBox[ StyleBox["private", FontSize->24, FontColor->RGBColor[1, 0, 1]], TraditionalForm]]], " op een speciale manier en wel zo, dat ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{ StyleBox["public", FontColor->RGBColor[0, 0.500008, 0]], StyleBox["\[CenterDot]", FontColor->RGBColor[0, 0.500008, 0]], StyleBox["private", FontColor->RGBColor[1, 0, 1]]}], "=", RowBox[{"1", " ", RowBox[{"(", RowBox[{"mod", " ", "d"}], ")"}]}]}], FontSize->24], TraditionalForm]]], ".\nGa na dat de volgende keuze hieraan voldoet:" }], "Text"], Cell[BoxData[{ StyleBox[ RowBox[{"public", "=", "3"}], FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], "\[IndentingNewLine]", StyleBox[ RowBox[{"private", "=", "7"}], FontSize->24, FontColor->RGBColor[1, 0, 1]]}], "Input"], Cell[TextData[{ "\nAls Alice een bericht ", Cell[BoxData[ FormBox["B", TraditionalForm]]], " wil coderen, dan gebruikt ze de ", Cell[BoxData[ FormBox[ StyleBox["public", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], TraditionalForm]]], " sleutel en berekent het codebericht ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["C", FontSize->18], StyleBox["=", FontSize->18], RowBox[{ StyleBox[ SuperscriptBox["B", StyleBox["3", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]], FontSize->18], StyleBox[" ", FontSize->18], RowBox[{"(", RowBox[{ StyleBox["mod", FontSize->18], StyleBox[" ", FontSize->18], StyleBox["10", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], StyleBox[")", FontSize->18]}]}]}], TraditionalForm]]], ".\nHet codebericht ", Cell[BoxData[ FormBox[ StyleBox["C", FontSize->18], TraditionalForm]]], " stuurt ze aan Bob.\nBob gebruikt de ", Cell[BoxData[ FormBox[ StyleBox["private", FontSize->24, FontColor->RGBColor[1, 0, 1]], TraditionalForm]]], " sleutel en", Cell[BoxData[ FormBox[" ", TraditionalForm]]], "berekent ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SuperscriptBox["C", StyleBox["7", FontSize->24, FontColor->RGBColor[1, 0, 1]]], " ", RowBox[{"(", RowBox[{"mod", " ", StyleBox["10", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], ")"}]}], FontSize->18], TraditionalForm]]], " en nu komt het grote moment, want:\n", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["C", StyleBox["7", FontSize->24, FontColor->RGBColor[1, 0, 1]]], " ", "=", RowBox[{ SuperscriptBox[ RowBox[{"(", StyleBox[ SuperscriptBox["B", StyleBox["3", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]], FontSize->18], " ", ")"}], StyleBox["7", FontSize->24, FontColor->RGBColor[1, 0, 1]]], " ", "=", RowBox[{ StyleBox[ SuperscriptBox["B", StyleBox[ RowBox[{ StyleBox["3", FontColor->RGBColor[0, 0.500008, 0]], StyleBox["\[CenterDot]", FontColor->RGBColor[0, 0.500008, 0]], StyleBox["7", FontColor->RGBColor[1, 0, 1]]}], FontSize->24]], FontSize->18], " ", "=", RowBox[{ StyleBox[ SuperscriptBox["B", StyleBox["21", FontSize->18]], FontSize->18], "=", RowBox[{ RowBox[{ StyleBox[ SuperscriptBox["B", RowBox[{ StyleBox["1", FontSize->18], " "}]], FontSize->18], "\[CenterDot]", SuperscriptBox[ RowBox[{"(", StyleBox[ SuperscriptBox["B", StyleBox["5", FontSize->18]], FontSize->18], " ", ")"}], StyleBox["4", FontSize->18]]}], " ", "=", RowBox[{ StyleBox["B", FontSize->18], StyleBox[ RowBox[{ StyleBox[" ", FontSize->18], " "}]], "(", RowBox[{"mod", " ", StyleBox["10", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], ")"}], " "}]}]}]}]}], TraditionalForm]], FontSize->18], ". \nDe laatste gelijkheid, die volgt uit bovenstaande stelling van Euler, \ hoeft niet waar te zijn als ", Cell[BoxData[ FormBox[ StyleBox["B", FontSize->18], TraditionalForm]]], " toevallig deelbaar is door de priemgetallen ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " of ", Cell[BoxData[ FormBox["q", TraditionalForm]]], ". Voor praktisch bruikbare (en dus grote) ", Cell[BoxData[ FormBox["p", TraditionalForm]]], " en ", Cell[BoxData[ FormBox["q", TraditionalForm]]], " is het algoritme zo in te richten dat dat nooit gebeurt.", Cell[BoxData[ FormBox[Cell[""], TraditionalForm]]], " \n\nProbeer nu zelf uit voor verschillende \"berichten\" kleiner dan 10:" }], "Text"], Cell[BoxData[{ RowBox[{"bericht", "=", "7"}], "\[IndentingNewLine]", RowBox[{"codebericht", "=", RowBox[{"PowerMod", "[", RowBox[{"bericht", ",", StyleBox["public", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]], StyleBox[",", FontSize->24], StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"decodeer", "=", RowBox[{"PowerMod", "[", RowBox[{"codebericht", ",", StyleBox["private", FontSize->24, FontColor->RGBColor[1, 0, 1]], ",", StyleBox["n", FontSize->24, FontColor->RGBColor[0, 0.500008, 0]]}], "]"}]}]}], "Input"], Cell["", "Text"] }, Open ]] }, Open ]] }, AutoGeneratedPackage->None, WindowSize->{1012, 714}, WindowMargins->{{0, Automatic}, {4, Automatic}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"Magnification"->1, "PaperOrientation"->"Portrait", "PaperSize"->{597.562, 842.375}, "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "user1", "home4", "Beukers", "math"}, "stedelijk.nb.ps", CharacterEncoding -> "iso8859-1"]}, FrontEndVersion->"7.0 for Linux x86 (32-bit) (February 25, 2009)", StyleDefinitions->Notebook[{ Cell[ CellGroupData[{ Cell["Style Definitions", "Subtitle"], Cell[ "Modify the definitions below to change the default appearance of all \ cells in a given style. 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CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell[ "The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Hyperlink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookLocate[#2]}]& ), ButtonNote -> ButtonData}], Cell[ StyleData["Hyperlink", "Presentation"], FontSize -> 16], Cell[ StyleData["Hyperlink", "Condensed"], FontSize -> 11], Cell[ StyleData["Hyperlink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ "The following styles are for linking automatically to the on-line \ help system.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["MainBookLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["MainBook", #]}]& )}], Cell[ StyleData["MainBookLink", "Presentation"], FontSize -> 16], Cell[ StyleData["MainBookLink", "Condensed"], FontSize -> 11], Cell[ StyleData["MainBookLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["AddOnsLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier", FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["AddOns", #]}]& )}], Cell[ StyleData["AddOnsLink", "Presentation"], FontSize -> 16], Cell[ StyleData["AddOnsLink", "Condensed"], FontSize -> 11], Cell[ StyleData["AddOnsLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["RefGuideLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier", FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["RefGuide", #]}]& )}], Cell[ StyleData["RefGuideLink", "Presentation"], FontSize -> 16], Cell[ StyleData["RefGuideLink", "Condensed"], FontSize -> 11], Cell[ StyleData["RefGuideLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["GettingStartedLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["GettingStarted", #]}]& )}], Cell[ StyleData["GettingStartedLink", "Presentation"], FontSize -> 16], Cell[ StyleData["GettingStartedLink", "Condensed"], FontSize -> 11], Cell[ StyleData["GettingStartedLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["OtherInformationLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0, 0, 1], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["OtherInformation", #]}]& )}], Cell[ StyleData["OtherInformationLink", "Presentation"], FontSize -> 16], Cell[ StyleData["OtherInformationLink", "Condensed"], FontSize -> 11], Cell[ StyleData["OtherInformationLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[ StyleData["Header"], CellMargins -> {{0, 0}, {4, 1}}, DefaultNewInlineCellStyle -> "None", LanguageCategory -> "NaturalLanguage", StyleMenuListing -> None, FontSize -> 10, FontSlant -> "Italic"], Cell[ StyleData["Footer"], CellMargins -> {{0, 0}, {0, 4}}, DefaultNewInlineCellStyle -> "None", LanguageCategory -> "NaturalLanguage", StyleMenuListing -> None, FontSize -> 9, FontSlant -> "Italic"], Cell[ StyleData["PageNumber"], CellMargins -> {{0, 0}, {4, 1}}, StyleMenuListing -> None, FontFamily -> "Times", FontSize -> 10]}, Closed]], Cell[ CellGroupData[{ Cell["Palette Styles", "Section"], Cell[ "The cells below define styles that define standard ButtonFunctions, \ for use in palette buttons.", "Text"], Cell[ StyleData["Paste"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, After]}]& )}], Cell[ StyleData["Evaluate"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], SelectionEvaluate[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["EvaluateCell"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["CopyEvaluate"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["CopyEvaluateCell"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[], All]}]& )}]}, Closed]], Cell[ CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell[ "The cells below define styles useful for making placeholder objects \ in palette templates.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Placeholder"], Placeholder -> True, StyleMenuListing -> None, FontSlant -> "Italic", FontColor -> RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions -> { Editable -> False, Selectable -> False, StripWrapperBoxes -> False}], Cell[ StyleData["Placeholder", "Presentation"]], Cell[ StyleData["Placeholder", "Condensed"]], Cell[ StyleData["Placeholder", "Printout"]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["PrimaryPlaceholder"], StyleMenuListing -> None, DrawHighlighted -> True, FontSlant -> "Italic", Background -> RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions -> { Editable -> False, Selectable -> False, StripWrapperBoxes -> False}], Cell[ StyleData["PrimaryPlaceholder", "Presentation"]], Cell[ StyleData["PrimaryPlaceholder", "Condensed"]], Cell[ StyleData["PrimaryPlaceholder", "Printout"]]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["FormatType Styles", "Section"], Cell[ "The cells below define styles that are mixed in with the styles of \ most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.", "Text"], Cell[ StyleData["CellExpression"], PageWidth -> Infinity, CellMargins -> {{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel -> False, ShowSpecialCharacters -> False, AllowInlineCells -> False, Hyphenation -> False, AutoItalicWords -> {}, StyleMenuListing -> None, FontFamily -> "Courier", FontSize -> 12, Background -> GrayLevel[1]], Cell[ StyleData["InputForm"], InputAutoReplacements -> {}, AllowInlineCells -> False, Hyphenation -> False, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["OutputForm"], PageWidth -> Infinity, TextAlignment -> Left, LineSpacing -> {0.6, 1}, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["StandardForm"], InputAutoReplacements -> { "->" -> "\[Rule]", ":>" -> "\[RuleDelayed]", "<=" -> "\[LessEqual]", ">=" -> "\[GreaterEqual]", "!=" -> "\[NotEqual]", "==" -> "\[Equal]", Inherited}, LineSpacing -> {1.25, 0}, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["TraditionalForm"], InputAutoReplacements -> { "->" -> "\[Rule]", ":>" -> "\[RuleDelayed]", "<=" -> "\[LessEqual]", ">=" -> "\[GreaterEqual]", "!=" -> "\[NotEqual]", "==" -> "\[Equal]", Inherited}, LineSpacing -> {1.25, 0}, SingleLetterItalics -> True, TraditionalFunctionNotation -> True, DelimiterMatching -> None, StyleMenuListing -> None], Cell[ "The style defined below is mixed in to any cell that is in an \ inline cell within another.", "Text"], Cell[ StyleData["InlineCell"], LanguageCategory -> "Formula", ScriptLevel -> 1, StyleMenuListing -> None], Cell[ StyleData["InlineCellEditing"], StyleMenuListing -> None, Background -> RGBColor[1, 0.749996, 0.8]]}, Closed]], Cell[ CellGroupData[{ Cell["Automatic Styles", "Section"], Cell[ "The cells below define styles that are used to affect the display \ of certain types of objects in typeset expressions. For example, \ \"UnmatchedBracket\" style defines how unmatched bracket, curly bracket, and \ parenthesis characters are displayed (typically by coloring them to make them \ stand out).", "Text"], Cell[ StyleData["UnmatchedBracket"], StyleMenuListing -> None, FontColor -> RGBColor[0.760006, 0.330007, 0.8]]}, Closed]]}, Open]]}, Visible -> False, FrontEndVersion -> "7.0 for Linux x86 (32-bit) (February 25, 2009)", StyleDefinitions -> "Default.nb"] ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[567, 22, 271, 9, 163, "Title"], Cell[CellGroupData[{ Cell[863, 35, 32, 0, 54, "Section"], Cell[898, 37, 70, 0, 49, "Text"], Cell[971, 39, 207, 7, 45, "Input", InitializationCell->True], Cell[1181, 48, 504, 15, 112, "Input", InitializationCell->True], Cell[1688, 65, 1161, 35, 156, "Input", InitializationCell->True], Cell[2852, 102, 290, 9, 45, "Input", InitializationCell->True], Cell[3145, 113, 193, 5, 45, "Input", InitializationCell->True], Cell[3341, 120, 229, 7, 45, "Input", InitializationCell->True], Cell[3573, 129, 230, 6, 45, "Input", InitializationCell->True] }, Open ]], Cell[CellGroupData[{ Cell[3840, 140, 40, 0, 54, "Section"], Cell[3883, 142, 477, 11, 93, "Text"], Cell[4363, 155, 156, 6, 49, "Text"], Cell[4522, 163, 131, 4, 45, "Input"], Cell[4656, 169, 31, 0, 49, "Text"], Cell[4690, 171, 102, 3, 45, "Input"], Cell[4795, 176, 27, 0, 49, "Text"], Cell[4825, 178, 131, 4, 59, "Input"], Cell[4959, 184, 34, 0, 49, "Text"], Cell[4996, 186, 193, 6, 45, "Input"], Cell[5192, 194, 42, 0, 49, "Text"], Cell[5237, 196, 79, 2, 45, "Input"], Cell[5319, 200, 43, 0, 49, "Text"], Cell[5365, 202, 75, 2, 45, "Input"], Cell[5443, 206, 276, 7, 71, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[5756, 218, 40, 0, 54, "Section"], Cell[5799, 220, 131, 3, 49, "Text"], Cell[5933, 225, 82, 2, 45, "Input"], Cell[6018, 229, 415, 9, 93, "Text"], Cell[6436, 240, 159, 3, 68, "Input"], Cell[6598, 245, 160, 3, 49, "Text"], Cell[6761, 250, 68, 1, 45, "Input"], Cell[6832, 253, 899, 30, 115, "Text"], Cell[7734, 285, 494, 13, 134, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[8265, 303, 31, 0, 54, "Section"], Cell[8299, 305, 290, 5, 71, "Text"], Cell[8592, 312, 322, 9, 90, "Input"], Cell[8917, 323, 276, 9, 71, "Text"], Cell[9196, 334, 400, 14, 83, "Input"], Cell[9599, 350, 491, 14, 124, "Text"], Cell[10093, 366, 353, 12, 55, "Input"], Cell[10449, 380, 584, 17, 149, "Text"], Cell[11036, 399, 130, 3, 45, "Input"], Cell[11169, 404, 157, 3, 49, "Text"], Cell[11329, 409, 72, 1, 45, "Input"], Cell[11404, 412, 163, 3, 49, "Text"], Cell[11570, 417, 325, 11, 55, "Input"], Cell[11898, 430, 117, 3, 49, "Text"], Cell[12018, 435, 76, 1, 45, "Input"], Cell[12097, 438, 127, 3, 49, "Text"], Cell[12227, 443, 285, 9, 55, "Input"], Cell[12515, 454, 65, 0, 49, "Text"], Cell[12583, 456, 72, 1, 45, "Input"], Cell[12658, 459, 1617, 51, 93, "Text"], Cell[14278, 512, 66, 1, 45, "Input"], Cell[14347, 515, 118, 3, 49, "Text"], Cell[14468, 520, 293, 7, 71, "Text"] }, Open ]], Cell[CellGroupData[{ Cell[14798, 532, 41, 0, 54, "Section"], Cell[14842, 534, 250, 5, 71, "Text"], Cell[15095, 541, 101, 2, 68, "Input"], Cell[15199, 545, 30, 0, 49, "Text"], Cell[15232, 547, 142, 5, 45, "Input"], Cell[15377, 554, 449, 16, 61, "Text"], Cell[15829, 572, 459, 18, 71, "Text"], Cell[16291, 592, 165, 6, 45, "Input"], Cell[16459, 600, 829, 32, 105, "Text"], Cell[17291, 634, 219, 6, 68, "Input"], Cell[17513, 642, 813, 30, 83, "Text"], Cell[18329, 674, 241, 8, 80, "Input"], Cell[18573, 684, 4196, 158, 295, "Text"], Cell[22772, 844, 670, 21, 109, "Input"], Cell[23445, 867, 16, 0, 49, "Text"] }, Open ]] }, Open ]] } ] *) (* End of internal cache information *)