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[{"key": "dc.contributor.advisor", "value": "V\u00e4h\u00e4kangas, Antti", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Rehn, Rasmus", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-05-10T05:31:14Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-05-10T05:31:14Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/63874", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoitus on tutustuttaa lukija salakirjoituksen maailmaan lukuteorian n\u00e4k\u00f6kulmasta. Tutkielma sis\u00e4lt\u00e4\u00e4 salausmenetelmiin tarvittavat matemaattiset pohjatiedot, Diffie-Hellmanin salausmenetelm\u00e4n ja RSA-salausmenetelm\u00e4n, sek\u00e4 molemmille salausmenetelmille soveltuvat purkumenetelm\u00e4t. Monet esitetyist\u00e4 tuloksista perustuvat algebraan, mutta t\u00e4h\u00e4n tutkielmaan tulokset on pyritty muuttaaman lukuteorian piiriin siten, ett\u00e4 lukija ei v\u00e4ltt\u00e4m\u00e4tt\u00e4 edes tied\u00e4 lukevansa algebraan perustuvia tuloksia. Ensimm\u00e4isess\u00e4 luvussa tutustutaan jaollisuuteen, lukuteoriaan ja kokonaislukujen tekij\u00f6ihinjakoon. Tarkoituksena on esitt\u00e4\u00e4 riitt\u00e4v\u00e4n kattava pohja salausmenetelmi\u00e4 varten, jotta kaikki niiss\u00e4 k\u00e4ytetty matematiikka olisi hyvin perusteltua. Toisessa luvussa k\u00e4sitell\u00e4\u00e4n lis\u00e4\u00e4 jaollisuustuloksia modulaariaritmetiikan n\u00e4k\u00f6kulmasta. Kolmannessa luvussa k\u00e4sitell\u00e4\u00e4n primitiivisi\u00e4 juuria, jotka ovat t\u00e4rke\u00e4ss\u00e4 roolissa Diffe-Hellmanin salausmenetelm\u00e4n turvallisuuden perustelemisessa. Ensin osoitetaan, miten primitiivisten juurten moninkertojen avulla saadaan esitetty\u00e4 salausmenetelm\u00e4ss\u00e4 k\u00e4yt\u00f6ss\u00e4 oleva lukujoukko. T\u00e4m\u00e4n j\u00e4lkeen osoitetaan, ett\u00e4 salausmenetelm\u00e4ss\u00e4 k\u00e4ytetyss\u00e4 lukujoukossa on aina olemassa primitiivinen juuri. Salausmenelm\u00e4t esitet\u00e4\u00e4n nelj\u00e4nness\u00e4 ja viidenness\u00e4 luvussa siten, ett\u00e4 niiden toimivuus perustellaan aiemmin esiteltyjen tuloksien avulla ja niist\u00e4 n\u00e4ytet\u00e4\u00e4n havainnollistavat esimerkit. Nelj\u00e4nness\u00e4 luvussa k\u00e4sitell\u00e4\u00e4n Diffie-Hellmanin salausmenetelm\u00e4\u00e4, jonka tarkoituksena on pysty\u00e4 luomaan informaation l\u00e4hett\u00e4j\u00e4lle ja vastaanottajalle yhteinen salausavain, jota muut ulkopuoliset eiv\u00e4t p\u00e4\u00e4se lukemaan. Sen j\u00e4lkeen tutustutaan Shankin algoritmiin, jonka avulla ulkopuolinen henkil\u00f6 voi yritt\u00e4\u00e4 selvitt\u00e4\u00e4 t\u00e4t\u00e4 salausavainta. Viidenness\u00e4 luvussa k\u00e4sitell\u00e4\u00e4n RSA-menetelm\u00e4\u00e4, jonka tarkoituksena on pysty\u00e4 l\u00e4hett\u00e4m\u00e4\u00e4n viesti kahden henkil\u00f6n v\u00e4lill\u00e4 siten, ett\u00e4 ulkopuolinen ei pysty lukemaan sen sis\u00e4lt\u00f6\u00e4. Lopuksi esitell\u00e4\u00e4n Pollardin p-1 -menetelm\u00e4, jolla ulkopuolinen henkil\u00f6 voi yritt\u00e4\u00e4 avata salattua viesti\u00e4. K\u00e4ytt\u00e4m\u00e4ll\u00e4 tutkielmassa esitettyj\u00e4 salausmenetelmi\u00e4 yhdess\u00e4, voidaan tietojen vaihtaminen toteuttaa turvallisesti esimerkiksi internet-sivustoilla. Liitteiss\u00e4 on laskettu Maxima-ohjelmalla esimerkit kaikista k\u00e4ytetyist\u00e4 salaus- ja purkumenetelmist\u00e4. Liitteet on pyritty tekem\u00e4\u00e4n mahdollisimman helppolukuisiksi Maximaa k\u00e4ytt\u00e4m\u00e4tt\u00f6mille henkil\u00f6ille siten, ett\u00e4 he voisivat toteuttaa itsen\u00e4isesti samat esimerkit.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-05-10T05:31:14Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-05-10T05:31:14Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "54", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "RSA", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Diffie-Hellman", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Shankin algoritmi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Pollardin menetelm\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "primitiivinen juuri", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "jaollisuus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "modulaariaritmetiikka", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Lukuteoriaan perustuvia salausmenetelmi\u00e4", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201905102529", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", 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