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[{"key": "dc.contributor.advisor", "value": "Lehtonen, Ari", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Pehkonen, Salla", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-06-20T06:53:59Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-06-20T06:53:59Z", "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/123456789/64748", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa k\u00e4sitell\u00e4\u00e4n \u00e4\u00e4rellisi\u00e4 kuntia ja tarkastellaan joitakin niiden sovelluksia salakirjoituksiin liittyen. Aluksi m\u00e4\u00e4ritell\u00e4\u00e4n polynomit sek\u00e4 polynomin juuri ja tutkitaan niiden erilaisia ominaisuuksia. Lis\u00e4ksi todistetaan ensimm\u00e4isen luvun ehk\u00e4 t\u00e4rkein lause, polynomien jakoyht\u00e4l\u00f6, jota k\u00e4ytet\u00e4\u00e4n my\u00f6hemmin muun muassa polynomien jaollisuuden tarkastelussa. Jaollisuuden yhteydess\u00e4 k\u00e4sitell\u00e4\u00e4n polynomien suurinta yhteist\u00e4 tekij\u00e4\u00e4, jaottomia polynomeja sek\u00e4 juurien monikertaisuutta.\nSeuraavaksi m\u00e4\u00e4ritell\u00e4\u00e4n, mit\u00e4 ovat \u00e4\u00e4relliset kunnat ja tarkastellaan niiden olemassaoloa sek\u00e4 sit\u00e4, miten ja mill\u00e4 ehdoilla niit\u00e4 voidaan muodostaa, esimerkiksi millainen on oltava alkioiden lukum\u00e4\u00e4r\u00e4 \u00e4\u00e4rellisessa kunnassa. M\u00e4\u00e4ritell\u00e4\u00e4n my\u00f6s \u00e4\u00e4rellisiin kuntiin liittyvi\u00e4 muita k\u00e4sitteit\u00e4 ja niiden ominaisuuksia. Lis\u00e4ksi tarkastellaan erilaisia \u00e4\u00e4rellisiin kuntiin liittyvi\u00e4 kuvauksia, kuten Eulerin \u03c6-funktio, joka kertoo alkioiden lukum\u00e4\u00e4r\u00e4n tietylle kertaluvulle. Er\u00e4s t\u00e4rkeimmist\u00e4 toisen luvun tuloksista on, ett\u00e4 jokaiselle luvulle q, joka on muotoa p^n, miss\u00e4 p on alkuluku ja n positiivinen kokonaisluku, voidaan muodostaa yksi ja vain yksi \u00e4\u00e4rellinen kunta.\nTutkielman kaksi viimeist\u00e4 lukua liittyv\u00e4t \u00e4\u00e4rellisten kuntien soveltamiseen. Kolmannessa luvussa m\u00e4\u00e4ritell\u00e4\u00e4n salakirjoituksen periaatteet ja k\u00e4sitteet sek\u00e4 esitell\u00e4\u00e4n joitakin tunnettuja salausmenetelmi\u00e4, kuten RSA. Tarkastellaan my\u00f6s toistetun neli\u00f6innin ideaa, joka siis nopeuttaa huomattavasti luvun b^n mod m laskemista, kun luku n on todella suuri.\nViimeisess\u00e4 luvussa keskityt\u00e4\u00e4n yhteen t\u00e4rkeimp\u00e4\u00e4n \u00e4\u00e4rellisiin kuntiin liittyv\u00e4\u00e4n ongelmaan, nimitt\u00e4in diskreettiin logaritmiin. Diskreetti logaritmi nousee \u00e4\u00e4rellisten kuntien yhteydess\u00e4 oleelliseksi ongelmaksi, kun k\u00e4sitell\u00e4\u00e4n salausj\u00e4rjestelmi\u00e4. Esimerkiksi Diffien ja Hellmanin avaintenvaihdossa ja ElGamalin salausj\u00e4rjestelm\u00e4ss\u00e4 hy\u00f6dynnet\u00e4\u00e4n t\u00e4t\u00e4 ongelmaa. Lis\u00e4ksi tarkastellaan er\u00e4st\u00e4 algoritmia, joka on kehitelty diskreettien logaritmien ratkaisemiseksi \u00e4\u00e4rellisiss\u00e4 kunnissa. T\u00e4t\u00e4 algoritmia varten todistetaan my\u00f6s t\u00e4rke\u00e4 kiinalainen j\u00e4\u00e4nn\u00f6slause.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-06-20T06:53:59Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-06-20T06:53:59Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "51", "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.title", "value": "\u00c4\u00e4relliset kunnat ja niiden sovelluksia", "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-201906203336", "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", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "polynomit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "salakirjoitus", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "logaritmit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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