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[{"key": "dc.contributor.advisor", "value": "Kurittu, Lassi", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Sormunen, Lauri", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2016-02-17T18:36:22Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2016-02-17T18:36:22Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2016", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1522472", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/48817", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tavoitteena on esitt\u00e4\u00e4 tunnetuimmat alkulukutestit niin matemaattiselta\r\nperustoiltaan kuin k\u00e4yt\u00e4nn\u00f6n toteutuksiltaan ohjelmakoodin muodossa.\r\nAlkulukutestit jaotellaan yleisesti deterministisiin ja probabilistisiin testeihin; deterministiset\r\ntestit antavat t\u00e4ysin varman vastauksen, mutta ovat suurille luvuille huomattavasti\r\nprobabilistia testej\u00e4 hitaampia. Probabilistiset testit ovat nopeita tehd\u00e4,\r\nmutta saattavat antaa v\u00e4\u00e4r\u00e4n vastauksen. Testien suoritusaikaa mitataan karkeasti\r\nniiden suorittamiseksi vaadittavien laskutoimitusten lukum\u00e4\u00e4ar\u00e4ll\u00e4.\r\nTutkielmassa k\u00e4sitell\u00e4\u00e4n deterministisist\u00e4 testeist\u00e4 jakolaskumenetelm\u00e4, Wilsonin\r\nlause, Prothin testi, Lucasin ja Lehmerin testi ja viimeisen\u00e4 suhteellisen uusi AKStesti.\r\nYleisesti deterministiset testit eiv\u00e4t ole kovin k\u00e4ytt\u00f6kelpoisia, sill\u00e4 niiden suoritusaika\r\nkasvaa eksponentiaalisesti testattavan luvun kasvaessa tai ne toimivat ainoastaan\r\ntietty\u00e4 muotoa oleville luvuille. Probabilistisiin testeihin p\u00e4\u00e4dyt\u00e4\u00e4n hy\u00f6ynt\u00e4m\u00e4ll\u00e4\r\nFermat\u2019n pient\u00e4 lausetta k\u00e4\u00e4nteisesti, ja t\u00e4t\u00e4 kutsutaan Fermat\u2019n alkulukutestiksi.\r\nKyseinen testi ei kuitenkaan toimi kovin luotettavasti: on olemassa Carmichaelin\r\nlukuja, jotka hyvin todenn\u00e4k\u00f6isesti toteuttavat Fermat\u2019n pienen lauseen yht\u00e4l\u00f6n,\r\nvaikka ovatkin yhdistettyj\u00e4 lukuja. Fermat\u2019n alkulukutestin johdannainen Millerin ja\r\nRabinin testi on kuitenkin eritt\u00e4in k\u00e4ytetty sen vuoksi, ett t\u00e4lle pystyt\u00e4\u00e4an m\u00e4\u00e4rittelem\u00e4\u00e4n\r\nvirheraja, jolla testi antaa v\u00e4\u00e4ri\u00e4 vastauksia. Solovayn ja Strassenin testi on\r\nhyvin l\u00e4hell\u00e4 kahta viimeksi mainittua testi\u00e4, muttei ole aivan yht\u00e4 luotettava kuin\r\nMillerin ja Rabinin testi.\r\nAlkulukutesti\u00e4, kuten monia algoritmeja yleens\u00e4kin, nimitet\u00e4\u00e4n polynomiaikaiseksi,\r\njos siin\u00e4 vaadittavien laskutoimitusten lukum\u00e4\u00e4r\u00e4 on polynomi testattavan luvun\r\nnumeroiden lukum\u00e4\u00e4r\u00e4n suhteen. Pitk\u00e4\u00e4nn alkulukutestaus ei ollut polynomisessa\r\najassa ratkaistava ongelma, ennen kuin vuonna 2002 julkaistiin AKS-testi, joka on\r\nensimm\u00e4inen deterministinen ja polynomiaikainen alkulukutesti. Tutkielmassa todistetaan\r\nlopulta my\u00f6s t\u00e4m\u00e4n testin toimivuus sek\u00e4 todetaan, ett\u00e4 kyseess\u00e4 todellakin\r\non polynomiaikainen algoritmi.", "language": "", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Lauri Sormunen (lakrsorm) on 2016-02-17 18:36:21.811990. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2016-02-17T18:36:22Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201602171600.pdf: 597704 bytes, checksum: b7455772a53d021968d843bd97c4f39a (MD5)\nlicense.html: 4776 bytes, checksum: b4fa1934eb45d05602da5b368677299a (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2016-02-17T18:36:22Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201602171600.pdf: 597704 bytes, checksum: b7455772a53d021968d843bd97c4f39a (MD5)\nlicense.html: 4776 bytes, checksum: b4fa1934eb45d05602da5b368677299a (MD5)\n Previous issue date: 2016", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (77 sivua)", "language": null, "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": "Alkuluvut", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Alkulukutesti", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Miller-Rabin", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Fermat", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Lucas-Lehmer", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "AKS", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "algoritmi", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Alkulukutesteist\u00e4", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, 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