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1、AComputationalIntroductiontoNumberTheoryandAlgebra(BETAversion2)VictorShoupiiCopyright°c2003byVictorShoupAllrightsreserved.Nopartofthisbookmaybereproducedinanyformbyanyelectronicormechanicalmeanswithoutpermissioninwritingfromtheauthor.Theonly
2、exceptiontothisarethatindividualsarefreetostoreelectroniccopiesandprintpapercopiesforpersonaluseonly.Pleasedonotpostelectroniccopiesofthisdocumentonpubliclyaccessiblewebservers;instead,postalinktowww.shoup.net/ntb.Ifyouwantacopyshoptoprintanumberofcopiesoftheb
3、ookforuseinaclass,pleasecontacttheauthorforpermission.PrefaceNumbertheoryandalgebraplayanincreasinglysigni¯cantroleincomputingandcommunications,asevidencedbythestrikingapplicationsofthesesubjectstosuch¯eldsascryptographyandcodingtheory.Mygoalinwritingthisbookw
4、astoprovideanintroductiontonumbertheoryandalgebra,withanemphasisonalgorithmsandapplications,thatwouldbeaccessibletoabroadaudience.Inparticular,Iwantedtowriteabookthatwouldbeaccessibletotypicalgraduatestudentsincomputersciencewhohaveareasonableamountofgeneralma
5、thematicalexperience,whilenotpresupposinganythinginparticularbeyondastandardundergraduatecalculussequence.Thestructureofthebookissomewhatunique.Allofthemathematicsrequiredbeyondatypicalcalculussequenceisdevelopedfromscratch."More-over,thebookgenerallyalternat
6、esbetweentheory"andapplications":oneortwochaptersonaparticularsetofpurelymathematicalconceptsarefollowedbyoneortwochaptersonalgorithmsandapplications
7、themathematicspro-videsthetheoreticalunderpinningsfortheapplications,whiletheapplicationsbothmotivateandillu
8、stratethemathematics.Ofcourse,thisdichotomybe-tweentheoryandapplicationsisnotperfectlymaintained:thechaptersthatfocusmainlyonapplicationsincludethedevelopmentofsomeofthemathemat-icsthatisspecializedforaparticularapplication,andveryoccasionally,someofthechapter
9、sthatfocusonmathematicsincludeadiscussionofrelatedalgo-rithmicideasaswell.Indevelopingthemathematicsrequiredtodiscusscertainapplications,Itriedtostrikeareasonablebalancebetween,ont