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Editor-in-DCehiiredfLr·ye n ch SeniAocrq uisitEidointsoW ri:l liHaomff man AssociEadtiet oCra:r oliCneel ano MarketiMnagn ageJre:ff W eidenaar MarketiAnsgs istaKnetn:d rBaa ssi SenioMra naginEgd itoKra:r eWne rnholm ProductPiroonj eMcatn ageBre:t hH ouston ProjeMcatn agePra:u Cl. A nagnostopoulos CompositainodnI llusitorantW:i ndfaSlolf twaursei,n Zgz TX E ManufacitnugMr anagerE:v elyBne aton PhotRoe searcMha:u reeRna ymond SenioCro veDre signBeert:hP aquin CoveDre sigNna:n cyG ouleStt,u dio;wink CoveIrm ageG:r ayN umber1s9,5 8( colla©g Jea)spJeorh n(sb 1.9 30I)P rivate CollectILi iocne nsbeydV AGA,N ewY orkN,. Y. PhotCor editGsr:a tefualc knowledgmiesmn atd et ot hec opyrihgohltd eorfst he biographpihcoatlol si,s toenpd a g7e5 2,w hicihsh erebmya dep arto fth isc opyripgahgte . Manyo ft hed esignatuisoenbdsy m anufacturaenrdss e llteord si stingtuhiespihrr o ducts are claimaesdt rademarWksh.e ret hosdee signataipopnesari nt hibso oka,n dA ddison Weslewya sa waroef a trademarckl aitmh,e d esignathiaovnebs e enp rintienid n itial caposr a lcla ps. LibraroyfC ongress Cataloging-inD-aPtuab lication RosenK,e nnetHh. Elementnaruym bethre orayn di tasp plicatIKi eonnnse tHh.R osen-. 6the d. p. cm. Includbeisb liograrpehfeirceanlc aensdi ndex. ISBN-1937:8 -0-321-50(0a3l1pk-a.1p er) ISBN-100-:3 21-500(3a1l-pk8a. p er) 1.Numbetrh eory-TextboIo.Tk ist.l e. QA241.R6270 11 512.7--'-d2c22 2010002572 Copyri©g h2t0 112,0 052,0 00b yK ennetHh. R osenA.l lr ighrtess ervNeod p.a rt oft hipsu blicamtaiyob ne r eproducsetdo,r ienda retrievsayls teomr,tr ansmitted, ina nyfo rm orb ya nym eanse,l ectronmiecc,h anicpahlo,t ocopyriencgo,r dionrg , otherwiwsiet,h otuhtep riowrr ittpeenr missoifot nh ep ublishPerri.n tientd h eU nited StatoefsA mericFao.ri nformatioonon b tainpienrgm issfoirou ns eo fm ateriinatl h is workp,l eassueb miatw rittreenq uetsoPt e arsoEnd ucatiIonnc,R. i,g htansd Contracts Departme5n0t0B, o ylstSotnr eeStu,i t9e0 0B,o stoMnA, 02116f,a xy ourre quetsot (6178)4 8-704o7re, - maaitlh ttp://www.pear.scoonm/leedgal/permissions.htm. 1 2 3 4 5 61 0-7C \V8- 194 131 2 11 10 Addison-Wesley isa ni mprionft PEARSON ISBN1 0:0 -321-50031-8 ------- www.pearsonhighered.com ISBN1 3:9 78-0-321-50031-1 Preface My goailn w ritithnigst exhta sb eetno w ritane a ccessiabnldie n vitiinntgr oducttoi on numbetrh eory. ForIwe amnotsettd,oc reaatnee ffectitvoeofo lr teachianndg l earning. Ih opetdo c aptuthree r ichneands sb eautoyft hes ubjeancdt i tusn expecutseedf ulness. Numbetrh eoirsyb otchl assiancda mlo dema,n da,tt hsea met imeb,o tphu raen da pplied. Int hitse xIth ,a vset rivetdoc aptutrhee sceo ntrasatsipnegc otfns u mbetrh eroyI.h ave workehdar dt oi ntegrthaetseea specitnsto on ec ohesitveex t. Thibso oki si defoarla nu ndergradnuuamtbee r theory actao nuyr slee Nvoefo lr.m al prerequibseiytoenscd o lleagleg ebarrea n eedefodr mosto ft hem aterioalth,e rt han somel eveolfm athematicmaalt uriTthyi.sb ooki sa lsdoe signteodb ea sourcbeo ok for elementnaurmyb etrh eoriytc; a ns ervaesa u sefuslu pplemfoernc to mputsecri ence coursaensda sa p rimeforr thosien tereisntn eedwd evelopmeinnnt usm ber thaenodr y cryptograBpehcya.u istei sc omprehensiitiv sde e,s ignteosd e rvbeo tha sa t extboanodk asa l ifetime refeforree nlceem entnaurmyb ethre orayn di twsi de-ranagpipnlgi cations. Thiesd iticoenl ebrattheess i lvaenrn iverosfat rhyib so okO.v ert hep as2t5 y ears, closteo1 00,0s0t0u dewnotrsl dwihdaev set udineudm bethre orfryo m previoeudsi tions. Eachs uccesseidviet iooftn h ibso okh asb enefifrtoemd fe edbacakn ds uggestfrioomn s manyi nstructsotrusd,e natnsd,r evieweTrhsi.sn ewe ditifoolnl owtsh es ameb asic approaacsha lplr evioeudsi tiobnustw, i tmha nyi mprovemeanntdse nhancemeIn ts. inviitnes trucutnofrasm ilwiiarth t hibso oko,r w hoh avneo tl ookeadta r eceendti tion, toc arefulelxyam ineth e sixtehd itiIoh na.v ceo nfidetnhcayeto uw ilalp precithaet rei ch exercsiestets h,efa scinatbiinogg raphanidc hails tornioctaelts h,eu p-to-dcaotvee rage, carefualn dr igorporuoso ftsh,me a nyh elpfuelx ampltehsre,i ch applitchasetu ipopnosr,t for computational seuncghai sMn aepsl ean d Mathematiacnadth, e manyr esources availaobnlth ee Web. Changeisn t heS ixtEhd ition Thec hangiensth es ixtedhi tihoanv bee edne signteomd a ket heb ooke asiteotr e acahn d learn from,m orei nteresantdi inngv itianngda, s u p-to-daaspt oes sibMlaen.y o ft hese changewse res uggestbeyud s erasn dr eviewoefrt sh efi ftehd itiTohne.fo llowilnigs t highlisghotmseo ft hem orei mportcanhta ngienst hiesd ition. ix x Preface Newd iscoveries • Thiesd ititorna crkesc ednits coveorfbi oetsah n umeriacnadla t heoretniactaulrA em.o ng thnee wc omputational discovienrth iees si xretedhfli etciatorenfoed u rM ersenpnrei mes andt hel ateesvti denscuep portmianngy o penc onjectuTrheesT .ao -Greethne orem provitnhgee xisteonfca er bitrlaorniaglry i thmeptriocg ressoifpo rnism eisso neo ft he recethneto retidciaslc ovedreisecsr iibnet dh iesd ition. Biographainedhs i storincoatle s • BiographoifTe erse ncTeao ,E tienBneez ouNto,r maMna cLeoFde rrerCsl,i ffoCrodc ks, andW aclaSwi erpinssukpip lemtehneatl reaedxyt ensciovlel ecotfib oino grapihnit ehse bookS.u rprisinign formatiaobno uste crBerti ticsrhy ptogradpihsicco veprrieedsa ting thew orko fR ivesSth,a miarn,d A dlemahna sb eena dded. Conjectures • Thet reatmoefnc to njecttuhrreosu gheoluetm entnaruym bethre orhya sb eeenx panded, particutlhaorslaeyb ouptr imneu mberasn dd iophanteiqnuea tioBnost.hr esolvaendd openc onjectaurreae dsd ressed. Combinatonruimable trh eory • A news ectioofnt heb ookc overpsa rtitiao fanssc,i natainndga ccessitbolpeii cn combinatonruimable trh eorTyh.i sn ews ectiionnt roduscuecshi mportatnotp iacss Ferrerdsi agrampsa,r titiidoenn tiaensdR, a manujawno'rsk o nc ongruenIcnet sh.i s sectipoartni,t ion idenintciltuideEisun,lg e ri'msp ortraenstu lartes p,r oveuds inbgo th generatfuinncgt ioannsdb ijections. Congruennutm berasn de llipctuircv es • A news ectiiosdn e votteodt hefa mousc ongruneunmtb eprr oblewmh,i cahs kwsh ich positiinvtee gaerrets h ea reoaf a righttr ianwgilterh a tiosniadlle e ngtThhsi.ss e ction contaaib nrsi ienft roducttoei lolni pctuircv aensd r elattheecs o ngruneunmtb eprr oblem tofi ndinrga tiopnoailn otnsc ertaeilnl ipctuircv eAsl.s ot,h isse ctiroenl attheceso ngruent numbeprr obletmoa rithmetpirco gressoiftho rnese s quares. Geometrrieca soning • Thise ditiionnt rodutcheeus s eo fg eometrrieca soniinntg h es tudoyf d iophantine probleInm sp.a rticnuelwarm ,a terisahlo wtsh afitn dinrga tiopnoailn otnst heu nicti rcle ise quivalteofin ntd inPgy thgaortriepalne sa,n dt hafitn dinrga tiotnraila ngwlietsah giveinn tegaesra reias e quivalteofin ntd inrga tiopnoailn otnsa na ssociaetleldi ptic curve. Cryptography • Thiesd itieolni minattheeus n necessraersyt rictthiawothn e nt heR SA cryptosyisst em usedt oe ncryap ptl aintmeexsts agteh imse ssagnee edtso b er elatipvreilmyte o t he moduluisnt hek ey. Preface xi Greatecsotm mond ivisors • Greatecsotm mond ivisaorrens o wd efineidn t hefi rscth aptaesri ,sw hati tm eanfosr Bezocuote fficients twoi ntegteorb ser elativelyT hpert iermm e. isn owi ntroduced and useidn th e book. Jacobsiy mbols • Morem otivatiisop nr ovidfoerd th e usefulneosfsJ acobsiy mbolIsn.p articulaarn, expanddeids cussoinot nh eu sefulneosfst heJ acobi syimnbe ovla luatLiengge ndre symbolissn owp rovided. Enhanceedx ercsiestes • Extenswiovrek h asb eend onet oi mproveex ercisseete sv enf artherS.e verhauln dred newe xercirsaensg,i frnogm routitnoce h allenghianvgeb, e ena ddedM.o reovenre,w computatiaonndea xlp loraetxoerryc icsaenbs e fo undi nth isn ewe dition. Accurancy • Morea ttenttihoanen v ebre forhea sb eenp aitdo e nsuritnhgea ccuraocfthy i se dition. Twoi ndependaecnctu racchye ckehrasv ee xamintehde e ntirtee xatn dt hea nswetros exercises. WebS itwew,w .pearsonhighered.com/rosen • TheW ebs itfoer thiesd itihoansb eenc onsidereaxbplayn deSdt.u denatnsdi nstructors wilfli nmda nyn ewr esourtcheescy a nu sei nc onjuncwtiitothnh ebo okA.m ongt hnee w featuraersea ne xpandceodl lecotfia opnp leatm sa,n ualfo ru sincgo mptutieonngailn es toe xplonruem betrh eo,ra ynda W ebp agede votteodn umbetrh eonreyw s. ExerciSseet s Becauseex erciasreess o i mportaanl tar,g ep ercentoafmg ye writianngdr eviswioornk hasb eedne votteodt hee xercsiesteSs t.u densthso ulkde eipn m indt hathte beswta yt o learn mathematiistc osw orka sm anye xerciassep so ssibIlw ei.lb lr iefldye scritbhee typeosfe xercifosuensd i nth isb ooka ndw herteo fi nda nswearnsd s olutions. StandaErxde rcises • Manyr outienxee rciasreeis n cludteodd e velboaps isck ilwlist,hc artea kesno t hat botho dd-numbearnedde ven-numbeerxeedr ciosfet sh itsy paree includAe dl.a rge numbeorf i ntermediateex-elrecviehslee lssp t udenptusts evercaoln cepttosg ethteor form newr esulMtasn.yo theerx erciasnedbs l ocokfse xerciasreeds e signteod develop newc oncepts. ExerciLseeg end • Challengeixnegr ciarsee isna mplseu pplayn dar e markewdi tohn es ta(*r ) i ndicaat ing difficuelxte rcainsdet wos ta(r s** ) i ndicaatnie nxgt remdeilfyfi ceuxletr ciTshee.ra er e xii Preface somee xercitsheacsto ntariens uulstesld a tienrt het extth;e saer em arkewdi taha rrow symbo(l> ) .T heseex ercissheosu bleda ssignbeyid n structwohresn evpeors sible. • ExercAisnes wers Thea nswetrosa lold d-numbeerxeedr ciares epsr ovidaettd h ee ndo ft het exMto.r e complestoel utitootn hse seex ercicsaenbs ef ounidnt hSe tudeSnotl'ust Miaonnusat lha t can befo undo nt heW ebs itfoer thibso okAl.l s olutihoanvsbe e ecna refullcyh ecked andr echectkoee dn suarcec uracy. • ComputatiEoxnearlc ises Eachs ectioinn clucdoemsp utatainodne sx ploratidoenssi gnteobd e d onew itah c om putatiopnraolg rasmu,c ahs M apleM,a thematPiARcIIaG,P, o rS ageo,ru sinpgr ograms writtebnyi nstructaonrd/sos rt udenTthse.ra er reo utinceo mputatieoxnearlc issteusd ents cand o lteoarn h owt oa pplbya sicco mmand(sa dse scriibnAe pdp endDi foxr M aplaen d Mathemaatnidoc nat hWe ebs itfoer P ARI/GP and Saagswe e)l,al s m oreo pen-ended questidoenssi gnfoerde xperimentaantidco rne atiEvaicthsy e.c tioanl sion cluads eesot f programmpirnogj edcetssi gnteobd e d oneb ys tudeunstisn agp rogrammlianngg uage ort hec omputatiopnraolg raomft hecihro icTeh.eS tudeMnatn'usa tloC omputations andE xploraotnit ohneWs eb s itper ovidaenss werhsi,n tasn,dg uidanthcaetw ilhle lp studeunstesc omputatitoonoatllosa ttatchke seex ercises. Web Site Studenatnsd i nstructwoirlsfl i nad comprehencsoilvlee ctoifro ens ourocne tsh is bookW'esb s itSet.u den(tasws e lals i nstructcoarnfins d) aw ider angeo fr esouracte s www. pearsonhigheredR.ecsoom/urroicsneetsne .n foderod n liyn struucsteo rcb aean c cessaetdw ww.pearsonhighereidn.sctroum/citcroacrno;s b tatihne piars swoforrdt hese resourfrcoemsP earson. ExternUanksl • TheW ebs itfoer t hibso ocko ntaaig nusi dper ovidainnngo taltiendtk osm anyW ebs ites relevtaonn tu mbetrh eorTyh.e ssei tarees k eyetdot hepa gien t heb ookw herree levant (J. materiisad li scussTehde.sl eo catiareo nmsar kedi nt heb ookw itthh ei con For convenieanl cieos,ftt hmeo sitm portWaenbts itreesl atteond u mbetrh eoirspy ro vided inA ppendDi.x • NumbeTrh eoryN ews TheW ebs itael scoo ntaais nesc tihoing hligthhtleia ntge dsits coveirnni uemsb etrh eory. • StudenSto'lsu tiMoannsu al Worked-out sotloau lttlih oeon dsd -numbeerxeedr ciisnte hset exatn ds ampleex ams can befo undi nt heo nliSnteu deSnotl'ust Miaonnu al. Preface xiii StudeMnatn'usfa olCr o mputaatnidEo xnpsl orations • A manuaplr ovidriensgo urscuepsp orttihnecg o mputatiaonndes x ploratciaonbn es foundo nt hWee bs itfoer t hibso okT.h imsa nuaplr ovides worked-ouotrp asrotliuatli ons solutitoomn asn yo fth esec omputatiaonndea xlp loraetxoerryc isewse,la lsa h si ntasn d guidanfocre a ttackoitnhge rTsh.i msa nuawli lslu pporttov, a ryidnegg reedsi,ff erent comptutieonnvailr onmeinntcsl,u dMianpgl eM,a thematiacnadP, A Rl/GP. Applets • Ane xtensciovlel ecotfia opnp leartesp rovidoentd h Wee bs itTeh.e saep plectasnb eu sed bys tudenfotr ss omec ommonc omputatiionn nusm bethre orayn dt oh elupn derstand concepatnsde xplocroen jectBureessi.da elsg oritfohrmc so mptutiionnn usm betrh eo,r y ac ollecotfic orny ptograapphpilcei tsas l spor ovidTehde.s ien cluadpep lefotrse ncyrp tiodne,c ryptciroynp,t analaynsdci rsy,p tographic protocoblost,ch l aadsdseircsasli ng ciphearnsdth eR SA cryptosysTtheemsc.er yptograapphpilcec tasnb eu sefodr i ndivid ualg,r ouapn,d c lassroaocmt ivities. SuggePsrtoedj ects • A usefuclo llecotfis oung gesptreodj eccatnas l sboe fo undo nth eW ebs itfoer th isb ook. Thesper ojeccatnss ervaesfi naplr ojefocrts st udeanntdsfo r grouposfs tudents. InstrucMtaonru'asl • Workesdo lutitoona sl elx erciisnet sh et exti,n cludthien ge ven-numbeerxeedc ises, anda varieotfoy t herre sourccaensb efo undo nt heW eb sitfoer instruc(twohrisc h isn ota vailatbosl teu denAtmso)n.g t hesoet herre souracreess a mplsey llaabdiv,i ocne planning whicht osc eocvteairno,dn a st esbta nk. How toD esigan C oursUes intgh iBso ok Thibso okc ans ervaest htee xfotr e lementnaruym betrh eocroyu rsweist mha nyd ifferent slanatnsd a tm anyd ifferelnetv eClosn.s equenitnlsytru,c towrislh la vea greadte aolf flexibidleistiyg nthienigsr y llawbiithth ist exMto.s ti nstrucwtiolwrlas n tt oc ovethre cormea teriianCl h apt1e r( asn eededS)e,c ti2o.n1( asn eededC)h,a pt3e,rS ections 4.1-4C.h3a,p t6e,rS ectio7n.s1 -7a.n3d,S ectio9n.s1 -9.2. Tofi lolu tt hesiyrl laibnis,t rucctaonar dsdm ateroinat lo piocfsi nterGeesnte.r ally, topiccasn b eb roadly claasssp iufireevd e rsuasp pliePdur.e t opiicnsc luMdoeb ius inversion (7S.e4c)it,ni toeng pearr titions 7(.S5e)pc,rt iimoint riovoet (sC hapt9e)r, continufreadc tio(nCsh apt1e2r) d,i ophanteiqnuea ti(oCnhsa pt1e3r) a,n dG uassian integ(eCrhsa pt1e4r) . Somei nstructwoirlwsla ntt oc ovearc cessaipbpllei catsiuocanhs sd ivibsiilitteys ts, thpee rpetcuaalle ndaanrdc, h ecdki gi(tCsh apt5e)Tr.h osien strucwthoowr asn tt os tress computaeprp licatainodcn rsy ptogrsahpohuyl cdo veCrh apt2e ra ndC hapt8e.rT hey maya lswoa ntt oi ncluSdeec tio9n.s3a nd9 .4C,h apt1e0r,a ndS ecti1o1n. 5. xiv Preface Aftedre cidiwnhgi ctho pitcosc oveirn,s tructmoarysw istho c onsutlhtefo llowing figurdei splaythien dge pendenocfcy h apters: /I1 � 2 3 12 I /i� 5 6 13 14 I 7 � ----- 8� /9 ----- 10 11 Although Ch2a mpatyeb re o mittiefdd esirietdd ,o ese xplathien b ig-n0o tation usetdh roughtohuett e xttod escribteh ec omplexoifta yl goritChhmasp.t 1e2rd epends onlyo nC hapt1e,ra ss howne,x cepfotr Theorem1 2.4w,h icdhe pendosn m aterial from Chapt9e.rS ecti1o3n. i4s t heo nlpyar to fC hapt1e3rth atd ependosn C hapter 12.Chapte1r1c anb es tudiweidt hocuotv eriCnhga pt9e ri ft heo ptioncaolm ments involvpirinmgi tirvoeo tisnS ecti9o.n1a reo mittSeedc.t i1o4n. s3h ouladl sboe c overed inc onjunctwiiotSnhe ctio1n3 .3. Forfu rthears sistainncsetr,u ctcoarncs o nsutlhtes uggesstyeldl afobric oursweist h differeenmtp hasperso vidientd h eI nstrucRteosro'usr Gcuei doen t heW ebs ite. Acknowledgments Ia pprecitahtece o ntinsutreodn sgu ppoarntd e nthusioafBm i lHlo ffmanm,y editaotr Pearsaonnd A ddison-Wefsallreo yn gtehra ann yo ft hme any ediwthoorh sa vper eceded hima,n dG regT obipnr,e sidoeftn htme a thematidcisv isoifoP ne arsoMny. s pecigarla ti tudgeo etso C aroliCneel anaos,s ocieadtiet foorra ,l hle ra ssistwaintcdhee velopmaenndt productoifot nh en ewe ditiMoyn a.p preciaatlisgooon e tso t hep roductmiaornk,e ting, andm editae abme hintdh ibso oki,n cludBientgHh o usto(nP roducPtrioojne Mcatn ager), MaureeRna ymon(dP hotRoe searcheCra)r,Cl o ttr(eMleld iPar oduceJre)ffW, e idenaar (ExecutMiarvkee tinMga nagerK)e,n drBaa ss(iM arketAisnsgi staanntdB) e,th Paquin (DesignaetrP )e arsoann,dP aulA nagnostopo(uplroosj meacnta gerJ)a,c quSic arlott (compositRiiocnkC) a,m p( copyedaintdop rr oofreadaenrd)L ,a ureMlu lle(rar tisatt) WindfaSlolf twaIra el.s woa ntt or eitermayt teh anktsoa ltlh osweh os upportmeyd worko nt hefi rsfitv ee ditioofnth si sb ooki,n cludthien mgu ltitoufdm ey previeoduis torastA ddisoWne sleayn dm y managemeantAt T &T BelLla borato(raineidst v sa rious incarnations). Pref ace xv Specitahla nkgsot oB artG oddardw hoh asp reparethde solutioofna sle lx erci ses int hibso oki,n cludthionsge f ounadt t hee ndo ft heb ooka ndo nt heW ebs itaen,d w ho hasr eviewtehdee ntibroeo kI.am alsgor atetfouJ le an-ClaEuvdared an d RogeLri psett fort heiherl cph eckianngd r echeckthien egn tirmea nuscript, itnhceanl suwdeirntsgo exercisIe wso.u ldal sloi kteo t hanDka viWd righfto rh ism anyc ontributtioto hn es Web siftoethr i sb ooki,n cludmiantge rioanlP ARI/GP,n umbethre orayn dc rpytogra phy appletthse,c omputatainodne xploramtiaonnu alan,d thes uggestperdo jecTthsa.n k s alsgoo es Ltaorr yW ashingtaonndK eitCho nrafdo trh ehierl pfsuulg gestcioonncse rning congruennutm berasn de llipctuircv es. Reviewe rs Ih avbee nefitfreodm the thoughtrfeuvli eanwds s uggestfrioomn su serosfp revioeudsi tionst,oa lolf w homI o fferh eartfthealntk sM.a nyo fth eiird eahsa vbee einn corporated int hiesd itiMoyn p.r ofouthnadn ksg ot ot her eviewwehrosh elpemde prepatrhees ixth edition: JenniBfeeirn ekWee,s teNrenw E nglanCdo lle ge DaviBdr adleUyn,i versoifMty a ine-Or ono FlavCioal onnGae,o rgMea sonU niver sity KeitCho nradU,n iversoifCt oyn nect icut PaveGlu erzhoUyn,i versoifHty a waii PauEl. G unnelUlnsi,v ersoifMty a ssachusetts- Amherst CharlePsa rryV,i rgiPnoilay tecIhnnsitci atnudtS et atUen iver sity HollSyw isheOrr,e goSnt atUen iver sity LawrenSczee C,a lifornSitaa tPeo lytecUhnniivce rsPiotmyo,n a 50 Ia lswoi sht ot hanakg aitnh ea pproximaterleyv iewoefrp sr evioeudsi tioofn s thibso okT.h eyh aveh elpeidm provteh ibso okt hroughiotulsti fFei.n allIyth ,a nki n advanacletl h osweh os enmde suggestaionndcs o rrectiontsh efi unt urYeo.um ays en d sucmha teritaom le inc aroef P earsoant m [email protected]. KennethH .R osen Middlet,oN wenwJ ersey This page intentionally left blank