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Springer Berlin Heidelberg NewY ork Barcelona HonKgo ng London Milan Paris Singapore Tokyo NeaKlo blitz AlgebrAasipce cts ofC ryptography Witha nA ppendoinxH yperellCiuprtviecs byA lfrJeMd.e nezeYsi,H- ongW u, andR oberJtZ. u ccherato With7 F igures �Sgperri n AlgoritahnmdsC omputation inM athematiVcosl um3e · Editors E.B eckeMr. B ronsteHi.Cn o hen D.E isenbuRd.G ilman NealK oblitz Yi-HongWu DepartmeonfMt a thematics DepartmeonfDt i scraentde UniversoifWt ays hington StatisStciiceanlc es SeattWlAe 9,8 19U5S,A AuburUnn iversity e-mail: AuburAnL,3 684U9S,A [email protected] RoberJtZ. u ccherato AlfrJeMd.e nezes EntruTsetc hnologies DepartmeonfCt o mbinatorics 750H eroRno ad andO ptimization OttawOan,t ario UniversoifWt ayt erloo CanadKa1 V1 A7 WaterlOonot,a rio e-mail: CanadNa2 L3 G1 [email protected] e-mail: [email protected] rloo.ca CorrecSteecdo nPdr int1i9n9g9 ThLei broafCr oyn grheacssas t altohogere idg pnrailn atsfi onlgl ows: LibraorfCy o ngreCsast aloging-in-PDuabtlai cation KobliNteza,l1 ,9 48.-A lgebraasipce cotfcs r yptogrIaN pehayKl o blitz. p.e m-.(Algorithamnsdc omputatiinmo ant hematIiScSsN1, 4 31-1v5.35 )0 ; "Witahn a ppendoinxh yperellciuprtvibecysAl freJdM .e nezeYsi,- HoWnug, a nd RoberJZt.u ccherato". Includbiebsl iograprhefiecraelan ncdei sn deIxS.B N3 -540-634(4h6a-rod coavlekpr.a: p er) 1.C odintgh eor2y.C. u rveEsl,l iptIi.Tc i.t lIeI.S. e ries. QA268.K518959 80 05.8'2'0159172--4d8c72C17I 9P MathematSiucbsj eCclta ssific(a1t9i9on1nT) 7:1 9,4 A6o6,8 P2n5Y,1 6n,Y 4o ISSN1 431-1550 ISBN3 -540-634S4p6r-ion ger-BVeerrllHiaengi delbNeerwgY ork Thiwso rki ss ubjetcoct o pyriAlglhr ti.g hatrser eservwehde,t htehrew holoer p arotf t hem ate riailsc oncernsepde,c ifictahlerl iyg hotfst ranslarteiporni,n trienugso,ef i llustrarteicointsa, tiobnr,o adcasrteipnrgo,d ucotnim oinc rofiolrmi na nyo thewra ya,n ds toraignde a tbaa nks. Duplicaotfito hni psu blicaotrip oanr ttsh ereiospf e rmitotendlu yn detrh ep rovisiooftn hse German CopyrigohftS epLtaewm b9e,1r 9 65i,ni tcsu rrent vearnsdpi eornm,i ssfioorun s e musta lwaybseo btainferdo mS pringVeerr-laVgi.o latairoenl si abfloerp rosecuutnidoentr h e GermaCno pyriLgahwt. ©SpringVeerr-lBaegr lHieni delb1e9r9g8 PrintienGd e rmany Theu seo fg enerdaels cripntaimveesr ,e gistenraemde s, tradeemtacir.nkt sh,ip su blication doenso ti mpleyv,e inn t haeb senocfea s pecisfitca temtehnatst,u cnha measr ee xempftr omt he relevparnott ecltaiwvasen dr egulatainodtn hse refforrefeeo rg eneruasle . Covedre sigMne:t aDespilgunGs m bH,B erlin. TypesettiTnygp:e sientT EXb yt hea uthoarn dr eformabttyAe dda mL einKza,r lsruuhsei,nag SpringTEeXr m acrpoa-ckage. SPIN1 0706488 46/314534-3 2o -1P rintoenda cid-fpraepee r Preface Thibso oiksi ntenadsae t de fxotar c ourosnce r yptogwriatephmh pyhi asos n algebmreatihcoI dtis sw. r itstoae snt ob ea ccestsoig brlaed uoarat dev anced undergrsatduudaeatnsewt esla,ls t os cienitnoi tshtfieser l dTsh.efi rstth ree chapftoerramss e lf-coinnttariondteuobdc a tsicioocnn caepnttdes c hniHqeurees . mya pproiaiscn ht uaintidin vfeo rFmoearxl a.m ptlhteer, e atomfce onmtp utational compleixnCi htayp 2t,we hri llaec kfionrgm arliigseotmrip,ch astihazese pse cts oft hseu bjtehcaatrtm e o sitm poritnca rnytp tography. Chapt4-e6ra sntd h Aep pencdoinxtm aaitne trhifaaotltr h meo spta hratns o t previouslyi nt aepxptebaoroeAkd n ofvoferelma .t iustr heie n cluosfti horne e typoefcs r ypto-g"rhaipdhmdyoe nno misayls"tc eommsb,i natorisayls-algebraic temasn,hd y pereslylsitp-ettmihscaa rta eta ne arsltya ogfde e velopImtie sn t. tosoo otnok nowwh icihfa, n yo,ft hecsrey ptoswyislutlle tmism baeto efl y practuisceBa.ul it n t hrea pigdrloyw fiienlgod fc ryptogirtia swp ohryt hwhile toc ontineuxapllnloeyrwo e n e-cwoanys truccotmiifonrngods mi ffearreenoatfs mathemPaetrihcsasop.ms oe ft hree adweirclsol n trtiotb hurete es etahrasctth i ll neetdobs e d one. Thibso oiksd esignnoaetsd a c omprehreenfseirvweeon rcbkeu,r t a thaesr as electteixvtebT ohoemka .n eyx erc(iwsiaetnshs waettr hsbe a cokft hbeo ok) makiest u itfaoburls eien a m atohrc ompustceirec nocuero srie na p rogroafm indepesntduednyt. Iw isthot hatnhkpe a rticiintp haMena ttsh emaStciiceanlc es IRnessteiarch tutSeu'msm Gerra duSattuedP ernotg rianAm l gebArsapiecoc fCt rsy ptography (Berk1e6-l2eJ7yu ,n1 e9 9f7o)gr i vmient gh oep porttuotn eistty-p taeroatftcs hh e manustco rfti hpbioso akn fdo firn dienrgr aonrudsn clatrhinatetie edsfie xdi ng. Ia me specgiraaltlteyoAf luflr Meedn ezfeocsra refruelaldtyih nmega nuscript anmda kimnagn vya lucaobrlree catnisdou ngsg esFtiinoanIlsw l.oy u,ll di kteo thaJnakc ques foPlrae ttatmrieirn neg p orhti wsoo rn(k s oomfei nto yte ptu b lishienCd h)a p4t;ae nrAd l fred MYein-eHzoWeuns,ga , n Rdo beZrutc cherato foarg reetioln emgte i ncltuhdeeeil re menttraerayto mfhe ynpte reclulrivpetsi c asa nA ppendix. SeatStelpet,e 1m9b9e7r NeaKlo blitz Contents Chapt1e.Cr r yptogra.p.h.y. .................... .............. §1 .E arHliys t.o r.y . . . . . . . . . . . 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §2T.h Ied eoafP ublKiecyC ryptog.r a.p h.y . . . . . 2. . . . . . . . . . . . . . . . . . §3T.h ReS AC ryptos.y. s t.e m. . . . . . . . .5 . . . . . . . . . . . . . . . . . . . . . . . . . §4D.i ffie-Haenltdlh mDeai ng iStiagln aAtlugroer i.t h.m . . . 8. . . . . . . . . §5S.e crSehta rCionigFn,l ippainnTdgi ,mSe p eonntH omewo.r k. .1 0. . . §6P.a sswoSridgsn,a taunCrdie psh,e .r s. . . . . . . 1.2 . . . . . . . . . . . . . . . . . . . §7P.r actCircyaplt osaynsUdts eemfIsum lp racOtniecs.a l. . . .1 3. . . . . . . . Exerc.i s.e s. . . . . . . . . . . . . 1. 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapt2e.Cr o mplexoifCt oym putations 18 §1T.h eB ig-N0o tat.i o.n . . . . . . . . . .1 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . . 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §2L.e ngotfNh u mbe.r s. . . . . . . . . . .2 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .2 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3T.i mEes tima.t e.s . . . . . . . . . . .2 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . . 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §4P., NaPnN,dP -Comple.t. e n.e s.s . . . . . . .3 4. . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . . 41. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §5P.r omise P.r o.b l.e m.s . . . . . . . . 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §6R.a ndomAilzgeodr iatnhCdmo sm pleCxliatsy.s es. . . . .4 5. . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .4 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §7S.o mOet hCeorm pleCxliatsys. e s. . . . . . . . 4.8 . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . . 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapt3e.Ar l gebra 53 §1 .F iel.d s. . . . . . . . . . . . . .5 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .5 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §2F.i nFiiteel .d s. . . . . . . . . . . . 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .6 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3T.h eE ucliAdlegaonr fiotPrho ml ynom.i a.l s. . . . .6 3. . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . . 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §4P.o lynoRmiinag.ls . . . . . . . . . . . 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . . 7.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII Contents §5G.r obBnaesre s 70 Exerci.s e.s . . . . . . . . . . . . 7.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapt4e.Hr i ddeMno nomiCarly ptosystems 80 §1 .T hIem ai-MatSsyusmto.et mo. . . . . . . . .8 0. . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . . 8.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §2P.a tarLiintD'trsla eg o.n . . . . . . . . . . 8.7 . . . . . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . . 9.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3S.y stTehmasMt i gBhetM orSee cu.r e. . . . . . . 96. . . . . . . . . . . . . . . . . . Exercises 102 Chapt5e.Cr o mbinatorial-CArlygpetborsayisct ems 103 §1 .H ist.o r.y . . . . . . . . . . . . 1.0 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §2I.r releovfBa rnacses Tahredo'rse. m . . . . . . . 1.0 4. . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .1 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3C.o ncrCeotmeb inatoriaSly-sAtle.gm es.b r.a i.c . . 1.0 5. . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .1 0 .9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §4T.h Bea sCiocm putaAtligoenPbarrloa b l.e m. . . . . .1 1.1 . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .1 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §5C.r yptogVrearpshoiifIoc dn e Maelm bers.h i.p . . . . 1.1 2. . . . . . . . . . . . . . §6L.i neAalrg eAbtrtaa c.k s. . . . . . . . . .1 13. . . . . . . . . . . . . . . . . . . . . . . . . . . §7D.e siganS iencguS ryes te.m . . . . . . . . .1 1.4 . . . . . . . . . . . . . . . . . . . . . . . Chapt6e.Er l lipatnidHc y perellCirpytpitco systems 117 §1 .E lliCputrivc.e s. . . . . . . . . . . .1 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .1 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §2E.l liCputriCvcre y ptosy.s t.e m.s . . . . . . .1 3.1 . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .1 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3E.l liCputriAvcne a loogfuC elsa ssNiucmablTe hre oPrryo ble.m s.1 3.7 . Exerci.s e.s . . . . . . . . . . . .1 3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §4C.u ltBuarcaklg rCoounnjde:c otnEu lrleisCp utrivce s anSdu rprRiesliantgwi iotOnhts h Perro bl.e ms. . . . .1 3.9 . . . . . . . . . . . . §5H.y pereClulrivp.et si. c . . . . . . . . . 1.4 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .1 4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §6. Hyperellipt.i c. C.r y.p t.o s.y s.t e.m s. .1 4.8 . . . . . . . . . . . . . . . . . . . . . . . Exerci.s e.s . . . . . . . . . . . .1 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AppendAinxE .l ementIanrtyr odutcotH iyopne relCluirpvteisc byA lfreJ.dM enezYie-sH,o nWgu ,a nRdo beJr.Zt u cchera.t o. . 1.5 5. . . . . . §1B.a sDiecfi nitainPodrn osp e.r ti.e s. . . . . . .1 5.6 . . . . . . . . . . . . . . . . . . . . . §2P.o lynoamniRdaa lt iFounnaclt i.o n.s . . . . . . 1.5 9. . . . . . . . . . . . . . . . . . . §3Z.e raonsPd o le.s . . . . . . . . . . . 1.6 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §4D.i vis.o r.s . . . . . . . . . . . . 1.6 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents IX §5R.e presSeenmtii-nRge Ddiuvciesd.o r.s . . . . . .1 6.9 . . . . . . . . . . . . . . . . . §6R.e ducDeidv is.o r.s . . . . . . . . . .1 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §7A.d diRnegd ucDeidv is.o r.s . . . . . . . . 1.7 2. . . . . . . . . . . . . . . . . . . . . . . . . Exerc.i s.e s. . . . . . . . . . . . .1 7.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AnswetrosE xerci.s es. . . . . . . . . . .1 7.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography 193 SubjeIcntd ex 201

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Algebraic Aspects of Cryptography (Algorithms and Computation in Mathematics) PDF

217 Pages·2004·3.82 MB·English

by Neal Koblitz

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