uhongGaoandDanielPtofMathematicalSciences,ClemsonUnivClemson,SouthCaro - PDF document

uhongGaoandDanielPtofMathematicalSciences,ClemsonUnivClemson,SouthCarolina29634-1907,USAtofComputerScience,UnivyofTto,CanadaM5S-1A4paperfocuspolynomialsrithmprovidingavtofitthatimproesRabin'scostestimatebalogfactor.Wegiveapreciseanalysisoftheprobabilitythataran-dompolynomialofdegreetainsnoirreduciblefactorsofdegreeless).ThisprobabilityisnaturallyrelatedtoBen-Or's(1981)algorithmfortestingirreducibilityofpolynomialsoerniteelds.Walsocomputetheprobabilityofapolynomialbeingirreduciblewhenithasnoirreduciblefactorsoflowdegree.Thisprobabilityisusefulintheanalysisofvariousalgorithmsforfactoringpolynomialsoerniteelds.epresentanexperimentalcomparisonoftheseirreducibilitymethodswhentestingrandompolynomials.importantinimplementingcryptosystemsanderrorcorrectingcodes.Onewpolynomialndingirreduciblepolynomialsandtestingtheirreducibilityofpolynomialsaretalproblemsinniteelds.Aprobabilisticalgorithmforndingirreduciblepolynomialsthatworkswinpracticeispresentedin[26].Thecentralideaistotakepolynomialsatrandomandtestthemforirreducibilit.LetbethenberofirreduciblepolynomialsofdegreeeraniteeldI.Itiswwn(see[21],p.142,Ex.3.26&3.27)that n�q(qn=2�1) (q�1)nInqn�q Thismeansthatafraction1ofthepolynomialsofdegreeareirreducible,andsowendonaerageoneirreduciblepolynomialofdegreetries.Inordertotransformthisideaintoanalgorithmonehastoconsiderirreducibilitytests. Insections2and3,wefocusontestsforirreducibilit.LetLetx],degapolynomialtobeforirreducibilit.Assumethatarethehesforthisproblem:Butler(1954):isirreducibleifandonlyifdimk)=1,whereistherobeniusmaponIIx]=(f)thatsendsh2IFq[x]=(f)tohq2IFq[x]=(f),andIistheidenymaponIIx]=(f)(see[4]);f;x,and0mod(see[26Otherirreducibilitytestscanbefoundin[14],[31],and[12Inthispaper,weconcentrateonRabin'stest,andavtpresentedin[1].Insection2,wereviewRabin'sandBen-Or'sirreducibilityalgorithms.Westatesection3,efocusonBen-Or'salgorithm.Thisleadsusbehaviorofpolynomials,polynomialsanalysisisexpressedasanasymptoticformin,thedegreeofthepolynomialtobetestedforirreducibilit.First,wexaniteeldI,andthenwestudystudy],probabilisticpropertiespolynomialserniteeldsfrequentlyhaeashapethatresemblescorrespondingproper-tiesofthecycledecompositionofpermutationstowhichtheyreducewhenthegoespolynomialoferI),when.ThisprobabilityrelatesnaturallywithBen-Or'salgorithm.Theprobabilityofapolynomialbeingirreduciblewhenithasnoirreduciblefactorsoflowdegreeprovidesusefulinformationforfactoringpoly-nomialsoerniteelds(seeforinstance[126).WeprovidetheprobabilitofapolynomialbeingirreduciblewhenithasnoirreduciblefactorsofdegreeatInsection4,wegiveanexperimentalcomparisononthealgorithmsdiscussedsection2.polynomialhasamhbetteraeragetimebehaviorthanothers,eventhoughitswyisthewerysparseirreduciblepolynomialsareusefulforseveralapplications:pseu-dorandomnbergeneratorsusingfeedbackshiftregisters([15]),discreteloga-rithmoerI([6],[23]),andecientarithmeticinniteelds(Shoupprivunication,1994).Hoer,fewresultsareknownaboutthesepolynomialsbeyondbinomialsandtrinomials(see[22],Chapter3,andthereferencesthere).section5,polynomialswithupto(1)nonzeroterms(notnecessarilytheloestcoecienforinnitelymanydegreeseassumethatarithmeticinIisgiven.ThecostmeasureofanalgorithmwillbethenberofoperationsinI.Thealgorithmsinthispaperusebasicpolynomialoperationslikeproductsandgcds.Weconsiderinthispaperexclu- loglogpolynomialsofdegreeatmostusing\fast"arithmetic([28],[27],[5])canbeenas),andthecostofagcdbeteentopolynomialsofdegreeatcanbetakenas)operationsinI.Thenberofmcationsneededtocomputemodymeansoftheclassicaldsquaringmethodd],p.441{442),wherepolynomialerIbertationof.Therefore,thecostofcomputingmodythismethod)operationsinIusingFFTbasedmethods.Inthissection,wereviewRabin'sandBen-Or'stests,andwepresentavofRabin'smethod.Rabinirreducibilitytestandan RabinIrreducibilitAmonicpolynomialolynomialx]ofdegreeallthedistinctprimedivisorsofEither\isirreducible"or\isreducible". ],p.275,Lemma1).;:::;palllx]ofdeisirrducibleininx]ifandonlyifgcf;x,for,andThebasicideaofthisalgorithmistocomputemodindependentlyforeacyrepeatedsquaring,andthentotakethecorrespondentgcd.Theworst-caseanalysisgivenin[26]is)log)operationsinIer,itcanbeshownthat)loglog)isanupperboundofthe berofoperationsinIforthisalgorithm.Indeed,rstnotethatthenberofdistinctprimefactorsofisatmostlog.Thecostofexponentiationsis )log )logistheharmonicsum.Usingthewwnapprotionoftheharmonicsum([16],p.452),=loglog ),weobtain)loglog),whichdominatesthecost)log)ofcomputinggcd's.Therefore,thetotalcostofRabin'salgorithmis)loglogAsanimprot,weproposethefollowingvtforthecomputationofmod,for1 tofRabinIrreducibilitAmonicpolynomialolynomialx]ofdegree;:::;pallthedistinctprimedivisorsofEither\isirreducible"or\isreducible".:: Theorem2.2nomialirrducibility,anduses)logationsinThecorrectnessofthealgorithmfollowsfromthecorrectnessofthepocomputations.WeproethatmodyinductiononBasis:when=1,modestep:forsomemod.Then,mod Withthisvt,intheworst-case,thenberofpolynomialminRabin'salgorithmtocomputeallpoersusingrepeatedsquaringis)log)logexponen)logber)log)logBen-Orirreducibilitytest Ben-OrIrreducibilitAmonicpolynomialolynomialx]ofdegreeEither\isirreducible"or\isreducible". ThecorrectnessofBen-Or'sprocedureisbasedonthefollowingfact(see[21p.91,Theorem3.20).,theppx]istheprductofallmonicduciblepolynomialsininx]whosedeedividesmodf;x;:::; .Thepolynomialisreducibleifandonlyifoneofthegcd'sisdierenfrom1.Intheworstcase,thisalgorithmcomputes timesathpoerandagcdofpolynomialsofdegreeatmost.Recallingthecostoftheseoperationsfromsection1,beha)log(usingFFTbasedmultiplicationalgorithms,andtherefore,itisworsethanourRabin'svt.Hoer,ascanbeseenfromourtheoreticalandexperimenresultsinSections3and4,Ben-Or'salgorithmisveryecient.Themainreasonfortheeciencyofthisalgorithmisthatrandompolynomialsoflargedegreeareerylikelytohaeanirreduciblefactorofsmalldegree,andBen-Or'salgorithmklydiscardsthesepolynomials(seealso[19poinnotknown.Ben-Or'saerage-caseanalysisisonlyknownwhengoestoinnitexpected amongtheirreduciblefactorsof;thentheexpectedcostofBen-Or'salgorithm)log(Ben-Or([1Theorem2)esan).Infact,herelatesthefactorialdecompositionofpolynomialswiththecyclicdecompositionofpermutations.Theresultfollowsfromthestudyoftheexpectedlengthoftheshortestcycleinarandompermutation([30]).Hothisrelationbeteenirreduciblefactorsofpolynomialsandcyclesofpermtionsjustholdswhenthesizeoftheeldislarge,asitwasobservedin[7].polynomialsolynomialswithoutirreduciblefactorsoflowdegreemakeBen-Or'sirreducibil-ytesttoexecutealargenberofiterations.Theprobabilitythatarandompolynomialofdegreetainsnofactorsoflowdegreegivesmeaningfulinfor-mationonthebehaviorofBen-Or'salgorithm.Wecallapolynomialithasnoirreduciblefactorsofdegrees.Inthissectionweareinterestedinthedistributionofroughpolynomials.Thefollowingtheoremisproedin[2]whenisxed.Theorem3.1Denotebyn;mtheprabilityofarandommonicpofdeough.Thenwhenn;m (1+uniformlyforbethecollectionofallmonicirreduciblepolynomialsinI,thesummationofallmonicpolynomialswithallirreduciblefactorswith(1+beaformalvariable,andthedegreeof.Thesubstitutionproducesthegeneratingfunction)ofpolynomialswithallirreduciblefactorshavingdegrees�km Notethatarywhen,andthuswecannotapplythetransferlemmasin[8,24Asusual,[)representsthecoecientof),andobservethatn;mmzn]Pm(z)=qn.Inordertoestimaten;m),weapplyTheorem10.8in[24)presentsapoleoforder1at withresidue qmYk=1(1�q�k)Ik:6 productSuppose q�1 .ByOdlyzko([24],Theorem10.8), qgq(m)1 q�n�1!�n+r�1 q�1r�n1 =max.Therefore,n;m q�11 qgq(m)!()�n=w+1 b�1gq(m)nw+1 independenm;qonlyneedtoestimateintermof.When 1,and.Consideringthateobtain j1�jmYkj1�zkjIk1 (1+ log(1+ b�1mYk=1exp(Ikrk)1 b�1mYk=1exp(bk1 b�1 mXk=1bk!1 b�1expbm+1 b�11 b�1expb b�1bclogn=1 b�1expb b�1nclogb:Hence,w1 b�1expb ,and b�1=bn2 b�1expb ByTheorem3.2belo em1 n;m gq(m)�12 b�1eclognexpb Aslog0and1,theright-handof(2)approachesto0asSincethequanyontheright-handof(2)isindependentofeseethatn;m)approachesto1uniformlyforThiscompletestheproof. Inthenexttheoremweestimatethefunction Theorem3.2oranyprimepandpositiveinte,wehave emexp �mXk=11 k!mYk=1(1�q�k)Ik1�1 p q�q q�1
exp �mXk=11 ,wehave qkIk!e�Hme� istheEuler'sconstant,andNotethatlog(1+1,and k=2=�1�1 p By(1),weha qkmYk=1exp�Ik =exp qk!exp0@�mXk=1qk k�q(q2� (q�1)k qk1Aexp �mXk=11 k!
exp q q�1mXk=1q2�1 k! �mXk=11 k!
exp 1Xk=11 2q q�1=1�1 p q�q q�1
exp �mXk=11 upperboundbounderboundspolynomialsandforthedensityofnormalelements(see[11]).Sinceitissimple,ereproduceithere.Asand01,weha qkIkmYk=11�1 qkqk�1 k=mYk=1 1 qk�1qk�1!�1 kmYkexp�1 =exp k!:8 betheharmonicsum,i.e.,.Then=xdx1+log,andth).When wnapproximationof=log ),weha ,thisresultisinaccordancewiththecorrespondentoneofpermwithnocyclesoflengthorless(see[29eprovideinTable1belowthevaluesofn;m)andtheirratio,=logforsev1000.Thiswsthatgenceofn;m)to)isveryfast.Moreoer,asn;m)quicdecreases.Forinstance,forarandompolynomialofdegree900,thereisaprob-yofmorethan0.9ofhavingafactorofdegreeatmost9.Thisisanotherhforirreduciblepolynomialsofdegreeatmost)inordertohahighprobabilityofndingafactor.polynomialbeirreducibleifithasnoirreduciblefactorsoflowdegree.Manyal-gorithmsforfactoringpolynomialsoerniteeldscomprisethefollowingthreeefactorization(replaceapolynomialbyasquarefreeonewhicpolynomialexponenreducedto1);efactorization(splitasquarefreepolynomialintoaproductofpolynomialswhoseirreduciblefactorshaeallthesamedegree);andefactorization(factorapolynomialwhoseirreduciblefactorshaethesamedegree).bottlenecpolynomialfactorizationproblem(see[14],[18],[13]).Thisstepofthefactoriza-tionprocessworksasfollows:atanypoin,alltheirreduciblefactorsofdegreeuptoebeenfound,andalltheirreduciblefactorsofdegreegreaterthanremaintobedeterminedfromafactorAnaturalwyofimprovingthedistinct-degreefactorizationstepisbytestingasymptoticscenario,thecostoftheirreducibilitytestisaboutthesameasthedistinct-degreefactorizationalgorithm.Analternativetooercomethisproblemproblem](x6).Thecentralideaistoruntheirreducibilitytestandthedistinct-degreefactorizationalgorithminparallel,feedingtheformerwithpartialinformationobtainedbythelatter(seethedetailsin[12Itisclearthattheprobabilityofamonicpolynomialbeingirreduciblewhenithasnoirreduciblefactorsoflowdegreeprovidesusefulinformationintheaboprocess.Inthefollowing,wederiveanasymptoticformulaforthisprobabilit n m n;m gq(m) P/g 2 1 .25000000000000000000000000 .25000000000000000000000000 1.0000 3 1 .25000000000000000000000000 .25000000000000000000000000 1.0000 4 2 .18750000000000000000000000 .18750000000000000000000000 1.0000 5 2 .18750000000000000000000000 .18750000000000000000000000 1.0000 6 2 .18750000000000000000000000 .18750000000000000000000000 1.0000 7 2 .18750000000000000000000000 .18750000000000000000000000 1.0000 8 3 .14062500000000000000000000 .14355468750000000000000000 .97963 9 3 .14453125000000000000000000 .14355468750000000000000000 1.0068 10 3 .14355468750000000000000000 .14355468750000000000000000 .99997 20 4 .11828613281250000000000000 .11828541755676269531250000 1.0000 30 4 .11828541755676269531250000 .11828541755676269531250000 1.0000 40 5 .09776907367631793022155762 .09776907366723652792472876 1.0000 50 5 .09776907366723719405854354 .09776907366723652792472876 1.0000 60 5 .09776907366723652792472876 .09776907366723652792472876 1.0000 70 6 .08484899050039278356888779 .08484899050039278175814854 1.0000 80 6 .08484899050039278175860054 .08484899050039278175814854 1.0000 90 6 .08484899050039278175814857 .08484899050039278175814854 1.0000 100 6 .08484899050039278175814854 .08484899050039278175814854 1.0000 200 7 .07367738498865927351164168 .07367738498865927351164168 .99999 300 8 .06551498664534958936373162 .06551498664534958936373162 1.0000 400 8 .06551498664534958936373162 .06551498664534958936373162 1.0000 500 8 .06551498664534958936373162 .06551498664534958936373162 1.0000 600 9 .05872097388539275926570230 .05872097388539275926570230 1.0000 700 9 .05872097388539275926570230 .05872097388539275926570230 1.0000 800 9 .05872097388539275926570230 .05872097388539275926570230 1.0000 900 9 .05872097388539275926570230 .05872097388539275926570230 1.0000 able1.aluesofn;m)and),with=log=2.Theorem3.3n;m.Then,asachtoinnity,n;m istheEuler'scconsideringthepolynomialsofdegreeerIinsidethesetofpolynomialsofdegreeerIwithoutirreduciblefactorsofdegreelessthanorequaltoUsing(1),Theorems3.1and3.2,whenhtoinniteobtainn;m n;m n e� m=e m n:10 ExperimendescribeSection2.WeprovidearunningtimecomparisonofthealgorithmsforrandompolynomialsonaSunSparc20computer.ThealgorithmswereimplementedonaC++softareduetoShoup.Thispacagecontainsclassesforniteeldsandpolynomialsoerniteeldswithimplementationsforbasicoperationssuchasultiplication,takinggcd,andsoon(foradescriptionofthesoftaresee[32polynomials.able2andTable3.ThedegreesinTable2andinTable3werechosensuchasmanyprimedivisorsandfewprimedivisors,respectiv.Thenberpolynomialstestedas10forTable2andforTable3,wherethedegreeofthepolynomialsbeingtestedforirreducibilit.ItcanbeseenfromcaseofdivisorsBen-Orhasbestbehaalgorithmshappenswhentestingirreduciblepolynomials.Weincludeacolumnwiththenberofirreduciblepolynomialsthatweretestedforeachdegree. n Rabin Rabin's Ben-Or berof Vt Irreducible 105 0.7990 0.6133 0.2000 9 210 2.7652 2.6942 0.9938 14 330 10.5672 17.4147 2.3443 8 420 10.5702 4.3971 1.5189 7 able2.eragetimeinsecondsfortesting10polynomialsoerIofdegreewithmanyprimedivisors. n Rabin Rabin's Ben-Or berof Vt Irreducible 101 4.1564 8.9188 0.5901 4 256 20.8451 46.9867 2.7950 5 331 32.7156 71.8685 2.7719 9 eragetimeinsecondsfortesting5polynomialsoerIofdegreewithfewprimedivisors. polynomialsarepresentedinTable4belo n Rabin Rabin's Ben-Or berof Vt Irreducible 101 186.1383 280.2633 13.0198 4 105 78.2952 61.8381 11.2629 5 210 227.5800 174.7400 23.0000 3 able4.eragetimeinsecondsfortesting5polynomialsofdegreeerIalsotestedcaseoferylargeelds.eragetimeinsecondsofCPUfortesting315polynomialsofdegree105oerIaprimewith100bits,wRabin'svInthiscase,5irreduciblepolynomialswerefound.betterbehaviorthanothers,eventhoughitsorst-casecomplexityisamongthem.Avtoftheaboealgorithmsthateconomizesgcd'scomputa-tionscanbegivenusingBen-Or'sideasupto)iterationsandourRabin'stafterthatpoint.Forthisalgorithmtogetherwithmoreexperimentalre-sults,see[25polynomialsytestsSection2,toexperi-tourprogramsonvariouspolynomialsoflargedegrees.Forreduciblepoly-Section3,programsterminatealmostimmediately.Hoer,whentestinganirreduciblepolynomial,wedonothaeapriorianyideaofwlongitetocom-pletethetask.Itisdesirabletohaesomesimplepolynomialswhiceknowinanceareirreduciblesothatwecantestthecorrectnessofourprogramsandwtheapproximatetimeourcomputerneedsonvariousdegrees.Thiswalsohelpusindecidingtherangeofdegreestocomparethetests.By\simple",emeanpolynomialsthatcaneitherbeconstructedeasily(withouttestingfororhaeonlyafewnonzeroterms.problemofsparseirreduciblepolynomialsisalsoofindependentinwnopenproblemistoconstructirreduciblepolynomialsoerIofdegreewithatmost)nonzerotermsinitsloestcoecients.These polynomialsareusefulinthediscretelogarithmproblem([6],[23]).Shoup(pri-atecommunication,1994)pointsoutthatifIIx]=(f)withirreducibleandandx]ofsmalldegree,saydeg2log,thenexponenoperationserations)Experimenwthatsuchpolynomialexistsfor1000takingdeg2+logShparlinski([33])givesaconstructionofirreduciblepolynomialswithdegreesoftheform4erIerI,and2erI,foran3,andk;leintegers.Inthefollowing,werstgeneralizehisconstructionandthenconstructexplicitlyseveralinnitefamiliesofirreduciblepolynomials.Theorem5.1;:::;p;:::;egativeintegers.Then,overanyniteeldwhosecharacteristicisdis-tinctfr,thereisanirrduciblepolynomialofdewithatmost1)+1oterms.ifallareodd.Ifoneof,sa,is2and3mod4,let=lcm(2).Then,1)for1,and41)ifisevbeanelementinIthatisnotathpoerinIfor1.ThepolynomialerI([21],p.124,Theorem3.75),isirreducibleoerIforallnonnegativein.ThenisirreducibleoerIofdegree.Thepolynomial(4)hasatmost+1nonzeroterms.polynomialsTheorem5.1berdependsexponenpolynomialshaeonlynonzerotermstheloestcoecients).Thiscanbeseenfromthefollowingexamples.Example5.21mod4esidue.Thenisirrducibleoverforallorinstanc3mod8isaprime,thentake5mod12isaprime,thentake2mod5isaprime,thentakeTheseresultsarefrom[17](Prosition5.1.3forthersttwo,andTheem2,p.54,forthethir Example5.33mod4em5.1.following[3].isoddand3mod4isaprime.L+1)+1).Constructativelyasfollows: 2p (modv; 2p (mode,ateachstep,onecantakeanyofthesignsarbitrarily.L.Thenisirrducibleover,andoveraswell,forallExample5.4em5.1followingolynomialsoverforallk;`;m;nconstructionsofpolynomials,olynomials,](Chap-ter3),and[10ethankBruceRichmondforhelpfuldiscussionsandVic-authorwhilevisitingtheUnivyofWaterloowhosehospitalityandsupportaregratefullyacBen-Or,M.Probabilisticalgorithmsinniteelds.c.22ndIEEESymp.oundationsComputerScienc(1981),pp.394{398.polynomialsoerniteelds.InInformationTheoryandApplic,A.GullivN.Secord,ol.793erlag,1994,pp.1{23. Blake,I.,Gao,S.,andMullin,R.Explicitfactorizationof+1oerIwithprime(mod4).Appl.Alg.Eng.Comm.Comp.4(1993),89{94.Butler,M.Onthereducibilityofpolynomialsoeraniteeld.Quart.J.Math.(1954),102{107.polynomialsarbitraryalgebras.cta.Inform.28(1991),693{701.IEEETans.Info.Theory30(1984),587{594.Flajolet,P.,Gourdon,X.,andPanario,D.Randompolynomialsandpoly-ol.1099NotesinComputerScienc,Springer-Verlag,pp.232{243.SIAMJournalonDiscreteMathematics32(1990),216{240.onzurperiodsSubmittedtobolicComputationtendedabstractinProc.LAol.911ofLectureNotesinComputerScience311{322),1995.o,S.,andMullen,G.ksonpolynomialsandirreduciblepolynomialsoniteelds.J.NumberTheory49(1994),118{132.SubmittedtoFieldsandtheirApplic(abstractinAMSAall1995,#904-68-227,p.798),1995.onzurGathen,J.,andGerhard,J.Arithmeticandfactorizationofpolyno-mialsoerIc.ISSAC'96,Zurich,Switzerland(1996),L.Y.N.,Ed.,Apress,pp.1{9.onzurGathen,J.,andPanario,D.Asurveyonfactoringpolynomialsoniteelds.SubmittedtothespecialissueoftheMAGMAconferenceinJ.Symb.,1996.onzurGathen,J.,andShoup,V.ComputingFrobeniusmapsandfactoringpolynomials.Computcomplexity2(1992),187{224.Golomb,S.W.ShiftrgisterseAegeanParkPress,LagunaHills,Cali-fornia,1982.2nded.,Reading,MA,1994.Ireland,K.,andRosen,M.AClassicalIntrductiontoModernNumberThe2nded.erlag,Berlin,1990.tofen,E.,andShoup,V.Subquadratic-timefactoringofpolynomialsoniteelds.c.27thACMSymp.TheoryofComputing(1995),pp.398{406.cher,J.,andKnopfmacher,A.tingirreduciblefactorsofpoly-nomialsoeraniteeld.Theartofcvol.2:seminumericalalgorithms2nded.,ReadingMA,1981.Lidl,R.,andNiederreiter,H.Finiteeldsol.20ofdiaofMathe-maticsanditsApplic,ReadingMA,1983.Menezes,A.,Blake,I.,Gao,X.,Mullin,R.,Vanstone,S.,andYt,Lancaster,1993. ol.209NotesinComputerScienc,Springer-Verlag,pp.224{314.o,A.Asymptoticenumerationmethods.okofCombinatoricsR.Graham,M.Grhel,andL.Loasz,Eds.Elsevier,1996.anario,D.binatorialandalgebraicaspectsofpolynomialsoerniteelds.PhDThesis,inpreparation,1996.Rabin,M.O.Probabilisticalgorithmsinniteelds.SIAMJ.Comp.9ge,A.hnelleMultiplikationvonPolynomenuberKorpernderCharak-teristik2.ctaInf.7(1977),395{398.ge,A.,andStrassen,V.hnelleMultiplikationgroerZahlen.puting7(1971),281{292.Sedgewick,R.,andFlajolet,P.nIntrductiontotheAnalysisofA,ReadingMA,1996.Shepp,L.,andLloyd,S.Orderedcyclelengthsinarandompermmer.Math.Soc.121(1966),340{357.Shoup,V.astconstructionofirreduciblepolynomialsoerniteelds.J.Symb.Comp.17(1995),371{391.Shoup,V.AnewpolynomialfactorizationalgorithmanditsimplemenSymb.Comp.20(1996),363{397.arlinski,I.Findingirreducibleandprimitivepolynomials.Appl.Alg.Eng.Comm.Comp.4(1993),263{268..ThispaperhasappearedinoundationsofComputationalMathematicsF.CucerandM.Shub(Eds.),Springer1997,346{361.ThisarticlewasprocessedusingtheLXmacropacagewithLLNCSst