uoagr Abstract Pseudorandom sequences have many applications in cryp tography and spread spectrum communications In this dissertation on one hand we develop tools for assessing the randomness of a sequence and on the other hand we propose new constru ID: 26735
Download Pdf The PPT/PDF document "Study on Pseudorandom Sequences with App..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Prof.NicholasKaloupsidiswasthesupervisorofthisthesis. si+m1 si+m2 si nonlinearfeedbackfunctionFig.1.Theblockdiagramofafeedbackshiftregister.orderofisdenedtobethemaximumoftheordersoftheproducttermsappearinginthealgebraicnormalformwithnonzerocoecients.DeÞnition1.ThelengthoftheshortestFSRhavingafeedbackfunctionoforderthatgeneratesdeÞnesthethordercomplexityofsequence,andisdenotedbyDeÞnition2.ThelengthoftheshortestFSRthatgeneratesiscalledmumnonlinearcomplexityofsequence,andisgivenbyThesecondordercomplexityofasequenceisreferredtoasthequadraticspanofthesequenceanditisdenotedby).Incasethefeedbackfunctionislinearand,...,0)=0,wedenethelinearcomplexity.Itisclearthat······andbetwoperiodicbinarysequencesofperiod.Theirperiodiccrosscorrelationfunctionisdenedas N.When,theautocorrelationfunctionisdened.DeÞnition3.Twosequencesofthesameperiodaresaidtobecyclicallyequiv-alent,iftheyarerelatedbyaleft(orright)cyclicshift.Otherwise,theyarecyclicallydistinct.3FirstOrderApproximationofBinarySequencesbeabinarysequenceofperiod1anditslinearspan.TheBerlekamp Masseyalgorithmneeds2sequencedigitsinordertodetermineandthelinearfeedbackshiftregister(LFSR)associatedtotheleastorderhomogeneouslinearrecursion([8]).Therefore,thelinearspanisacriticalindex 4OntheQuadraticSpanofBinarySequencesInChapter4,weinvestigatethequadraticspanofnitebinarysequences([14],[15]).In[1],thequadraticspanofthedeBruijnsequenceswasstudied,andapartialgeneralizationoftheBerlekamp Masseyalgorithm,basedonGaus-sianelimination,wasproposed.Twomoreecientalgorithmforcalculatingthequadraticspanofasequencearedescribedinthischapter.Therstonetakesadvantageofthespecialstructureofthecorrespondinglinearsystemsofequations.Letn,m,andlet)bethequadraticspanofthe.Inconnectionwiththealgebraicnormalformweintroducethevector(3)whichcontainsthecoecientsoftheunknownquadraticfeedbackfunctionFrom(1),thecalculationofaquadraticfeedbackfunctionthatgeneratesagivenisequivalenttosolvingthesystemoflinearequationsn,mn,m)(4)wheren,m)isthe(+1)2matrixn,mBasedonthefollowingTheorems,wedevelopedaniterativealgorithmthatcomputesthequadraticspanofanitebinarysequence.Theorem1.Letbethegreatestintegersuchthatrank))=rankn,d+1)=rank))=rankotherwiseTheorem2.Let+1)=,where.Thenitholds+2++1),foralll,1].Afterthecomputationofthequadraticspan,wesolvethesystemoflinearequa-tions(4),inordertondthefeedbackfunctionofthecorrespondingFSR.Thesecondalgorithmisamodiedversionofthefundamentaliterativeal-gorithm(FIA).FIAwasintroducedin[3]forsolvingthemulti-sequenceshift registersynthesisproblem.Thegoalofthealgorithmistondthesmallestinitialsetofcolumns,inagivenmatrix,whicharelinearlyindependent. Inordertocomputeabooleanfeedbackfunctionof)variablesthatgeneratesthesequence,wehavetosolvethesystem(1).Duetothespecialstructureofn,m),the2possibledierentrowsofthematrixformabase)of(2)over(2),whichcanalwaysbewrittenasalowertri-angularmatrix.Thus,usingappropriateoutputsofthespanalgorithm,weshowthatthesystem(5)canbeeasilyreducedtoalowtriangularsystemofrankn,sp)))equationsandvariablesthatcanbeeasilysolvedbybacksubstitution.Theother2variablesof))thatdonotappearinthereducedsystemaresetequaltozero.Thesystemoflinearequations(5)has2degreesoffreedom.Thus,thereare2functionswith)variablesthatcanproducethesame.Inthecaseofperiodicsequencesofperiod,itholdsFinally,westudythecardinalityofn,SP),thesetofbinarysequencesoflengthwithspan,asvaries.Themainresultsonthespandistributionfollow.Let,SPSP,SP 2+)|=|Z( ,foreven.6ConstructionofSequenceswithfour-valuedAutocorrelationfromGMWSequencesOneofthemostimportantfamiliesofpseudorandomsequencesareGordon,Mills,Welch(GMW)sequences([10]).TheGMWsequencesandtheirgener-alizationcalledcascadedGMWsequenceshavebeenextensivelystudiedintheInChapter6,wedescribetheconstructionofalargeclassofbalancedbi-narysequenceswithfour valuedautocorrelationfunction.Binarysequenceswithgoodautocorrelationpropertiesplayanimportantroleincommunicationsys-temsemployingphase reversalmodulationtechniques.Theconstructionisbasedonthemodulo2additionoftwoGMWsequenceswithrelativelyprimeperiods.Theresultingsequenceshaveperiodequaltotheproductoftheperiods.Addi-tionally,othercharacteristicsoftheclassmembers,suchasthelinearspanandtheperiodiccrosscorrelation,areinvestigated([18]).DeÞnition4([10]).Letn,kbetwointegerssuchthat,andbeanintegerintherangerelativelyprimeto.Considerthebinarysequencegivenby=tr(6)whereisaprimitiveelementof,andisanintegerintherangerelativelyprimeto.Then,isaGMWsequence.TheabovedenitionimpliesthatGMWsequencesareperiodicwithleastperiod1.SomeofthepropertiesofaGMWsequencearethefollowing[10]: Theorem5.ThespectrumoftheautocorrelationfunctionofsequencedeÞnedasdescribedabove,isgivenby0(mod0(mod0(mod(0)otherwiseOfspecialinterestisthecasewherethecomponentGMWsequenceshaverelativeprimeperiods,i.e.=1.WeintroducethesetsandwhichcontainallcyclicallydistinctGMWsequenceswithperiods1andrespectively,andthesetandforallwith0wheregcd()=1.RecallthatGMWfor2.Clearly,|·|Corollary1.ThespectrumoftheautocorrelationfunctionofsequenceisfourÐvaluedandisgivenby0(mod0(mod0(modotherwisewhere.Moreover,thevalue1occursonetime,occurstimes,occurstimesandoccurstimes.ThelinearspanofasequencedependsonitscomponentsequencesandasthefollowingLemmaindicates.Lemma1.Let.Then,(12)Finallywecomputethecrosscorrelationfunctionoftwomembersofthefamily.Theaboveresultscanbeeasilyextendedinthecaseofanyfamilyofsequenceswithidealautocorrelation.References1.A.H.ChanandR.A.Games,OnthequadraticspansofdeBruijnsequences,IEEETrans.Inform.Theory,vol.IT 36,pp.822 829,Jul.1990.2.T.W.Cusick,C.Ding,andA.Renvall,StreamCiphersandNumberTheoryNorth HollandMathematicalLibrary.ElsevierScience,1998.