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Information Integrity

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Title:
Information Integrity
Physical Description:
1 online resource (140 p.)
Language:
english
Creator:
Graves, Eric S
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Wong, Tan Foon
Committee Co-Chair:
Shea, John Mark
Committee Members:
Fang, Yuguang
Smith, Rick L

Subjects

Subjects / Keywords:
byzantine -- detection -- information -- integrity -- relay -- secrecy
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre:
Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
This thesis presents the concept of Information Integrity, or the mathematical concept of determining modifications by a Byzantine   relay. As networks continue to grow in complexity, information theoretic security has failed to keep pace. As a result many parties whom want to communicate, referred to as nodes, do not have direct contact with one another, which gives rise to the very basic need to ensure that information you received has not been modified on its journey. Thus the need for information integrity is paramount to secure communications in a modern age.     In order to fully demonstrate what is and is not possible with information integrity we presented two types of integrity that we refer to as weak and strong integrity. From there we apply these concepts to simple networks with practical relaying scenarios. Without information integrity, protocols must be restricted to relying on trusted paths and not involving possibly malevolent relays. The ability for a channel to provide a naturally arising integrity allows for more possibilities, and more importantly faster and securer communication systems.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Eric S Graves.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Wong, Tan Foon.
Local:
Co-adviser: Shea, John Mark.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045197:00001

MISSING IMAGE

Material Information

Title:
Information Integrity
Physical Description:
1 online resource (140 p.)
Language:
english
Creator:
Graves, Eric S
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Wong, Tan Foon
Committee Co-Chair:
Shea, John Mark
Committee Members:
Fang, Yuguang
Smith, Rick L

Subjects

Subjects / Keywords:
byzantine -- detection -- information -- integrity -- relay -- secrecy
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre:
Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
This thesis presents the concept of Information Integrity, or the mathematical concept of determining modifications by a Byzantine   relay. As networks continue to grow in complexity, information theoretic security has failed to keep pace. As a result many parties whom want to communicate, referred to as nodes, do not have direct contact with one another, which gives rise to the very basic need to ensure that information you received has not been modified on its journey. Thus the need for information integrity is paramount to secure communications in a modern age.     In order to fully demonstrate what is and is not possible with information integrity we presented two types of integrity that we refer to as weak and strong integrity. From there we apply these concepts to simple networks with practical relaying scenarios. Without information integrity, protocols must be restricted to relying on trusted paths and not involving possibly malevolent relays. The ability for a channel to provide a naturally arising integrity allows for more possibilities, and more importantly faster and securer communication systems.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Eric S Graves.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Wong, Tan Foon.
Local:
Co-adviser: Shea, John Mark.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045197:00001


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INFORMATIONINTEGRITYByERICGRAVESADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013EricGraves 2

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Toanyonewhohadadreamandwastoostubborntoletgo. 3

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ACKNOWLEDGMENTS IwouldliketothankrstandforemostmyLordandSaviorJesusChrist,withoutwhomIwouldstillbestuckatsquareone.IwouldalsoliketothankMomandDadfordoingalltheMomandDadstufftheydidtoraiseme.ToDr.TanWongwhochallengedme,andyethelpedineverystepoftheway,despitebothnotationalnightmaresandmyconstantdesiretomaketheproblemmorecomplicated.Dr.Sheaforgivingmeachanceintherstplace.Kareem,forbeingthebestfriendIcouldaskforandbeingunderstandinginawaynooneelsecouldbe.JulieandDanielforbeinginspirationincarnate,andforthegoodfood.David,Fatima,DamiemandSarahwhoprovidedthestressreliefonlyagroupoffriendscouldwhentheburdensweretoomuch.CatandPapageno,twokittieswhichwouldmakemetakebreaksfromworkastheywouldwalkacrossmykeyboardandchewthepaperIwaswritingon.Iwouldliketothankaswell,Dr.RickSmithandDr.MichaelFangforagreeingtoserveonmycommitteeandfortheirsuggestions,commentsandconcernsnomatterhowharshIfelttheywereatthetime.FurthermoreDr.LeenhapatNavaravongforbeingaWINGLabpartneraswellasafriend.Dr.LiwhogavemeaninternshipdespitethefactIwaswoefullysuitedforit.AndnallyIwouldliketothankHeather,wholovedmedespitemythesis. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ............................... 4 LISTOFTABLES ................................... 7 LISTOFFIGURES .................................. 8 ABSTRACT ...................................... 9 CHAPTER 1INTRODUCTION ................................ 10 1.1History ................................... 12 1.2MotivatingExample ........................... 13 1.3InformationTheoryOverview ...................... 15 1.4OrganizationofDissertation ....................... 19 2MANIPULABILITY ............................... 20 2.1NotationsandAssumptions ....................... 20 2.1.1Notation .............................. 20 2.1.2ManipulabilityVariables ..................... 22 2.1.3VectorTypeNotation ....................... 23 2.2TheoremandResults .......................... 24 2.3TheoremandLemmaProofs ...................... 26 3CHANNELDEGREDATIONATTACK .................... 49 3.1SystemModel .............................. 51 3.1.1Notation .............................. 51 3.1.2Channelmodel .......................... 52 3.1.3Maliciousnessofrelay ...................... 54 3.2TheoremsandResults .......................... 55 3.3Examples ................................. 57 3.3.1Motivatingexample ........................ 57 3.3.2I.i.d.attacks ............................ 58 3.3.3Non-ergodicattacks ....................... 60 3.3.4Higherorderexample ...................... 61 3.3.5Counter-example ......................... 63 3.4ProofsofTheoremsandLemmas .................... 64 3.4.1Converse ............................. 68 4STRONGINFORMATIONINTEGRITYINTHETWOWAYTWOHOPRELAY ..................................... 70 4.1Notation .................................. 71 5

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4.2ChannelModel .............................. 72 4.3Achievability ................................ 77 4.3.1Theorem .............................. 77 4.3.2CodeConstruction ........................ 77 4.3.3Erroranalysisunderminimalmodication ........... 80 4.3.4Erroranalysisundersignicantmodication .......... 83 4.3.5Decodinganddetectionerroranalysis ............. 87 4.4TechnicalLemmas ............................ 88 5INTEGRITYINTHETWOWAYTWOHOPRELAY ............. 103 5.1NotationConventions .......................... 104 5.2Achievability ................................ 105 5.2.1Theorem .............................. 105 5.2.2CodeConstruction ........................ 106 5.2.3Integrity .............................. 106 5.2.4Secrecy .............................. 108 5.3LemmasandTheorems ......................... 112 6CONCLUSIONSANDFUTUREWORK ................... 131 6.1Conclusions ................................ 131 6.2FutureWork ................................ 132 REFERENCES .................................... 136 BIOGRAPHICALSKETCH ............................. 140 6

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LISTOFTABLES Table page 1-1PossibleoutcomesofthesysteminFig.1-1. ................ 14 7

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LISTOFFIGURES Figure page 1-1Motivatingexampleofabinary-inputadditionchannel. ........... 13 3-1Amplify-and-forwardrelayingmodel. ..................... 53 3-2Plotofempiricalcdfsofk^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1obtainedinthemotivatingexamplewiththefourrelaymanipulationmapscorrespondingto1=I,2,3,and4respectively. .............................. 58 3-3Plotofempiricalcdfsofk^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1forvariousi.i.d.attacksconsideredinSections3.3.4and3.3.5. ........................... 61 4-1Two-way,half-duplex,amplify-and-forwardrelaymodel. .......... 72 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyINFORMATIONINTEGRITYByEricGravesAugust2013Chair:TanWongMajor:ElectricalandComputerEngineeringThisdissertationintroducestheconceptofInformationIntegrity,whichisthedetectionandpossiblecorrectionofinformationmanipulationbyanyintermediarynodeinacommunicationsystem.Asnetworkscontinuetogrowincomplexity,informationtheoreticsecurityhasfailedtokeeppace.Asaresultmanypartieswhomwanttocommunicate,referredtoasnodes,donothavedirectcontactwithoneanother,whichgivesrisetotheneedtoensurethatanyandallinformationreceivedhasnotbeenmodiedbyacorruptthirdparty.Thustheneedforinformationintegrityisparamounttosecurecommunicationsinamodernage.Wedeveloptheconceptofinformationintegrityinaninformationtheoreticmannersothatthemethodsdevelopedwithincanbeappliedtodifferentsystems.Fromthesemethodsweestablishtwouniquetypesofintegrity,onebaseduponchannelobservationsandonebaseduponcoding.Thesemethodsshowthatintegritycanbeguaranteedwithlittletonomodicationofthemethodsestablishedforreliablecommunications.Fromthisweareabletodeterminetheconceptoftheintegrityregionofabroadcastinrespecttothearelay. 9

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CHAPTER1INTRODUCTIONInorderfortwopartiestocommunicatethroughanintermediarynode,itisrequiredthattheintermediary(heretoreferredtoasarelay)mustfaithfullyforwardtheinformation.Inacommunicationnetwork,suchcooperativebehaviorisnotguaranteedasrelaysmayhavereasonsforforwardingfalseinformationinordertofooltheintendedparticipants.Suchattacks,oftenreferredtoasByzantineattacks,havemajorramicationsonprotocolsthatoperatewithinthenetwork.Clearlyanyprotocolthatmustaccountforthepossibilityofthistypeofintrusionisatthedisadvantageofaprotocolthatmayoperatewithoutregardstothisrestrictionwhencomparedunderanymetric.Fromthiswededucetheessentialquestionregardingintegrity:Whatisthebestprotocolgiventheneedtoaccountforthistypeofattack?Whenconsideringatheoreticalcommunicationsystem(i.e.,onewherewemaydenetheentirecumulationofactionsbyasetofstrictmathematicalstatements),thereareonlyafewmajorconcernsgenerallyconsidered.Themostcommonoftheseistherateofcommunication,oralternativelyhowmuchinformationinhowshortofatimecanbepassedfromnodeAtonodeB.ThusitshouldbeinherentlyobviousthatinnetworksconsistingofmanynodesbesideAandBcooperationbythesemultiplenodescangenerallyimprovethecommunicationratebetweennodesAandB.Thisiswheretheneedforintegritybecomesapparent,insuchanetworkwherecooperationisdemanded,cooperationmustalsobeguaranteed.Tobestunderstandtheexactneedandpurposeofthemethodsandmathematicsexpressedwithinthisdissertation,onemayconsiderthefollowingcompletelyimpracticalbutunderstandablescenario.Inourscenariothereexiststwogoodfriends,AliceandBob,andforanygivenreasonBobneedstodepositmoneyintoAlice'sbankaccount.However,Bobdoesnotknowthebankaccountnumberneededtodepositthemoney.WhichwouldbeofmoredireconsequencesaveforthatAliceisnearenoughsuchthat 10

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shemayemploythehelpofapasserbytorelaysaidinformationtoBobwhilenotbeingsocloseashavingtheabilitytoestablishdirectcommunication.Thisdemonstratesoneoftheprinciplespreviouslymentioned,thatcooperationinanetworkcanhelptoincreasetherateofcommunication,forwithouthelpofsomepasserbyitisclearthattherewillbenocommunicationbetweenAliceandBobandthusarateofzero.Continuingwithourclearlyonlyhypotheticallypossiblesituation,supposethatAliceandBob'sgoodfriendRomeohappenstopassbyatthisclearlymostfortuitousmoment.Romeobeingafriend,andonewhohappenstoplacefriendshipabovegreedonceagainforanyarbitraryreason,listensintentlytoAliceandsubsequentlyrelaystheinformationtoBobwhothendepositsthemoneyinAlice'sbankaccount.Thesystemworkedasitwasintendedandallparties,withthepossibleexceptionofBob,aresatisedwiththetransactionthattookplace.Nowconsideringthealternativehypotheticalscenario,whereinsteadofRomeotheonlypasserbyisne'er-do-wellRichard.Aspreviously,RichardlistenstoAlice,butrecognizingtheopportunityfornancialgain,decidestorelayhisbankaccountnumbertoBob.BobthendepositsthemoneyintoRichard'sbankaccountmuchtothechagrinofAliceandpresumablyBob.Whileclearlyanabstraction,thisscenariodictatesreasonablywelltheunderlyingproblemthatourworksolves.Fortheremainderofthischapter,abriefoverviewregardingpreviousattemptstosolvethisorrelatedproblemispresentedinsection 1.1 .Subsequently,wewillpresentaquickmotivatingexamplein 1.2 tosuggesthowaproblemsolutioncanrelyuponchannelparametersandprobabilitiesaswellasaquickintroductiontoinformationtheoryinsection 1.3 .Thiswillinturnbecomethemajormathematicaltoolusedtohandlethisproblem.Finally,section 1.4 willpresenthowthispaperwasintendedtobeorganizedandwhattheoverarchinggoalofeachchapterisintendedtobe. 11

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1.1HistoryThegenesisofthisproblemhasrootsincryptography[ 1 ]wherecodeslike[ 2 3 ]havebeenstudiedasameansofdeterminingByzantineattacks.Whilemorerecentworkhasfocusedontheintegrityofanetworkusinglinearnetworkcoding[ 4 5 ],aswellasthestudyofusingcodingtodeterminemanipulationinbasicchannelswhicharesupportedbyarelay[ 6 7 ].Unfortunatelynoneofthesedevelopanunderlyingframeworkforintegrity.DuetoShannon[ 8 ],weknowthattheonlyprobabilisticallysecurecryptographicencryptionistheonetimepad.Whileonlyconsideringnetworkswhichemploylinearnetworkcodingisproblematicduetotheassumptionthatthereexistsmultiplepathstothedestination.Originallytheproblemofguaranteeinginformationintegritywastreatedbytheuseofanon-observablekey.Gilbert[ 2 ]originallymotivatedbyadishonestgamblingpitboss,studiedtheproblemwithuseofplanarcodesandrandomcodes.WhileCrameretal.[ 3 ],showthatanyrandomlineartransformationintoaspaceofgreaterdimensioniswithhighprobabilityinvertible.Thusifoneweretomodifythetransformationorthesymbols,itiswithhighprobabilitythatthemodiedsequencewouldnotbeonecorrespondingtoanyvalidinputsequence.Theresultingcodesareknownasalgebraicmanipulationdetection(AMD)codes.Inordertoextendtheseideastothephysicallayer,additionalredundancyisrequiredtocopewiththepossibilityofchannelerrors.MaoandWu[ 7 ]posedtheproblemoftryingtodeterminewhichrelayinamultiplerelaytwo-hopnetworkwasmanipulatingthedata.Across-layermethodissetforthinwhichacryptographickeyisinsertedintothesignal,bywhichtheintendeddestinationdeterminesfromthephysicallayererrorrateifmanipulationhasoccurred.Inslightcontrast,themorerecentworkofHeandYener[ 6 ],mainlyfocusedontheprobleminthetwo-waytwo-hopchannelstudiedinthispaper,doesnotrequireuseofasharedsecretkey.Instead,anLDPCcodeisemployedtosupportsecrettransmissionwhichinturnallowsthe 12

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Figure1-1.Motivatingexampleofabinary-inputadditionchannel. useofanAMDcodetodetectattacksbytherelaykey.Themajordrawbackistheadherencetoaparticulartypeofcode,andnoclearwaytogeneralizetoanyarbitrarychannel.Correspondingly,theworkdoesnotprovideadeeperunderstandingofwhattheimplicationsofmodicationare,makingextensionstobeyondtheadditionchannelHeconsidereddifcult.Incontrastweproposeadifferentstrategybyseparatingtheconceptsofsecrecyandintegritytoensurethatcommunicationcanbeveriedwithoutusinganykeyoranysecrettransmission.Usingrandomcodingtechniques,weobtainaninnerboundonthecapacityregionwithguaranteedinformationintegrityinthegeneralscenariooftwonodeswhichmustcommunicatethroughaByzantinerelaynode. 1.2MotivatingExampleTomotivatethemaliciousnessdetectabilitymodelandresultsinlatersections,letusrstconsiderthesimplebinary-inputadditionchannelshowninFig. 1-1 ,inwhichtwosourcenodes(Alice&Bob)communicatetooneanotherthrougharelaynode(Romeo)indiscretetimeinstants.ThesourcealphabetsofAliceandBobarebothbinaryf0,1g.ThechannelfromAliceandBobtoRomeoisdenedbythesummationofthesymbolstransmittedbyAliceandBob.Romeoissupposedtobroadcasthisobservedsymbol,withoutmodication,backtoAliceandBob.BothAliceandBobobservethesymboltransmittedbyRomeoperfectly.ThustheinputandoutputalphabetsofRomeoandthe 13

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Table1-1.PossibleoutcomesofthesysteminFig. 1-1 Alice Bob Romeo Romeo DetectionOutcome In Out 0 0 0 0 Notmalicious 1 Notdetected 2 Alice&Bobbothdetect 0 1 1 0 Bobdetects 1 Notmalicious 2 Alicedetects 1 0 1 0 Alicedetects 1 Notmalicious 2 Bobdetects 1 1 2 0 Alice&Bobbothdetect 1 Notdetected 2 Notmalicious observationalphabetsofbothAliceandBobareallternaryf0,1,2g.Alice,forinstance,canobtainBob'ssourcesymbolbysubtractingherownsourcesymbolfromthesymboltransmittedbyRomeo.NowconsiderthepossibilitythatRomeomaynotfaithfullyforwardthesymbolthatheobservestoAliceandBobinanattempttoimpedethecommunicationbetweenthem.ThemainquestionthatweareinterestediniswhetherAliceandBobareabletodiscern,fromtheirrespectiveobservedsymbols,ifRomeoisactingmaliciouslybyforwardingsymbolsthataredifferentthanthosehehasreceived.Toproceedansweringthisquestion,letusrstconsiderasingleroundoftransmission,i.e.,AliceandBobtransmittheirsourcesymbolstoRomeoandthenRomeobroadcastsasymbol(maybedifferentthanwhathehasreceived)backtoAliceandBob.SupposethatAlicesendsa0andreceivesa2backfromRomeo.ThenitwillbecleartoherthatRomeomusthavemodiedwhathehasreceived.Ontheotherhand,ifAlicereceivesa1back,thenshewillnotbeabletotellwhetherRomeohasactedmaliciouslyornot.OnecancontinuethislineofsimpledeductiontoobtainallthepossibleoutcomesinTable 1-1 .ItisclearfromthetablethatneitherAlicenorBobwillbeabletodetermineifRomeoismaliciousingeneralfromasingleroundoftransmission. 14

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HoweverthesituationchangesifAliceandBobknowthesourcedistributionsofoneanotherandareallowedtodecideonthemaliciousnessofRomeoovermultipleroundsoftransmission.Tofurtherelaborate,supposethatthesourcesymbolsofBobandAlicearei.i.d.Bernoullirandomvariableswithparameter1 2.ThentheprobabilitiesoftheeventsthatRomeo'sinputsymboltakesonthevalues0,1,and2are1 4,1 2,and1 4,respectively.Inparticular,outofmanyroundsoftransmissiononewouldexpecthalfofthesymbolstransmittedbyRomeobe1's.FromTable 1-1 ,weseethatforRomeotobemaliciousandremainedundetectedbyneitherAlicenorBob,hecanonlychangea0toa1anda2toa1inanysingleroundoftransmission.Butifhedoessooften,thenumberof1'sthathesendsoutwillbemorethanhalfofthenumberoftransmissionrounds,asexpectedfromnormaloperation.Ontheotherhand,RomeocanfooloneofAliceandBobbychanginga1toeithera0or2.ButheisnotabletodetermineineachchangewhetherAliceorBobisfooled.Henceovermanysuchchangestheprobabilityofnotbeingdetectedbecomedecreasinglysmall.Insummary,AliceandBobmayindividuallydeduceanymaliciousnessofRomeobyobservingthedistributionofRomeo'soutputsymbolconditionedontheirrespectiveowninputsymbols.ThiscapabilityisinducedbytherestrictionsonwhatRomeocandothatareimposedbythecharacteristicoftheadditionchanneldepictedinFig. 1-1 .ItisimportanttonoticethatAliceandBobdonotneedtopossessanysharedsecretthatisnotprivytoRomeo. 1.3InformationTheoryOverviewFirstaquickdiscussiononwhatpurposethetoolsofinformationtheoryserve.Thegoalofinformationtheoryistodeterminetheamountofinformationcontainedinanysequence.Butwhatdenesinformation?Whileeverydayusesofthetermmayhavedenitionthatisseeminglyuid,theconceptswhichperpetuatethemathematicalfoundationsarequiterigid.Informationisinshort,thestudyofrandomnessandhowrandomvariablesrelatetooneanother.Appropriatelythemeasureofrandomnessisdenedbyentropy. 15

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Denition1.1. Thediscreteentropy,heretoreferredtoasentropy,ofarandomvariableXwithalphabetXisH(X)=)]TJ /F9 11.955 Tf 11.29 11.36 Td[(Xx2Xp(x)log2p(x).TheconditionalentropyofXgivenUisH(XjU)=)]TJ /F9 11.955 Tf 20.3 11.36 Td[(Xx2X,u2Up(x,u)log2p(xju).Whilemaybenotdirectlyintuitivelyobviousentropycanbeseenasameasureoftheaverageminimumlengthsequencerequiredinordertodescribesomeevent.Consider,asmanyhave,twoeventsrelatedtowinningthelottery.Inanygivenlotterytheidearemainsthesame,onemustguessasequenceofnumbersinordertowinaprize.Assumingeverysequenceisequallylikelyandweobtaintheentropyofthesequencesisequaltothelogofthenumberoftotalsequences.Fortherstevent,considerthesituationinwhichthereare6numbersdrawn,withreplacement,fromapotentialeldof64,thentheentropyoftheeventofthesequence(substitutingalogarithmwithbase64insteadof2forthepurposesofthisexplanation)ofthesequenceislog64646=6.Itrequires6observations,or6numberstodeterminethewholesequenceoflotterynumbers.Alternativelyconsidertheeventofactuallywinningthelottery,ofwhichtheprobabilityisnearly0,andbecausebyaxiomaticdenitionwemaydene0log0=0,theentropyisalso0.Meaningitwouldnotnearlynotrequireanysortofobservationtodeterminethatsomeonehadnotwonthelottery.Thenextimportanttermisthatofmutualinformation,orsimplyinformation. Denition1.2. ThemutualinformationbetweenrandomvariablesXandUisI(X;U)=H(X))]TJ /F5 11.955 Tf 11.96 0 Td[(H(XjU)=H(U))]TJ /F5 11.955 Tf 11.96 0 Td[(H(UjX)=Xx2X,u2Up(x,u)log2p(x,u) p(x)p(u)TheconditionalmutualinformationofXandUgivenVisI(X;UjV)=H(XjV))]TJ /F5 11.955 Tf 11.95 0 Td[(H(XjU,V)=Xx2X,u2U,v2Vp(x,u,v)log2p(x,ujv) p(xjv)p(ujv). 16

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Information,intermsofentropy,canbeseenasacomparisonoftherandomnessXandtherandomnessofXwhenUisknown.Asanexample,onemayconsiderthestateofcleanlinessofaneighbor'shouse.IfthisneighborhaschildrenofacertainagethecleanlinessofthishouseXmaydependsignicantlyonanobservationofiftheparentsarehomeU.Conditionalinformationthatisconstantregardlessofobservation,suchastheamountoftroublethechildrenwillbein(V)mayalsoplayarolebutonlyinsomuchasmakingitacomparisonbetweenknowingthestateofcleanlinessofthehousegiventhetrouble(XjV)versusthestateofthecleanlinessofthehousegiventhetroubleandiftheparentsarehome(XjU,V).Obviouslyasdescribedabove,mutualinformationandentropyhavenophysicalmeaningotherthanassomearbitrarycomparisonofanebulousvalue1.Thenexttwodenitions,thatoftypesandtypesetswillactuallygiveamoreconcretephysicaljusticationforentropyandmutualinformation. Denition1.3. ThetypeofasequencexnisthedistributionPXdenedbyPX(x)=1 nnXi=11(xn(i)=x).Thetypeisliterallythedistributiondenedbythestatisticalmeanofxn.Forinstancethesequence1,0,1,1,1,0hasastatisticalmeanforXi=1of4 6=2 3andastatisticalmeanforXi=0of1 3. Denition1.4. TheTypeSetofdistributionPX,referredtoasTnX,isthesetofallxnwhichhavetypePX.TheTypicalSetofdistributionPXwithsomeisthedenedasTn[X]=[P^X:jPX)]TJ /F4 7.97 Tf 6.59 0 Td[(P^Xj
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Typeandtypesetshaveallofthefollowingproperties2n(H(X))]TJ /F17 7.97 Tf 6.59 0 Td[()
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Denition1.5. Thedeltaconventionisadeltanthatsatisesthesepropertiesn!0,p nn!1.Whichbasicallyallowsforustolookatvalueswhichtechnicallydiverge,buttheirlogarithmsconverge. 1.4OrganizationofDissertationTherewillbevechapterswithcontentandresults,andanalconclusiontalkingabouttheimplicationsandtheimpactoftheworkandmethodsderivedwithin.Inchapter 2 webeginwithdevelopingtheconceptofmanipulability,whichisdirectlytiedthedeterminationofmodicationbyobservationofinputandoutputsymbolspairswithoutregardstoanycodingstructure.Inthischapterwewilldenewhenamatrixismanipulable,andhowtochecksousinglinearprogramming.Wewillalsodevelopdistanceboundingtechniquesgivenanamountofmanipulationthatwillplayamoredenedroleinsubsequentchapters.Following,inchapter 3 wewillpresentthechanneldegradationattack,andtherolemanipulabilityplaysinthedetectionthereof.Afterconcludingourdiscussionofcode-lesssystemsandtechniques,wewillmoveontolookatinformationtheoreticmethodsforintegrity,whichwewillrefertoassimplyinformationintegrity.Therstofthesewillbeamethodwherethechannelprimaryprovidestheintegrityinchapter 4 ,whichwillberefereedtoasstrongintegrity.Whilechapter 5 willpresentamethodwhichtakesadvantageofbothcodingandthechanneltoprovideintegrity.Finallychapter 6 willdescribethefuturedirectionofthework. 19

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CHAPTER2MANIPULABILITYThepurposeofthischapteristodevelopthemathematicaltoolsnecessaryfordetectionofmanipulation.Inadditionweestablishguidingprinciplesandassumptionsunderwhichdetectionispossible.Asevidentbythedenitionofinformation,thereisaninherentneedforrandomnessincommunications.Onemayunderstandthisbyrecognizingthatanythingalreadyknownneednotbecommunicated.Fromthisunderstandingofthenatureofcommunicationonemayeasilydeterminethatinanysystememployingarelay,therelayalwayshasapossibilityofmanipulatingtheinformationwithoutbeingdetected.Thuswesearchforprobabiliticsolutiontothisproblem.Usingthisguidingprinciplewemodelthesymbolsreceivedandtransmitted,whichisthetotalcumulationofassetsavailabletoanynode,bytheirassociatedprobabilitiesandattempttodeterminewhateffectmodicationhasonthesedistributions.Thiswillbedonebyviewingtheprobabilisticrelationshipsasmatricesandthendevelopingrelationshipsbetweenthesematriceswhicharecrucialfordetection.Todothiswerstmustestablishalitanyofnotationandunderlyingassumptionsinsection 2.1 beforestatingthethoeremsandresultsinsection 2.2 .Tohelpfacilitatetheimportanceofthetheoremswehaveremovedtheproofsnecessarytosection 2.3 ,weadvisethatthissectionbeskippedbyallofthoseexceptforwhomtheresultsareofdirectinteresttoasthemathisquitesubstantialandtheveilofintuitivereasoningishardtopenetrate. 2.1NotationsandAssumptions 2.1.1NotationLetabea1mrowvectorandAbeamnmatrix.Fori=1,2,...,mandj=1,2,...,n,aiwilldenotethei-thelementofa,andAi,jwilldenotethe(i,j)-thelementofA.Moreover,wewritetheithrowofAasAi.Forreferencetotherowthatistheresultofsomematrixoperationwewilluse[BA]i,j.LetthetransposeofAbedenoted 20

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byAT.ThenthejthcolumnofAis(ATj)T.Itisourconventionthatallcolumnvectorsarewrittenasthetransposesofrowvectors.Forinstance,aisarowvectorandaTisacolumnvector.TheL1-normofaiskak1=Pmi=1jaij,whiletheEuclideannormofaiskak2=p aaT.Theoperationvec(A)vectorizesthematrixAbystackingitscolumnstoformamn1columnvector.WedenekAk1,kvec(A)k1andkAk2,kvec(A)k2.NotethatkAk2istheFrobeniusnormofA.TheidentityandzeromatricesofanydimensionaredenotedbythegenericsymbolsIand0,respectively.LetXbeadiscreterandomvariable.WeusejXjtodenotethesizeofthealphabetofX.Ourconventionistousethecorrespondinglowercaseletter,suchasx,todenotetheelementsinthealphabetofarandomvariable.TodistinguishbetweendifferentelementsofX,thenotationofximaybeused;inwhichcaseitshouldbenotedthatthechosenorderingshouldremainconsistentthroughouttheproblemandthatX=fx1,x2,...,xjXjg.Wedenotetheprobabilitymassfunction(pmf)Pr(X=x)byp(x).WedenotetheconditionalpmfofarandomvariableYgivenXbyPr(Y=yjX=x)byp(yjx)forsimplicity.LetxndenoteasequenceofnsymbolsdrawnfromthealphabetofX.Thecountingfunction(xjxn)denotesthenumberofoccurrencesofxinthesequencexn.Let1(xn(i)=x)betheindicatorfunctionoftheconditionthattheithsymbolinthesequencexnisx.Thenweclearlyhave (xjxn)=nXi=11(xn(i)=x).(2)Thecountingfunctionalsotriviallyextendstogivethenumberofoccurrencesofatupleofsymbolsdrawnfromthecorrespondingtupleofalphabetsofrandomvariables.Forexample,ifxnandynarelength-nsequencesofsymbolsdrawnfromthealphabetsofXandY,respectively,then(x,yjxn,yn)=nXi=11(xn(i)=x)1(yn(i)=y). 21

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Wheneverthereisnoconfusion,wewriteforinstancen(x)and1ni(x)inplaceof(x;xn)and1(xn(i)=x),respectively,tosimplifynotation.Finally,letusdenethesymbolindexingmapsn(xi),i=1,2,...,n,byassigningtheindexvaluejton(xi)whentheithsymbolinthesequencexnisxj,namely,thejthelementinthealphabetofX. 2.1.2ManipulabilityVariablesForthegeneralformulationofmanipulabilityweareconcernedwithfourmainareasofinterest,theobservationsindependentoftherelay,theobservationsfromtherelay,whattherelayreceivesandwhattherelaysends.Wedenotetheserandomvariables,respectively,asX,Y,U,V.ItshouldbenotedthatthevariableVdoesnotnecessarilyhaveaxed,stationary,orergodicprobabilitydensityfunction.InsteadweconsiderthejUjjUjmatrixnwhose(i,j)thelementisdenedby ni,j,n(vi,uj) n(uj).(2)Itisobviousthatnisastochasticmatrixdescribingavalidconditionalpmfofactitiouschannel,whichwewillrefertoastheattackchannel,andwilltaketheplaceofp(vju).PrimarilyweareconcernedwiththepossibilityoftherelayappendingachanneltoU,andthuswanttoobservetheequationPYjUPUjX=PYjVnPUjX.Forbetterbookkeeping,wewillwritethetwoconditionalpmfsintermsofthejUjjX1jmatrixAandjY1jjUjmatrixBwhoseelementsarerespectivelydenedbyAi,j,p(uijx1,j)Bi,j,p(~y1,ijvj).Tocompletethebookkeepingprocess,wedenethejY1jjX1jmatrix)]TJ /F4 7.97 Tf 6.78 4.33 Td[(nas )]TJ /F4 7.97 Tf 6.77 4.94 Td[(n,BnA,(2) 22

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whichcanbeinterpretedastheconditionalpmfofthenode1'sobservationiftherelayweretoactinani.i.d.mannerdescribedbytheattackchanneln.WemayalsoassumethatAcontainsnoall-zerorowsandthatp(x1,j)>0forallx1,j2X1.Iftheseassumptionsdonothold,wecanreinforcethembyremovingsymbolsfromthealphabetsofX1andUanddeletingthecorrespondingrowsandcolumnsfromA,withoutaffectingthesystemmodel.Inaddition,notethatrelabelingofelementsinthealphabetsofX1andY1amountstopermutingthecolumnsofAandrowsofB,respectively.RelabellingofelementsinthealphabetofU,andhencethecorrespondingelementsinthealphabetofV,requiressimultaneouspermutationoftherowsofAandcolumnsofB.Itisobviousthatalltheserelabellings,andhencethecorrespondingpermutationsofrowsandcolumnsofAandB,donotchangetheunderlyingsystemmodel.Thereforewewillimplicitlyassumeanysuchconvenientpermutationsintherestofthepaper.Apessimisticviewisrequired,andforthatreasonweconsiderthemostmalicious,inthesenseoftotalvalueschange,suchthatBA=PYjX.Tothatendwedene, ^n=argmax^:B^A2[)]TJ /F16 5.978 Tf 4.82 2.27 Td[(n]k^)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1(2)whereisanarbitrarilysmallpositiveconstant.Itisimportanttonotethatbecauseofhow)]TJ /F4 7.97 Tf 6.78 4.34 Td[(nisdened,thereexistsatleastonesuchthatB^nA2[)]TJ /F4 7.97 Tf 6.77 4.34 Td[(n],whichcanbeseenbytaking^n=n.Furthermore,becauseB^nA2[)]TJ /F4 7.97 Tf 6.77 4.34 Td[(n],weknowthatB^nA)]TJ /F5 11.955 Tf 11.95 0 Td[(BnA1
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Denition2.3. (Polarizedvectors)Forb0and0b,avector!issaidtobe(b,)-polarizedatjif!i28>><>>:[b,1)fori=j(,]fori6=j.Further!issaidtobe(b,)-doublepolarizedat(j,k)if!i28>>>>>><>>>>>>:[b,1)fori=j(,)]TJ /F5 11.955 Tf 9.3 0 Td[(b]fori=k[)]TJ /F13 11.955 Tf 9.3 0 Td[(,]fori6=j,k. 2.2TheoremandResults Denition2.4. (Manipulableobservationchannel)Theobservationchannel(A,B)ismanipulableifthereexistsajUjjUjnon-zeromatrix,whosejthcolumn,foreachj=1,2,...,jUj,isbalancedand(0,0)-polarizedatj,withthepropertythatallcolumnsofAareintherightnullspaceofB.Otherwise,(A,B)issaidtobenon-manipulable.Thisdentionisgivensignicancebythefollowingtheorem: Theorem2.1. LetAandBbestochasticmatriciesofsizesjUjjXjandjYjjUjrepsectively,additionallysupposethat(A,B)isnonmanipulable.ThenforalljUjjUjmatriciessuchthatkBA)]TJ /F5 11.955 Tf 11.96 0 Td[(BAk1thereexistsaconstant,dependentonlyonthechannel,suchthatk)]TJ /F5 11.955 Tf 11.96 0 Td[(Ikc.Similarlywecandeneaconversestatement,whichdealswiththescenarioformanipulablesituation. Theorem2.2. LetAandBbestochasticmatriciesofsizesjUjjXjandjYjjUjrepsectively,additionallysupposethat(A,B)ismanipulable.ThenthereexistsajUjjUjmatrix6=IsuchthatBA=BA. Checkingfornon-manipulability. 24

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Themanipulabilityoftheobservationchannel(A,B)canbecheckedbysolvingalinearprogramasshownbelow: Algorithm1. (Non-manipulable?) 1. Let,,andbeajX1jjY1jmatrix-valuedvariableandtwo1jUjvector-valuedvariables,respectively. 2. Solvethefollowinglinearprogram:min,,jUjXk=1k)]TJ /F13 11.955 Tf 11.96 0 Td[(k)]TJ /F3 11.955 Tf 11.95 0 Td[([AB]k,ksubjectto1)]TJ /F13 11.955 Tf 11.95 0 Td[(k0k=1,2,...,jUj,k+[AB]k,k)]TJ /F13 11.955 Tf 11.95 0 Td[(k0k=1,2,...,jUj,k+[AB]k,l0k6=l=1,2,...,jUj. 3. Iftheoptimalvaluein 2 )is0,thenconcludethat(A,B)isnon-manipulable.Otherwise(i.e.,theoptimalvalueispositive),concludethat(A,B)ismanipulable.ForcaseswheretherightnullspaceofBistrivial,checkingmanipulabilityof(A,B)ismadesimplebyTheorem 2.3 below.Fornotationclarityinexpressingthetheorem,letusdenethefollowingconstantsthatdependonlyonA:Amin,miniXjAi,jamin,Amin jUj(jX1j+Amin).NotethatbothAminandaminarepositivesinceAdoesnotcontainanyall-zerorow. Theorem2.3. SupposethattherightnullspaceofBistrivial.Then(A,B)isnon-manipulableifandonlyiftheleftnullspaceofAdoesnotcontainanynormalized,(amin,0)-doublepolarizedvectors.Weremarkthattheconditionofnon-existenceofnormalized,(amin,0)-doublepolarizedvectorsintheleftnullspaceofAisrelativelyeasytocheckbyforinstanceemployingthefollowingalgorithm: 25

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Algorithm2. (Doublepolarizedvectorinleftnullspace?)Letn=jUj)]TJ /F7 11.955 Tf 17.66 0 Td[(rank(A).ThefollowingstepscanbeemployedtocheckwhethertheleftnullspaceofAcontainsanynormalized,(amin,0)-doublepolarizedvectors: 1. Ifn=0,thentheleftnullspaceofAmustnotcontainanynormalized,(amin,0)-doublepolarizedvector. 2. Ifn=jUj)]TJ /F3 11.955 Tf 18.84 0 Td[(1,thentheleftnullspaceofAmustcontainanormalized,(amin,0)-doublepolarizedvector. 3. If1njUj)]TJ /F3 11.955 Tf 17.93 0 Td[(2: (a) FindanjUjmatrixwhoserowsformabasisfortheleftnullspaceofA. (b) Performelementaryrowoperations,permutingcolumnsifnecessary,tomakeintotherow-reducedechelonform=(I~),where~isan(jUj)]TJ /F5 11.955 Tf 17.93 0 Td[(n)block. (c) Foreachi=1,2,...,n,ifallelementsof~i,exceptforasinglenegativeelement,arezero,thengoto 3f ). (d) Foreachi,j2f1,2,...,ngandi6=j,if~i=c~jforsomec>0,thengoto 3f ). (e) ConcludethattheleftnullspaceofAdoesnotcontainanynormalized,(amin,0)-doublepolarizedvector,andterminate. (f) ConcludethattheleftnullspaceofAcontainsanormalized,(amin,0)-doublepolarizedvector.Practicallyspeaking,thetrivialityoftherightnullspaceofBguaranteesthatthepmfofVcanbeunambiguouslyobtainedbynode1fromobservingY1.Thisrequirementisreasonableifnode1isexpectedtobeabletoobservethebehavioroftherelay,andisoftensatisedinpracticalscenarios.TherequirementoftheleftnullspaceofAnotcontaininganynormalizeddouble-polarizedvectorisnotover-restrictive,andcanbesatisedinmanycasesbyadjustingthesourcedistributionofnode2. 2.3TheoremandLemmaProofsInordertoprovethevariousresultsinSection 2.2 ,wewillneedtoextendthenotionofpolarizationofvectorsgiveninDenition 2.3 tomatrices: 26

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Denition2.5. (Polarizedmatrices)Let1njUj.Forb0and0b,wesaythatanjUjmatrixis(b,)-polarizedifi,j28>><>>:[b,1)forj=i(,]forj6=i.If,inaddition,i,j=0foralli,j=1,2,...,nandi6=j,wesaythatis(b,)-diagonalpolarized.Moreover,bysayinganormalizedisintheleftnullspaceofA,wemeanallrowsofarenormalizedvectorsintheleftnullspaceofA.Inadditiontothenotionofpolarizedvectorsandmatrices,wewillalsoemploythefollowinggeneralizationoflineardependence: Denition2.6. (s-dependent)Avector!iss-dependent(s0)uponasetofvectors1,2,...,nofthesamedimensionifthereexistsasetofcoefcientsc1,c2,...,cnsuchthat!)]TJ /F4 7.97 Tf 18.3 14.95 Td[(nXi=1cii1s.Withthesedenitionsinplace,wewillrstestablishafewimportantandinterestingpropertiesofpolarizedvectorsandmatricesintheleftandrightnullspacesofAandB,respectively.Inaddition,wewillshowthattheconditionoftheobservationchannel(A,B)beingnon-manipulableissufcientinguaranteeingthevalidityoftheseproperties.Thenwewillapplysomeofthesepropertiestoboundthedistancebetweenanestimateoftheattackchannelandthetrueattackchannel.TheaforementionedpropertiesofpolarizedvectorsintheleftnullspaceofAwillalsobeusedtoproveAlgorithm 2 andTheorem 2.3 Propertiesconcerningpolarizedvectorsandmatricesinnullspaces.LetusrststudytheleftnullspaceofA.ThefollowingsimplelemmaaboutnormalizedvectorsintheleftnullspaceofAiscriticaltomanyotherresultsinthissection: 27

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Lemma1. Supposethatisanon-zeronormalized1jUjvectorintheleftnullspaceofA.Thenmustcontainatleastonepositiveelementandonenegativeelement,andmaxi:i>0iaminmini:i<0i)]TJ /F5 11.955 Tf 21.92 0 Td[(amin. Proof. Writea=maxi:i>0iforconvenience,andnotethata=0bydenitionifcontainsnopositiveelement.Sinceisnormalized,wehave Xi:i0jij=1)]TJ /F9 11.955 Tf 14.95 11.36 Td[(Xi:i>0jij1)]TJ /F5 11.955 Tf 11.96 0 Td[(ajUj.(2)BecauseisintheleftnullspaceofA,PiiAi,j=0forallj.Thatimplies Xi:i>0iAi,j=)]TJ /F9 11.955 Tf 14.28 11.35 Td[(Xi:i0iAi,j=Xi:i0jijAi,j.(2)But,because0Ai,j1,wecanmakethefollowinginequalityXi:i>0iAi,jXi:i>0iajUj.Substituting( 2 )backin,wegetajUjXi:i0jijAi,jwhichmustholdforallj.Therefore,ajUjjX1jXjXi:i0jijAi,jXi:i0jijAminwhichcausesacontradictionwhena=0sinceAmin>0.Hence,amustnotbe0,andmusthaveatleastonepositiveelement.Further,by( 2 ),ajUjjX1jAmin(1)]TJ /F5 11.955 Tf 11.95 0 Td[(ajUj) 28

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whichgivesthedesiredlowerboundona.Theproofofexistenceofanegativeelementandthefactthattheminimumnegativeelementmustbenolargerthan)]TJ /F5 11.955 Tf 9.3 0 Td[(aminissimilar. Animmediate,butimportantlaterinprovingTheorem 2.6 ,consequenceofthelemmaisthefollowingobservation: Lemma2. Let0
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First,sinceiss-dependentupontherowsin,weknowthatthereexistsasetofcoefcientsc1,c2,...,cn)]TJ /F6 7.97 Tf 6.59 0 Td[(1suchthatforanyj=1,2,...,jUj, j)]TJ /F13 11.955 Tf 11.95 0 Td[(sn)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi=1cii,jj+s.(2)Inparticular,fori=1,2,...,n)]TJ /F3 11.955 Tf 12.03 0 Td[(1,wehavei)]TJ /F13 11.955 Tf 12.04 0 Td[(scii,ii+s.Usingthefactsthati,iaandthat)]TJ /F3 11.955 Tf 9.3 0 Td[(1i,wecandeterminethat )]TJ /F3 11.955 Tf 9.29 0 Td[(1)]TJ /F13 11.955 Tf 11.95 0 Td[(s aci+s a.(2)FromLemma 1 ,weknowthatthereexistsanindexmsuchthatm)]TJ /F5 11.955 Tf 23.71 0 Td[(amin.Toproceed,wewanttoshowthatmnandconsider( 2 )withj=m.Separatingthetermswithpositiveandnegativeci'sinthesummationandusingtheupperboundin( 2 ),wehaveXi:ci<0jciji,m)]TJ /F13 11.955 Tf 21.92 0 Td[(m)]TJ /F13 11.955 Tf 11.95 0 Td[(s+Xi:ci>0cii,mamin)]TJ /F13 11.955 Tf 11.96 0 Td[(s)]TJ /F3 11.955 Tf 11.96 0 Td[((n)]TJ /F3 11.955 Tf 11.95 0 Td[(1)+s a (2)wherethesecondinequalityisobtainedbyusingtheupperboundonciin( 2 ).Butbecausem>n,wehavei,mn)]TJ /F6 7.97 Tf 6.58 0 Td[(1.Thenbyusingthelowerboundonciin( 2 ),wegetPi:ci<0jciji,m(n)]TJ /F3 11.955 Tf 11.96 0 Td[(1)n)]TJ /F6 7.97 Tf 6.59 0 Td[(11+s a.Thuswearriveattheconclusionthat(n)]TJ /F3 11.955 Tf 11.96 0 Td[(1)n)]TJ /F6 7.97 Tf 6.58 0 Td[(11+s aamin)]TJ /F13 11.955 Tf 11.96 0 Td[(s)]TJ /F3 11.955 Tf 11.95 0 Td[((n)]TJ /F3 11.955 Tf 11.96 0 Td[(1)+s a,whichclearlyviolatesthecondition0nn,Xi:ci<0jciji,j)]TJ /F13 11.955 Tf 21.92 0 Td[(j)]TJ /F13 11.955 Tf 11.95 0 Td[(s+Xi:ci>0cii,j)]TJ /F13 11.955 Tf 21.92 0 Td[()]TJ /F13 11.955 Tf 11.96 0 Td[(s)]TJ /F3 11.955 Tf 11.96 0 Td[((n)]TJ /F3 11.955 Tf 11.96 0 Td[(1)+s a (2) 30

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wherethesecondinequalityisduetothefactthatj.Furtherseparatingthetermswithpositiveandnegativei,jinthesumontheleftsideof( 2 ),forj>n,wehaveXi:ci<0,i,j<0jcijji,jj+s+(n)]TJ /F3 11.955 Tf 11.95 0 Td[(1)+s a+Xi:ci<0,i,j>0jciji,j+s+(n)]TJ /F3 11.955 Tf 11.95 0 Td[(1)+n)]TJ /F6 7.97 Tf 6.58 0 Td[(1+s+sn)]TJ /F6 7.97 Tf 6.58 0 Td[(1 a (2)wherethesecondinequalityresultsfromtheboundPi:ci<0,i,j>0jciji,j(n)]TJ /F3 11.955 Tf 11.11 0 Td[(1)n)]TJ /F6 7.97 Tf 6.58 0 Td[(11+s a,asshownabove.Becausecm<0,weknowfrom( 2 )thatifm,j<0,thenjcmjjm,jj+s+(n)]TJ /F3 11.955 Tf 12.64 0 Td[(1)+n)]TJ /F15 5.978 Tf 5.76 0 Td[(1+s+sn)]TJ /F15 5.978 Tf 5.76 0 Td[(1 a.Further,bytheabovederivedresultthatjcmjamin)]TJ /F13 11.955 Tf 11.95 0 Td[(s,wegetjm,jj+s amin)]TJ /F13 11.955 Tf 11.96 0 Td[(s+(n)]TJ /F3 11.955 Tf 11.95 0 Td[(1)+n)]TJ /F6 7.97 Tf 6.58 0 Td[(1+s+sn)]TJ /F6 7.97 Tf 6.59 0 Td[(1 a(amin)]TJ /F13 11.955 Tf 11.95 0 Td[(s)=0n,ifm,j<0.Therefore,jm,jj0forallj>n.Combiningallaboveresults,weobservethatm,ma,jm,jj0n,andm,j=0forallotherindexvaluesofjexceptn.FromLemma 1 ,wemusthavem,n)]TJ /F5 11.955 Tf 21.92 0 Td[(amin.Thusmis(amin,0)-doublepolarizedat(m,n). Lemma4. Letamina1and2njUj.Let0
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constructanormalized,1jUjvector^thatis(amin,00)-polarizedatnwiththeadditionalpropertythat^j=0forallj=1,2,...,n)]TJ /F3 11.955 Tf 12.2 0 Td[(1.Thenweapplyelementaryrowoperationsandnormalizationtotherowsof[T^T]Ttoobtainthedesirednormalized,njUj,(amin,00)-diagonalpolarizedmatrix.First,set~=)]TJ /F4 7.97 Tf 12.8 14.94 Td[(n)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1i i,ii.Becauseisnots-dependentupontherowsof,wemusthavek~k1>s.Hencewecannormalize~toobtain^,i.e.,^=~=k~k1.Clearly,forjn,write ~j=j)]TJ /F9 11.955 Tf 14.95 11.36 Td[(Xi:i>0i i,ii,j+Xi:i<0jij i,ii,j.(2)Butsincei,iaandjforj>n,wehaveXi:i>0i i,ii,jXi:i>0 a()]TJ /F3 11.955 Tf 9.3 0 Td[(1))]TJ /F3 11.955 Tf 21.92 0 Td[((n)]TJ /F3 11.955 Tf 11.96 0 Td[(1) aXi:i<0jij i,ii,jXi:i<01 an)]TJ /F6 7.97 Tf 6.58 0 Td[(1(n)]TJ /F3 11.955 Tf 11.96 0 Td[(1)n)]TJ /F6 7.97 Tf 6.58 0 Td[(1 a.Applyingthesetwoboundsto( 2 ),weget,forj>n,~j+(n)]TJ /F3 11.955 Tf 11.95 0 Td[(1)+n)]TJ /F6 7.97 Tf 6.59 0 Td[(1 a,whichimplies^j1 s+(n)]TJ /F3 11.955 Tf 11.96 0 Td[(1)+n)]TJ /F6 7.97 Tf 6.59 0 Td[(1 a,"00n.Further,notethat^isanormalizedvectorintheleftnullspaceofA.HencebyLemma 1 andthefactthat00n
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Clearly,~i,n=0and~i,j=i,jforjn,~i,j=i,j)]TJ /F3 11.955 Tf 13.15 8.09 Td[(i,n ^n^jn)]TJ /F6 7.97 Tf 6.59 0 Td[(1+maxfn)]TJ /F6 7.97 Tf 6.59 0 Td[(1,"g amin=n)]TJ /F6 7.97 Tf 6.59 0 Td[(1+" aminwherethesecondinequalityisdueto)]TJ /F3 11.955 Tf 9.3 0 Td[(1i,nn)]TJ /F6 7.97 Tf 6.59 0 Td[(1,)]TJ /F3 11.955 Tf 9.29 0 Td[(1^j",and^namin,andthelastequalityresultsfromthefactthatjUjs<1.Hence^i,jn)]TJ /F6 7.97 Tf 6.59 0 Td[(1 a+" amina=00n.Asdiscussedbefore,^i,j=0foralli=1,2,...,n)]TJ /F3 11.955 Tf 12.07 0 Td[(1andj6=in.Hence,againusingLemma 1 ,wemusthave^i,iaminsince00
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n)]TJ /F6 7.97 Tf 6.59 0 Td[(1
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1jUjmatrixintheleftnullspaceofA.Inductively,supposethatwehaveconstructed,fromtherst(i)]TJ /F3 11.955 Tf 13.27 0 Td[(1)rowsof,the(i)]TJ /F3 11.955 Tf 13.27 0 Td[(1)jUjnormalizedmatrix^(i)]TJ /F6 7.97 Tf 6.58 0 Td[(1)thatis(amin,i)]TJ /F6 7.97 Tf 6.59 0 Td[(1)-diagonalpolarizedandintheleftnullspaceofA.Iftheithrowofiss-dependentontherowsof^(i)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(i.e.,therst(i)]TJ /F3 11.955 Tf 12.37 0 Td[(1)rowsof),thenbyLemmas 3 and 5 ,thereexistsanormalized,(amin,0i)-doublepolarized(at(m,i)withm0,from( 2 ),thereexistsasmallenoughsuchthat0na2mins (amin)]TJ /F17 7.97 Tf 6.59 0 Td[(s)~an)]TJ /F15 5.978 Tf 5.76 0 Td[(1+.Choosing=0)]TJ /F4 7.97 Tf 28.85 6.48 Td[(a2mins (amin)]TJ /F17 7.97 Tf 6.58 0 Td[(s)~an)]TJ /F15 5.978 Tf 5.76 0 Td[(1andusingpart1)ofLemma 5 giveus0203...0n0.NowapplyingTheorem 2.4 givesusthedesiredresult. Thenextlemmastatesthattheobservationchannel(A,B)beingnon-manipulableissufcientfortheconditionofnon-existenceofanynormalized(amin,0)-doublepolarized,1jUjvectorintheleftnullspaceofArequiredinCorollary 1 : 35

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Lemma6. If(A,B)isnon-manipulable,thenthereexistsapairofconstantssandArespectivelysatisfying0
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WenowturnourattentiontotherightnullspaceofB.ThefollowingresultstatesthatnormalizedvectorsintherightnullspaceofBhavesimilarpropertiesofnormalizedvectorsintheleftnullspaceofAasdescribedinLemma 1 : Lemma7. TherightnullspaceofBconsistsonlyofbalancedvectors.Supposethat!Tisanon-zeronormalizedjUj1vectorintherightnullspaceofB.Letbmin,1 jUj(jY1j+1).Thenmaxj:!j>0!jbminminj:!j<0!j)]TJ /F5 11.955 Tf 21.92 0 Td[(bmin. Proof. Let!TbeavectorintherightnullspaceofB.ThenjY1jXi=1jUjXj=1Bi,j!j=0.SwappingtheorderofthetwosumsandusingthefactthatPjY1ji=1Bi,j=1,wegetPjUjj=1!j=0.Furthermoresuppose!Tisnon-zeroandnormalized,itmustthenhaveatleastonepositiveelementandonenegativeelement.Recognizingtheprecedingfact,wecanemployessentiallythesameargumentintheproofofLemma 1 toshowmaxj:!j>0!jbminandminj:!j<0!j)]TJ /F5 11.955 Tf 21.92 0 Td[(bmin. BasedonLemma 7 ,itiseasytocheckthattheresultsfromLemma 3 toCorollary 1 allapplytoBTwithaminreplacedbybminand~andenedinLemma 5 replacedby~bn,b2min1+n bmin)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Thatis,theaboveresultsareallapplicabletonormalized,polarizedvectorsandmatricesintherightnullspaceofBwiththecorrespondingmodications.Finally,thenexttheoremstatesthatnon-manipulabilityoftheobservationchannel(A,B)guaranteestheexistenceofacounterpartofCorollary 1 fortherightnullspaceof 37

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BthatistobeusedintheproofofTheorem 2.6 .Tosimplifynotationinstatementofthelemma,letAdenotethesetofalljUj21columnvectorsoftheformvec(A),whereisajUjjUjmatrixwhosejthcolumn,forj=1,2,...,jUj,isbalancedand(0,0)-polarizedatj,andkk22jUj.ItiseasytocheckthatthesetAisclosed,bounded,andconvex.Let~AdenotetheconehullofA.Thenitcanbereadilycheckedthat~AisthesetofvectorsofthesameformthatmakeupAwiththenormboundkk22jUjremoved. Theorem2.5. SupposethattherightnullspaceofBisnon-trivial,andtheobservationchannel(A,B)isnon-manipulable.Thenthereexistsapositive,whichdependsonlyonAandB,satisfyingthepropertythatifisajUjjUjmatrixwhosecolumnsarevectorsintherightnullspaceofBandkk1=1,thenthereisanormalizedvectorsimultaneouslygivingvec()and!T0forall!T2A. Proof. Letn=jUj)]TJ /F3 11.955 Tf 20.21 0 Td[(rank(B)1bethedimensionoftherightnullspaceofB.LetbeajUjnmatrixwhosecolumnsformanorthonormalbasisoftherightnullspaceofB.Byippingthepolaritiesofthecolumnsof(i.e.,thebasisvectors),weobtain2ndifferentbasesfortherightnullspaceofB.FixanormalizedwithcolumnsintherightnullspaceofB.Itissimpletocheckthat1 jUj2kk221.Foreachj=1,2,...,jUj,employingone,say(j),amongthe2nbasesabovewecandecomposeTjasTj=Pni=1Tj((j)Ti)T(j)TiwithTj((j)Ti)T0foralli.LetBbetheconvexhullofthesetofvectorsoftheformvec(),whereisanyjUjjUjmatrixsuchthatTj=Pni=1bi,j(j)Tiforj=1,2,...,jUjwith1 jUj2Pni=1PjUjj=1b2i,j1andbi,j0foriandj.Obviouslyvec()2B.Bygeometricreasoning,Bisaboundedsetthatdoesnotcontaintheorigin.Since(A,B)isnon-manipulable,~Amustintersecttriviallywiththesetofvectorsoftheformvec(~),where~isanyjUjjUjmatrixwhosecolumnsarevectorsintherightnullspaceofB(i.e.,theintersectioncontainsonlythezerovector).HenceBand~Aaredisjoint.Belowweemployaslightlystrongerversionoftheargumentgivenin[ 10 ,pp. 38

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48]toshowthatBand~Acanbestrictlyseparatedbyahyperplanethatpassesthroughtheorigin.Giventheabove,weconcludethatthereexistTa2~AandTb2Bthatachievetheminimum(positive)Euclideandistancebetween~AandB.NowletT1beanarbitraryvectorin~A.Since~Aisaconvexcone,Ta+t(T1)]TJ /F13 11.955 Tf 11.24 0 Td[(Ta)2~Aforall>0and00and00.Further,letting!1gives (a)]TJ /F13 11.955 Tf 11.96 0 Td[(b)T10.(2)Similarly,ifT2beanarbitraryvectorinB,thenTb+t(T2)]TJ /F13 11.955 Tf 12.11 0 Td[(Tb)2BbytheconvexityofB.Thenkb+t(2)]TJ /F13 11.955 Tf 11.95 0 Td[(b))]TJ /F13 11.955 Tf 11.96 0 Td[(ak22ka)]TJ /F13 11.955 Tf 11.96 0 Td[(bk22forall0
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manipulatedsignal^nasdenedin( 2 ).Inordertoshowtheorem 2.1 ,werstprovethefollowingtheorem,ofwhichtheorem 2.1 isaspecialcaseof. Theorem2.6. Let)]TJ /F4 7.97 Tf 6.78 4.34 Td[(n=PYjVnPUjX,>0,and^n=argmax^:B^A2[)]TJ /F16 5.978 Tf 4.82 2.27 Td[(n]k^)]TJ /F5 11.955 Tf 12.48 0 Td[(Ik1.If(A,B)isnon-manipulable,thenkn)]TJ /F3 11.955 Tf 13.35 2.66 Td[(^nk1c1+c2kn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1forsomepositiveconstantsc1andc2thatdependonlyonAandB. Proof. LetAandBdenotetheorthogonalprojectorsontotheleftnullspaceofAandrightnullspaceofB,respectively.Wedecomposen)]TJ /F3 11.955 Tf 13.87 2.66 Td[(^nintothreecomponentsasbelow:n)]TJ /F3 11.955 Tf 13.35 2.66 Td[(^n=(n)]TJ /F3 11.955 Tf 13.35 2.66 Td[(^)A| {z }AA+B(n)]TJ /F3 11.955 Tf 13.35 2.66 Td[(^)(I)]TJ /F3 11.955 Tf 11.96 0 Td[(A)| {z }B+(I)]TJ /F3 11.955 Tf 11.96 0 Td[(B)(n)]TJ /F3 11.955 Tf 13.35 2.66 Td[(^)(I)]TJ /F3 11.955 Tf 11.95 0 Td[(A)| {z } (2)wheretherowsofAarenormalized,andAisajUjjUjdiagonalmatriceswhosestrictlypositivediagonalelementsarethenormalizationconstantsfortherowsofA.NotethattherowsofAarevectorsintheleftnullspaceofA.MoreoverthecolumnsofBarevectorsintherightnullspaceofB.AlsonotethatwehaveassumedthatAdoesnotcontainanyall-zerorowswithoutanylossofgenerality(see( 2 )below).BecausetherowsofAarenormalized,wehavefrom( 2 ), kn)]TJ /F3 11.955 Tf 13.35 2.66 Td[(^nk1kB+k1+jUjXi=1Ai,i.(2)ThusitsufcestoboundkB+k1andthediagonalelementsofA.WerstboundthediagonalelementsofA.Todothis,rewrite( 2 )as I)]TJ /F3 11.955 Tf 13.35 2.65 Td[(^n=I)]TJ /F3 11.955 Tf 11.96 0 Td[(n+AA+B+.(2)Because^nisavalidstochasticmatrix,alldiagonalelementsofI)]TJ /F3 11.955 Tf 13.51 2.66 Td[(^nmustbegreaterthanorequalto0andalloff-diagonalelementsmustbelessthanorequal0.Thus 40

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( 2 )gives Ai,iAi,j8>><>>:kI)]TJ /F3 11.955 Tf 11.95 0 Td[(nk1)-222(kB+k1ifj=ikI)]TJ /F3 11.955 Tf 11.95 0 Td[(nk1+kB+k1ifj6=i.(2)NowbyLemma 6 ,wehavesandArespectivelysatisfy( 2 )and( 2 )suchthattheleftnullspaceofAdoesnotcontainany(amin,A)-doublepolarizedvectors.Hencewecanchooseapositive
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jUjXi=1ji,jj+jUj )]TJ /F2 11.955 Tf 5.48 -9.69 Td[(kI)]TJ /F3 11.955 Tf 11.95 0 Td[(nk1+kB+k1 (2)wherethesecondinequalityisobtainedbyusing( 2 )andthefactthattherowsofAarenormalized.For2njUj,Corollary 1 suggeststhatforeachi2S,jUjXj=1Xk2SAk,kAk,j=Ai,ijUjXj=1Ai,j+Xk2S,k6=iAk,k Ai,iAk,j>Ai,is. (2)Notethattheboundin( 2 )isalsotriviallyvalidforthecaseofn=1sinces<1andtherowsofAarenormalized.Substituting( 2 )into( 2 ),weobtainthatforeachi2S, Ai,i0.Withoutlossofgenerality,supposethelatteristruebelow.Becausethelinearmapping()AwithdomainrestrictedtotheorthogonalcomplementoftheleftnullspaceofAisinvertible,wehave kBk1cAkBAk1=cAB(2)forsomeconstantcAthatdependsonlyonA.ThusboundingBissufcient.Tothatend,rightmultiplybothsidesof( 2 )byAtoobtain (I)]TJ /F3 11.955 Tf 13.35 2.66 Td[(^)A| {z }=BB+(I)]TJ /F3 11.955 Tf 11.95 0 Td[(n)A+A.(2)Since^isstochastic,vec()2A,whereAisthesetofjUj21columnvectorsdenedjustrightbeforeTheorem 2.5 inSection 2.3 .As(A,B)isnon-manipulableandthe 42

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left-nullspaceofBisnon-trivial,Theorem 2.5 guaranteestheexistenceofapositiveconstantandanormalizedvectorgivingvec()0andvec(B).NotethatdependsonlyonAandB.Henceleft-multiplyingbothsidesofthevectorizedversionof( 2 )byyieldsBjfvec((I)]TJ /F3 11.955 Tf 11.95 0 Td[(n)A)+vec(A)gj k(I)]TJ /F3 11.955 Tf 11.95 0 Td[(n)Ak1+kAk1 p jUjjX1j2 (kI)]TJ /F3 11.955 Tf 11.96 0 Td[(nk1+kk1).ApplyingthisboundonBbackto( 2 ),weobtain kBk1cAp jUjjX1j2 (kI)]TJ /F3 11.955 Tf 11.96 0 Td[(nk1+kk1).(2)Tocompletetheproof,weneedtoboundkk1.NotethatBA1=B^nA)]TJ /F5 11.955 Tf 11.95 0 Td[(BnA<. (2)NotethatthelinearmappingB()AwithdomainrestrictedsimultaneouslytotheorthogonalcomplementsoftherightnullspaceofBandleftnullspaceofAisinvertible.Combiningthisfactand( 2 ),thereexistsapositiveconstantcAB,whichdependsonlyonAandB,suchthat kk1cABp jX1jjY1j.(2)Finally,substituting( 2 ),( 2 ),and( 2 )backinto( 2 ),weobtainthedesiredboundonkn)]TJ /F3 11.955 Tf 13.35 2.65 Td[(^k1giveninthestatementofthetheoremwithc0=cABjUj s+cAB1+jUj3 s 1+cAp jUjjX1j2 !c1=p jX1jjY1jc0c2=jUj3 s+cAp jUjjX1j2 1+jUj3 s. 43

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Tocompletetheproofoftheorem( 2.1 ),letn=Iandnotethatforany^suchthatkB^A)]TJ /F5 11.955 Tf 12.34 0 Td[(BAk1,willhavethepropertythatkI)]TJ /F3 11.955 Tf 13.74 2.66 Td[(^k1kI)]TJ /F3 11.955 Tf 13.73 2.66 Td[(^nk1.Then,usingtheorem( 2.6 )weknowthatkI)]TJ /F3 11.955 Tf 13.41 2.66 Td[(^nk1c1.Furthermore,because^nisdenedasthemaximumdistancefromIforagiven[)]TJ /F4 7.97 Tf 6.77 4.33 Td[(n],all^suchthatB^A2[)]TJ /F4 7.97 Tf 6.78 4.33 Td[(n]musthavethepropertythatkI)]TJ /F3 11.955 Tf 13.35 2.65 Td[(^k1c1.ProofofTheorem 2.2 Tocompletetheproofoftheconverse,weneedtoshowtheexistenceofastochastic06=IwiththepropertythatB0A=BAwhentheobservationchannel(A,B)ismanipulable.Tothatend,rstnotethatthemanipulabilityof(A,B)impliestheexistenceofanon-zerowiththepropertythatallcolumnsofAareintherightnullspaceofB.Inaddition,(Tj)Tisbalancedand(0,0)-polarizedatj,foreachj=1,2,...,jUj.Let~= maxjj,jand0=I)]TJ /F3 11.955 Tf 14.08 2.65 Td[(~.Ittheneasytocheckthatthis0isavalidstochasticmatrix,06=1,andB0A=BA.ProofofTheorem 2.3 IntheproofofLemma 6 ,wehaveshownthattheconditionof(A,B)beingnon-manipulableimpliesthatnonormalized,(amin,0)-doublepolarizedvectorcanbeintheleftnullspaceofA.Itremainstoshowthereverseimplicationhere.GiventhattheleftnullspaceofAdoesnotcontainany(amin,0)-doublepolarizedvector,weneedtoshowthat(A,B)isnon-manipulablewhentherightnullspaceofBistrivial.Tothatend,letussupposeonthecontrarythat(A,B)ismanipulable.SincetherightnullspaceofBistrivial,thereexistsajUjjUjnon-zeromatrixintheleftnullspaceofA,withitsjthcolumn,(Tj)T,foreachj=1,2,...,jUj,isbalancedand(0,0)-polarizedatj.Letmbethenumberofnon-zerocolumnsof.Byproperpermutationofrowsandcolumnsofifnecessary,wecanassumewithnolossofgeneralitythattherstmcolumnsarenon-zerowhiletheremainingjUj)]TJ /F5 11.955 Tf 17.91 0 Td[(mcolumnsarezero. 44

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First,weclaimthatm2.Indeed,supposethatm=1andonlytherstcolumn(T1)Tisnon-zero.Thenwehave1,1>0and1,1A1=0.SinceitisourassumptionthatAcontainsnozerorows,wemusthave1,1=0,whichcreatesacontradiction.Next,let~1=1 k1k1~2=2 k2k1...~m=PjUji=mi kPjUji=mik1.PutthesemrowvectorstogethertoformthemjUjmatrix~.Usingthefactthat(Tj)Tisbalancedand(0,0)-polarizedatjforj=1,2,...,mtogetherwithLemma 1 ,itiseasytocheckthat~isanormalized,(amin,0)-polarizedmatrixintheleftnullspaceofA.Moreover,itisalsotruethat m)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi=1kik1~i+jUjXi=mi1~m=0.(2)Now,asarguedintheproofofLemma 6 ,theremustexistsandA,whichrespectivelysatisfy( 2 )and( 2 ),suchthattheleftnullspaceofAcontainsno(amin,A)-doublepolarizedvector.HencewecanapplyCorollary 1 to~todeducethatnorowofitcanbes-dependentupontheotherrows.However,thisconclusioncontradicts( 2 ).Therefore(A,B)mustnotbemanipulable. JusticationforAlgorithms 1 and 2 .Algorithm 1 LetbeajUjjUjmatrix-valuedvariableandsbeajUj1vector-valuedvariable.Considerthefollowingconvexoptimizationproblem:mins,minf)]TJ /F5 11.955 Tf 15.27 0 Td[(s1,)]TJ /F5 11.955 Tf 9.3 0 Td[(s2,...,)]TJ /F5 11.955 Tf 9.3 0 Td[(sjUjg 45

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subjecttoBA=0jUjXk=1k,l=0l=1,2,...,jUj,sk)]TJ /F3 11.955 Tf 11.95 0 Td[(k,k0k=1,2,...,jUj,k,k)]TJ /F3 11.955 Tf 11.95 0 Td[(10k=1,2,...,jUj,k,l0k6=l=1,2,...,jUj. (2)Itisclearthattheoptimalvalueof( 2 )isattainedandliesinsidetheinterval[)]TJ /F3 11.955 Tf 9.3 0 Td[(1,0].Thisfurtherimpliesthat(A,B)isnon-manipulableifandonlyiftheoptimalvalueof( 2 )is0.Now,letandbethejX1jjY1jmatrix-valuedandjUj1vector-valuedLagrangemultipliersfortheequalityconstraintsshowninthethirdandfourthlinesof( 2 ).Further,letbethejUjjUjmatrix-valuedLagrangemultipliermatrix.Thediagonalelementsofcorrespondtotheinequalityconstraintsshowninthefthlineof( 2 ),whiletheoff-diagonalelementscorrespondtotheinequalityconstraintsshowninthelastlineof( 2 ).Atlast,letbethejUj1vector-valuedLagrangemultiplierfortheinequalityconstraintsshowninsecondlineof( 2 ).Followingthedevelopmentin[ 10 ,Ch.5],weobtaintheLagrangedualfunctionof( 2 )asbelow:g(,,,)=8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:)]TJ /F9 11.955 Tf 11.29 8.97 Td[(PjUjk=1kifk,k1k=1,2,...,jUj,k+k+[AB]k,k=k,kk=1,2,...,jUj,k+[AB]k,l=)]TJ /F3 11.955 Tf 9.3 0 Td[(k,lk6=l=1,2,...,jUjotherwise. 46

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ConsidertheLagrangedualproblemof( 2 ):max,,,g(,,,)subjecttok0k=1,2,...,jUj,k,l0k,l=1,2,...,jUj. (2)Sincetheprimalproblem( 2 )satisestheSlater'scondition[ 10 ,Ch.5],theoptimaldualitygapbetweentheprimalanddualproblemsiszero.Thatis,theoptimalvalueof( 2 )isthesameastheoptimalvalueof( 2 ).Hence(A,B)isnon-manipulableifandonlyiftheoptimalvalueof( 2 )is0.Finally,itiseasytoverifythatthenegationoftheoptimalvalueofthelinearprograminStep 2 ofAlgorithm 1 istheoptimalvalueof( 2 )andviceversa.Asaresult,Algorithm 1 determineswhether(A,B)ismanipulable.Algorithm 2 WeuseLemma 1 toshowthattheAlgorithm 2 canbeusedtocheckfordoublepolarizedvectorsintheleftnullspaceofA.Tothatend,recallthatn=jUj)]TJ /F1 11.955 Tf 18.99 0 Td[(rank(A)isthedimensionoftheleftnullspaceofA.Ifn=0,theleftnullspaceofAistrivialandhenceitcannotcontainanynormalized,(amin,0)-doublepolarizedvector.Forn>0,let=(I~)betherow-reducedechelonbasismatrixasstatedinStep 2 )ofthealgorithm.Whenn=jUj)]TJ /F3 11.955 Tf 18.73 0 Td[(1,~iisascalar,fori=1,2,...,n.ByLemma 1 ,~imustbenegativeandthenormalizedversionofimustbea(amin,0)-doublepolarizedvectorintheleftnullspaceofA.Nowassume1njUj)]TJ /F3 11.955 Tf 20.1 0 Td[(2andconsiderthestatedsteps.Firstnotethatany(amin,0)-doublepolarizedvectorintheleftnullspaceofAcanonlybealinearcombinationofatmosttworowsof.IfinStep 3c )thereisa~ithatcontainsallbutonezeroelement,Lemma 1 againforcesthenon-zeroelementbenegativeandthenormalizedversionofibea(amin,0)-doublepolarizedvectorintheleftnullspace 47

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ofA.Otherwiseno(normalized)rowsofcanbe(amin,0)-doublepolarized.Henceitremainstocheckwhetheranypairofrowsofcanbelinearlycombinedtoformadoublepolarizedvector.Withoutlossofgenerality,supposethatthenormalizedversionofc11+c22is(amin,0)-doublepolarizedforsomenon-zeroconstantsc1andc2.Thenitiseasytoseethatc1andc2mustbeofoppositesignsand~1=)]TJ /F4 7.97 Tf 10.49 4.89 Td[(c2 c1~2.Henceifnopairsofrowsof~satisfytheconditioncheckedinStep 3d ),theleftnullspaceofAcannotcontainanynormalized,(amin,0)-doublepolarizedvector. 48

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CHAPTER3CHANNELDEGREDATIONATTACKInthischapterweinvestigatetheimmediatepracticalapplicationsofchapter 2 ,inwhichwedeterminedtheeffectmanipulationhadontheconditionaldistributionsofreceivedandtransmittedsymbols.Toapplythismathematicalframework,wecasttheprobabilisticoutputasathatofatrustmetricintrustmanagement.Trustmanagement[ 11 ][ 13 ]isawidelyresearchedapproachtoaddressthisquestion.Inessence,trustmanagementpertainstotheestablishment,distribution,andmaintenanceoftrustrelationshipsamongnodesinanetwork.Baseduponsuchrelationships,itisexpectedthattrustednodeswillfaithfullyoperateaccordingtosomeprotocolsthattheyhaveagreedupon.Theaforementionedtrustrelationshipsareprimarilyquantiedusingtrustmetricsthatareevaluatedthroughnodesinteractingwithandobservingthebehaviorsofeachother[ 14 19 ].Fornodesthatdonotdirectlyinteractwitheachother,trustrelationshipscanbeestablishedandmaintainedviainference[ 20 ].Itisclearthatthevaluationoftrustmetricsiscriticallyimportantinthistrustmanagementapproach.Manydifferentwayshavebeenproposedtoevaluatethetrustmetricsbasedonauthenticationkeys[ 21 23 ],reputation[ 24 25 ],andevidencecollectedfromnetworkaswellasphysicalinteraction[ 26 28 ].Giventhecomplexityanddifcultyinvolvedinquantifyingthevaguenotionoftrust,onewouldexpectthesevaluationschemesarenaturallyadhoc.Tomoresystematicallyconstructatrustmetric,oneneedstospecifytheclassofmaliciousactionsagainstwhichthemetricmeasures.Inthischapter,weconsidertheclassofdatamanipulationattacksinwhichintermediatenodesmayalterthechannelsymbolsthattheyaresupposedtoforward.Wecastthetrustmetricvaluationproblemasamaliciousnessdetectionproblemagainstthisclassofattacks,forwhichchapter 2 isofimmediateconsequence.Moreprecisely,anode(oranothertrustednodecalledawatchdog[ 29 ])detectsifanothernoderelaysmanipulatedsymbolsthataredifferent 49

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fromthoseoriginallysentoutbythenodeitself.Observationsfordetectionagainstthemaliciousattackcanbemadeinthephysicaland/orhigherlayers.Mostexistingmaliciousnessdetectionmethodsarekey-based,requiringattheminimumthesourceanddestinationnodestoshareasecretkeythatisnotprivytotherelaynodebeingexamined.In[ 30 ],keys,whichcorrespondtovectorsinanullspace,aregiventoallnodesinanetwork.Anymodicationtotheencrypteddatabyarelaynodecanbecheckedbydeterminingwhethertheobserveddatafallsintothenullspaceornot.In[ 31 ],symmetriccryptographickeysareappliedtothedataatsourceanddestinationforthepurposeofcheckingthemaliciousnessofrelaynode(s).Anotherkey-basedapproachisconsideredin[ 32 ]bymeasuringwhetheronlyasmallamountofpacketsaredroppedbyrelaynodes.In[ 7 ],across-layerapproachbasedonmeasurementofchannelsymbolsistaken.Twokeysareemployedinthatscheme;onetocreateasetofknowndataandanothertomakethedataindistinguishablefromthekey.Whenthedestinationreceivesthemessage,theprobabilityoferrorofthetransmittedvaluesofthekeycanbeusedtodetermineiftherelaynodeisactingmaliciously.Inall,thekey-basedmaliciousnessdetectionschemesdescribedabovearefarfromdesirableastheyrequirethesupportofsomekeydistributionmechanism,whichinturnpresumestheexistenceofinherentlytrustednodesinthenetwork.Moreoversomekey-basedmethodshavebeenshowninsecurein[ 33 ]whennondeterminismandbit-levelrepresentationofthedataisconsidered.Fortheclassofsymbolmanipulationattacks,itisintuitivethatmaliciousnessofanodeshouldbedetectedbythenearbynodesbasedonmeasurementsobtainedatthelowerlayers,sincesuchmeasurementsaremorereliablethanthosemadebyfarawaynodesandatthehigherlayersastherearefewerchancesforpotentialadversariestotamperwiththeformermeasurements.Henceweinvestigatethemaliciousnessdetectabilityproblemfromaphysical-layerperspectivebyconsideringamodelinwhichtwosourceswanttoshareinformationthroughapotentiallyuntrustworthyrelaynode 50

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thatissupposedtorelaytheinformationintheamplify-and-forwardmanner.In[ 34 ],weprovideapreliminarystudyontheproblemunderarestrictivecaseinwhichtherelaymayonlymodifythechannelsymbolsbasedonsomeindependentandidenticallydistributed(i.i.d)attackmodel,andthesourcenodescanperfectlyobservethesymbolsforwardedbytherelay.Inthispaper,weextendthetreatmenttoageneralchannelmodelandaneffectivelygeneralclassofsymbolmanipulationattacks.ThedetailsofthechannelandattackmodelsareprovidedinSection 3.1 .InSection 3.2 ,westateourmainresult,whichisanecessaryandsufcientconditiononthechannelthatguaranteesasymptoticdetectionofmaliciousnessindividuallybybothsourcenodesusingempiricaldistributionsoftheirrespectiveobservations.TheresultspresentedinSection 3.2 makeclearthatmaliciousnessdetectabilityunderourmodelisaconsequenceofthestochasticcharacteristicsofthechannelandthesources.Itworkssolelybasedonobservationsmadebythesourcenodesaboutthesymbolssentbytherelaynodetogetherwithknowledgeaboutthechannel.Nopresumedsharedsecretbetweenanysetofnodesisrequiredorused.Thustheproposedmaliciousnessdetectionapproachcanbeusedindependentoforinconjunctionwiththekey-basedmethodsdescribedabovetoprovideanotherlevelofprotectionagainstadversaries. 3.1SystemModel 3.1.1NotationWealsoadoptthenotationof[ 35 ]fordistributions,types,typeclasses,andtypicalsets.Forinstance,weusePX(x)andPXjY(xjy)todenotethedistributionofXandconditionaldistributionofXgivenY,respectively.Wheneverappropriate,wewilltreatPXasadiagonalmatrixwithPX(x)asthediagonalelements,andPXjYasastochasticmatrixwhoseelementsarespeciedbyPXjY(xjy)withxandyindexingtherowsandcolumns,respectively.Forasequenceofrandomvariables,wewillusetheshorthandformtodenoteitsdistribution;e.g.,p(xn)istheprobabilitymassfunction(pmf)ofXn. 51

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Inaddition,thecountingfunction(xjxn)denotesthenumberofoccurrencesoftheelementx2Xinthesequencexn.ThetypeofthesequencexnisdenotedbyPxn(x),1 n(xjxn),whiletheconditionaltypeofxngivenynis1Pxnjyn(xjy),Pxn,yn(x,y) Pyn(y).ForanytypePofsequencesinXn,thetypeclassofPisdenotedbyTnP,fxn:Pxn(x)=P(x)g.IfQisaconditionaltypeofsequencesinXngivenyn,thentheconditionaltypeclassofQ(orQ-shell)givenynisTnQ(yn),fxn:Pxnjyn(xjy)=Q(xjy)g.ForagivendistributionPonXand>0,denethefollowingcollectionoftypesthatareclosetoP:[P]n,fPxn:jPxn(x))]TJ /F5 11.955 Tf 11.96 0 Td[(P(x)jandPxn(x)=0wheneverP(x)=0g.TheP-typicalsetisthendenedasTn[P],fxn:Pxn2[P]ng.WhenPisthedistributionofX,wemayalsouse[X]nandTn[X]todenotetheabovecollectionoftypesandtypicalset,respectively.IfPXjYisaconditionaldistribution,thePXjY-(orXjY-)typicalsetgeneratedbyynisdenedasTn[PXjY](yn)=Tn[XjY](yn),fxn:Pxnjyn2[PXjY]n(yn)g,where[PXjY]n(yn)=[XjY]n(yn),Pxnjyn:Pxnjyn(xjy))]TJ /F5 11.955 Tf 11.95 0 Td[(PXjY(xjy)Pyn(y)andPxnjyn(xjy)=0wheneverPXjY(xjy)=0isthecollectionofconditionaltypesclosetoPXjYgivenyn.Wemaydropthepart(yn)fromthenotationabovewheneverthedependenceisclearfromthecontext. 3.1.2ChannelmodelConsiderthechannelmodelshowninFig. 3-1 ,whichservesasageneralizationofthemotivatingexamplepresentedidescribedinsection 1.2 .Twonodes(1and2)simultaneouslyforwardtheirsourcesymbolstoarelaynode.Therelaynodeissupposedtoforwarditsreceivedsymbolsbacktothetwonodes(or 1ForPyn(y)=0,Pxnjyn(jy)canbechosenarbitrarily. 52

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(a)Multiple-accesschannel(timeinstants1,2...,n) (b)Broadcastchannel(timeinstantsn+1,n+2,...,2n) Figure3-1.Amplify-and-forwardrelayingmodel. tosomeothernodes)intheamplify-and-forwardmanner.Thereissomepossibilitythattherelaymaymodifyitsreceivedsymbolsinanattempttodegradetheperformanceofthetransmission.Ourgoalistodetermineifandwhenitispossiblefornodes1and2todetectanymaliciousactoftherelaybyobservingthesymbolsbroadcastfromtherelayinrelationtothesymbolsthattheyindividuallytransmitted.Notethattheabovemodelalsocoverstheperhapsmorecommonscenarioinwhichonlyonenodehasinformationtosendwhilethetransmissionbytheothernodeisregardedasintentionalinterference.Morespecically,letX1andX2betwoindependentdiscreterandomvariablesthatspecifythegenericdistributionsofthesymbolstransmittedbynodes1and2,respectively.Attimeinstants1,2,...,n,nodes1and2transmiti.i.d.symbolsrespectivelydistributedaccordingtoX1andX2.Thetransmissiongoesthroughamemorylessmultiple-accesschannel(MAC)withtherandomvariableUdescribingitsgenericoutputsymbol.TheMACisspeciedbytheconditionalpmfp(ukjx1,i,x2,j).Therelaynode,duringtimeinstants1,2,...,n,observestheoutputsymbolsoftheMAC,processes(ormanipulates)them,andthenbroadcaststheprocessedsymbolsouttonodes1and2attimeinstantsn+1,n+2,...,2nviaamemorylessbroadcastchannel(BC)withtherandomvariablesVdescribingitsgenericinputsymbolandY1andY2describingthegenericoutputsymbolsatnodes1and2,respectively.TheBCisspeciedbytheconditionalpmfp(y1,i,y2,jjvk).Inaddition,becausetherelayissupposed 53

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toworkintheamplify-and-forwardmanner,weadoptthereasonablemodelthatthealphabetsofUandVareofthesamesize,andthereisaone-to-onecorrespondenceui$vi,i=1,2,...,jUj,betweenelementsofthealphabets.LetundenotethesequenceofnoutputsymbolsoftheMACobservedbytherelayduringtimeinstants1,2,...,n,andvndenotethesequenceofninputsymbolsoftheBCtransmittedbytherelayduringtimeinstantsn+1,n+2,...,2n.Thenthemappingvn=n(un)representsthemanipulationperformedbytherelay.Themanipulationmapnisallowedtobearbitrary,deterministicorrandom,andknowntoneithernode1nor2.TheonlyrestrictionweimposeistheMarkovityconditionthatp(vnjun,xn1,xn2)=p(vnjun),wherexn1andxn2denotethesymbolsequencestransmittedbynodes1and2,respectively,duringtimeinstants1,2,...,n.Thatis,therelaymaypotentiallymanipulatethetransmissionbasedonlyontheoutputsymbolsoftheMACthatitobserves. 3.1.3MaliciousnessofrelayRecallequation( 2 ).Despiteomittedfromitsnotation,theattackchannelndependsonthesequencesunandvn.RatherthandirectlyactingontheMACoutputsymbolstoproduceinputsymbolsfortheBC,theattackchannelnextractsthestatisticalpropertiesofthemanipulationmapnthatarerelevanttoourpurposeofdening(andlaterdetecting)themaliciousnessoftheactionoftherelay: Denition3.1. (Maliciousness)Therelayissaidtobenon-maliciousifkn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1!0inprobabilityasnapproachesinnity.Otherwise,therelayisconsideredmalicious.Inthestrictestsense,normalamplify-and-forwardrelayoperationshouldrequiren=Iforallunandvnandforeachn.NeverthelessitturnsouttobebenecialtoconsidertherelaxationinDenition 3.1 whentheprimaryfocusistocheckwhethertherelayisdegradingthechannelratherthanattackingaspecicpartofthetransmission.Inparticular,theprobabilisticandlimitingrelaxationinDenition 3.1 allowsustoobtaindeniteresults(seeSection 3.2 )fortheverygeneralclassofpotentialmanipulation 54

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mapsdescribedabovebytoleratingactions,suchasmanipulatingonlyanegligiblefractionofsymbols,thathaveessentiallynoeffectontheinformationrateacrosstherelay.Wepointoutthatitispossibletodevelopsimilarresultsbasedontheabove-mentionedstrictestsenseofmaliciousnessforsomemorerestrictedclassesofmanipulationmaps(see[ 34 ]forinstance). EstimatesofVariables,andtheGoodEstimatorSet.Firstnode1obtainstheconditionalhistogramestimator^)]TJ /F4 7.97 Tf 6.77 4.34 Td[(nof)]TJ /F4 7.97 Tf 6.78 4.34 Td[(ndenedbyits(i,j)thelement: ^)]TJ /F4 7.97 Tf 6.77 4.94 Td[(ni,j,n(y1,i,x1,j) n(x1,j).(3)Thenitconstructstheestimator^nfrom^)]TJ /F4 7.97 Tf 6.78 4.34 Td[(naccordingtoequation( 2 ).In( 2 ),isasmallpositiveconstantandG(^\isthesetofjUjjUjstochasticmatrices,foreachofwhich(saydenotedby^),thereexistsajY1jjUjstochasticmatrix~)]TJ /F1 11.955 Tf 10.1 0 Td[(suchthatkB(~)]TJ /F2 11.955 Tf 9.43 0 Td[()]TJ /F3 11.955 Tf 12.2 2.66 Td[(^\Ak1andB^A=B~\005A,whereAandBdenotetheorthogonalprojectorsontotherowspaceofAandcolumnspaceofB,respectively.Wewillemploytheestimator^nspeciedin( 2 )toobtainthedetectabilityresultsinthefollowingsections. 3.2TheoremsandResultsLetDn=Dn(yn1,xn1)denoteadecisionstatisticbasedontherstnobservations(yn1,xn1)thatisemployedformaliciousnessdetection.Thefollowingtheoremstatesthatmaliciousnessdetectabilityisequivalenttonon-manipulablilityoftheobservationchannel: Theorem3.1. (Maliciousnessdetectability)Whenandonlywhentheobservationchannel(A,B)isnon-manipulable,thereexistsasequenceofdecisionstatisticsfDngwiththefollowingproperties(assuming>0below): 1. Iflimsupn!1Pr(kn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1>)>0,thenlimsupn!1PrDn>kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>=1. 55

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2. Ifliminfn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1)>0,thenlimn!1PrDn>ckn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1=0forsomepositiveconstantcthatdependsonlyonAandB.Thetheoremveriesthepreviousclaimthattheattackchannelnprovidesustherequiredstatisticalcharacterizationfordistinguishingbetweenmaliciousandnon-maliciousamplify-and-forwardrelay.Inaddition,asshownintheproofofthetheoremtobeprovidedlaterinSection 3.4 thisdistinguishabilityis(asymptotically)observablethroughthedecisionstatistick^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1,where^nistheestimatorofndescribedin( 2 ),fornon-manipulablechannels.Therequirementof(A,B)beingnon-manipulableisnotover-restrictiveandissatisedinmanypracticalscenarios.Theorem 3.1 canfurtherbeemployedtocharacterizedetectabilityofmaliciousnessoftherelayinthecontextofDenition 3.1 : Corollary2. Giventhat(A,B)isnon-manipulable,thesequenceofdecisionstatisticsfDnginTheorem 3.1 alsosatisesthefollowingproperties: 1. Iftherelayisnotmalicious(i.e.,kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1!0inprobability),thenlimn!1Pr(Dn>)=0forany>0. 2. Iftherelayismalicious(i.e.,kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1doesnotconvergeto0inprobability),thenlimsupn!1Pr(Dn>)limsupn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)forany>0. 3. IftherelayismaliciousandthereisasubsequencefnMgofattackchannelssatisfyingknM)]TJ /F3 11.955 Tf 11.96 0 Td[(k1!0inprobabilityforsomestochastic6=I,thenthereexists>0suchthatlimsupn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)=1,andforeverysuch,limsupn!1Pr(Dn>)=1. 4. Iftherelayismaliciouswithkn)]TJ /F3 11.955 Tf 11.95 0 Td[(k1!0inprobabilityforsomestochastic6=I,thenthereexists>0suchthatlimn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)=1,andfor 56

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everysuch,limn!1Pr(Dn>)=1.NotethatProperties1and2ofthecorollarytogetherstatethatDn!0inprobabilitywhenandonlywhentherelayinnotmalicious.Properties3and4provideprogressivelystrongermaliciousnessdetectiondifferentiationwhenmorerestrictionsareplacedontheattackchanneln. 3.3Examples 3.3.1MotivatingexampleToillustratetheuseofTheorem 3.1 ,letusrstreconsiderthemotivatingexampleinSection 1.2 .InthenotationofSection 3.1.2 ,X1andX2havethesamebinaryalphabetf0,1g,andtheMACisthebinaryerasureMACdescribedbyU=X1+X2.Thatis,thealphabetsofUandVarebothf0,1,2g.TheBCisidealdenedbyY1=VandY2=V.Inaddition,weassumetheusualequallylikelysourcedistributions,i.e.,X1andX2arei.i.d.equallylikelybinaryrandomvariables.Physically,thismodelapproximatesthescenarioinwhichtwoequal-distanceEthernetnodessendsignals(collision)toabridgenode,orthescenarioinwhichtwopower-controlledwirelessnodessendphasesynchronizedsignals(collision)toanaccesspoint.Inbothscenarios,thesignal-to-noiseratioisassumedtobehigh.ItiseasytocheckthatinthiscaseA=0BBBB@.50.5.50.51CCCCAandB=I33.HenceAmin=1 2,amin=1 15,andbmin=1 12.NotethattheleftnullspaceofAhasdimension1,andtherow-reducedechelonbasismatrixinAlgorithm 2 is(1)]TJ /F3 11.955 Tf 11.77 0 Td[(11).ThusAlgorithm 2 givesthefactthattheleftnullspaceofAdoesnotcontainanynormalized,(amin,0)-doublepolarizedvector.ByTheorem 3.1 anditsproofinSection 3.4 ,weknowthatthesequenceofdecisionstatisticsfk^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1g,where^nisdescribedin 57

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(a)i.i.dattacks,n=103 (b)i.i.dattacks,n=104 (c)i.i.dattacks,n=105 (d)non-ergodicattacks,n=105 Figure3-2.Plotofempiricalcdfsofk^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1obtainedinthemotivatingexamplewiththefourrelaymanipulationmapscorrespondingto1=I,2,3,and4respectively. ( 2 ),satisesproperties1)and2)statedinTheorem 3.1 .Thusanymaliciousrelaymanipulationisdetectableasymptotically. 3.3.2I.i.d.attacksTodemonstratetheasymptoticmaliciousnessdetectionperformancepromisedbyTheorem 3.1 ,andtoinvestigatetheperformancewithniteobservations,weperformedsimulationsforfourdifferentmanipulationmapswhichcorrespondtotherelayrandomlyandindependentlyswitchingitsinputsymbolbysymbolaccordingtotheconditional 58

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pmfsp(vjjui)speciedbythematrices1=I33,2=0BBBB@.99.005.005.005.99.005.005.005.991CCCCA,3=0BBBB@.99.0050.01.99.010.005.991CCCCA,and4=0BBBB@.9900.011.0100.991CCCCA. (3)Obviouslythecaseof1correspondstoanon-maliciousrelayandtheothercasescorrespondtoamaliciousrelay.Theparticularmalicious'swerechosentorepresentdifferentwaysarelaynodemaychoosetoattack.Forthecaseof2,therelaynodechanges1%ofthesymbolsreceivedwithoutregardstowhetherornotthiswillmakethemanipulationobvioustothesourcenodes.Incontrast,theattackof3ismorereservedinwhatitwilldo.Itcanbeseenthattherelay'smanipulationcanonlybeinstantlydetectedbyoneofthesourcenodesatanygiventransmittedvalue.Thatis,therelaytherelayswitches1%ofthereceivedsymbolsinwayslistedoutinTable. 1-1 exceptthoselabeledwithAlice&Bobbothdetect.Finallytheattackof4isthemostcautiousandwillonlytakeanactionthatneithersourcenodecanrecognizeasmanipulationwithoutlookingatmultipleobservations.Forthiscase,therelayswitches2%ofthereceivedsymbolswithvalues0or2to1.ThiscorrespondstotheNotdetectedoutcomesinTable 1-1 .Notethatkn)]TJ /F3 11.955 Tf 11.95 0 Td[(ik1!0inprobabilityasn!1ineachcase.Henceproperty1)ofCorollary 2 appliesforthecaseof1andproperty4)appliesforthecasesof2,3,and4.Indifferentsimulationruns,wesetn=103,104,and105.Fivethousandtrialswererunineachsimulation.Theempiricalcumulativedistributionfunctions(cdfs)ofk^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1obtainedfromthe5000trialsforeachsimulationareplottedinFigs. 3-2(a) 3-2(b) ,and 3-2(c) forthecasesofn=103,104,and105,respectively.Forthese 59

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threecases,thevaluesofchosenindeningtheestimator^nare0.2,0.1,and0.05,respectively.FromFigs. 3-2(b) and 3-2(c) respectivelywithn=104andn=105,aspredictedbyparts1)and4)ofCorollary 2 ,thedecisionstatistick^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1succeedsindifferentiatingbetweenthenon-maliciouscaseof1andthemaliciouscasesof2,3,and4withveryhighcondence.Forinstance,byselectingthedecisionthresholdat=0.065and=0.004respectivelyforthecasesofn=104andn=105,weareabletoobtainverysmallmissandfalsealarmprobabilitiesfordetectingmaliciousnessoftherelay.Forn=103,wecanseefromFig. 3-2(a) thatthereisstilldifferentiationbetweentheempiricalcdfsobtainedforthenon-maliciousandmaliciouscases.Howeverthemaliciousnessdifferentiationcondenceachievedismuchweakerthanthedetectorswiththelargervalueofn.Thissimulationexerciseillustratesthefactthatthedecisionstatisticbasedonthemaximum-normestimatorof( 2 ),whileisconvenientforprovingtheasymptoticdistinguishabilityresultinTheorem 3.1 ,maynotbeasuitablechoiceforconstructingapracticaldetectorwhenthenumberofobservations,n,isnotlarge.Othermoreefcientnite-observationdetectorsmaybeneeded. 3.3.3Non-ergodicattacksTodemonstratepart2)ofCorollary 2 withnon-i.i.dattacks,wesimulatedafewnon-ergodicattacksandconsideredagainthedecisionstatistick^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1.Inthesenon-ergodicattacks,therelaydecideswhetherornottomanipulatethesymbolsdependingonifthechecksumofallobservedsymbolsisevenornot.Conditioningonanevenchecksum,therelaymanipulatesthesymbolsi.i.d.accordingto2,3,and4asdescribedin( 3 ).Notethatthefortheseattacks,limn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)=0.5forall>0.Theresultsforthissimulationwithn=105and=0.01areplottedinFig. 3-2(d) .Fromthegure,therstimportantnoteisthattheempiricalcdfsofthedecisionstatisticexhibitstaircaseshapeswithastepat0.5aspredictedbypart2)ofCorollary 2 .When 60

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A=0BBBB@1 3001 31 301 31 31 301 31 3001 31CCCCA,B=0BB@1.50000.5.70000.3.50000.511CCA,1=I55,2=0BBBB@.99.0025.0025.0025.0025.0025.99.0025.0025.0025.0025.0025.99.0025.0025.0025.0025.0025.99.0025.0025.0025.0025.0025.991CCCCA,3=0BBBB@.99.01 3.002500.005.99.0025.01 30.005.01 3.99.01 3.0050.01 3.0025.99.00500.0025.01 3.991CCCCA,and4=0BBBB@.9850000.0075.985000.0075.0151.015.0075000.985.00750000.9851CCCCA. (3) (a)(A,B)non-manipulable (b)(A,B)manipulable Figure3-3.Plotofempiricalcdfsofk^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1forvariousi.i.d.attacksconsideredinSections 3.3.4 and 3.3.5 therelayisbeingmalicious,itisclearthatbychoosing=0.07,wecanagainobtainsmallmissandfalsealarmprobabilitiesfordetectingmaliciousnessoftherelay. 3.3.4HigherorderexampleLetusconsidertheadditionchannelasshowninFig. 1-1 withbothAliceandBobchoosingtheirsourcesymbolsuniformlyovertheternaryalphabetf0,1,2ginstead. 61

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HencetheinputandoutputalphabetsofRomeoisf0,1,2,3,4ginthiscase.ItiseasytoverifythatthecorrespondingAmatrixisgivenasin( 3 )onthenextpage.Furthermore,supposethatBCfromRomeobacktoAliceandBobisnotideal.Inparticular,letusmodelthemarginalBCfromRomeobacktoAlicebythematrixBgivenin( 3 ).NoticethatthisBhasanon-trivialrightnullspace.Firstweneedtodetermineifthepair(A,B)isnon-manipulable.Todothis,weusedtheAlgorithm 1 presentedinSection 2.2 .Inparticular,weemployedthelinearprogrammingsolverlinprogintheoptimizationtoolboxinMATLABtosolvethelinearprograminstep 2 ofAlgorithm 1 .Theoptimalvaluereturnedwasoftheorderof10)]TJ /F6 7.97 Tf 6.58 0 Td[(16,whichiscloseenoughto0forustodecide(A,B)asnon-manipulable.ThusagainTheorem 3.1 andCorollary 2 applytogivethatthedecisionstatistick^n)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1providesmaliciousnessdetectabilityforthischannel.AsinSection 3.3.2 ,wesimulatedi.i.d.attacksbyRomeo.Thefourdifferent'sshownin( 3 )werethecasesthatweconsideredinthesimulationstudy.Theattackof1correspondstothecaseinwhichRomeotruthfullyforwardsthereceivedsymbols.Fortheattacksof2,3,and4,Romeoalters1%ofthesymbolsthatitreceives.EachofthesethreecaseswasonceagainchosenforaparticularlevelofmaliciousnessasinSection 3.3.2 .Thecaseof2correspondstoanattackinwhichRomeoreturnsvaluesthatheknowswillinstantlyguaranteedetection.Theattackof3onlysendsbackvaluesforwhichitispossibletonotbeinstantlydetected.Finally4correspondstothecaseinwhichRomeoisthemostcautious,andwillnotsendbackanysymbolwhichisinstantlydetectable.Theempiricalcdfsofk^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1obtainedforthefourdifferenti'sareplottedinFig. 3-3(a) forthesimulationrunwithn=105and=0.05.Asbefore,bychoosingadecisionthresholdat=0.07wecanobtainverysmallmissandfalsealarmprobabilitiesfordetectingmaliciousnessofRomeo. 62

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3.3.5Counter-exampleTodemonstratetheconsequenceofhavingamanipulableobservationchannel(A,B),reconsidertheternary-inputexampleofSection 3.3.4 withthemarginalBCfromRomeobacktoAlicespeciedbythefollowingmatrixB=0BBBBBBBBBB@100000.50.3.200.5.2.30.3.2.500.2.30.51CCCCCCCCCCA.Tocheckwhether(A,B)ismanipulable,Algorithm 1 wasagainemployed.Theoptimalvalueofthelinearprograminstep 2 obtainedwas3,thusalertingusthat(A,B)ismanipulable.Indeeditcanbereadilycheckthatforany 2(0,1],thematrix=0BBBBBBBBBB@000000 00000 0)]TJ /F13 11.955 Tf 9.3 0 Td[( 0)]TJ /F13 11.955 Tf 9.3 0 Td[( 00000)]TJ /F13 11.955 Tf 9.3 0 Td[( 0 1CCCCCCCCCCA,isonethatsatisesBA=0requiredinDenition 2.4 tomake(A,B)manipulable.ThereforeTheorem 3.1 tellsusthatmaliciousnessdetectabilityisimpossibleforthischannel.AsinSections 3.3.2 and 3.3.4 ,ani.i.d.attackwith2=I)]TJ /F3 11.955 Tf 12.48 0 Td[(wassimulatedforn=105and=0.05.Thevalueof =1waschoseninthesimulation.Thischoicecorrespondstoanaverageof5 9ofthesymbolsarechangedbyRomeo.Clearlyhavingthismanysymbolschangedwouldbecatastrophicinmostpracticalcommunicationsystems,andisthereforeundesirable.Theempiricalcdfsofthenon-maliciouscaseandthemaliciouscaseof2obtainedfromthesimulationareplottedinFig. 3-3(b) 63

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ItisclearfromthegurethatthecasesforwhichRomeoisbeingmaliciousandnotmaliciousareindistinguishable.Hencethesevereattackof2cannotbedetected. 3.4ProofsofTheoremsandLemmasWeprovethedetectabilityportionofthetheorembyshowingthatthesequenceofdecisionstatisticsfDn=k^n)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1g,where^nistheestimatorforthestochasticmatrixndenedin( 2 ),satisesthetwodesiredpropertiesiftheobservationchannel(A,B)isnon-manipulable.Tothatend,rstnotethat^nisobtainedfromtheconditionalhistogramestimator^)]TJ /F4 7.97 Tf 6.78 4.34 Td[(nofthestochasticmatrix)]TJ /F4 7.97 Tf 6.78 4.34 Td[(ndenedin( 3 ).Weneedthefollowingconvergencepropertyofthesequencef^)]TJ /F4 7.97 Tf 6.78 4.34 Td[(ng: Lemma8. k)]TJ /F4 7.97 Tf 6.77 4.34 Td[(n)]TJ /F3 11.955 Tf 12.21 2.66 Td[(^)]TJ /F4 7.97 Tf 6.77 4.34 Td[(nk1!0inprobabilityasnapproachesinnity. Proof. First,denethejUjjX1jstochasticmatrixnbyits(i,j)thelementas ni,j,n(vi,x1,j) n(x1,j).(3)Forany>0,itisclearthatPrk)]TJ /F4 7.97 Tf 6.78 4.93 Td[(n)]TJ /F3 11.955 Tf 12.2 2.65 Td[(^)]TJ /F4 7.97 Tf 6.78 4.93 Td[(nk1>PrkBn)]TJ /F3 11.955 Tf 12.2 2.65 Td[(^)]TJ /F4 7.97 Tf 6.78 4.93 Td[(nk1> 2+PrkBn)]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F4 7.97 Tf 6.77 4.93 Td[(nk1> 2PrkBn)]TJ /F3 11.955 Tf 12.2 2.65 Td[(^)]TJ /F4 7.97 Tf 6.78 4.93 Td[(nk1> 2+Pr kn)]TJ /F3 11.955 Tf 11.96 0 Td[(nAk1> 2p jX1jjY1jkBk2!. (3)Thusthelemmaisprovedifwecanshowthatthetwoprobabilitiesontherighthandsideof( 3 )convergeto0asnapproachesinnity.Tothatend,noticerstthatkBn)]TJ /F3 11.955 Tf 12.21 2.66 Td[(^)]TJ /F4 7.97 Tf 6.77 4.94 Td[(nk1=jY1jXi=1jX1jXj=1j[Bn]i,j)]TJ /F3 11.955 Tf 12.21 2.66 Td[(^)]TJ /F4 7.97 Tf 6.77 4.94 Td[(ni,jjjY1jXi=1jX1jXj=1jUjXk=1p(y1,ijvk)n(vk,x1,j) n(x1,j))]TJ /F13 11.955 Tf 13.15 8.08 Td[(n(y1,i,vk,x1,j) n(x1,j)| {z }Hi,j,k 64

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wheretheinequalityaboveisduetothefactthatn(y1,i,x1,j)=Pkn(y1,i,vk,x1,j).ThisimpliesthatPrkBn)]TJ /F3 11.955 Tf 12.2 2.65 Td[(^)]TJ /F4 7.97 Tf 6.78 4.93 Td[(nk1> 2jY1jXi=1jX1jXj=1jUjXk=1PrjHi,j,kj> 2jUjjX1jjY1j. (3)Butforanyi,j,andk,PrjHi,j,kj> 2jUjjX1jjY1j4jUj2jX1j2jY1j2 2EH2i,j,k4jUj2jX1j2jY1j2 2Pr(xn1=2Tn[X1],)+E[X1],[H2i,j,k], (3)whereE[X1],[]denotesthatconditionalexpectationE[jxn12Tn[X1],].Further,foranysmallenoughpositive,E[X1],[H2i,j,k]E[X1],(p(y1,ijvk)n(vk,x1,j))]TJ /F13 11.955 Tf 11.95 0 Td[(n(y1,i,vk,x1,j))2 n2(p(x1,j))]TJ /F13 11.955 Tf 11.96 0 Td[()2=Pnm=1E[X1],E(p(y1,ijvk))]TJ /F3 11.955 Tf 11.95 0 Td[(1m(y1,i))2jvn,xn11m(vk,x1,j) n2(p(x1,j))]TJ /F13 11.955 Tf 11.95 0 Td[()2p(y1,ijvk) n(p(x1,j))]TJ /F13 11.955 Tf 11.95 0 Td[()2 (3)wheretheequalityonthethirdlineaboveresultsfromthefactthatp(yn1jvn,xn1)=p(yn1jvn)andtheelementsofyn1areconditionallyindependentgivenvn.Combining( 3 )andthewellknownfact,forexamplesee[ 36 ,Theorem6.2],thatPr(xn1=2Tn[X1],)!0asn!1,wegetfrom( 3 )thatPrjHi,j,kj> 2jUjjX1jjY1j!0asn!1.Using( 3 ),wefurthergetPrkBn)]TJ /F3 11.955 Tf 12.21 2.65 Td[(^)]TJ /F4 7.97 Tf 6.77 4.34 Td[(nk1> 2!0asn!1.Next,notethatwecanrewrite( 3 )asni,j=n(ui,x1,j) n(x1,j))]TJ /F9 11.955 Tf 11.96 11.35 Td[(Xk6=in(vk,ui,x1,j) n(x1,j)+Xk6=in(vi,uk,x1,j) n(x1,j).Similarly,wehave[nA]i,j=p(uijx1,j))]TJ /F9 11.955 Tf 11.95 11.36 Td[(Xk6=in(vk,ui) n(ui)p(uijx1,j)+Xk6=in(vi,uk) n(uk)p(ukjxj). 65

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LetH1i,j=n(ui,x1,j) n(x1,j))]TJ /F5 11.955 Tf 11.95 0 Td[(p(uijx1,j)H2i,j=Xk6=in(vk,ui,x1,j) n(x1,j))]TJ /F13 11.955 Tf 13.15 8.09 Td[(n(vk,ui) n(ui)p(uijx1,j)H3i,j=Xk6=in(vi,uk,x1,j) n(x1,j))]TJ /F13 11.955 Tf 13.15 8.09 Td[(n(vi,uk) n(uk)p(ukjx1,j).Thenwehave,foreachiandj,Pr ni,j)]TJ /F3 11.955 Tf 11.96 0 Td[([nA]i,j> 2jUjjX1jp jX1jjY1jkBk2!3Xl=1Pr jHli,jj> 6jUjjX1jp jX1jjY1jkBk2!. (3)Byemployingthefactthatp(xn1jun,vn)=p(xn1jun)andtheconditionalindependenceoftheelementsofungivenxn1,eachofthethreeprobabilitiesontherighthandsideof( 3 )canbeshowntoconvergeto0asnapproachesinnitybyusingtypicalityargumentssimilartotheonethatshowsPrjHi,j,kj> 2jUjjX1jjY1j!0above.Thus,theprobabilityonthelefthandsideof( 3 )alsoconvergesto0asnapproachesinnity.Finally,Pr kn)]TJ /F3 11.955 Tf 11.96 0 Td[(nAk1> 2p jX1jjY1jkBk2!jUjXi=1jX1jXj=1Pr ni,j)]TJ /F3 11.955 Tf 11.95 0 Td[([nA]i,j> 2jUjjX1jp jX1jjY1jkBk2!!0asn!1. NowweproceedtoshowdetectabilityinTheorem 3.1 .Foranyxed>0,choose^naccordingto( 2 ).ItisclearfromthedenitionofG(^)]TJ /F4 7.97 Tf 6.77 4.34 Td[(n)inSection 2.1.2 thatn2G(^)]TJ /F4 7.97 Tf 6.78 4.34 Td[(n)wheneverk)]TJ /F4 7.97 Tf 6.77 4.34 Td[(n)]TJ /F3 11.955 Tf 12.21 2.66 Td[(^)]TJ /F4 7.97 Tf 6.77 4.34 Td[(nk1 p jX1jjY1j.ThusLemma 8 impliesthatPr(n2G(^)]TJ /F4 7.97 Tf 6.77 4.33 Td[(n))!1asn!1,andhencetheprobabilitythatG(^)]TJ /F4 7.97 Tf 6.78 4.33 Td[(n)isnon-empty 66

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approaches1.Weemploythispropertybelowtoshowthatthesequenceofdecisionstatisticsfk^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1gsatisesbothProperties1and2statedinthetheorem.ToshowProperty1ofTheorem 3.1 ,rstnotethatPrk^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>\kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>Prn2G(^)]TJ /F4 7.97 Tf 6.77 4.94 Td[(n)\k^n)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1>\kn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1>=Prn2G(^)]TJ /F4 7.97 Tf 6.77 4.94 Td[(n)\kn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1>Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>))]TJ /F3 11.955 Tf 11.96 0 Td[(Pr(n=2G(^)]TJ /F4 7.97 Tf 6.77 4.94 Td[(n)) (3)wheretheequalityonthethirdlineisduetothedenitionof^nin( 2 ).Hence,iflimsupn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)>0,( 3 )implieslimsupn!1Pr(k^n)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1>jkn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)=1.ToshowProperty2ofTheorem 3.1 ,considertheconstantsc1,c2,andc3giveninTheorem 2.6 .Letc=2+c3.Bychoosing0< 2c1andmakinguseofTheorem 2.6 ,itiseasytocheckthatPr k)]TJ /F4 7.97 Tf 6.78 4.93 Td[(n)]TJ /F3 11.955 Tf 12.2 2.65 Td[(^)]TJ /F4 7.97 Tf 6.77 4.93 Td[(nk1 p jX1jjY1j\kn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1!Prk^n)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1c\kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1whichinturnimpliesPrk^n)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1>c\kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1Pr k)]TJ /F4 7.97 Tf 6.78 4.94 Td[(n)]TJ /F3 11.955 Tf 12.2 2.66 Td[(^)]TJ /F4 7.97 Tf 6.78 4.94 Td[(nk1> p jX1jjY1j!. (3)HencebyLemma 8 ,ifliminfn!1Pr(kn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1)>0,( 3 )giveslimn!1Pr(k^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>cjkn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1)=0. 67

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Corollary 2 .NowwecanusetheresultsabovetoproveCorollary 2 byshowingthesequenceofdecisionstatisticsfk^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1gsatisesthefourstatedpropertiesunderthecorrespondingspecialcases: kn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1!0inprobability.SincePr(k^n)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1>)Prk^n)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1>kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1 cPrkn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1 c+Prkn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1> cforany>0,Property2ofTheorem 3.1 implieslimn!1Pr(k^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)=0. kn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik190inprobability.Iflimsupn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)=0,thenthereisnothingtoshow.Otherwise,Lemma 8 and( 3 )givelimsupn!1Pr(k^n)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)limsupn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>). knM)]TJ /F3 11.955 Tf 11.96 0 Td[(k1!0inprobabilityand6=I.Notethatthereexists>0suchthatlimsupn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)=1inthiscase.Foranysuch,Property2justaboveimpliestherequiredresult. kn)]TJ /F3 11.955 Tf 11.95 0 Td[(k1!0inprobabilityand6=I.Inthiscase,thereexists>0suchthatlimn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)=1.UsingLemma 8 and( 3 )givesthedesiredresultforanysuch. 3.4.1ConverseRecallthatxn1andyn1arethesequencesofsymbolstransmittedandreceivedbynode1attimeinstants1,2,...,nandn+1,n+2,...,2n,respectively.LetDn=Dn(xn1,yn1)bethenthdecisionstatisticinthesequence,assumedtoexistinthestatementofTheorem 3.1 ,thatsatisesProperties1and2.Supposethatthereexistsastochastic06=IsuchthatB0A=BA,)]TJ /F1 11.955 Tf 6.78 0 Td[(.Fortheidentityrelaymanipulationmap,i.e.,n(un)=un,itiseasytocheckthatp(vnjun)=Qnm=1Im(vn),m(un),n=I,andp(ynjxn)=Qnm=1)]TJ /F17 7.97 Tf 6.78 -1.86 Td[(m(yn1),m(xn1).Sincen=I 68

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foralln,wehaveforevery>0,X(xn1,yn1):Dn(xn1,yn1)>"nYm=1)]TJ /F17 7.97 Tf 6.77 -1.86 Td[(m(yn1),m(xn1)p(x1,m(xn1))#| {z }q(yn1,xn1)=Pr(Dn>)=PrDn>kn)]TJ /F5 11.955 Tf 11.95 0 Td[(Ik1 c!0 (3)wheretheconvergenceonthelastlineisduetotheassumptionthatProperty2ofTheorem 3.1 holds,andcistheconstantdescribedintheproperty.Considernowtherandomrelaymanipulationmapthatresultsinp(vnjun)=Qnm=10m(vn),m(un).Forthisrandommanipulationmap,itisagaineasytocheckthatp(ynjxn)=Qnm=1)]TJ /F17 7.97 Tf 6.77 -1.86 Td[(m(yn1),m(xn1),whichisaconsequenceoftheassumptionthatB0A=)]TJ /F1 11.955 Tf 19.46 0 Td[(,andkn)]TJ /F3 11.955 Tf 11.95 0 Td[(0k1!0inprobability.Since06=I,thereexistsa>0suchthatlimn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)=1.Forthis,limsupn!1X(xn1,yn1):Dn(xn1,yn1)>q(yn1,xn1)=limsupn!1Pr(Dn>)limsupn!1PrDn>kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>limn!1Pr(kn)]TJ /F5 11.955 Tf 11.96 0 Td[(Ik1>)=1 (3)wheretheequalityonthelastlineisduetotheassumptionthatProperty1ofTheorem 3.1 holds.Theconclusionsin( 3 )and( 3 )areclearlyinconictandcannotbesimultaneouslytrue.Thereforetherecannotexistastochastic06=IsuchthatB0A=BAandProperties1and2hold.Tonishtheproofuse 2.2 69

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CHAPTER4STRONGINFORMATIONINTEGRITYINTHETWOWAYTWOHOPRELAYInchapters 2 and 3 theoremsandresultswerepresentedthatdealtwithscenariosthatlackedanyerrorcontrolcoding.Thisassumptionisseverelyrestrictingasallbutperfectchannelsmayuseerrorcontrolcodingtoincreasetheirrateoftransmission.Thepurposeofthischapteristointroduceerrorcontrolcodingintoourmodelandstudytheaffects.Inparticularwewishtodeterminethemaximumpossiblerateatwhichcommunicationispossiblewhileguaranteeingintegrity,thisratewillbereferredtoastheintegrityrateandthecorrespondingsetofratesforwhichintegrityispossiblewillbetheintegrityrateregion.1Itiswellknownthatreducingtherateofacodewillgenerallyimprovetheprobabilitythatadestinationdecodesthemessagecorrectly.Inthecontextofintegritythoughitalsomayleakinformationtotherelay,informationwhichitcouldthenusetodisruptcommunication.Thuswehavetwoantipodalgoalsinthatwedesirethecodetohaveahighenoughrateastonotleakinformationtotherelayobtainwhilemaintainingalowenoughrateforthedestinationtodecode.Thesegoalswillhavedirectconsequenceontheintegrityrateregionofasystem,whereinsection 4.3 theupperboundderivedwillbeduetotherequirementofdecoding,whilealowerboundonthetotalnumberofcodewordswillbeobtainedfromourneedtonotleakinformation.Webeginbyseparatingthedecoderanddetector.WewillrefertointegrityachievedinsuchamannerasStrongIntegrity.Notsurprisingly,thesemethodswillendupbeingheavilydependentupontheresultspresentedinchapter 2 ,andmanipulabilitywillonceagainbepivotalintheassuranceofintegrity.Aspreviouslywerstbeginwiththerelevantnotationinsection 4.1 ,beforeonceagainpresentingthechannel 1Becauseintegrityisabinaryevent,thereisnoambiguityinthemeaningofintegrityrate. 70

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modelinsection 4.2 .Subsequentlywewillpresentthetheoremsandstreamlinedproofsthereofinsection 4.3 .Theseproofswillcenteraroundnotonlytheachievablerateregionrelatingtothenaldestination,buttotheachievabledetectionprobabilitygivenmodication.Itshouldbenotedthattheseresultsaregeneralinthesensethatnoassumptionismadeonwhattherelaymaydo.Finallywepresentthelemmasinsection 4.4 whichjustifythestepstakeninsection 4.2 4.1NotationWeadoptthefollowingnotationalconvention.Weuseupper-casescriptletterstodenotediscretealphabets.Foranalphabetdenotedbyanupper-casescriptletter,weusethecorrespondinglower-caseandupper-caseletterstoindicateanelementinthealphabetandarandomvariabletakingvaluefromthealphabet,respectively.Forexample,letXdenoteanalphabet.ThenxandXareanelementinXandarandomvariableoverX,respectively.Inmanyoccasionsbelow,wewillhavestatementsthatarevalidovereveryelementinthealphabet.Insuchcases,wewillemploysanseriffacefontsinplaceoftheregularfacefontstoimplythatthequantitiesinconsiderationrangeovertheircorrespondingalphabets.Forexample,xmeansforallx2X.Further,weusexn2XndenoteasequenceofnsymbolsdrawnfromthealphabetofX.Theithsymbolofxnwillbedenotedasxi.WewillusetheboldfaceletterscandCtodenoteorderedcollectionsofsequencesfromanalphabet,i.e.,codebooks.Forexample,CnX1isarandomcodebookoflength-nsequenceswithsymbolsdrawnfromX1,whilecnX1isaxedone.Tosimplifynotation,wewilldropthealphabetsubscript,i.e.,therandomandxedcodebookswillberespectivelydenotedbyCn1andcn1.MoreoverCn1(i)denotestheithcodewordinCn1.Generally,weemploylower-caseGreekletters,suchasn,todenotesequencesofsmallpositivenumbers.WewillemploytheBachmannLandaunotationtodescribetheasymptoticorderofgrowthofsequencesofpositivenumbers.Inaddition,wewillinmanyoccasionsapply[ 35 ,Lemma2.13]toboundthecardinalityofatypicalset.For 71

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(a)Multiple-accesschannel (b)Broadcastchannel Figure4-1.Two-way,half-duplex,amplify-and-forwardrelaymodel. example,Tn[X]n2n[H(X)+n],wheren2)]TJ /F6 7.97 Tf 6.67 -4.65 Td[(logn n)]TJ /F13 11.955 Tf 11.96 0 Td[(nlogn.Wewillreservethesymbolforthispurposethroughoutthepaper.Finally,weuse1()todenotetheindicatorfunction.Wealsoneedthefollowingdenitionsaboutvectors. 4.2ChannelModelConsiderthetwo-way,half-duplexrelaychannelmodelshowninFig. 4-1 ,inwhichtwonodes(1and2)simultaneouslysendsymbolstoarelaynodethroughadiscrete,memorylessmultiple-accesschannel(MAC).Thehalf-duplexrelaynodeisthensupposedtobroadcastitsreceivedsymbolsbacktothetwonodesintheamplify-and-forward(AF)manner.Forsimplicity,weassumethatthebroadcastchannel(BC)fromtherelaybacktothenodesisperfect.Thatis,bothnodesperfectlyobservethesymbolssentoutbytherelay.Thereissomepossibilitythattherelaymaymodifyitsreceivedsymbolsinanattempttoconductasubstitutionattack.Thedesigngoalisforeachnodetoatleastdetectanymaliciousactoftherelayintheeventthatitcannotdecodetheinformationsentbytheothernode;inotherwords,toguaranteetheintegrityoftheinformationforwardedbytherelay. 72

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Morespecically,letX1,X2,andUdenotethediscretealphabetsofnode1'sinput,node2'sinput,andtheoutputoftheMAC.Overtimeinstants1,2,...,n,supposethatnodes1and2transmitthesymbolsequencesxn1andxn2,respectively,throughthememorylessMAC.TheoutputsequenceUnisconditionallydistributedaccordingto p(unjxn1,xn2)=nYi=1PUjX1,X2(uijx1,i,x2,i)(4)wheretheconditionalpmfPUjX1,X2(ujx1,x2)speciestheMAC.Therelaynode,duringtimeinstants1,2,...,n,observestheoutputsymbolsequenceUnoftheMAC,processes(ormanipulates)it,andthenbroadcaststheprocessedsymbolsequencetonodes1and2attimeinstantsn+1,n+2,...,2nviatheperfectBC.LetVbethealphabetoftherelay'sprocessedsymbols.BecausetherelayissupposedtoworkintheAFmanner,theremustbeaone-to-onecorrespondencebetweentheelementsofUandV.Thus,withoutlossofgenerality,wemayassumethatV=U.LetVndenotetherelay'soutputsequence.TheassumptionofperfectBCfromtherelaytothenodesimpliesthatY1=Y2=U,PY1jV=PY2jV=I,and p(yn1,yn2jvn)=1(yn1=yn2=vn).(4)Forconveniencehereafter,wesimplymakeVnthesymbolsequenceobservedbybothnodes.LetR1andR2betwopositiverates.Foreachpositiveintegern,considertheencoder-decoderquadruple(Cn1,Cn2,gn1,gn2):Cn1:f1,2,...,2nR1g!Xn1Cn2:f1,2,...,2nR2g!Xn2gn1:Unf1,2,...,2nR1g!f1,2,...,2nR2g[f!ggn2:Unf1,2,...,2nR2g!f1,2,...,2nR1g[f!g 73

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whereCn1andgn1aretheencoderanddecoderusedbynode1,andCn2andgn2aretheencoderanddecoderusedbynode2.NotethatweallowtheencodersCn1andCn2toberandom.Thesymbol!inthedecoderoutputalphabetsdenotesthedecisionthatthereceivedsequenceisdeemeduntrustworthy,i.e.,therelayhaspossiblybeenmalicious.LetW1andW2beindependentmessagesofnodes1and2thatareuniformlydistributedoverf1,2,...,2nR1gandf1,2,...,2nR2g,respectively.ThenXn1=Cn1(W1)andXn2=Cn2(W2)arethecodewordssentbynodes1and2totherelaythroughtheMAC.ThepotentialmanipulationbytherelayisspeciedbytheconditionaldistributionofVngiventheotherrandomquantitiesmentionedabove.WeimposetheMarkovityrestrictionontheconditionaldistributionthat p(vnjun,xn1,xn2,w1,w2,cn1,cn2)=p(vnjun,cn1,cn2)(4)whichmeansthattherelaymaypotentiallymanipulatethetransmissionbasedonlyontheoutputsymbolsoftheMACthatitobservesaswellasitsknowledgeaboutthecodebooksusedbythenodes.Ifp(vnjun,cn1,cn2)=1(vn=un),thenweregardtherelayasnon-malicious.Otherwisetherelayismalicious.Withtheschedulingandcodingschemedescribedabove,wesaythattheratepair(R1,R2)isachievablewithinformationintegrityifthereexistsasequenceofencoder-decoderquadruplesf(Cn1,Cn2,gn1,gn2)gsuchthat:2Givenany">0and>0,Prfgn1(Vn,W1)6=W2[gn2(Vn,W2)6=W1jd(Vn,Un)=0g" 2Theerroreventprobabilitiesemployedheretodeneachievabilityareonesthataverageoverallmessages.NotethatthecapacitytheoremspresentedinSection 4.3 remainunchangediftheaveragederrorprobabilitiesusedinthedenitionherearereplacedbythemaximalconditionalerrorprobabilitiesconditionedoveranymessages.Theclaimcanbeeasilyveriedbythestandardcodewordexpurgationprocedure[ 9 ]. 74

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wheneverPrfd(Vn,Un)=0gPrfgn1(Vn,W1)=2fW2,!g[gn2(Vn,W2)=2fW1,!gjd(Vn,Un)>0g"wheneverPrfd(Vn,Un)>0g (4)forallsufcientlylargen.Notethatthesecondrequirementaboveforcesthedecoderstoeitherdetectthesubstitutionattackbytherelayorcorrectthesymbolsmodied.Thereishowevernorequirementforthedecoderstodetectanymaliciousnessoftherelaywhentheycandecodetheintendedmessage.Toenforcemaliciousnessdetection,weconsiderthefollowingpairofdetectionfunctions:hn1:Unf1,2,...,2nR1g!fX,!ghn2:Unf1,2,...,2nR2g!fX,!gemployedbynodes1and2,respectively.ThesymbolXindicatesthatthedetectorsthinktherelayisnon-malicious,whilethesymbol!correspondstomaliciousnessasbefore.Inordertoavoidconictingmaliciousnessdecisionsmadebythedecodinganddetectionfunctions,weimposetherestrictionsonh1andh2that 1. ifg1(vn,w1)=!,thenh1(vn,w1)=! 2. ifg2(vn,w2)=!,thenh2(vn,w2)=!forallw1,w2,andvn.Incorporatingtheadditionalmaliciousnessdetectionrequirement,wesaythattheratepair(R1,R2)isachievablewithf(n)-stronginformationintegrityifthereexistsasequenceofencoder-decoder-detectorsextuplesf(Cn1,Cn2,gn1,gn2,hn1,hn2)gsatisfying( 4 )andthefollowingrequirements:2Givenany">0,>0,and'n2!(f(n))\o(1),thereexistsapositiveconstantc1suchthatPrfhn1(Vn,W1)=hn2(Vn,W2)=Xjd(Un,Vn)c'ng1)]TJ /F13 11.955 Tf 11.95 0 Td[(" 75

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wheneverPrfd(Vn,Un)c'ngPrfhn1(Vn,W1)=hn2(Vn,W2)=!jd(Un,Vn)>'ng1)]TJ /F13 11.955 Tf 11.95 0 Td[("wheneverPrfd(Vn,Un)>'ng (4)forallsufcientlylargen.Thecapacityregionwith(f(n)-strong)informationintegrityisthendenedastheclosureofthesetofachievableratepairswith(f(n)-strong)informationintegrity.NotethatwehavenotcountedtheuseoftheperfectBCfromtherelaybacktothenodesintheratedenitionabove.Ifthatistobecounted,thefactorof0.5shouldbeaddedtoallratesbecausethexedtransmissionscheduledescrivedabove.Notethatfrom( 4 ),( 4 ),and( 4 ),thejointdistributionoftherandomvariables(Vn,Un,Xn1,Xn2,W1,W2,Cn1,Cn2)isgivenbyp(vn,un,xn1,xn2,w1,w2,cn1,cn2)=p(vnjun,cn1,cn2)p(unjxn1,xn2)1(cn1(w1)=xn1)1(cn2(w2)=xn2)p(w1)p(w2)p(cn1,cn2). (4)Asin[ 37 ]toavoiddealingwithtrivialcases,weassumethroughthepaperthatjUj2.Inaddition,foranyPX1(x1)>0andPX2(x2)>0,neitherPUjX1norPUjX2containsanyall-zerorows,wherebothmatricesareinducedfromPU,X1,X2(u,x1,x2)=PUjX1,X2(ujx1,x2)PX1(x1)PX2(x2).Ifthisassumptiondoesnothold,wecanreinforceitbyremovingsymbolsthatviolatetheassumptionfromUanddeletingthecorrespondingrowsinPUjX1andPUjX2,withoutaffectingthesystemmodel. 76

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4.3Achievability 4.3.1Theorem Theorem4.1. Aninnerboundonthelogn p n-stronginformationintegritycapacityregionistheclosureoftheconvexhullofall(R1,R2)satisfyingR1
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condition:I(X1;U)I(X1,X2;U), (4)independentlyanduniformlypick2nR1codewordsCn1(1),Cn1(2),...,Cn1(2nR1)fromthetypicalsetTn[X1]n,wherenisapositivesequencethatwillbespeciedlater.Similarly,pick2nR2independentcodewordsCn2(1),Cn2(2),...,Cn2(2nR2)fromthetypicalsetTn[X2]n.Then,inthiscase,instantiationsofCn1andCn2denethedeterministicencodingfunctionsfornodes1and2,respectively.Ontheotherhand,if(R1,R2)doesnotsatisfytheconditionin( 4 ),chooseanotherpair(R01,R02),withR01R1andR02R2,thatdoes.Randomlypick2nR01and2nR02codewordsfromTn[X1]nandTn[X2]nasabovetoformcodebooksfornodes1and2,respectively.Pickindependent(ofallotherrandomquantities)randomnumbersW01uniformlyfromf1,2,...,2n(R01)]TJ /F4 7.97 Tf 6.58 0 Td[(R1)gandW02uniformlyfromf1,2,...,2n(R02)]TJ /F4 7.97 Tf 6.58 0 Td[(R2)g.ThenmapW1toCn1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((W1)]TJ /F3 11.955 Tf 11.96 0 Td[(1)2n(R01)]TJ /F4 7.97 Tf 6.59 0 Td[(R1)+W01andW2toCn2)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((W2)]TJ /F3 11.955 Tf 11.95 0 Td[(1)2n(R02)]TJ /F4 7.97 Tf 6.58 0 Td[(R2)+W02.Inotherwords,thecodebookofeachuserispartitionedintoeachsizesubsets,eachofwhichisassociatedtoauniquemessage.Theactualcodewordsentisuniformlychosenfromthesubsetassociatedwiththemessagetobesent.Notethattheresultingencodingfunctionsarerandomlyinthiscase.Clearlyifwecandecodeto(W1)]TJ /F3 11.955 Tf 12.07 0 Td[(1)2n(R01)]TJ /F4 7.97 Tf 6.59 0 Td[(R1)+W01,wecanalsoobtainW1,andifwecandecodeto(W2)]TJ /F3 11.955 Tf 12.5 0 Td[(1)2n(R02)]TJ /F4 7.97 Tf 6.59 0 Td[(R2)+W02,wecanalsoobtainW2.Therefore,wemaysimplyassumebelow(R1,R2)satises( 4 ),andemploythedeterministicencoderswithoutlossofanygenerality.Foreaseofdescribingthedecodinganddetectionfunctionsaswellasforsimplernotationintheerroranalysislater,letusrstdenethefollowingsubsetsofUn:U(xn1,xn2),nun:(un,xn1,xn2)2Tn[U,X1,X2]2o, 78

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V(xn1;w2,cn2),(un:[^w26=w2(un,xn1,cn2(^w2))\Tn[U,X1,X2]26=;),~V(xn1;cn2),(un:[w2(un,xn1,cn2(w2))\Tn[U,X1,X2]26=;).Forachosencodebookpair(cn1,cn2),nodes1and2employthetypicalitydecodersanddetectorsasrespectivelydenedbelow:gn1(vn,w1)=8>>>>>><>>>>>>:^w2ifvn2Un+n(cn1(w1),cn2(^w2))andthereisnow026=^w2suchthatvn2Un(cn1(w1),cn2(w02)),!otherwise,gn2(vn,w2)=8>>>>>><>>>>>>:^w1ifvn2Un+n(cn1(^w1),cn2(w2))andthereisnow016=^w1suchthatvn2Un(cn1(w01),cn2(w2)),!otherwise,hn1(vn,w1)=8>><>>:Xifgn1(vn,w1)6=!!ifgn1(vn,w1)=!hn2(vn,w2)=8>><>>:Xifgn2(vn,w2)6=!!ifgn2(vn,w2)=!,wherendependsonn,n'n,andnn+n.Thedetailedchoicesofthesequencesn,n,andnwillbespeciedlaterinSection 4.3.5 .Clearlythedetectionfunctionsdenedaboveareconsistentwiththecorrespondingdecodingfunctions.Tosimplifynotation,wewillusebelowthefollowing(n)constants:0n,2(1+jX1j)n,00n,2(1+jX2j)n,~n,2(1+jX1jjX2j)n,~0n,2(1+jX1j)jX2jn,~00n,2(1+jX2j)jX1jn,^0n=2jX2jn,^n=2jX1jjX2jn,_n,2(1+jUj)n,and_0n,2(1+jUj)jX2jnintheerroranalysisofSections 4.3.3 and 4.3.4 below.Similarsetsofconstantswiththesymbolreplacedbyand,respectively,willalsobeused. 79

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4.3.3ErroranalysisunderminimalmodicationConsiderboundingPrfgn1(Vn,W1)6=W2,d(Vn,Un)ngasfollows:Prfgn1(Vn,W1)6=W2,d(Vn,Un)ngPrfUn=2Un(Cn1(W1),Cn2(W2))g+PrfUn2Un(Cn1(W1),Cn2(W2)),gn1(Vn,W1)6=W2,d(Vn,Un)ng(a)PrfUn=2Un(Cn1(W1),Cn2(W2))g+PrfUn2Un(Cn1(W1),Cn2(W2)),Vn2Un+n(Cn1(W1),Cn2(W2))\Vn(Cn1(W1);W2,Cn2),d(Vn,Un)ng (4)whereinequality(a)isaconsequenceofLemma 11 thatif(un,vn)satisesun2Un(cn1(w1),cn2(w2))andd(vn,un)n,thenvn2Un+n(cn1(w1),cn2(w2)).Thuswehavetoboundtheprobabilitiesofbotheventsontherighthandsideof( 4 ).ToboundPrfUn=2Un(Cn1(W1),Cn2(W2))gnotethatPrfUn=2Un+n(Cn1(W1),Cn2(W2))gPrn(Cn1(W1),Cn2(W2))=2Tn[X1,X2]no+Prn(Cn1(W1),Cn2(W2))2Tn[X1,X2]n,Un=2Un(Cn1(W1),Cn2(W2))o. (4)By( 4 ),wehavePrn(Cn1(W1),Cn2(W2))=2Tn[X1,X2]no(a)=Xw1,w224Xcn1,cn21(cn1(w1),cn2(w2))=2Tn[X1,X2]np(cn1,cn2)35p(w1)p(w2)(b)(n+1)2jX1jjX2j2nnlogn (4)where,notingthatcodewordsarechosenuniformlyfromthetypicalsets,inequality(b)isobtainedbyapplyingLemma 10 tothebracketedinnersumin(a)withX1asX,X2asY, 80

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andnchosenaccordingto( 4 ).Ontheotherhand,againby( 4 ),wehavePrn(Cn1(W1),Cn2(W2))2Tn[X1,X2]n,Un=2Un(Cn1(W1),Cn2(W2))o(a)=Xw1,w2Xcn1,cn224Xun=2Un(cn1(w1),cn2(w2))p(unjcn1(w1),cn2(w2))351(cn1(w1),cn2(w2))2Tn[X1,X2]np(cn1,cn2)p(w1)p(w2)2jUjjX1jjX2je)]TJ /F6 7.97 Tf 6.59 0 Td[(2n2n (4)wherewehaveboundedthebracketedinnersumin(a)by2jUjjX1jjX2je)]TJ /F6 7.97 Tf 6.59 0 Td[(2n2nbythecombinationof( 4 ),part1)ofLemma 9 ,and[ 35 ,Lemma2.12].Puttingtheboundsin( 4 )and( 4 )backinto( 4 ),weget PrfUn=2Un(Cn1(W1),Cn2(W2))g(n+1)2jX1jjX2j2nnlogn+2jUjjX1jjX2je)]TJ /F6 7.97 Tf 6.59 0 Td[(2n2n.(4)NexttoboundPrfUn2Un(Cn1(W1),Cn2(W2)),Vn2Un+n(Cn1(W1),Cn2(W2))\Vn(Cn1(W1);W2,Cn2),d(Vn,Un)ng,recallfrom( 4 ),wehavePrfUn2Un(Cn1(W1),Cn2(W2)),Vn2Un+n(Cn1(W1),Cn2(W2))\Vn(Cn1(W1);W2,Cn2),d(Vn,Un)ngXvn,un,cn1,cn2,w1,w2,^w21(un,cn1(w1),cn2(w2))2Tn[U,X1,X2]2n1(d(vn,un)n)1(^w26=w2)1(vn,cn1(w1),cn2(w2))2Tn[U,X1,X2]2(n+n)1(vn,cn1(w1),cn2(^w2))2Tn[U,X1,X2]2np(vnjun,cn1,cn2)p(unjcn1(w1),cn2(w2))p(w1)p(w2)p(cn1,cn2)(a)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n[H(UjX1,X2))]TJ /F17 7.97 Tf 6.59 0 Td[(_n]Xvn,un,w1,w2,^w2,cn11un2Tn[UjX1]_0n(cn1(w1))1(^w26=w2)1(d(vn,un)n)p(w1)p(w2)p(cn1)Xcn21(vn,cn1(w1),cn2(w2))2Tn[U,X1,X2]2(n+n)1(vn,cn1(w1),cn2(^w2))2Tn[U,X1,X2]2np(cn2) 81

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(b)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n[H(UjX1,X2))]TJ /F17 7.97 Tf 6.59 0 Td[(_n]Xw1,w2,^w2,cn11(^w26=w2)p(w1)p(w2)p(cn1)Xun1un2Tn[UjX1]_0n(cn1(w1))Xvn1(d(vn,un)n)1(vn,cn1(w1))2Tn[U,X1](^0n+^0n)Xcn2p(cn2)1cn2(w2)2Tn[X2jU,X1](00n+00n)(vn,cn1(w1))1cn2(^w2)2Tn[X2jU,X1]00n(vn,cn1(w1))(c)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n[H(UjX1,X2))]TJ /F17 7.97 Tf 6.59 0 Td[(_n]2)]TJ /F6 7.97 Tf 6.59 0 Td[(2n[I(X2;UjX1))]TJ /F6 7.97 Tf 6.58 0 Td[(200n]Xw1,w2,^w2,cn11(^w26=w2)p(w1)p(w2)p(cn1)Xun1un2Tn[UjX1]_0n(cn1(w1))Xvn1(d(vn,un)n)(d)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n[I(X2;UjX1))]TJ /F17 7.97 Tf 6.59 0 Td[(nlogjUj)]TJ /F6 7.97 Tf 17.37 0 Td[(2_0n)]TJ /F6 7.97 Tf 6.59 0 Td[(400n]Xw1,w2,^w2,cn11(^w26=w2)p(cn1)p(w1)p(w2)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n[I(X2;UjX1))]TJ /F4 7.97 Tf 6.59 0 Td[(R2)]TJ /F17 7.97 Tf 6.59 0 Td[(nlogjUj)]TJ /F6 7.97 Tf 17.37 0 Td[(2_0n)]TJ /F6 7.97 Tf 6.59 0 Td[(400n] (4)whereinequality(a)isobtainedfromapplyingtheupperbound2)]TJ /F4 7.97 Tf 6.59 0 Td[(n[H(UjX1,X2))]TJ /F17 7.97 Tf 6.58 0 Td[(_n]top(unjcn1(w1),cn1(w1))usingpart2)ofLemma 9 and[ 35 ,Lemma2.13andProblem2.5],inequality(b)isobtainedbyapplyingpart2)ofLemma 9 repeatedly,andinequality(d)resultsfromcountingthenumberofsequencesvn'sthatareataHammingdistancennfromeachunandapplying[ 35 ,Lemma2.13]toboundthesumoverunin(c)by2n[H(UjX1)+_0n].Moreover,inequality(c)isduetothedesignthatdifferentcodewordsareindependentlyselectedineachcodebookandthefactthatforevery(vn,cn1(w1))2Tn[U,X1]^0n+^0n,PrnCn2(^w2)2Tn[X2jU,X1]00n(vn,cn1(w1))o=Tn[X2jU,X1]00n(vn,cn1(w1)) Tn[X2]n2n[H(X2jU,X1)+00n] 2n[H(X2))]TJ /F17 7.97 Tf 6.59 0 Td[(n]2)]TJ /F4 7.97 Tf 6.59 0 Td[(n[I(X2;UjX1))]TJ /F6 7.97 Tf 6.58 0 Td[(200n]wherewehaveused[ 35 ,Lemma2.13].NotethatthesameboundalsoappliestoPrnCn2(w2)2Tn[X2jU,X1]00n+00n(vn,cn1(w1))osince00n00n+00n.Nowapplyingthebounds 82

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( 4 )and( 4 )to( 4 ),weobtainPrfhn1(Vn,W1)=!,d(Vn,Un)ngPrfgn1(Vn,W1)6=W2,d(Vn,Un)ng(n+1)2jX1jjX2j2nnlogn+2jUjjX1jjX2je)]TJ /F6 7.97 Tf 6.58 0 Td[(2n2n+2)]TJ /F4 7.97 Tf 6.58 0 Td[(n[I(X2;UjX1))]TJ /F4 7.97 Tf 6.59 0 Td[(R2)]TJ /F17 7.97 Tf 6.59 0 Td[(nlogjUj)]TJ /F6 7.97 Tf 17.37 0 Td[(2_0n)]TJ /F6 7.97 Tf 6.59 0 Td[(400n]. (4) 4.3.4ErroranalysisundersignicantmodicationUnderthiscase,wearetoboundPrfgn1(Vn,W1)=2fW2,!g,d(Vn,Un)>ngandPrfhn1(Vn,W1)=X,d(Vn,Un)>'ng.Tothatend,foreachpairofsequence(un,vn)denethetripleofrandomvariables(X1,U,V)asbelow:(X1,U,V)=argmax(~X1,~U,~V)H(~X1j~U,~V)subjecttoP~U,~V(u,v)=Pun,vn(u,v)P~X1j~U(x1ju))]TJ /F5 11.955 Tf 11.96 0 Td[(PX1jU(x1ju)~0nP~X1j~V(x1jv))]TJ /F5 11.955 Tf 11.95 0 Td[(PX1jU(x1jv)~0n+~0n (4)wheretherandomvariables~X1,~U,~Vthatappearsintheconstraintsof( 4 )respectivelytakevaluesoverX1,U,andU.Foranysequencen,denethefollowingsubsetofUnVn: En,f(un,vn):H(X1jU)>H(X1jU,V)+ng.(4)ThedetailedchoiceofnwillbegiveninSection 4.3.5 .FirstnotethatPrfgn1(Vn,W1)=2fW2,!g,d(Vn,Un)>ngPrfUn=2Un(Cn1(W1),Cn2(W2))g 83

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+PrfUn2Un(Cn1(W1),Cn2(W2)),gn1(Vn,W1)=2fW2,!g,d(Vn,Un)>ng(a)PrfUn=2Un(Cn1(W1),Cn2(W2))g+PrfUn2Un(Cn1(W1),Cn2(W2)),Vn2Vn+n(Cn1(W1);W2,Cn2),(Un,Vn)2EngPrfUn=2Un(Cn1(W1),Cn2(W2))g+PrfUn2Un(Cn1(W1),Cn2(W2)),Vn2~Vn+n(Cn1(W1);Cn2),(Un,Vn)2Eng. (4)Toobtaininequality(a)in( 4 ),rstnoticethatgn1(vn,w1)=2fw2,!gimpliesvn2Vn+n(cn1(w1);w2,cn2).Thenforanypair(un,vn)suchthatun2Un(cn1(w1),cn2(w2))andgn1(vn,w1)=2fw2,!g,itcanbeseenbyapplyingpart2)ofLemma 9 repeatedlythatunandvnareinTn[U]^nandTn[U]^n+^n,respectively.Inaddition,if(un,vn)=2En,thenpart2)ofLemma 13 impliesthatvn2Un(cn1(w1),cn2(w2)),wherenisgivenby( 4 )intheproofoflemma.Thus,bychoosingthedecodingfunctiongn1inSection 4.3.2 withthisn,wecanconcludethatfUn2Un(Cn1(W1),Cn2(W2)),gn1(Vn,W1)=2fW2,!g,(Un,Vn)=2Eng=;.Similarly,bychoosingninrelationto'n(cf.Section 4.3.5 )asprovidedbypart1)ofLemma 13 ,wehavePrfhn1(Vn,W1)=X,d(Vn,Un)>'ngPrfUn=2Un(Cn1(W1),Cn2(W2))g+PrfUn2Un(Cn1(W1),Cn2(W2)),Vn2~Vn+n(Cn1(W1);Cn2),(Un,Vn)2Eng, (4)sincehn1(vn,w1)=Ximpliesvn2~Vn+n(cn1(w1);cn2)andfUn2Un(Cn1(W1),Cn2(W2)),hn1(Vn,W1)=X,(Un,Vn)=2En,d(Vn,Un)>'ng=;.NowthetermPrfUn=2Un(Cn1(W1),Cn2(W2))gintheboundsof( 4 )and( 4 )ishandledin( 4 )before.ItthusremainstoboundPrfUn2Un(Cn1(W1),Cn2(W2)),Vn2~Vn+n(Cn1(W1);Cn2),(Un,Vn)2Eng.Todoso,denethefollowingsetsforconvenience:Q1(un,vn;cn1),nw1:cn1(w1)2Tn[X1jU]~0n(un)\Tn[X1jU]~0n+~0n(vn),(un,cn1(w1))2Tn[U,X1]^0no 84

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Q2(un,vn;cn1,cn2),8<:w2:cn2(w2)2[w12Q1(un,vn;cn1)Tn[X2jU,X1]00n(un,cn1(w1))9=;Q3(un;xn2;cn1),nw1:cn1(w1)2Tn[X1jU,X2]0n(un,xn2)o. (4)ClearlywemusthavejQ1(un,vn;cn1)j2nR1,jQ2(un,vn;cn1,cn2)j2nR2,andjQ3(un;xn2;cn1)j2nR1.Fixany>1,andlet(),ln)]TJ /F13 11.955 Tf 12.95 0 Td[(+1.From( 4 ),wehavePrfUn2Un(Cn1(W1),Cn2(W2)),Vn2~Vn+n(Cn1(W1);Cn2),(Un,Vn)2Eng=Xun,vn,cn1,cn2,w1,w21((un,vn)2En)1(un2Un(cn1(w1),cn2(w2)))1)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(vn2~Vn+n(cn1(w1);cn2)p(vnjun,cn1,cn2)p(unjcn1(w1),cn2(w2))p(w1)p(w2)p(cn1,cn2)(a)2)]TJ /F4 7.97 Tf 6.58 0 Td[(n[H(UjX1,X2)+R1+R2)]TJ /F17 7.97 Tf 6.59 0 Td[(_n]Xun,vn,cn1,cn21((un,vn)2En)p(vnjun,cn1,cn2)p(cn1,cn2)Xw1,w21(un2Un(cn1(w1),cn2(w2)))1)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(vn2~Vn+n(cn1(w1);cn2)(b)2)]TJ /F4 7.97 Tf 6.58 0 Td[(n[H(UjX1,X2)+R1+R2)]TJ /F17 7.97 Tf 6.59 0 Td[(_n]Xun,vn,cn1,cn21un2Tn[U]^n1vn2Tn[U]^n+^n1((un,vn)2En)p(vnjun,cn1,cn2)p(cn1,cn2)Xw1,w21cn1(w1)2Tn[X1jU]~0n(un)\Tn[X1jU]~0n+~0n(vn)1(un,cn1(w1))2Tn[U,X1]^0n1cn2(w2)2Tn[X2jU,X1]00n(un,cn1(w1))1(un,cn2(w2))2Tn[U,X2]^00n1cn1(w1)2Tn[X1jU,X2]0n(un,cn2(w2))2)]TJ /F4 7.97 Tf 6.58 0 Td[(n[H(UjX1,X2)+R1+R2)]TJ /F17 7.97 Tf 6.59 0 Td[(_n]Xun,vn,cn1,cn21un2Tn[U]^n1vn2Tn[U]^n+^n1((un,vn)2En)p(vnjun,cn1,cn2)p(cn1,cn2)Xw210@cn2(w2)2[w12Q1(un,vn;cn1)Tn[X2jU,X1]00n(un,cn1(w1))1A1(un,cn2(w2))2Tn[U,X2]^00nXw11cn1(w1)2Tn[X1jU,X2]0n(un,cn2(w2))2)]TJ /F4 7.97 Tf 6.58 0 Td[(n[H(UjX1,X2)+R1+R2)]TJ /F17 7.97 Tf 6.59 0 Td[(_n]Xun,vn1un2Tn[U]^n1vn2Tn[U]^n+^n1((un,vn)2En) 85

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Xcn1,cn2jQ2(un,vn;cn1,cn2)j0B@maxw2:(un,cn2(w2))2Tn[U,X2]^00njQ3(un;cn2(w2);cn1)j1CAp(vnjun,cn1,cn2)p(cn1,cn2)(c)2)]TJ /F4 7.97 Tf 6.58 0 Td[(n[H(UjX1,X2)+R1+R2)]TJ /F17 7.97 Tf 6.59 0 Td[(_n] 2n[R1+R2]Xun,vn1un2Tn[U]^n1vn2Tn[U]^n+^n"PrjQ1(un,vn;Cn1)j>2nhjR1)]TJ /F4 7.97 Tf 6.58 0 Td[(H(X1)+H(X1jU,V)j++n+ni+PrjQ2(un,vn;Cn1,Cn2)j>2nR2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;UjX1)+jR1)]TJ /F4 7.97 Tf 6.58 0 Td[(H(X1)+H(X1jU,V)j+++n+300n,jQ1(un,vn;Cn1)j2nhjR1)]TJ /F4 7.97 Tf 6.59 0 Td[(H(X1)+H(X1jU,V)j++n+ni+Prmaxw2:(un,Cn2(w2))2Tn[U,X2]^00nQ3(un;Cn2(w2);Cn1)>2n[n+20n]#+Xun,vn1un2Tn[U]^n1vn2Tn[U]^n+^n1((un,vn)2En)2n[n+20n]2nR2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;UjX1)+jR1)]TJ /F4 7.97 Tf 6.58 0 Td[(H(X1)+H(X1jU,V)j+++n+300nXcn1,cn2p(vnjun,cn1,cn2)p(cn1,cn2)!(d)2)]TJ /F4 7.97 Tf 6.58 0 Td[(n[H(UjX1,X2)+R1+R2)]TJ /F17 7.97 Tf 6.59 0 Td[(_n]n2n[2H(U)+R1+R2+2^n+^n]3e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n"n+Xun,vn,cn1,cn21un2Tn[U]^n1((un,vn)2En)p(vnjun,cn1,cn2)p(cn1,cn2)22nR2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;UjX1)+jR1)]TJ /F4 7.97 Tf 6.59 0 Td[(H(X1)+H(X1jU,V)j+++2n+20n+300n)(e)3e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2n"n+n[H(U)+I(X1,X2;U)+_n+2^n+^n]ln2+22)]TJ /F4 7.97 Tf 6.59 0 Td[(n[R1+R2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X1,X2;U))]TJ /F6 7.97 Tf 6.58 0 Td[(2n)]TJ /F17 7.97 Tf 6.59 0 Td[(_n)]TJ /F6 7.97 Tf 6.58 0 Td[(20n)]TJ /F6 7.97 Tf 6.59 0 Td[(300n]+2)]TJ /F4 7.97 Tf 6.58 0 Td[(n[H(UjX1,X2)+R1+R2)]TJ /F17 7.97 Tf 6.59 0 Td[(_n]Xun,vn,cn1,cn21un2Tn[U]^n1((un,vn)2En)p(vnjun,cn1,cn2)p(cn1,cn2)22nhjR2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X2;UjX1)+R1)]TJ /F4 7.97 Tf 6.58 0 Td[(H(X1)+H(X1jU,V)j++2n+20n+300ni3e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2n"n+n[H(U)+I(X1,X2;U)+_n+2^n+^n]ln2+22)]TJ /F4 7.97 Tf 6.58 0 Td[(n[R1+R2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X1,X2;U))]TJ /F6 7.97 Tf 6.58 0 Td[(2n)]TJ /F17 7.97 Tf 6.59 0 Td[(_n)]TJ /F6 7.97 Tf 6.59 0 Td[(20n)]TJ /F6 7.97 Tf 6.59 0 Td[(300n]+2)]TJ /F4 7.97 Tf 6.58 0 Td[(n[R1+R2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X1,X2;U))]TJ /F17 7.97 Tf 6.58 0 Td[(_n)]TJ /F17 7.97 Tf 6.59 0 Td[(^n]22nhjR2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X2;UjX1)+R1)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X1;U))]TJ /F17 7.97 Tf 6.59 0 Td[(nj++2n+20n+300ni(f)=3e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2n"n+n[H(U)+I(X1,X2;U)+_n+2^n+^n]ln2+22)]TJ /F4 7.97 Tf 6.58 0 Td[(n[R1+R2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X1,X2;U))]TJ /F6 7.97 Tf 6.58 0 Td[(2n)]TJ /F17 7.97 Tf 6.59 0 Td[(_n)]TJ /F6 7.97 Tf 6.58 0 Td[(20n)]TJ /F6 7.97 Tf 6.59 0 Td[(300n] 86

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+22)]TJ /F4 7.97 Tf 6.59 0 Td[(n[n)]TJ /F6 7.97 Tf 6.59 0 Td[(2n)]TJ /F17 7.97 Tf 6.59 0 Td[(_n)]TJ /F6 7.97 Tf 6.59 0 Td[(30n)]TJ /F6 7.97 Tf 6.59 0 Td[(300n]. (4)Inequalities(a)and(b)in( 4 )areobtainedliketheircounterpartsin( 4 ).Inequality(c)resultsfromthefactsthatTn[U]^n2n[H(U)+^n](cf.[ 35 ,Lemma2.13]),andthatjQ2(un,vn;cn1,cn2)j2nR2andmaxw2jQ3(w2;un;cn1,cn2)j2nR1.Inequality(d)isaconsequenceofLemma 16 ,and"n2()]TJ /F13 11.955 Tf 9.3 0 Td[(nlogn)(providedthatn2)]TJ /F6 7.97 Tf 6.74 -4.97 Td[(1 n\o(1))andn2)]TJ /F6 7.97 Tf 6.67 -4.65 Td[(logn naretheonesgiveninthelemma.Inequality(e)isobtainedbyusingtheboundTn[U]^n2n[H(U)+^n]asin(c).Finally,equality(f)isvalid,aslongasn2o(1),whennissufcientlylargebecauseR1+R2>I(X1,X2;U).Finally,puttingtheboundsin( 4 )and( 4 )backinto( 4 )and( 4 ),weobtainmaxnPrfgn1(Vn,W1)=2fW2,!g,d(Vn,Un)>ng,Prfhn1(Vn,W1)=X,d(Vn,Un)>'ngo(n+1)2jX1jjX2j2nnlogn+2jUjjX1jjX2je)]TJ /F6 7.97 Tf 6.59 0 Td[(2n2n+3e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2n"n+n[H(U)+I(X1,X2;U)+_n+2^n+^n]ln2+22)]TJ /F4 7.97 Tf 6.59 0 Td[(n[R1+R2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X1,X2;U))]TJ /F6 7.97 Tf 6.58 0 Td[(2n)]TJ /F17 7.97 Tf 6.59 0 Td[(_n)]TJ /F6 7.97 Tf 6.58 0 Td[(20n)]TJ /F6 7.97 Tf 6.59 0 Td[(300n]+22)]TJ /F4 7.97 Tf 6.58 0 Td[(n[n)]TJ /F6 7.97 Tf 6.59 0 Td[(2n)]TJ /F17 7.97 Tf 6.59 0 Td[(_n)]TJ /F6 7.97 Tf 6.58 0 Td[(30n)]TJ /F6 7.97 Tf 6.58 0 Td[(300n]. (4) 4.3.5DecodinganddetectionerroranalysisChoosen=3jX1jjX2j n,naccordingto( 4 ),andn=3n+_n+30n+300n,wherenisthesequencein( 4 ).Itiseasytocheckthenthat2q logn nandn2(logn)2 n.Thus,forany'n2!logn p n\o(1),wecanobtainn2('n)suchthatLemma 13 applies,andhencealsoimpliesthechoiceofn2('n).Withthesechoicesofn,n,n,andn,itiseasytochecktheupperboundsonthedetectionanddecodingerroreventsoftheresultingdecodersanddetectorsgivenin( 4 )and( 4 )allconvergetozeroasnapproachesinnity. 87

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Fixany">0and>0.GiventhatPrfd(Vn,Un)=0g,from( 4 )andbysymmetry,theconditionalprobabilityofdecodingerroreventisthusPrfgn1(Vn,W1)6=W2g[fgn2(Vn,W2)6=W1gd(Vn,Un)=01 Prfgn1(Vn,W1)6=W2,d(Vn,Un)ng+1 Prfgn2(Vn,W2)6=W1,d(Vn,Un)ng"forallsufcientlylargen.Ontheotherhand,giventhatPrfd(Vn,Un)>0g,by( 4 ),( 4 ),andsymmetry,wehavePrgn1(Vn,W1)=2fW2,!g[gn2(Vn,W2)=2fW1,!gd(Vn,Un)>0"forallsufcientlylargen.Inaddition,sincen2('n),thereexistsc1suchthatnc'nforallsufcientlylargen.GiventhatPrfd(Vn,Un)c'ng,by( 4 )andsymmetry,wehavePrfhn1(Vn,W1)=!g[fhn2(Vn,W2)=!gd(Vn,Un)c'n1 Prfhn1(Vn,W1)=!,d(Vn,Un)ng+1 Prfhn2(Vn,W2)=!,d(Vn,Un)ng"forallsufcientlylargen.Finally,giventhatPrfd(Vn,Un)>'ng,by( 4 )andsymmetry,wehavePrfhn1(Vn,W1)=Xg[fhn2(Vn,W2)=Xgd(Vn,Un)>'n1 Prfhn1(Vn,W1)=X,d(Vn,Un)>'ng+1 Prfhn2(Vn,W2)=X,d(Vn,Un)>'ng"forallsufcientlylargen. 4.4TechnicalLemmas Lemma9. Let(X,Y,U)beatripleofrandomvariablesjointlydistributedaccordingtop(x,y,u): 1. Ifxn2Tn[X]1andyn2Tn[YjX]2(xn),then(xn,yn)2Tn[X,Y]1+2andyn2Tn[Y](1+2)jXj. 88

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2. If(xn,yn)2Tn[X,Y],thenyn2Tn[Y]jXjandxn2Tn[XjY](1+jXj)(yn). Proof. 1. Thisis[ 35 ,Lemma2.12]. 2. Let(xn,yn)2Tn[X,Y].Thenpart1)ofthelemmaimpliesyn2Tn[Y]jXj.Toseexn2Tn[XjY](1+jXj)(yn),noticethatPxn,yn(x,y))]TJ /F5 11.955 Tf 11.95 0 Td[(Pyn(y)PXjY(xjy)=(Pxn,yn(x,y))]TJ /F5 11.955 Tf 11.95 0 Td[(PX,Y(x,y))+(PY(y))]TJ /F5 11.955 Tf 11.95 0 Td[(Pyn(y))PXjY(xjy)+jXjPXjY(xjy)(1+jXj). Lemma10. SupposethatXandYareindependentrandomvariables.Foranyse-quencen2o(1),thereexistsanothersequencen2)]TJ /F2 11.955 Tf 5.48 -.49 Td[(p )]TJ /F13 11.955 Tf 9.3 0 Td[(nlognsuchthatnnandTn[X]nTn[Y]nnTn[X,Y]n Tn[X]nTn[Y]n(n+1)2jXjjYj2nnlogn. Proof. Notethatthislemmaappearsasaproblemin[ 35 ,Problem2.9].Weprovideaproofheretogivetheexplicitexponentialbound.WeusethegenericnotationS(x,y)todenotethejointtypeofasequencepairinXnYn,P(x)todenotethetypeofasequenceinXn,andQ(y)todenotethetypeofasequenceinYn.Foreachpair(P,Q)oftypes,deneS(P,Q),(S:XyS(x,y)=P(x),XxS(x,y),Q(y)).Fixany0n>n.ThenitiseasytoverifythatTn[X]nTn[Y]nnTn[X,Y]0ncanbepartitionedintodisjointsubsetsasbelow: Tn[X]nTn[Y]nnTn[X,Y]0n=[P2[X]nn[Q2[Y]nn[S2S(P,Q)n[X,Y]n0nTnS.(4) 89

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ButforeachS2S(P,Q),(xn,yn)2TnS,xn2TnS=Q(yn)andyn2TnQ.Hence jTnSj=Xyn2TnQTnS=Q(yn).(4)Further,foreachyn2TnQ,TnS=Q(yn)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n[D(PkPX)+H(P)]=Xxn2TnS=Q(yn)PnX(xn)=PnXjY)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(TnS=Q(yn)yn2)]TJ /F4 7.97 Tf 6.59 0 Td[(nD(S=QkPXjQ),whichimpliesTnS=Q(yn)2n[D(PkPX)+H(P))]TJ /F4 7.97 Tf 6.59 0 Td[(D(S=QkPXjQ)].Puttingthatbackinto( 4 ),weobtain jTnSjTnQ2n[D(PkPX)+H(P))]TJ /F4 7.97 Tf 6.59 0 Td[(D(S=QkPXjQ)]2n[H(Q)+D(PkPX)+H(P))]TJ /F4 7.97 Tf 6.59 0 Td[(D(S=QkPXjQ)].(4)NowifP2[X]nn,Q2[Y]nn,andS2S(P,Q)n[X,Y]n0n,thenH(Q)+D(PkPX)+H(P))]TJ /F5 11.955 Tf 11.95 0 Td[(D(S=QkPXjQ)=H(Y)+H(Q))]TJ /F5 11.955 Tf 11.95 0 Td[(H(Y)+H(X))]TJ /F9 11.955 Tf 23.76 11.36 Td[(Xx:PX(x)>0[P(x))]TJ /F5 11.955 Tf 11.96 0 Td[(PX(x)]log2PX(x))]TJ /F9 11.955 Tf 21.92 11.36 Td[(Xy:Q(y)>0Q(y)D(S(,y)=Q(y)kPX)(a)H(X)+H(Y))-222(jYjnlog2n)]TJ /F13 11.955 Tf 11.96 0 Td[(nlog20@Yx:PX(x)>0PX(x)1A)]TJ /F3 11.955 Tf 22.72 8.09 Td[(1 2ln2Xy:Q(y)>01 Q(y)"XxjS(x,y))]TJ /F5 11.955 Tf 11.95 0 Td[(PX(x)Q(y)j#2(b)H(X)+H(Y))-222(jYjnlog2n)]TJ /F13 11.955 Tf 11.96 0 Td[(nlog20@Yx:PX(x)>0PX(x)1A)]TJ /F3 11.955 Tf 13.15 8.08 Td[((0n)]TJ /F13 11.955 Tf 11.95 0 Td[(n)2 2ln2 (4) 90

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whereinequality(a)isobtainedfromtheupperboundonjH(Q))]TJ /F5 11.955 Tf 11.95 0 Td[(H(Y)jgivenin[ 35 ,Lemma1.2.7]andthelowerboundonD(k)givenin[ 9 ,Lemma11.6.1],andinequality(b)isduetothefactthattheremustexistapair(x0,y0)satisfyingjS(x0,y0))]TJ /F5 11.955 Tf -449.05 -23.91 Td[(PX(x0)PY(y0)j>0nasS2S(P,Q)n[X,Y]n0n.Applying( 4 )and( 4 )to( 4 )givesTn[X]nTn[Y]nnTn[X,Y]0n(n+1)2jXjjYj2nH(X)+H(Y)jYjnlog2n)]TJ /F17 7.97 Tf 6.59 0 Td[(nlog2Qx:PX(x)>0PX(x))]TJ /F15 5.978 Tf 7.78 4.52 Td[((0n)]TJ /F14 5.978 Tf 5.76 0 Td[(n)2 2ln2 (4)wheretheboundsonthenumbersofpossibletypesP,Q,andS(cf.[ 35 ,Lemma2.2])havebeenusedtoobtaintheinequality.By[ 35 ,Lemma2.13],Tn[X]n2n[H(X))]TJ /F17 7.97 Tf 6.59 0 Td[(n]andTn[Y]n2n[H(Y))]TJ /F17 7.97 Tf 6.59 0 Td[(n].Nowchoose0n=n,where n,n+r 2ln2h)]TJ /F13 11.955 Tf 9.3 0 Td[(nlogn+2n)-222(jYjnlog2n)]TJ /F13 11.955 Tf 11.95 0 Td[(nlog2Qx:PX(x)>0PX(x)i.(4)Itisthenobviousthatn2)]TJ /F2 11.955 Tf 5.48 -.49 Td[(p )]TJ /F13 11.955 Tf 9.29 0 Td[(nlogn.From( 4 ),wealsohaveTn[X]nTn[Y]nnTn[X,Y]n Tn[X]nTn[Y]n(n+1)2jXjjYj2nnlognasdesired. Lemma11. Foranyn,nandpair(un,vn)withthepropertythatd(vn,un)n,un2Un(cn1(w1),cn2(w2))impliesvn2Un+n(cn1(w1),cn2(w2)). Proof. Forsimplernotationbelow,writexn1=cn1(w1)andxn1=cn2(w2).Since XvXu6=vPvnjun(vju)Pun(u)=d(vn,un)n,(4)wehavePu6=vPvn,un(v,u)n.ThusXx1,x2Xu6=vPvn,un,xn1,xn2(v,u,x1,x2)=Xu6=vPvn,un(v,u)n, 91

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whichfurtherimplies Xu6=vPvn,un,xn1,xn2(v,u,x1,x2)n(4)Ontheotherhand,againbecauseof( 4 ),Xx1,x2XvPun,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.96 0 Td[(Pvn,un,xn1,xn2(v,v,x1,x2)=Xv[Pun(v))]TJ /F5 11.955 Tf 11.95 0 Td[(Pvn,un(v,v)]=1)]TJ /F9 11.955 Tf 11.95 11.36 Td[(XvPvn,un(v,v)n,whichfurtherimplies Pun,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.95 0 Td[(Pvn,un,xn1,xn2(v,v,x1,x2)n.(4)Using( 4 )and( 4 ),wegetPvn,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.96 0 Td[(Pun,xn1,xn2(v,x1,x2)=Xu6=vPvn,un,xn1,xn2(v,u,x1,x2))]TJ /F9 11.955 Tf 11.95 9.68 Td[(Pun,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.95 0 Td[(Pvn,un,xn1,xn2(v,v,x1,x2)2n.Consequently,Pvn,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.96 0 Td[(PU,X1,X2(v,x1,x2)(a)Pvn,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.96 0 Td[(Pun,xn1,xn2(v,x1,x2)+Pun,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.95 0 Td[(PU,X1,X2(v,x1,x2)2n+2nwherethesecondtermin(a)isboundedby2nsinceun2Un(xn1,xn2). Lemma12. Foranypair(un,vn)suchthatun2Tn[U]^nandvn2Tn[U]^n+^n,let(~X1,~U,~V)beatripleofrandomvariablesthatsatisestheconstraintsin( 4 ).IfPX1(x1)>0,thenP~Uj~X1(ujx1))]TJ /F5 11.955 Tf 11.95 0 Td[(PUjX1(ujx1)n 92

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P~Vj~X1(vjx1))]TJ /F5 11.955 Tf 11.95 0 Td[(PUjX1(vjx1)n+nwheren,(1+jUj)(^n+~0n) minx01:PX1(x01)>0PX1(x01)n,(1+jUj)(^n+~0n) minx01:PX1(x01)>0PX1(x01). (4) Proof. Itobviouslysufcestoshowthesecondbound.Sincevn2Tn[U]^n+^n,P~X1,~V(x1,v))]TJ /F5 11.955 Tf 11.96 0 Td[(PX1,U(x1,v)jP~V(v))]TJ /F5 11.955 Tf 11.96 0 Td[(PU(v)jP~X1j~V(x1jv)+P~X1j~V(x1jv))]TJ /F5 11.955 Tf 11.95 0 Td[(PX1jU(x1jv)PU(v)^n+^n+~0n+~0n. (4)Animmediateconsequenceof( 4 )isP~X1(x1))]TJ /F5 11.955 Tf 11.95 0 Td[(PX1(x1)jUj(^n+^n+~0n+~0n).IfPX1(x1)>0,thenP~Vj~X1(vjx1))]TJ /F5 11.955 Tf 11.95 0 Td[(PUjX1(vjx1)1 PX1(x1)hP~X1(x1))]TJ /F5 11.955 Tf 11.96 0 Td[(PX1(x1)P~Vj~X1(vjx1)+P~X1,~V(x1,v))]TJ /F5 11.955 Tf 11.96 0 Td[(PX1,U(x1,v)i(1+jUj)(^n+^n+~0n+~0n) PX1(x1)n+n. Lemma13. Letn2o(1).Foranynandn,supposethatthepair(un,vn)satisesthefollowingtwoconditions: un2Tn[U]^nandvn2Tn[U]^n+^n H(X1jU)H(X1jU,V)+n,where(X1,U,V)isasdenedin( 4 ).If(PUjX1,I)isnon-manipulable,then 1. d(un,vn)'nforsomen'n2O)]TJ /F13 11.955 Tf 5.48 -9.69 Td[(n+p n)]TJ /F13 11.955 Tf 11.96 0 Td[(nlogn,and 2. un2Un(cn1(w1),cn2(w2))impliesvn2Un(cn1(w1),cn2(w2))forsomen2('n)satisfyingnn+n. 93

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Proof. FirstnotethatI(X1;VjU)=H(X1jU))]TJ /F5 11.955 Tf 11.95 0 Td[(H(X1jU)+H(X1jU))]TJ /F5 11.955 Tf 11.96 0 Td[(H(X1jU,V)jH(X1jU))]TJ /F5 11.955 Tf 11.95 0 Td[(H(X1jU)j+nXuH(X1jU=u)jPU(u))]TJ /F5 11.955 Tf 11.95 0 Td[(PU(u)j+XujH(X1jU=u))]TJ /F5 11.955 Tf 11.96 0 Td[(H(X1jU=u)jPU(u)+n(a)^njUjlogjX1j)-222(jX1j~0nlog~0n+nwhere(a)isaconsequenceof[ 35 ,Lemma2.7].Continuingon,wehavejUjlogjX1j^n)-222(jX1j~0nlog~0n+nI(X1;VjU)=XuD)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(PX1,VjU(x1,vju)PVjU(vju)PX1jU(x1ju)PU(u)(a)Xu1 2ln2"Xx1,vPX1,VjU(x1,vju))]TJ /F5 11.955 Tf 11.95 0 Td[(PVjU(vju)PX1jU(x1ju)#2PU(u)(b)1 2ln2"Xx1,u,vPVjX1,U(vjx1,u))]TJ /F5 11.955 Tf 11.96 0 Td[(PVjU(vju)PX1,U(x1,u)#21 2ln2"Xx1,uPVjX1,U(vjx1,u))]TJ /F5 11.955 Tf 11.95 0 Td[(PVjU(vju)PX1,U(x1,u)#2 (4)whereinequality(a)isdueto[ 9 ,Lemma11.6.1],andinequality(b)resultsfromtheconvexityofthesquarefunction.From( 4 )and( 4 ),weclearlyhavePVjX1,U(vjx1,u))]TJ /F5 11.955 Tf 11.96 0 Td[(PVjU(vju)p 2ln2(jUjlogjX1j^n)-222(jX1j~0nlog~0n+n)"min(x01,u0):PX1,U(x01,u0)>0PX1,U(x01,u0)#)]TJ /F6 7.97 Tf 6.59 0 Td[(1p 2ln2(jUjlogjX1j^n)-222(jX1j~0nlog~0n+n)min(x01,u0):PX1,U(x01,u0)>0PX1,U(x01,u0))]TJ /F3 11.955 Tf 12.67 0 Td[(~0n)]TJ /F3 11.955 Tf 12.67 0 Td[(^n)]TJ /F6 7.97 Tf 6.59 0 Td[(1 (4) 94

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forall(x1,u)suchthatPX1,U(x1,u)>0andsufcientlylargen.Forbookkeepingpurposeswelabel'0n,p 2ln2(jUjlogjX1j^n)-222(jX1j~0nlog~0n+n)264min(x01,u0):PX1,U(x01,u0)>0PX1,U(x01,u0))]TJ /F3 11.955 Tf 12.66 0 Td[(~0n)]TJ /F3 11.955 Tf 12.66 0 Td[(^n375)]TJ /F6 7.97 Tf 6.59 0 Td[(1.ThereforeifPX1(x1)>0,thenforsufcientlylargen,PUjX1(vjx1))]TJ /F9 11.955 Tf 11.96 11.35 Td[(XuPVjU(vju)PUjX1(ujx1)PVjX1(vjx1))]TJ /F5 11.955 Tf 11.96 0 Td[(PUjX1(vjx1)+PVjX1(vjx1))]TJ /F9 11.955 Tf 11.95 11.35 Td[(XuPVjU(vju)PUjX1(ujx1)+XuPVjU(vju)PUjX1(ujx1))]TJ /F5 11.955 Tf 11.95 0 Td[(PUjX1(ujx1)(a)n+XuPVjX1,U(vjx1,u))]TJ /F5 11.955 Tf 11.95 0 Td[(PVjU(vju)PUjX1(ujx1)+jUjnn+jUjn+XuPVjX1,U(vjx1,u))]TJ /F5 11.955 Tf 11.95 0 Td[(PVjU(vju)PUjX1(ujx1))]TJ /F5 11.955 Tf 11.95 0 Td[(PUjX1(ujx1)+PUjX1(ujx1)(b)n+jUjn+jUjn+'0n,~'n (4)whereinequality(a)isduetoLemma 12 withnandngivenby( 4 ),and(b)resultsfromLemma 12 and( 4 ).Now,byanargumentessentiallythesameastheoneprovidedintheproofof[ 37 ,Thm.5],if(PUjX1,I)isnon-manipulable,then( 4 )impliesd(vn,un)=XuXv6=uPVjU(vju)PU(u)XuXv6=uPVjU(vju)Xu,v1(v=u))]TJ /F5 11.955 Tf 11.96 0 Td[(PVjU(vju)c~'n,'n (4)forsomeconstantcthatdependsonlyonPUjX1.Wemayassumec1byloosingtheboundin( 4 )ifnecessary.Obviously,n'n2O)]TJ /F13 11.955 Tf 5.48 -9.68 Td[(n+p n)]TJ /F13 11.955 Tf 11.96 0 Td[(nlogn 95

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Nowforsimplernotationbelow,writexn1=cn1(w1)andxn1=cn2(w2).Foranyv6=u,wehave,becauseof( 4 ),Xx1,x2Pvn,un,xn1,xn2(v,u,x1,x2)=PV,U(v,u)'n,whichimplies Pvn,un,xn1,xn2(v,u,x1,x2)'n.(4)Ontheotherhand,againbecauseof( 4 ),Xx1,x2Pun,xn1,xn2(u,x1,x2))]TJ /F5 11.955 Tf 11.95 0 Td[(Pvn,un,xn1,xn2(u,u,x1,x2)=1)]TJ /F5 11.955 Tf 11.96 0 Td[(PVjU(uju)PU(u)'n,whichimplies Pun,xn1,xn2(u,x1,x2))]TJ /F5 11.955 Tf 11.95 0 Td[(Pvn,un,xn1,xn2(u,u,x1,x2)'n.(4)Using( 4 )and( 4 ),wegetPvn,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.96 0 Td[(Pun,xn1,xn2(v,x1,x2)=Xu6=vPvn,un,xn1,xn2(v,u,x1,x2))]TJ /F9 11.955 Tf 11.95 9.68 Td[(Pun,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.95 0 Td[(Pvn,un,xn1,xn2(v,v,x1,x2)jUj'n.Consequently,Pvn,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.96 0 Td[(PU,X1,X2(v,x1,x2)(a)Pvn,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.96 0 Td[(Pun,xn1,xn2(v,x1,x2)+Pun,xn1,xn2(v,x1,x2))]TJ /F5 11.955 Tf 11.95 0 Td[(PU,X1,X2(v,x1,x2)jUj'n+2nwherethesecondtermin(a)isboundedby2nsinceun2Un(xn1,xn2).Setting n,1 2jUj'n+n,(4) 96

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wehavevn2Un(xn1,xn2).Finally,itistrivialtoverifythatnn+n,andthatn2('n). Lemma14. FixanyconditionaldistributionsPXjUandPXjV.Forany~n,~n,andeverypair(un,vn)suchthatTn[XjU]~n(un)\Tn[XjV]~n(vn)6=;,thereexistsn2)]TJ /F6 7.97 Tf 6.67 -4.65 Td[(logn nsuchthat1 nlogTn[XjU]~n(un)\Tn[XjV]~n(vn))]TJ /F3 11.955 Tf 11.96 0 Td[(maxH(~Xj~U,~V)nwheretheconditionalentropyH(~Xj~U,~V)ismaximizedoveralltriples(~X,~U,~V)ofrandomvariablesthatsatisfyP~U,~V(u,v)=Pun,vn(u,v)P~Xj~U(xju))]TJ /F5 11.955 Tf 11.96 0 Td[(PXjU(xju)~nP~Xj~V(xjv))]TJ /F5 11.955 Tf 11.95 0 Td[(PXjV(xjv)~n. Proof. Asimilarresult,withtypeclassesratherthantypicalsets,appearsin[ 35 ,Problem2.10].Toprovethisversion,noterstthat Tn[XjU]~n(un)\Tn[XjV]~n(vn)=[Q2[XjU]n~nQ02[XjV]n~nTnQ(un)\TnQ0(vn).(4)FixanyQ2[XjU]n~nandQ02[XjV]n~n.Then,foreachxn2TnQ(un)TTnQ0(vn),itiseasytoverifythatTnPxnjun,vn(un,vn)TnQ(un)\TnQ0(vn).Hence,thesetTnQ(un)TTnQ0(vn)canbepartitionedintodisjointclasses,eachofwhichidentieswithaconditionaltypePxnjun,vnsatisfyingtheconstraintsPxnjun=QandPxnjvn=Q0.Letthesetofalltheseclass-identifyingconditionaltypesbedenotedby 97

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P(Q,Q0).ThenTnQ(un)\TnQ0(vn)=[P2P(Q,Q0)TnP(un,vn).Puttingthisbackinto( 4 ),wegetTn[XjU]~n(un)\Tn[XjV]~n(vn)=[Q2[XjU]n~nQ02[XjV]n~n[P2P(Q,Q0)TnP(un,vn).Becausethemaximumnumbersofconditionaltypesin[XjU]n~n,[XjV]n~n,andeachP(Q,Q0)arerespectively(n+1)jXjjUj,(n+1)jXjjVj,and(n+1)jXjjUjjVj[ 35 ,Lemma2.2],Tn[XjU]~n(un)\Tn[XjV]~n(vn)XQ2[XjU]n~nQ02[XjV]n~nXP2P(Q,Q0)jTnP(un,vn)j(n+1)3jXjjUjjVjmaxP2P(Q,Q0):Q2[XjU]n~n,Q02[XjV]n~njTnP(un,vn)j(a)(n+1)3jXjjUjjVj2nmaxP2P(Q,Q0):Q2[XjU]n~n,Q02[XjV]n~nH(PjPun,vn)whereinequality(a)isobtainedbyapplying[ 35 ,Lemma2.5]toboundjTnP(un,vn)jfromabove.Ontheotherhand,Tn[XjU]~n(un)\Tn[XjV]~n(vn)maxP2P(Q,Q0):Q2[XjU]n~n,Q02[XjV]n~njTnP(un,vn)j(n+1)jXjjUjjVj2nmaxP2P(Q,Q0):Q2[XjU]n~n,Q02[XjV]n~nH(PjPun,vn)where[ 35 ,Lemma2.5]isagainappliedtoboundjTnP(un,vn)jfrombelow.Thus,itremainstoestablishmaxH(~Xj~U,~V))]TJ /F13 11.955 Tf 11.96 0 Td[(0nmaxP2P(Q,Q0):Q2[XjU]n~n,Q02[XjV]n~nH(PjPun,vn)maxH(~Xj~U,~V) (4)forsome0n2)]TJ /F6 7.97 Tf 6.68 -4.65 Td[(logn n.Thenlettingn=0n+3jXjjUjjVj nlog2(n+1)givesthedesiredresult.Toshow( 4 ),rstnotethattheupperboundin( 4 )isobvioussincethejointdistributionPPun,vn,foreveryP2P(Q,Q0)whereQ2[XjU]n~nandQ02[XjV]n~n, 98

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satisestheconditionsfor(~X,~U,~V)giveninthestatementofthelemma.Toshowthelowerbound,observethatduetotheassumptionthatTn[XjU]~n(un)TTn[XjV]~n(vn)6=;,fP(Q,Q0):Q2[XjU]n~n,Q02[XjV]n~ngisdenseinthesetofconditionaldistributionsP~Xj~U,~V,witherrorof)]TJ /F6 7.97 Tf 6.74 -4.98 Td[(1 n.By[ 35 ,Lemma2.7],H(~Xj~U,~V)isacontinuousfunctionofP~Xj~U,~Vanditsmaximumisbounded,theexistenceof0n2)]TJ /F6 7.97 Tf 6.68 -4.65 Td[(logn ngivingthelowerboundin( 4 )isguaranteed. Lemma15. [ 35 ,Lemma17.9]Let>1.TheprobabilityinNindependenttrials,aneventofprobabilityatmostqhappeningmorethanqNtimesisboundedabovebye)]TJ /F17 7.97 Tf 6.58 0 Td[(()qN,where(),ln)]TJ /F13 11.955 Tf 11.96 0 Td[(+1. Lemma16. SupposethatCn1andCn2arerandomlychosenaccordingtothedescriptioninSection 4.3.2 .Let>1andn2)]TJ /F6 7.97 Tf 6.74 -4.97 Td[(1 n\o(1).Letn2)]TJ /F2 11.955 Tf 5.48 -.49 Td[(p )]TJ /F13 11.955 Tf 9.3 0 Td[(nlognbethesequenceobtainedinLemma 10 ,~0n,0n,00n,^00n,and~0nbeinthedenitionsofthesetsQ1,Q2,andQ3givenin( 4 ).Foranypairofsequences(un,vn),let(X1,U,V)beasspeciedin( 4 ).Thenthereexistsn2)]TJ /F6 7.97 Tf 6.67 -4.65 Td[(logn nsuchthat1)PrjQ1(un,vn;Cn1)j>2nhjR1)]TJ /F4 7.97 Tf 6.58 0 Td[(H(X1)+H(X1jU,V)j++n+nie)]TJ /F17 7.97 Tf 6.58 0 Td[(()2n[jR1)]TJ /F16 5.978 Tf 5.76 0 Td[(H(X1)+H(X1jU,V)j++n+n],2)Pr(jQ2(un,vn;Cn1,Cn2)j>2nR2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;UjX1)+jR1)]TJ /F4 7.97 Tf 6.59 0 Td[(H(X1)+H(X1jU,V)j+++n+300n,jQ1(un,vn;Cn1)j2nhjR1)]TJ /F4 7.97 Tf 6.59 0 Td[(H(X1)+H(X1jU,V)j++n+ni)e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2njR2)]TJ /F16 5.978 Tf 5.76 0 Td[(I(X2;UjX1)+jR1)]TJ /F16 5.978 Tf 5.76 0 Td[(H(X1)+H(X1jU,V)j+j++n+300n,3)Prmaxw2:(un,Cn2(w2))2Tn[U,X2]^00njQ3(un;Cn2(w2),Cn1)j>2nhjR1)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X1;UjX2)j++20nie)]TJ /F17 7.97 Tf 6.59 0 Td[(()2njR1)]TJ /F16 5.978 Tf 5.76 0 Td[(I(X1;UjX2)j++20n+nR2ln2.Thus,thethreeprobabilitiesin1)3)canbefurtherdominatedbythedoubleexponen-tialterme)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n"n,forsome"n2()]TJ /F13 11.955 Tf 9.3 0 Td[(nlogn). 99

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Proof. 1.)ByLemma 14 ,weknowthat,foreachw1,PrnCn1(w1)2Tn[X1jU]~0n(un)\Tn[X1jU]~0n+~0n(vn),(un,Cn1(w1))2Tn[U,X1]^0noPrnCn1(w1)2Tn[X1jU]~0n(un)\Tn[X1jU]~0n+~0n(vn)o=Tn[X1jU]~0n(un)\Tn[X1jU]~0n+~0n(vn) Tn[X1]n2n[H(X1jU,V)+n] 2n[H(X1))]TJ /F17 7.97 Tf 6.59 0 Td[(n]=2)]TJ /F4 7.97 Tf 6.59 0 Td[(n[H(X1))]TJ /F4 7.97 Tf 6.59 0 Td[(H(X1jU,V))]TJ /F17 7.97 Tf 6.58 0 Td[(n)]TJ /F17 7.97 Tf 6.59 0 Td[(n]. (4)SincethecodewordsinCn1arei.i.d.,applyingLemma 15 with( 4 )gives PrnjQ1(un,vn;Cn1)j>2n[R1)]TJ /F4 7.97 Tf 6.59 0 Td[(H(X1)+H(X1jU,V)+n+n]oe)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n[R1)]TJ /F16 5.978 Tf 5.76 0 Td[(H(X1)+H(X1jU,V)+n+n](4)whichisguaranteedtovanishonlyifR1>H(X1))]TJ /F5 11.955 Tf 13.2 0 Td[(H(X1jU,V).Ontheotherhand,ifR1H(X1))]TJ /F5 11.955 Tf 13.37 0 Td[(H(X1jU,V),wecanfurtherrelaxtheupperboundonPrnCn1(w1)2Tn[X1jU]~0n(un)\Tn[X1jU]~0n+~0n(vn)oin( 4 )to2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(R1)]TJ /F17 7.97 Tf 6.59 0 Td[(n)]TJ /F17 7.97 Tf 6.59 0 Td[(n)beforeapplyingLemma 15 toget PrnjQ1(un,vn;Cn1)j>2n(n+n)oe)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n(n+n)(4)instead.Combining( 4 )and( 4 )givesthedesiredresult.2.)Supposecn1satisesjQ1(un,vn;cn1)j2njR1)]TJ /F4 7.97 Tf 6.58 0 Td[(H(X1)+H(X1jU,V)j++n+n.Then,foreachw2,wehavePr8<:Cn2(w2)2[w12Q1(un,vn;cn1)Tn[X2jU,X1]00n(un,cn1(w1))9=;Xw12Q1(un,vn;cn1)PrnCn2(w2)2Tn[X2jU,X1]00n(un,cn1(w1))o=Xw12Q1(un,vn;cn1)Tn[X2jU,X1]00n(un,cn1(w1)) Tn[X2]n 100

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(a)jQ1(un,vn;cn1)j2n[H(X2jU,X1)+00n] 2n[H(X2))]TJ /F17 7.97 Tf 6.58 0 Td[(n]2)]TJ /F4 7.97 Tf 6.58 0 Td[(nhI(X2;UjX1))]TJ /F2 11.955 Tf 6.59 -.99 Td[(jR1)]TJ /F4 7.97 Tf 6.58 0 Td[(H(X1)+H(X1jU,V)j+)]TJ /F17 7.97 Tf 6.59 0 Td[(n)]TJ /F6 7.97 Tf 6.59 0 Td[(300ni (4)whereinequality(a)isdueto[ 35 ,Lemma2.13].SincethecodewordsinCn2arei.i.d.,applyingLemma 15 with( 4 )inthesamewayasinpart1)abovegivesPr(jQ2(un,vn;cn1,Cn2)j>2nR2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X2;UjX1)+jR1)]TJ /F4 7.97 Tf 6.59 0 Td[(H(X1)+H(X1jU,V)j+++n+300n)e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2njR2)]TJ /F16 5.978 Tf 5.76 0 Td[(I(X2;UjX1)+jR1)]TJ /F16 5.978 Tf 5.76 0 Td[(H(X1)+H(X1jU,V)j+j++n+300n. (4)BasedonthefactthatCn1andCn2areindependentlychosen,( 4 )impliesthedesiredresult.3.Forany(un,cn2(w2))2Tn[U,X2]^00n,[ 35 ,Lemma2.13]impliesPrnCn1(w1)2Tn[X1jU,X2]0n(un,cn2(w2))o=Tn[X1jU,X2]0n(un,cn2(w2)) Tn[X1]n2n[H(X1jU,X2)+0n] 2n[H(X1))]TJ /F17 7.97 Tf 6.58 0 Td[(n]2)]TJ /F4 7.97 Tf 6.59 0 Td[(nhI(X1;UjX2))]TJ /F6 7.97 Tf 6.58 0 Td[(20ni. (4)SincethecodewordsinCn1arei.i.d.,applyingLemma 15 with( 4 )likebeforegives PrjQ3(un;cn2(w2);Cn1)j>2nhjR1)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1;UjX2)j++20nie)]TJ /F17 7.97 Tf 6.58 0 Td[(()2njR1)]TJ /F16 5.978 Tf 5.76 0 Td[(I(X1;UjX2)j++20n.(4)Hence,using( 4 ),wegetPrmaxw2:(un,cn2(w2))2Tn[U,X2]^00njQ3(un;cn2(w2);Cn1)j>2nhjR1)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1;UjX2)j++20niXw2:(un,cn2(w2))2Tn[U,X2]^00nPrjQ3(un;cn2(w2);Cn1)j>2nhjR1)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1;UjX2)j++20ni2nR2e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2njR1)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X1;UjX2)j++20n, 101

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whichimpliesthedesiredresult.Finally,thedominatingbounde)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n"ncanbeobtainedbysetting"n=n,whennissufcientlylarge. 102

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CHAPTER5INTEGRITYINTHETWOWAYTWOHOPRELAYPreviouslywehaveshownthatchannelscouldsupportstrongintegrityifandonlyiftheywerenon-manipulable.Thisisproblematicformanipulablechannelswhichstrongintegritydoesnotcover,andclearlyneedbeconsideredaswell.Furthermoreifoneweretopaydetailedattentiontotheproofofstrongintegrity,itcanbeobservedthataloosenessoccurswhenitisassumedthatthereexistsacn2(^w2)2Tn[X2jUX1](vn,cn1(w1))forallpossiblevn.Clearlythisisnotthecase,infactonecanseefrombasicinformationtheoreticresultsthatoutofallTn[U]possiblevaluesofun,onlySw2Tn[UjX1X2](cn1(w1),cn2(w2))2nR22n(H(UjX1X2)+n)valuesofunaretypicalwithrespecttocn1(w1).Strongintegritythoughdoesnotconsiderthecodebooksindeterminingintegrity.Ingeneralintegrityneednotrelyuponthechannelalone,andthusthischapterspurposeistobuildthegeneralframeworkencompassingallavailableaspects.Thiswillleadtoamorestabledenitionofintegritythatislessdependentuponthechannelandmoredependentupontherate.Interestinglymanipulabilitywillstillplayarole,althoughmuchlesspronounced.Themajorcontributionsofthischapterarenotonlytheresult,butthemethodsaswell.Duetotheunknownnatureoftheactionsbytherelay,everypossibilitymustbeassumedandnonergodicmethodshadtobeconsidered.Thiscomplicationactuallyleadstosurprisinglystrongresultsthatareprimarilybaseduponprecisecountingofsetsizes.Duetothelargeredundancywithchapter 4 inproblemformationweneedonlypresenttwoquicknotationconventionsinsection 5.1 .Likewisewhenthemajortheoremofthischapteraswellasachievabilityproofsarepresentedinsection 5.2 ,wefocusononlypresentingthosewhicharedifferentthanthoseofchapter 4 .Notonlydowepresentproofsrelatingtointegrityinsection 5.2 wealsopresenttheachievablesecrecyaswellandshowthatitcanalwaysbecalculatedandformulatedasasubset 103

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ofintegrity,andinsuchawayleadtosemanticallystrongresults.Finallyweonceagainremovealargenumberofproofswhichdistractfromthebiggerpictureandplacetheminsection 5.3 wherethosewhoaremoreinterestedinthemoreminuteaspectsrelatingtointegritymayperusethem. 5.1NotationConventions Thesloppydeltaconvention.Forconveniencewedenenbythedeltaconventionsuchthatp nn!1andn!0.Notethatthevalueofn=log2(n+1) npreviouslyintegralalsosatisedthatdeltaconventionaswellasalllinearmultiples.Unlikeinstrongintegritywherethemanipulationofthesevanishingvaluesrequirecarefulattention,inthesubsequentsectionsallthatwillberequiredisthatanexistthatsatisesthedeltaconventionsuchthateverythingholdstrue.Asshouldbeplainfromchapter 4 ,thebookkeepingofexactvaluesofncanbecomequitelaborious,andonlyshouldbeattemptedinsituationswhereitiswarranted.Becauseoftheextremeburdenthisplacesonnotation,andconsideringthatmostoperationsonlylinearlyscalethisvalueofdelta,weadopttheconventionof[ 35 ],inwhichtheyonlypresentexplicitinformationaboutthedeltaandtheconstantmultiplyingifdeemednecessary.Thisconventionistakentoanextremehere.Becauseeveryvaluerelatingtoanpresentedwillbeontheorderofcn,wherecisdeterminedsolelybythesizeoftheparticularchannelalphabets,everyvaluewillendupsatisfyingthedeltaconvention.Assuchwedonotfeelitnecessarytomakenoteofthechangesinthescalarmultiplesofnevenonalinebylinevalue.Whilethismayseemsloppyorlazy,thealternativeistoeitherexplicitlystatethesefunctionswhichaftermultiplelinesofmanipulationwillgrowtosizestoolargefordisplayortoconstantlyupdatethevaluesofnwithsomenewexplicitreferencetothefactthevaluehaslinearlyscaled.Neitheralternativeactuallyaddsvaluetotheproofsandassuchwewillnotattempttodifferentiatethenewvaluesofnfromthepreviousvaluesofn.Inthiscontext,theexistenceofannonlyimpliesthatoneexiststhatissubjecttothepreviouslydeneddeltaconvention.Allsituations 104

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whichariseinwhichstrictobservationofthesevaluesisofimportancewillbenotedandthesevaluesdistinguished. max)]TJ /F8 11.955 Tf 9.29 0 Td[(tilde.Atmanytimesthroughouttheproofwewillmakeuseoflemma 14 ,referredtoastheslap-happylemma.InthislemmathemaxisreferredtooverthepartialdistributionsoforiginalvariablesX,U,andV.Onceagainbecauseofthelaboriousnatureofexplicitlystatingthisateveryinstanceofuse,weinsteadintroducethemax-tildeconvention.Wheneverthereisamax,withnorestrictingsetattached,thatisimmediatelyfollowedbysomevalueswhichdependonrandomvariableswithatildeattached(i.e.,~X),itistobeinterpretedthatthemaxistobetakenoverallrandomvariablesthathavepartialdistributionsequaltotheonesoftheoriginalvariablesassodenedinlemma 14 ,theslap-happy. 5.2Achievability 5.2.1Theorem Theorem5.1. Theintegritycapacityregionistheclosureoftheconvexhullofall(R1,R2)satisfyingR1
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anyarbitrarysequencefortheothersourceandsimulatethechannelandthusfooltheoriginalsource.Thisthusnecessitatestheexclusionofthisregion.Furthermoreitisclearthatiftheinputsareforcedtobeindependent,thenthisrateregionmatchesthatofthecapacityregioninallcasesbutthosethatmustbeexcludedaspreviouslymentioned. 5.2.2CodeConstructionThecodeconstructionisnearlyidenticaltothatpresentedinchapter 4 saveforthefactthatwenolongerneedthesecondringandthuschoosethedenitionofourdecodersasgn1(vn,w1)=8>><>>:^w2ifthereexistsaunique(vn,cn1(w1),cn2(^w2))2Tn[U,X1,X2]!otherwise,gn2(vn,w2)=8>><>>:^w1ifthereexistsaunique(vn,cn1(^w1),cn2(w2))2Tn[U,X1,X2]!otherwise.Becausereliabletransmissionwasachievedpreviously,thereisnoneedtorepeattheargumentshere. 5.2.3IntegrityRecallfrompreviouslythattoboundtheprobabilitythatmanipulationwillgoundetectedwastoessentiallyboundPrfUn2U(Cn1(W1),Cn2(W2))\V(Cn1(W1);W2,Cn2)g.ThoughnowtakingthecodingintoaccountweseePrfUn2U(Cn1(W1),Cn2(W2))\V(Cn1(W1);W2,Cn2)gXun,vn,w1,w2,^w2,cn1,cn2p(vn,un,cn1,cn2,w1,w2)1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[((un,cn1(w1),cn2(w2))2Tn[UX1X2]1)]TJ /F3 11.955 Tf 5.47 -9.68 Td[((vn,cn1(w1),cn2(^w2))2Tn[UX1X2]2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(R1+R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1,X2;U))2)]TJ /F4 7.97 Tf 6.59 0 Td[(nH(U)Xun,vn,w1,w2,^w2,cn1,cn2p(vnjun,cn1,cn2)p(cn1,cn2)1)]TJ /F3 11.955 Tf 5.48 -9.69 Td[((un,cn1(w1),cn2(w2))2Tn[UX1X2]1)]TJ /F3 11.955 Tf 5.47 -9.69 Td[((vn,cn1(w1),cn2(^w2))2Tn[UX1X2] 106

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2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(R1+R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1,X2;U))2)]TJ /F4 7.97 Tf 6.59 0 Td[(nH(U)Xun,vn,w1,w2,^w2,cn1,cn2p(vnjun,cn1,cn2)p(cn1,cn2)1)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(un2Tn[U]1)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(vn2Tn[U]1)]TJ /F27 11.955 Tf 5.48 -9.68 Td[(cn2(w2)2Tn[X2jU](un)1)]TJ /F27 11.955 Tf 5.48 -9.68 Td[(cn2(^w2)2Tn[X2jU](vn)1)]TJ /F27 11.955 Tf 5.48 -9.69 Td[(cn1(w1)2Tn[X1jUX2](un,cn2(w2))\Tn[X1jUX2](vn,cn2(^w2))(a)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(R1+R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1,X2;U))2)]TJ /F4 7.97 Tf 6.59 0 Td[(nH(U)Xun,vn,cn1,cn2p(vnjun,cn1,cn2)p(cn1,cn2)1)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(un2Tn[U]1)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(vn2Tn[U]"maxw1,unXw21)]TJ /F27 11.955 Tf 5.48 -9.68 Td[(cn2(w2)2Tn[X2jUX1](un,cn1(w1))#2Xw1,w2,^w21)]TJ /F27 11.955 Tf 5.48 -9.68 Td[(cn1(w1)2Tn[X1jUX2](un,cn2(w2))\Tn[X1jUX2](vn,cn2(^w2))(b)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(R1+R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1,X2;U)+H(U)+n)Xun,vn,cn1,cn2p(vnjun,cn1,cn2)p(cn1,cn2)1)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(un2Tn[U]1)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(vn2Tn[U]"maxw1,unXw21)]TJ /F27 11.955 Tf 5.48 -9.68 Td[(cn2(w2)2Tn[X2jUX1](un,cn1(w1))#2Xw1,PX2jUV,PX02jUVcn2(w2)2TnPX2jUV(un,vn)cn2(^w2)2TnPX02jUV(un,vn)1)]TJ /F27 11.955 Tf 5.48 -9.68 Td[(cn1(w1)2Tn[X1jUX2](un,cn2(w2))\Tn[X1jUX2](vn,cn2(^w2))(c)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(R1+R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1,X2;U)+H(U)+n)Xun,vn,cn1,cn2p(vnjun,cn1,cn2)p(cn1,cn2)1)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(un2Tn[U]"maxw1,unXw21)]TJ /F27 11.955 Tf 5.48 -9.69 Td[(cn2(w2)2Tn[X2jUX1](un,cn1(w1))#21)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(vn2Tn[U]"maxw2,unXw11)]TJ /F27 11.955 Tf 5.48 -9.68 Td[(cn1(w1)2Tn[X1jUX2](un,cn2(w2))#2XPX2jUVPX02jUVw110BBBBBB@cn1(w1)2[cn2(w2)2TnPX2jUV(un,vn)cn2(^w2)2TnPX02jUV(un,vn)hTn[X1jUX2](un,cn2(w2))\Tn[X1jUX2](vn,cn2(^w2))i1CCCCCCA(d)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(R1+R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1,X2;U)+H(U)+n)2nj2R2)]TJ /F6 7.97 Tf 6.59 0 Td[(2I(X2;UjX1)+jR1)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1;U,V)j+j++e)]TJ /F6 7.97 Tf 6.58 0 Td[(2nn. (5) 107

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Wherefor(a),(b)and(c)arefromrepeateduseoflemma 18 ,while(d)isaresultoftheorem 5.3 inconjunctionwithlemma 22 .Theprobabilityboundsinequation( 5 )canbealternativelyrepresentedgiventhatthenumberofcodewordsforsource1andsource2,representedby~R1and~R2,aresuchthat~R1>I(U;X1)+n~R2>I(U;X2)+n~R1+~R2>I(U;X1,X2)+nthenbyuseofequation( 5 ), PrfUn2U(Cn1(W1),Cn2(W2))\V(Cn1(W1);W2,Cn2)g2n(R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;UjX1)+n)2)]TJ /F4 7.97 Tf 6.59 0 Td[(nminI(~X1;~Vj~U).(5)ThusgivinginformationintegrityaslongasR2
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ourschemewecaneasilyseethatwithprobabilityapproaching1,bothreceiverswillagreeuponthekey.BecausewedesireuniformityofthekeywetaketheharmfulactionofannouncingtotherelaynodeknowledgeoftheinduceddistributionP~X1~X2~U.Whatremainsistoderivepropertiesconcerningthiskey,webeginwithdeterminingtheprobabilityofanygivenkey,foraparticularunandP~X1~X2~U,p(kjun,Tn~X1~X2~U)=Xcn1,cn2,w1,w2p(k,cn1,cn2,w1,w2,un) p(un)1(cn1(w1),cn2(w2),un)2Tn~X1~X2~U=Xcn1,cn2,w1,w21(cn1(w1),cn2(w2),un)2Tn~X1~X2~Up(unjcn1(w1),cn2(w2))p(w1,w2jcn1,cn2)p(cn1,cn2) p(un)(a)=2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(R1+R2)2)]TJ /F4 7.97 Tf 6.59 0 Td[(nDP~Uj~X1~X2(ujx1,x2)kp(ujx1,x2)+H(P~Uj~X1~X2(ujx1,x2))1 p(un)Xcn1,cn2,w1,w21(cn1(w1),cn2(w2),un)2Tn~X1~X2~Up(cn1,cn2). (5)Where(a)isbecausebothP~Uj~X1,~X2andPUjX1X2areknown,aswellasun2Tn~Uj~X1~X2(cn1(w1),cn2(w2))thenbydenitionPr(unjxn1,xn2)=2)]TJ /F4 7.97 Tf 6.58 0 Td[(nDP~Uj~X1~X2(ujx1,x2)kp(ujx1,x2)+H(P~Uj~X1~X2(ujx1,x2)),thestrictequalityallowsusthesurprisingbenetofdroppingthedependenceuponthecodewordsentirely.Fromequation( 5 )itcanbeeasilyseenthatforanygivenunandTn~X1~X2~Uthattheprobabilityofallkeysareequal.Becausethefwedenedtobeonetooneoverthesetsinquestionforanycodebookscn1,cn2andksuchthatthereexistsaw1,w2,unsuchthatk=f(un,w1,w2),then p(kjun,Tn~X1,~X2,~U)=1 Pw1,w21(cn1(w1),cn2(w2),un)2Tn~X1,~X2,~U,(5)andequivalentlyforanysetofkeysforparticularunandP~X1,~X2,~U,denotedK(un,P~X1,~X2,~U), Xk2K1)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(k2K(un,P~X1,~X2,~U)=Xw1,w21(cn1(w1),cn2(w2),un)2Tn~X1,~X2,~U.(5) 109

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Todeterminethenumberofkeysgenerated,whatisstillrequiredistolowerboundthesizeofthedenominatorinequation( 5 )whenaveragedoverallcodebooks.TothisgoalrecognizethatXcn1,cn2,w1,w21(cn1(w1),cn2(w2),un)2Tn~X1~X2~Up(cn1,cn2)=Xcn1,cn2,w1,w21cn1(w1)2Tn~X1j~U(un)1cn2(w2)2Tn~X2j~U~X1(un,cn1(w1))p(cn1,cn2)Xcn1,cn2,w210B@cn2(w2)2[w1:cn1(w1)2Tn~X1j~U(un)Tn~X2j~U~X1(un,cn1(w1))1CAp(cn1,cn2)Clearlyforanyvalueofunandcn1(w1)2Tn[X1jU](un)thereexistsadistributionP~Xj~Usuchthatcn1(w1)2Tn~Xj~U(un)andthatTn~Xj~U(un)>2n(H(X1jU))]TJ /F17 7.97 Tf 6.59 0 Td[(n).Using[ 35 ,p.419],again,wecandeterminethat Pr(Xw1Cn1(w1)2Tn~X1j~U(un)<2n(jR)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1;U)j+)]TJ /F17 7.97 Tf 6.58 0 Td[(n))e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n(jR)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X1;U)j+)]TJ /F14 5.978 Tf 5.75 0 Td[(n).(5)Ournexttaskistodeterminethenumberofw2whichpairwiththesew1,aslightproblemthoughexistsifthesetsofTn~X2j~X1,~U(cn1(w1),un),overlapfordifferentvaluesofw1,toaccountforthiswebeginbynotingthatXxn22Xn210B@xn22[w1:cn1(w1)2Tn~X1j~U(un)Tn~X2j~U,~X1(un,cn1(w1))1CAXxn2Pw1:cn1(w1)2Tn~X1j~U(un)1xn22Tn~X2j~U,~X1(un,cn1(w1)) maxxn2Pw1:cn1(w1)2Tn~X1j~U(un)1xn22Tn~X2j~U,~X1(un,cn1(w1)). (5)Butforanyparticularxn2,weknowthatPrnCn1(w1)2Tn~X1j~U,~X2(un,xn2)o2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(I(X1;U,X2)+n),andduetothelimitsofchannelcapacitywehavethatR1
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foranyR1>n>0,wehave e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2nnPr8<:2nR1XiZi>2nn9=;Pr(Xw11Cn1(w1)2Tn~X1j~U,~X2(un,xn2)2nn),(5)andconsequentlyPr(maxxn22Tn[X2]Xw11Cn1(w1)2Tn~X1j~U,~X2(un,xn2)2nn)Xxn22Tn[X2]Pr(Xw11Cn1(w1)2Tn~X1j~U,~X2(un,xn2)2nn)2n(H(X2)+n)e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2nn
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Finally,fromequations( 5 )and( 5 )andallun2Tn[UjX1,X2](cn1(w1),cn2(w2)thattheirmustexistannsatisfyingthedeltaconventionsuchthatXcn1,cn2,w1,w21(cn1(w1),cn2(w2),un)2Tn~X1,~X2,~Up(cn1,cn2)2njR2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X2;UjX1)+jR1)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X1;U)j+j+)]TJ /F17 7.97 Tf 6.59 0 Td[(n. (5)Thusforallun2Tn[UjX1,X2](cn1(w1),cn2(w2),equation( 5 )allowsustoseethattheminimumequalprobablekeygenerationrate,Rk,isRk=1 nlog2minXk2K1)]TJ /F5 11.955 Tf 5.47 -9.68 Td[(k2K(un,P~X1,~X2,~U)R2)]TJ /F5 11.955 Tf 11.96 0 Td[(I(X2;UjX1)+jR1)]TJ /F5 11.955 Tf 11.96 0 Td[(I(X1;U)j++)]TJ /F13 11.955 Tf 11.95 0 Td[(n, (5)wheretheminimumisoverallP~X1,~X2,~U2[X1,X2,U].WhencombinedwiththemaximumvaluesofR1andR2allowableforreliabletransmissionweobtainthatourschemeachieves Rk=I(X1;UjX2))]TJ /F5 11.955 Tf 11.25 0 Td[(I(X1;U))]TJ /F13 11.955 Tf 11.25 0 Td[(n=I(X2;UjX1))]TJ /F5 11.955 Tf 11.25 0 Td[(I(X2;U))]TJ /F13 11.955 Tf 11.25 0 Td[(n=I(X1;X2jU))]TJ /F13 11.955 Tf 11.25 0 Td[(n,(5)whichalsohappenstobethesecretkeycapacity. SecretTransmissions.Thepreviousschememayalsobeusedinconjunctionwithbinningwherealltransmittedsequencesaredividedinto2n(Rk)]TJ /F17 7.97 Tf 6.58 0 Td[(n)binstoachievesecrecyinthesenseofprivatetransmissions.Unfortunatelytheuniformitywillbelostregardlessofhowthemessageischosen,becauseofthebinningprocedure.Although,itisclearthatthereexists(~W1,~W2)(W1,W2)suchthatI(~W1,~W2;Un)=0and1 nlog2(~W1~W2))]TJ /F3 11.955 Tf 13.25 8.09 Td[(1 nlog2j(W1,W2)j
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Ifa>b,theboundabovecanbefurtherloosenedto (1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)]TJ /F4 7.97 Tf 6.58 0 Td[(na)2nb1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b+n)(5)foranyn>0whenevernissufcientlylarge. 2. Forany0nminf2na,2g,wehave (1)]TJ /F13 11.955 Tf 11.96 0 Td[(n2)]TJ /F4 7.97 Tf 6.59 0 Td[(na)2nb(0ifab)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b)2ifa>b.(5) Proof. 1.)Theboundin( 5 )isadirectconsequenceoftheinequality(1)]TJ /F5 11.955 Tf 12.51 0 Td[(xy)m1)]TJ /F5 11.955 Tf 11.96 0 Td[(x+e)]TJ /F4 7.97 Tf 6.59 0 Td[(ymfor0x,y1andm>0givenin[ 9 ,Lemma10.5.3].Supposenowa>bandn>0.Simplecalculusshowsthattheinequalitye)]TJ /F4 7.97 Tf 6.58 0 Td[(x1)]TJ /F3 11.955 Tf 12.19 0 Td[(2)]TJ /F17 7.97 Tf 6.58 0 Td[(nxisvalidfor0xnln2.Applyingthisinequalitytoexp()]TJ /F3 11.955 Tf 9.3 0 Td[(2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(a)]TJ /F4 7.97 Tf 6.58 0 Td[(b))forn)]TJ /F6 7.97 Tf 7.99 0 Td[(ln(nln2) (a)]TJ /F4 7.97 Tf 6.59 0 Td[(b)ln2,theupperboundin( 5 )isloosenedto( 5 ).2.)First,itisobviousthat(1)]TJ /F13 11.955 Tf 12.41 0 Td[(n2)]TJ /F4 7.97 Tf 6.59 0 Td[(na)2nb0.Supposenowa>b.Ifn=0,thelowerboundin( 5 )istriviallytrue.Considern>0.Thenlog(1)]TJ /F13 11.955 Tf 11.95 0 Td[(n2)]TJ /F4 7.97 Tf 6.58 0 Td[(na)2nb)]TJ /F3 11.955 Tf 11.95 0 Td[(log(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b))=1 ln22nb1Xk=1kn2)]TJ /F4 7.97 Tf 6.58 0 Td[(kna k)]TJ /F10 7.97 Tf 16.35 14.94 Td[(1Xk=12)]TJ /F4 7.97 Tf 6.58 0 Td[(kn(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b) k1 ln21Xk=12)]TJ /F4 7.97 Tf 6.59 0 Td[(kn(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b) kkn2)]TJ /F4 7.97 Tf 6.59 0 Td[(nb(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(11 ln21Xk=12)]TJ /F4 7.97 Tf 6.59 0 Td[(kn(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b) k=)]TJ /F3 11.955 Tf 11.29 0 Td[(log)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b) (5)wheretheinequalityinthesecondlastlineisduetotheassumptionthatn2.Itistheneasytocheckthat( 5 )givesatrivialupperboundandanon-triviallowerboundon(1)]TJ /F13 11.955 Tf 11.96 0 Td[(n2)]TJ /F4 7.97 Tf 6.59 0 Td[(na)2nb.Thelowerboundisindeedtheonein( 5 ). Corollary3. (ModiedFreshmanBinomialTheorem)Ifa>b0,then1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b)]TJ /F17 7.97 Tf 6.58 0 Td[(n))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)]TJ /F4 7.97 Tf 6.58 0 Td[(na2nb1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b+n), 113

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and1+2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(a)]TJ /F4 7.97 Tf 6.58 0 Td[(b+2n))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(1+2)]TJ /F4 7.97 Tf 6.58 0 Td[(na2nb1+2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b)]TJ /F6 7.97 Tf 6.59 0 Td[(2n),forallsufcientlylargen. Proof. Toprovetherstassertionnotethatchoosen1inpart1)ofLemma 17 .Thenforanyn1,nnn,andthus1)]TJ /F3 11.955 Tf 12.08 0 Td[(2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b+n)1)]TJ /F3 11.955 Tf 12.08 0 Td[(2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b+n)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 6.58 0 Td[(na)2nbbypart1)ofLemma 17 .Furtherthelowerboundintheassertioncanbeobtainedbylooseningthelowerboundinpart2)ofLemma 17 asfollows:(1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.58 0 Td[(b))21)]TJ /F3 11.955 Tf 11.95 0 Td[(22)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b)1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(a)]TJ /F4 7.97 Tf 6.58 0 Td[(b)]TJ /F17 7.97 Tf 6.59 0 Td[(n).Theupperboundinthesecondassertionfollowsfromtherstassertionasbelow:(1+2)]TJ /F4 7.97 Tf 6.59 0 Td[(na)2nb=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F6 7.97 Tf 6.59 0 Td[(2na)2nb (1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 6.59 0 Td[(na)2nb1)]TJ /F3 11.955 Tf 11.95 0 Td[(2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(2a)]TJ /F4 7.97 Tf 6.59 0 Td[(b+n) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b)]TJ /F17 7.97 Tf 6.59 0 Td[(n)=1+2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(a)]TJ /F4 7.97 Tf 6.58 0 Td[(b)]TJ /F17 7.97 Tf 6.59 0 Td[(n)1+2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(a+2n) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b)]TJ /F17 7.97 Tf 6.59 0 Td[(n)1+2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(a)]TJ /F4 7.97 Tf 6.59 0 Td[(b)]TJ /F6 7.97 Tf 6.58 0 Td[(2n)wherethelastinequalityisvalidforallsufcientlylargen.Thelowerboundfollowssimilarly. Lemma18. Foranyfunctionsf:An!Ynandg:Bn!Yn,andsetsA02An,B02Bn,Xb2B0Xa2A01(f(a)=g(b))"maxb2B0Xa2A01(f(a)=g(b))#Xb1 [a2A0ff(a)=g(b)g! Proof. Simply,Xb2B0Xa2A01(f(a)=g(b))=XbXa1 [a2A0ff(a)=g(b)g!1(f(a)=g(b))"maxb2B0Xa2A01(f(a)=g(b))#Xb2B01 [a2A0ff(a)=g(b)g!. 114

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Forthefollowingfewlemmas,itwillbehelpfultodenethefollowingE(w)=Cn(w)2[^w2=fwgTnPX0jU,V,X(un,vn,Cn(^w))N(w)=Cn(w)=2[^w2=fwgTnPX0jU,V,X(un,vn,Cn(^w)). Lemma19. LetW=TnXbethesetofallcodewords,forwhichCn(w)ischosenuniformlyfrom,andletW0W,thenforanyU,V,X0suchthatthereexistsaconstantcsuchthatPrnCn(w)2TnPX0jU,V,X(un,vn,Cn(^w))ojW0j2)]TJ /F4 7.97 Tf 6.58 0 Td[(ncn,thenthereexistsansatisfyingthedeltaconvention 1 nlog2PrEW0,NW=fW0g)]TJ /F3 11.955 Tf 13.25 8.08 Td[(1 nlog2PrnCn(w)2TnPX0jUVX(un,vn,Cn(^w))ojW0jn.(5) Proof. First,throughcarefulmanipulationPrEW0,NW=fW0g=PrEW0jNW=fW0gPrNW=fW0g(a)=PrEW0jNW=fW0gY~w2W=W0PrfN~w(w)g=PrEW0jNW=fW0gY~w2W=W0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(PrfE~w(w)g)(b)=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(PrfNW0g)Y~w2W=W0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(PrfE~w(w)g)= 1)]TJ /F9 11.955 Tf 16.4 11.36 Td[(Y^w2W0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(PrfE^wg)!Y~w2W=W0(1)]TJ /F3 11.955 Tf 11.95 0 Td[(PrfE~w(w)g)=Y~w2W=W0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(PrfE~w(w)g))]TJ /F9 11.955 Tf 24.8 11.36 Td[(Y^w2W=fwg(1)]TJ /F3 11.955 Tf 11.95 0 Td[(PrfE^wg).Where(a)and(b)areduetotheindependenceofthecodewords.Usingtheboundsfromcorollary 3 establishes1 nlog2PrEW0,NW=fW0g)]TJ /F3 11.955 Tf 13.25 8.09 Td[(1 nlog2PrnCn(w)2TnPX0jUVX(un,vn,Cn(^w))ojW0j
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Lemma20. LetallcodewordsbechosenuniformlyoverW=TnX,andletW0W.FurthermoresupposethatthereexistsrandomvariablesU,V,X0thataredistributedsuchthatPX0jUVX(x0ju,v,x)=PXjUVX0(x0ju,v,x),andPrnCn(w)2TnPX0jU,V,X(un,vn,Cn(^w))ojW0j2)]TJ /F4 7.97 Tf 6.58 0 Td[(ncn,thenthereexistsanthatsatisesthedeltaconventionssuchthatPr8<:\w2W0EW(w),\~w2W=W0NW(~w)9=;jW0jPrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nnjW0j 2. (5) Proof. FirstnotethatE~w(w)\Ew(~w)=Ew(~w)=E~w(w).Fromhereitcanbeseenthroughprobabilitymanipulation,thatPr8<:\w2W0EW(w),\~w2W=W0NW(~w)9=;=Pr8<:\w2W0[^w2W0=fwgE^w(w),\~w2W=W0NW(~w)9=;=Pr8>><>>:\w2W0EW0(w),\~w2W=W0^w2W=f~wgN^w(~w)9>>=>>;(a)=XW00W0=fw00gPr8>><>>:\^w002W00E^w00(w00)\~w002W0=fW00,w00gN~w00(w00)\w2W0=fW00,w00gEW0=fw00g(w)\~w2W=W0^w2W=f~wgN^w(~w)9>>=>>;(b)=XW00W0=fw00gPr8>><>>:\^w002W00E^w00(w00)\~w002W=fW00,w00gN~w00(w00)\w2W0=fW00,w00gEW0=fw00g(w)\~w2W=W0^w2W=f~w,w00gN^w(~w)9>>=>>; 116

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(c)=XW00W0=fw00gPr8<:\w2W0=fW00,w00gEW0=fw00g(w)\~w2W=W0NW=fw00g(~w)9=;Pr8<:\^w002W00E^w00(w00)\~w002W=fW00,w00gN~w00(w00)9=;. (5)Line(a)isdueto\w2W0[w02W0=fwgEw0(w)=[^w002W0=fw00gE^w00(w00)\w2W0=fw00g[w02W0=fwgEw0(w)=[W00W0=fw00g24\^w002W00E^w00(w00)\~w002W0=fW00,w00gN~w00(w00)\w2W0=fW00,w00gEW0=fw00g(w)35,andthefactthatthelawoftotalprobabilityuniquelydividesthesetssothattheyaredisjointandthustheprobabilityoftheirunionisequaltothesumoftheindividualprobabilities.While(b)istheresultofremovingredundantsets,and(c)isduetocodewordindependence.Clearlynowanysubsetofasummationofprobabilitiesislessthanthewholesummation,andthuschoosingthesetwherejW00j=1fromequation( 5 )weobservePr8<:\w2W0EW(w),\~w2W=W0NW(~w)9=;XW00W0=fw00gjW00j=1Pr8<:\w2W0=fW00,w00gEW0=fw00g(w)\~w2W=W0NW=fw00g(~w)9=;Pr8<:\^w002W00E^w00(w00)\~w002W=fW00,w00gN~w00(w00)9=;(a)(jW0j)]TJ /F3 11.955 Tf 17.93 0 Td[(1)Pr8<:\w2W0=f~w00,w00gEW0=fw00,~w00g(w)\~w2W=W0NW=fw00g(~w)9=;PrfE~w00(w00)g2)]TJ /F4 7.97 Tf 6.59 0 Td[(nn. 117

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Here(a)isduetolemma 19 ,andP(A[B)>P(A).Alternatively,notethatifjW0jisoddwewillrequireknowledgeofwhatoccursinthecasejW00j=2,similarlyPr8<:\w2W0EW(w),\~w2W=W0NW(~w)9=;(jW0j)]TJ /F3 11.955 Tf 17.93 0 Td[(1) 2jW0jPr8><>:\w2W0=f~w00,^w00,w00gEW0=f~w00,^w00,w00g(w)\~w2W=W0NW=fw00g(~w)9>=>;(PrfE~w00(w00)g)22)]TJ /F4 7.97 Tf 6.59 0 Td[(nn.Ineithereventthisprocessofusingthelawoftotalprobabilitytoremoveredundancyandthenlowerboundtheprobabilityintermsofaprobabilitydependingonlyuponasubsetoftheoriginalcanberepeatedadterminusandthefollowingobtained;Pr8<:\w2W0EW(w),\~w2W=W0NW(~w)9=;8>><>>:(jW0j)]TJ /F3 11.955 Tf 17.93 0 Td[(1)!!PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nnjW0j 2ifjW0jisevenjW0j!! jW0j)]TJ /F6 7.97 Tf 8.94 0 Td[(2PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nnjW0j+1 2ifjW0jisodd. (5)Tofurtherlowerboundtheevenvalueinequation( 5 )associatedwithjW0jbeingeven,(jW0j)]TJ /F3 11.955 Tf 17.93 0 Td[(1)!!PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.58 0 Td[(nnjW0j 2(a)=2)]TJ 7.78 4.52 Td[(jW0j 2jW0j! jW0j 2!PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nnjW0j 2(b)2)]TJ 7.78 4.53 Td[(jW0j 2p 2jW0jjW0j+1 2ejW0j ejW0j 2jW0j 2+1 2e)]TJ 7.78 4.53 Td[(jW0j 2PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nnjW0j 2=p 4e)]TJ 7.78 4.53 Td[(jW0j 2)]TJ /F6 7.97 Tf 6.59 0 Td[(1jW0jjW0j 2PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nnjW0j 2=2p ejW0jPrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nn1 ejW0j 2. (5) 118

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Here(a)istheresultofthedoublefactorialformula,while(b)isduetoStirling'sbounds.Similarly,forthevalueofequation( 5 )associatedwithjW0jbeingodd,jW0j!! jW0j)]TJ /F3 11.955 Tf 17.94 0 Td[(2PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nnjW0j+1 2(a)2)]TJ 7.78 4.52 Td[(jW0j+1 2(jW0j+1)! jW0j+1 2!PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.58 0 Td[(nnjW0j+1 2(b)2)]TJ 7.78 4.52 Td[(jW0j+1 2p 2(jW0j+1)jW0j+1+1 2e)]TJ /F6 7.97 Tf 6.59 -.11 Td[((jW0j+1) ejW0j+1 2jW0j+1 2+1 2e)]TJ /F28 7.97 Tf 6.59 11.2 Td[(jW0j+1 2PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.58 0 Td[(nnjW0j+1 22p ejW0jPrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nn1 ejW0j+1 2, (5)where(a)and(b)reprisetheirpreviousexplanations.Toshowthatthereexistsan0nthatsatisesthedeltaconventionssuchthatitwillgiveustheformofequation( 5 )simultaneouslywemustshowthatforoddvaluesofjW0jtheadditional1 2intheexponentcanbemanipulatedintoavalueofnwithoutbreakingthedeltaconvention.ObservejW0jPrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(n0njW0j 2jW0jPrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nnjW0j+1 2occursifandonlyif 2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(0n)]TJ /F17 7.97 Tf 6.59 0 Td[(n)jW0j 2jW0jPrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(nn1 2.(5)RepresentingjW0j=2nandPrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o=2)]TJ /F4 7.97 Tf 6.59 -.01 Td[(n,aswillbetheircommonrepresentation,equation( 5 )takesonanequivalentformof0n2)]TJ /F4 7.97 Tf 6.59 0 Td[(n()]TJ /F13 11.955 Tf 11.95 0 Td[(+n)+n. 119

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Clearlyforanyxedand,thereexistsannsuchthat2)]TJ /F4 7.97 Tf 6.59 0 Td[(n()]TJ /F13 11.955 Tf 11.95 0 Td[(+n)+n<2n,andthereforebecausewemaychoose0n=2ntheremustexista0nsatisfyingthedeltaconventionandequation( 5 ). Lemma21. BoundingcodebooksLetallcodewordsbechosenuniformlyoverW=TnX,andletW0W.FurthermoresupposethatthereexistsrandomvariablesU,V,X0thataredistributedsuchthatPX0jUVX(x0ju,v,x)=PXjUVX0(x0ju,v,x),andPrnCn(w)2TnPX0jU,V,X(un,vn,Cn(^w))ojW0j2)]TJ /F4 7.97 Tf 6.58 0 Td[(ncn,thenPr(Xw2W1(EW(w))jWj2PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.58 0 Td[(nn)e)]TJ /F17 7.97 Tf 6.58 0 Td[((1 )jWj2PrCn(w0)2TnPX0jUVX(un,vn,Cn(~w))2)]TJ /F16 5.978 Tf 5.76 0 Td[(nn+e)]TJ /F17 7.97 Tf 6.59 0 Td[(()jWj2PrCn(w0)2TnPX0jUVX(un,vn,Cn(~w))2)]TJ /F16 5.978 Tf 5.75 0 Td[(nn. (5) Proof. LetZ1,...,ZjWjbedistributediidBernoulliwithPrfZi=1g=jWjPrfE~w(w)g2)]TJ /F4 7.97 Tf 6.58 0 Td[(n0n,forwhichitcaneasilybeseenthat,Pr8<:\w2W0Zw=1,\w2W=W0Zw=09=;jWjPrfE~w(w)g2)]TJ /F4 7.97 Tf 6.59 0 Td[(n0njW0j.ItcantheneasilybeshownthatforalljW0jjWj2PrfE~w(w)g2)]TJ /F4 7.97 Tf 6.58 0 Td[(n(20n)]TJ /F17 7.97 Tf 6.59 0 Td[(n)thatPr8<:\w2W0Zw=1,\w2W=W0Zw=09=;Pr8<:\w2W0EW(w),\~w2W=W0NW(~w)9=;.Now,forany>1choose0nsuchthat2)]TJ /F6 7.97 Tf 6.59 0 Td[(2n0n=2)]TJ /F15 5.978 Tf 5.75 0 Td[(2nn 2=2)]TJ /F4 7.97 Tf 6.59 0 Td[(n0n2)]TJ /F16 5.978 Tf 5.76 0 Td[(nn .Forthisparticularchoiceof0nitfollowsthatPr(Xw2W1(EW(w))jWj2PrfE~w(w)g2)]TJ /F4 7.97 Tf 6.59 0 Td[(n0n) 120

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1)]TJ /F3 11.955 Tf 11.95 0 Td[(Pr(Xw2W1(EW(w))1 jWj2PrfE~w(w)g2)]TJ /F4 7.97 Tf 6.59 0 Td[(n0n)=1)]TJ /F9 11.955 Tf 105.13 11.36 Td[(XW01 jWj2PrfE~w(w)g2)]TJ /F16 5.978 Tf 5.76 0 Td[(n0njWj2PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(n0n)e)]TJ /F17 7.97 Tf 6.58 0 Td[((1 )jWj2PrCn(w0)2TnPX0jUVX(un,vn,Cn(~w))2)]TJ /F16 5.978 Tf 5.76 0 Td[(n0n+e)]TJ /F17 7.97 Tf 6.59 0 Td[(()jWj2PrCn(w0)2TnPX0jUVX(un,vn,Cn(~w))2)]TJ /F16 5.978 Tf 5.75 0 Td[(n0n.Equation(a)istheresultofchoosing2)]TJ /F6 7.97 Tf 6.59 0 Td[(2n0n=2)]TJ /F4 7.97 Tf 6.58 0 Td[(n0n2)]TJ /F16 5.978 Tf 5.75 0 Td[(nn ,whichthenimpliesthattheprobabilitiesareboundedforalljW0jjWj2PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(20n)]TJ /F17 7.97 Tf 6.59 0 Td[(n)=jWj2PrnCn(w0)2TnPX0jUVX(un,vn,Cn(~w))o2)]TJ /F4 7.97 Tf 6.59 0 Td[(n0n1 Theorem5.3. Letallcodewordsofajointcodeforadiscretememorylesschannelp(unjxn1,xn2)=Qp(ujx1x2)bechosenuniformlyindependentlyovertypicalsetTn[X1]andtypeTn[X2].Thenforanyun,vndenethesetD(un,vn)=[cn2(w2)2TnPX2jUV(un,vn)cn2(^w2)2TnPX02jUV(un,vn)hTn[X1jUX2](un,cn2(w2))\Tn[X1jUX2](vn,cn2(^w2))i 121

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andaccompanyingA(un,vn)=ncn2:jD(un,vn)j22n(R2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X2;UX1)+n)2nmaxH(~X1j~V,~U)othenthereexistsa0nsatisfyingthedeltaconventionsuchthat PrfA(un,vn))ge)]TJ /F6 7.97 Tf 6.58 0 Td[(2n0n.(5) Proof. TobeginwenotethatjD(un,vn)jXcn2(w2)2TnPX2jUV(un,vn)cn2(^w26=w2)2TnPX02jUV(un,vn)max2nH(~X1j~U,~V,~X2,~X02). (5)Foreachcodewordfromcodebook2andeachunthesetofpossiblexn1shouldbenearlyunique,andthustheboundpresentedin( 5 )willnotbeloose.Itshouldbenotedthatthevalueofmax2nH(~X1j~U,~V,~X2,~X02)changesbaseduponthedifferentcodewordscn2(w2)andcn2(^w2)selected.Tocontinuewithanalysiswelookateverypossibilityofmax2nH(~X1j~U,~V,~X2,~X02),andupperboundthenumberofcodewordspairingsfortheseparticularvaluesofPX2,X02,U,V.FurthermoreweonlyneedconsiderdistributionsofX2,X02,U,VsuchthatR2>min(I(X2;U,V),I(X02;U,V)),elsetheexpectednumberofcodewordsisessentially0andwillnotfactorintotheimportantcasesofthebound.WemayalsoassumethatduetotheroleoftypicalitythefollowingaretrueaswelljH(X2))]TJ /F5 11.955 Tf 11.95 0 Td[(H(X02)jH(X2jU,V)!8>>>>>><>>>>>>:I(X2;U,V)>I(X02jU,V))]TJ /F13 11.955 Tf 11.96 0 Td[(nandI(X2;U,V,X02)>I(X02;U,V,X2))]TJ /F13 11.955 Tf 11.95 0 Td[(n. 122

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Wenextdividetheregionsasso 1. PX2jUV(x2ju,v)6=PX02jUV(x2ju,v) (a) R2max(I(X2;U,V),I(X02;U,V)) i. R2>min(I(X2;U,V,X02),I(X02;U,V,X2)) ii. R2min(I(X2;U,V,X02),I(X02;U,V,X2)) (b) R2min(I(X2;U,V,X02),I(X02;U,V,X2)) ii. R2min(I(X2;U,V,X02),I(X02;U,V,X2)) 2. PX2jUV(x2ju,v)=PX02jUV(x2ju,v)Condition(1.):PX2jUV(x2ju,v)6=PX02jUV(x2ju,v)Forallconditionsdesignatedby(1.)assumewithoutlossofgeneralitythatI(X2;U,V)I(X02;U,V),thealternativehypothesisissymmetric.Condition(1.a.):R2max(I(X2;U,V),I(X02;U,V))Itisimportanttonotethatfromlemma 15 ,wemaydeterminethatthereexistsansatisfyingthedeltaconventionsuchthat Pr(Xw21Cn2(w2)2TnPX2jUV(un,vn)>2n(R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;U,V)+n))e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2n(R2)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X2jU,V)+n),(5)andsymmetrically Pr(Xw21Cn2(w2)2TnPX02jUV(un,vn)>2n(R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X02;U,V)+n))e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2n(R2)]TJ /F16 5.978 Tf 5.76 0 Td[(I(X02jU,V)+n).(5)Condition(1.a.i.):R2>min(I(X2;U,V,X02),I(X02;U,V,X2))BecauseweassumedthatI(X2;U,V)>I(X02;U,V),wecandeterminethatmin(I(X2;U,V,X02),I(X02;U,V,X2))>I(X02;U,V,X2))]TJ /F13 11.955 Tf 12.79 0 Td[(n.Similartoequation( 5 ),becauseofthesizeofR2,foreveryCn2(w2)2TnPX2jUV(un,vn),thereexistsan0nsatisfyingthedeltaconventionsuchthatPr(X^w26=w21Cn2(^w2)2TnPX02jUVX2(un,vn,Cn2(w2))>2n(R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X02;U,V,X2)+0n)) 123

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e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n(R2)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X02;U,V,X2)+n). (5)Wecalltheunionoftheeventsthatareboundedinequation( 5 )and( 5 )asA1.a.i(un,vn),andbyuseoftheunionbound Pr(A1.a.i(un,vn))e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2n(R2)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X2;U,V)+n)+2nR2e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n(R2)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X02;U,V,X2)+0n).(5)ForcodebooksthatarenotinA1.a.i(un,vn)itisstraightforwardtoseethatXCn2(w2)2TnPX2jUV(un,vn)Cn2(^w2)2TnPX02jUVX2(un,vn,Cn2(w2))max2nH(~X1j~U,~V,~X2,~X02)2n(R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;U,V)+n)2n(R2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X02;U,V,X2)+0n)max2nH(~X1j~U,~V,~X2,~X02)22n(R2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X2;U,X1)+~n)max2nH(~X1j~U,~V), (5)where~n=n+0nandthussatisesthedeltaconvention.Condition(1.a.ii.):R2min(I(X2;U,V,X02),I(X02;U,V,X2))Inthisregion,byacombinationofdenitionandassumptionI(X2;U,V)R2I(X02;U,V,X2)
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Consequentlyduetoequation( 5 ),weobtainPr8><>:X^w26=w210B@Cn2(^w2)2[Cn2(w2)2TnPX2jUV(un,vn)TnPX02jUVX2(un,vn,Cn2(w2))1CA>2n(j2R2)]TJ /F4 7.97 Tf 6.58 0 Td[(I(X02;U,V,X2))]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;U,V)j+0n)9>=>;e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2n(j2R2)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X02;U,V,X2))]TJ /F16 5.978 Tf 5.76 0 Td[(I(X2;U,V)j+0n). (5)Compilingallprobabilitiesrequiredandover-boundingwiththeunionboundgivesPrfA1.a.ii(un,vn)g<2n(H(X2)+c2n)e)]TJ /F17 7.97 Tf 6.59 0 Td[(2nn+e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n(j2R2)]TJ /F16 5.978 Tf 5.76 0 Td[(I(X02;U,V,X2))]TJ /F16 5.978 Tf 5.76 0 Td[(I(X2;U,V)j+0n)+e)]TJ /F17 7.97 Tf 6.58 0 Td[(()2n(R2)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X2;U,V)+00n). (5)Next,duetoR2I(X2;U,V,X02)inconjunctionwithlemma 15 Pr(max^w2Xw26=^w21Cn2(w2)2TnX2jUVX02(un,vn,Cn2(^w2))>2nn)2n(H(X2)+0n)e)]TJ /F17 7.97 Tf 6.58 0 Td[(2nn.(5)Returningtoequation( 5 ),andbyuseofequations( 5 ),( 5 )and( 5 )thatifallcodebooksinA1.a.ii(un,vn)areremovedfromequation( 5 ),thenXCn2(w2)2TnPX2jUV(un,vn)Cn2(^w26=w2)2TnPX02jUVX2(un,vn,Cn2(w2))max2nH(~X1j~U,~V,~X2,~X02)2nj2R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X02;U,V,X2))]TJ /F4 7.97 Tf 6.58 0 Td[(I(X2;U,V)j+~nmax2nH(~X1j~U,~V,~X2,~X02)22n(R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;U,X1)+~n)max2nH(~X1j~U~V). (5)Condition(1.b.):R2I(X02;U,V),thenequation( 5 )isstillvalid.Condition(1.b.i)R2>min(I(X2;U,V,X02),I(X02;U,V,X2))Considerthatforthisset[Cn2(w2)2TnPX02jUV(un,vn)TnPX2jUVX02(un,vn,Cn2(w2)) 125

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=Xxn10BB@xn2[Cn2(w2)2TnPX02jUV(un,vn)TnPX2jUVX02(un,vn,Cn2(w2))1CCA=Xxn,Cn2(w2)1xn2TnPX2jUVX02(un,vn,Cn2(w2))1Cn2(w2)2TnPX02jUV(un,vn) PCn2(w2)1Cn2(w2)2TnPX02jUV(un,vn)1xn2TnPX2jUVX02(un,vn,Cn2(w2))=Xxn,Cn2(w2)1xn2TnPX2jUVX02(un,vn,Cn2(w2))1Cn2(w2)2TnPX02jUV(un,vn) PCn2(w2)1Cn2(w2)2TnPX02jUVX2(un,vn,xn) (5)Fromlemma 15 weobtainthatPr8<:XCn2(w2)1Cn2(w2)2TnPX02jUVX2(un,vn,xn)<1 2n(R)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X02;U,V,X2))]TJ /F17 7.97 Tf 6.59 0 Td[(n)9=;e)]TJ /F17 7.97 Tf 6.59 0 Td[((1 )2n(R)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X02;U,V,X2))]TJ /F14 5.978 Tf 5.76 0 Td[(n) (5)and Pr8<:XCn2(w2)1Cn2(w2)2TnPX02jUV(un,vn)>2n(R)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X02;U,V,X2)+n)9=;e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n(R)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X02;U,V)+n).(5)Labelingtheunionofeventsfromequation( 5 )and( 5 )asA1.b.i(un,vn),wemayobtainfromtheunionbound PrfCn22A1.b.i(un,vn)ge)]TJ /F17 7.97 Tf 6.59 0 Td[((1 )2n(R)]TJ /F16 5.978 Tf 5.76 0 Td[(I(X02;U,V,X2))]TJ /F14 5.978 Tf 5.76 0 Td[(n)+e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n(R)]TJ /F16 5.978 Tf 5.75 .01 Td[(I(X02;U,V)+n).(5)ForthecodebooksnotinA1.b.i(un,vn),equation( 5 )canbeupperboundedas[Cn2(w2)2TnPX02jUV(un,vn)TnPX2jUVX02(un,vn,Cn2(w2))2nH(X2jU,V,X02)2n(R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X02;U,V)+n)2)]TJ /F4 7.97 Tf 6.59 0 Td[(n(R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X02;U,V,X2))]TJ /F17 7.97 Tf 6.59 0 Td[(n)2n(H(X2jU,V)+2n). (5) 126

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Andthus Pr8>><>>:Cn2(w2)2[Cn2(w2)2TnPX02jUV(un,vn)TnPX2jUVX02(un,vn,Cn2(w2))>2nn9>>=>>;I(X2;U,V)=I(X02;U,V),andthusbothequations( 5 )and( 5 )arevalid.Furthermoreapartitioningofdistributionsleadstothefollowing10BBBB@Cn2(w)2[^w(6=w)Cn2(^w)2TnPX02jUV(un,vn)TnPX02jUVX2(un,vn,Cn(^w))1CCCCA=10BBBB@Cn2(w)2[PX02jUVX2[^w(6=w)Cn2(^w)2TnPX02jUVX2(un,vn,Cn2(w))TnPX2jUVX02(un,vn,Cn(^w))1CCCCA. (5)ButforanydistributionsuchthatPX02jUVX2(x02ju,v,x2)=PX2jUVX02(x02ju,v,x2)bydenitionforany^w6=w Cn2(^w)2TnPX02jUVX2(un,vn,Cn2(w))impliesCn2(w)2TnPX02jUVX2(un,vn,Cn2(^w)),(5)andalternativelyfordistributionsPX02jUVX2(x02ju,v,x2)6=PX2jUVX02(x02ju,v,x2), Cn2(^w)2TnPX02jUVX2(un,vn,Cn2(w))impliesCn2(w)=2TnPX02jUVX2(un,vn,Cn2(^w)).(5)Asaresultequation( 5 )isequivalentto10BBBB@Cn2(w)2[^w(6=w)Cn2(^w)2TnPX02jUV(un,vn)TnPX02jUVX2(un,vn,Cn(^w))1CCCCA 127

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=10BBBB@Cn2(w)2[PX02jUVX2:PX02jUVX2=PX2jUVX02[^w(6=w)Cn2(^w)2TnPX02jUVX2(un,vn,Cn2(w))TnPX2jUVX02(un,vn,Cn(^w))1CCCCA. (5)Furthermore,overanysummationwemaychooseonlyonedistributionPX02jUVX2=PX2jUVX02whileonlyincurringanratepenalty.Proceeding,wecanonceagainassumeequation( 5 ),whichinturnallowsustouselemma 21 toshowthat,Pr8>>>><>>>>:Xw2PX2jUV10BBBB@Cn(w)2[^w(6=w)Cn2(^w)2TnPX02jUVX2(un,vn,Cn2(w))TnPX2jUVX02(un,vn,Cn(^w))1CCCCA2n(2R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;U,V))]TJ /F4 7.97 Tf 6.58 0 Td[(I(X02;U,V,X2)+n)9>>>>=>>>>;2e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2n(2R2)]TJ /F16 5.978 Tf 5.75 0 Td[(I(X2;U,V))]TJ /F16 5.978 Tf 5.76 0 Td[(I(X02;U,V,X2)+n). (5)Therestoftheproofisidenticaltothosefoundincondition(1)andthusdonotneedtoberepeated.SummationofConditionsBecausetheexistsannthatsatisesdeltaconventionsthatcanbeusedforallboundspreviouslydetermined,exceptforadoubleexponentiallysmallnumberofcodebooks,wemaycharacterizeequation( 5 )as[Cn2(w2)2TnPX2jUV(un,vn)[Cn2(^w2)2TnPX02jUV(un,vn)hTn[X1jU,X2](un,Cn2(w2))\Tn[X1jU,X2](vn,Cn2(^w2))i22n(R2)]TJ /F4 7.97 Tf 6.59 0 Td[(I(X2;UjX1)+n)max2nH(~X1j~U,~V). (5)DenetheareasizebeingboundedasS(un,vn,Cn2),anddenethesetofcodebooksforwhichequation( 5 )isfalseas~C2(un,vn).InturnthenPr(Xw1Cn1(w1)2S(un,vn,Cn2)>22n2R2)]TJ /F6 7.97 Tf 6.58 0 Td[(2I(X2;UjX1)+jR1)]TJ /F6 7.97 Tf 6.59 0 Td[(minI(~X1;~U,~V)j+++n)Pr(Xw1Cn1(w1)2S(un,vn,Cn2)>22n2R2)]TJ /F6 7.97 Tf 6.58 0 Td[(2I(X2;UjX1)+jR1)]TJ /F6 7.97 Tf 6.59 0 Td[(minI(~X1;~U,~V)j+++n, 128

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Cn2=2~C2(un,vn)1 PrCn2=2~C2(un,vn)+PrCn22~C2(un,vn)e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2nj2R2)]TJ /F15 5.978 Tf 5.76 0 Td[(2I(X2;UjX1)+jR1)]TJ /F15 5.978 Tf 5.75 0 Td[(minI(~X1;~U,~V)j+j++n+6e)]TJ /F17 7.97 Tf 6.59 0 Td[(()2nn. (5) Lemma22. DoublyExponentiallySmallMadeEasyGivenconditionsA1andA2,ifthereexistsana>1suchthatPrA2 A1e)]TJ /F4 7.97 Tf 6.58 0 Td[(naPrfA1ge)]TJ /F4 7.97 Tf 6.58 0 Td[(na.thenPrfA1[A2g3e)]TJ /F4 7.97 Tf 6.59 0 Td[(na.Furthermore,letW0beasetpossiblydependentuponeventsA1andA2,andletjWj<2nc1forsomeconstantc1.IfXcn1)]TJ ET q .478 w 170.04 -345.89 m 184.52 -345.89 l S Q BT /F2 11.955 Tf 170.04 -355.74 Td[(A11)]TJ ET q .478 w 205.46 -345.89 m 219.94 -345.89 l S Q BT /F2 11.955 Tf 205.46 -355.74 Td[(A2Xw2W1(w2W0)
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b+Xcnp(cn)1(A1)+1(A2)1)]TJ ET q .478 w 241.32 -6.89 m 255.79 -6.89 l S Q BT /F2 11.955 Tf 241.32 -16.74 Td[(A12nc1b+2e)]TJ /F4 7.97 Tf 6.59 0 Td[(na2nc1 130

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CHAPTER6CONCLUSIONSANDFUTUREWORK 6.1ConclusionsInthisdissertation,weintroducetheconceptofinformationintegritywithaspecialfocusonsimplenetworks.Wehaveshowntheintegrityrateregionachievableforthetwowaytwohopnetworkwithperfectbackwardschannel.FurthermorewehavedevelopeddetailedandprecisetechniqueswhichdetectByzantineattackswithandwithoutthepresenceofcoding.Inchapter 2 wedevelopthemathematicsnecessaryfordetectionofByzantineattacksintheabsenceofcoding.Thesemethodsalsocanbetriviallyextendedtorate-distortionproblems.WehavedevelopedtheconditionsunderwhichforanytwostochasticmatricesofappropriatesizeAandB,thereexistsastochasticmatrix6=IsuchthatBA=BA.Additionally,incaseswhere=IistheonlysolutionwehaveshownthatthedifferencebetweenBAandBAcanbetreatedasalinearmultipleofthedifferencebetweenandI.Next,weeappliedtheworkinchapter 3 toatwo-waytwohoprelaywithouttheuseofcoding.Wealsoprovidedsimulationresultswithvariedparameterstoshowthatspeedandaccuracywithwhichthedetectorconverged.Inchapter 4 weintroducedstrongintegrity,whichwasamethodbywhichadetectoranddecodermaybeseparatedandstillconverge.AlsowithinweprovedthatforanynonmanipulabledistributionssuchthatPXjU=PXjVthatI(X;UjV)=nthenthehammingdistancebetweenunandvnwaslessthannn.Thishelpedtoestablishtherelationshipbetweenmanipulabilityandintegrity.Weappliedthesenoveltechniquestothechannelin 3 toshowthatintegritymaybeachieved.Finallyinchapter 5 wepresentedthemostgeneralformofintegrityforthechannel,onewhichalsoconsequentlyderivedthesecrecyratesnearsimultaneously. 131

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6.2FutureWorkGiventhatthisistherstrealattempttocreateanotionofinformationtheoreticintegritytherearemanypossibilitiesthatmaycomefromthework.Eachoftheseextensionswillbeimportantincreatingthisneweld. OneWayRelay.Ofimmediateconsequenceistoextendthesetechniquesandmethodstoothernetworks,therstoneofmajorimportancebeingtheone-wayrelay.Thereisanaddedlayerofdifcultythoughtodetectionintheonewayrelaythough,asingeneralthedestinationhasaccesstoonlyonevariable,Y,whichisingeneralaprobabilisticfunctionofthesource,X,andtherelay's,U,outputsymbols.ClearlyiftherelayvariableweretobeabletoseparateY=(Yx,Yu),whereYxdependedonlyonXandYuonlydependeduponUthiswouldbeasituationsimilartotheonepresentedinthisdissertation.Alternativelythough,onemayconsiderthetwobestknownrelaystrategiesfortheonewayrelay,inpartialdecodeandforwardandcompressandforward[ 38 ].Thesestrategiesareinterestingbecauseinbothofthesethereisaschemewhichreachesthebestknowninnerboundinwhichthedestinationmustalwaysdirectlydecodetherelay.Ifthedestinationmustalwaysdecodetherelayitseverelylimitstheoperationandtheabilityoftherelaytobemalicious.Eachschemedeservesuniqueattentioninhowintegrityshouldbederivedforthem. CompressandForwardIncompressandforwardtherelayrelayissupposedtosendasequencerelatedtothesequenceitobservedtothedestination.Correspondinglythecodebooksforcompressandforwardmaybecreatedinsuchamannerthatdestinationmayalwaysdecodetherelaystransmission.ThisgiveshopeforByzantinedetectionunderthecompressandforwardscheme.Ifthedestinationisabletodecodetherelaysmessageitallowsfortheseparationofsourceandrelay,thusrevertingthemodelbacktooneakintothosepresentedinthisdissertation. 132

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Therearemanycomplicationsthoughthatarisefromtheactionsoftherelay.Itmayintentionallyusethewrongalphabet,oruseadistributionthatwillimprovethechanneltoitself.Becauseoftheexactnatureandmethodspresentedinthisdissertationthough,theseproblemsmaybeovercome.Thiswouldleadtoresultsnotjustconcerningintegritybutsecrecyaswell. PartialDecodeandForwardInpartialdecodeandforward,thesourcenodesendsinformationtonotonlythedestinationbuttotherelay.Thistypeofoperationisusuallyperformedwhenthesourcetorelaychannelisexceptionallygood.Problematicallythough,thismeansthattherelayisactuallyrequiredtodecodepartofthesignal.Likecompressandforwardthough,codingschemesforpartialdecodeandforwardcanbederivedsuchthatthedestinationdecodestherelay'smessage.Allofthecomplicationswithcompressandforwardstillexistwithpartialdecodeandforward,butitisworthinvestigatingwhatintegrityrateregionmaybeachieved.Earlyinvestigationintothisproblemsuggeststhatitispossibletodetermineintegrityandsecrecyregardlessofdistributionchosenbytherelay,andregardlessofinteractionbetweenalltheassociatedrandomvariables.Thiswouldindeedbeaveryinterestingresultifearlyinvestigationprovestrue. SecrecyUnderManipulation.Inchapter 5 wepresentedaresultforsecrecywhichassumedthattherelayhadnotmodiedthesequences.Clearlyconsideringthescopeofthisdissertation,oneimmediateextensionisthedeterminationoftheamountofsecretinformationwhenintegrityiscompromised1,Rc.Whilenotderivedinthispaper,itisactuallyeasilyseen Rc<2(R2)]TJ /F5 11.955 Tf 11.96 0 Td[(I(X2;UjX1))+R1)]TJ /F5 11.955 Tf 11.38 0 Td[(I(X1;U))]TJ /F3 11.955 Tf 11.38 0 Td[(minI(~X1;~Vj~U)
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forcodingschemeandchannelpresentedinchapter 5 .Interestingly,theonlywayinwhichtherelaycangaininformationaboutthesourceistohavealargevalueofminI(~X1;~Vj~U),whichinturnimpliesahighprobabilityofdetection. Secrecy.Furthermorefromtheworkonsecrecypresentedinchapter 5 ,itshouldbeclearthatatleastinonecasesecrecyandintegritycanbederivedjointly.Thissuggestthattheremaybeawaytocreateauniedinformationtheoreticsecurity,wheretheconceptsofsecrecy,authenticationandintegrityaretreatedasvariationsofthesameproblemandthusleadingtosecurenetworkswhichareabletoprotectagainstbothactiveandpassiveattacks.Inapracticalsense,adestinationmustacceptwhateversequencesitsetsforthasvalid,allothersequencesindicateanattackofsomeunknownvarietyisoccurring.Byviewingtheactiveactionofanymaliciouspartybytheconsequencesonanynodesreceivedsequenceonecanformulateallactiveattacksinthewaydescribedinthisdissertation.Whenthenconsideringpassiveattackstobejointlyderivablewithactiveattackswouldallowsforlargernetworksasawholetohavetheirsecurityexamined. Compression.Inmostcommonnetworks,strategiessuchasphysicallayernetworkcodingareused.Inphysicallayernetworkcodingthealphabetsizethatrelayobservesislargerthanthealphabetsizethattherelaytransmits.Onceagainbecauseoftheexactnatureofthecountingarguments,extensiontoanycompressionschemewheretheoutputmapsmultiplereceivedsequencestoasingleoutputsequenceshouldbestraightforward.Clearlythoughtheexactnatureofthiscompressioninuencesintegrity.ButthismethodcouldleadtoamoregeneralversionoftheAMDcodediscussedintheintroduction. ContinuousDistributions.Becausedetectionanddecodingneednotbeconsideredtogetheronemaycreateadetectorbaseduponaquantizedversionofthereceivedinputvaluesfrom 134

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acontinuousdistribution.Alloftheassumptionswouldstillbevalidanddetectionwouldbepossible.Whilethismethodisstraightforward,itisalsoveryuglyandhefty.Ofmoreinterestwouldbeagroundupapproachthatre-castlemma 14 withuseofcontinuousdistributions.Furthermorebecausealloftheresultsregardingdetectionareformulatedaroundinclusionofthecodewordsincertainsets,analogousprobabilitiesforcontinuousdistributionswouldallowforlimitedmodicationoftheproofspresented. 135

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[13] L.ButtyanandJ.-P.Hubaux,SecurityandCooperationinWirelessNetworks.Cambridge,UK:CambridgeUniversityPress,2007. [14] R.Changiz,H.Halabian,F.Yu,I.Lambadaris,H.Tang,andC.Peter,Trustestablishmentincooperativewirelessnetworks,inProc.2010IEEE.MilitaryCommun.Conf.,pp.1074,Nov.2010. [15] S.Dehnie,H.Senear,andN.Memon,Detectingmaliciousbehaviorincooperativediversity,inProc.41stAnnualConf.onInform.Sci.andSyst.,pp.895,Mar.2007. [16] W.Gong,Z.You,D.Chen,X.Zhao,M.Gu,andK.-Y.Lam,TrustbasedmaliciousnodesdetectioninMANET,inProc.2009IEEEInt.Conf.E-BusinessandInform.Syst.Security,pp.1,May2009. [17] X.Jiang,C.Lin,H.Yin,Z.Chen,andL.Su,Game-basedtrustestablishmentformobileadhocnetworks,inProc.2009Int.Conf.Commun.andMobileComput.,vol.3,pp.475,Jan.2009. [18] T.JiangandJ.S.Baras,Trustevaluationinanarchy:Acasestudyonautonomousnetworks,inProc.INFOCOM2006,pp.1,Apr.2006. [19] G.TheodorakopoulosandJ.S.Baras,Trustevaluationinad-hocnetworks,inProceedingsofthe3rdACMworkshoponWirelesssecurity,WiSe'04,(NewYork,NY,USA),pp.1,ACM,2004. [20] C.Zhang,X.Zhu,Y.Song,andY.Fang,Aformalstudyoftrust-basedroutinginwirelessadhocnetworks,inProc.IEEEINFOCOM2010,2010. [21] S.Capkun,L.Buttyan,andJ.-P.Hubaux,Self-organizedpublic-keymanagementformobileadhocnetworks,IEEETransactionsonMobileComputing,vol.2,pp.52,Jan.-Mar.2003. [22] T.Beth,M.Borcherding,andB.Klein,Valuationoftrustinopennetworks,inComputerSecurityESORICS94(D.Gollmann,ed.),vol.875ofLectureNotesinComputerScience,pp.1,SpringerBerlin/Heidelberg,1994.10.1007/3-540-58618-0 53. [23] C.Zhang,Y.Song,andY.Fang,Modelingsecureconnectivityofself-organizedwirelessadhocnetworks,inProc.ofINFOCOM2008,(Phoenix,AZ),Apr.2008. [24] Y.-L.Sun,W.Yu,Z.Han,andK.Liu,Informationtheoreticframeworkoftrustmodelingandevaluationforadhocnetworks,IEEEJournalonSelectedAreasinCommunications,vol.24,pp.305,Feb.2006. 137

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[25] F.OlivieroandS.Romano,Areputation-basedmetricforsecureroutinginwirelessmeshnetworks,inProc.ofIEEEGLOBECOM2008,(NewOrleans,LA),Dec.2008. [26] A.Jsang,Analgebraforassessingtrustincerticationchains,inProc.NDSS99,(SanDiego,CA),Feb.1999. [27] G.TheodorakopoulosandJ.S.Baras,Ontrustmodelsandtrustevaluationmetricsforadhocnetworks,IEEEJournalonSelectedAreasinCommunications,vol.24,pp.318,Feb.2006. [28] S.Capkun,J.Hubaux,andL.Buttyan,Mobilityhelpssecurityinadhocnetworks,inProc.MobiHoc2003,(Annapolis,MD),June2003. [29] S.Marti,T.J.Giuli,K.Lai,andM.Baker,Mitigatingroutingmisbehaviorinmobileadhocnetworks,inProceedingsofthe6thannualinternationalconferenceonMo-bilecomputingandnetworking,MobiCom'00,(NewYork,NY,USA),pp.255,ACM,2000. [30] E.KehdiandB.Li,Nullkeys:Limitingmaliciousattacksvianullspacepropertiesofnetworkcoding,inProc.2009IEEEInt.Conf.ComputerCommun.,pp.1224,Apr.2009. [31] Y.-C.Hu,A.Perrig,andD.B.Johnson,Ariadne:Asecureon-demandroutingprotocolforadhocnetworks,Wirel.Netw.,vol.11,pp.21,Jan.2005. [32] P.PapadimitratosandZ.Haas,Securedatacommunicationinmobileadhocnetworks,IEEEJournalonSelectedAreasinCommunications,vol.24,pp.343,Feb.2006. [33] J.Mitchell,A.Ramanathan,A.Scedrov,andV.Teague,Aprobabilisticpolynomial-timecalculusforanalysisofcryptographicprotocols:(preliminaryreport),ElectronicNotesinTheoreticalComputerScience,vol.45,no.0,pp.280,2001.MFPS2001,SeventeenthConferenceontheMathematicalFoundationsofProgrammingSemantics. [34] E.GravesandT.F.Wong,Detectionofchanneldegradationattackbyintermediarynodeinlinearnetworks,inProc.IEEEINFOCOM,(Orlando,FL),Mar.2012. [35] I.CsiszarandJ.Korner,InformationTheory:CodingTheoremsforDiscreteMemorylessSystems.CambridgeUniversityPress,2nded.,2011. [36] R.W.Yeung,InformationTheoryandNetworkCoding.NewYork:Springer,2008. 138

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[37] E.GravesandT.F.Wong,Detectabilityofsymbolmanipulationbyanamplify-and-forwardrelay,CoRR,vol.abs/1205.2681,2012. [38] A.GamalandY.Kim,NetworkInformationTheory.CambridgeUniversityPress,2011. 139

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BIOGRAPHICALSKETCH EricGravesreceivedhisBachelorofSciencewithamajorofelectricalandcomputerengineeringandminorofmathematicsfromtheUniversityofFloridainspring2008.AfterwardshewasawardedtheAlumniFellowshiptoattendUniversityofFlorida,whichhedidandsubsequentlyreceivedhisMasterofScienceinelectricalandcomputerengineeringduringspring2011.Finally,beingsubjecttothelawofthrees,heobtainedhisDoctorofPhilosophyinelectricalandcomputerengineering,alsofromtheUniversityofFlorida,insummer2013.HisadvisorwasProfessorTanWongandco-advisorwasProfessorJohnShea.Hisresearchinterestsarecomparativepolitics,potteryandcreativewriting. 140