Secret sharing over fast-fading MIMO wiretap channels

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Secret sharing over fast-fading MIMO wiretap channels
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EURASIP Journal onWireless Communications and Networking
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Wong, Tan F.
Bloch, Matthieu
Shea, John M.
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Secret sharing over the fast-fading MIMO wiretap channel is considered. A source and a destination try to share secret information over a fast-fading MIMO channel in the presence of an eavesdropper who also makes channel observations that are different from but correlated to those made by the destination. An interactive, authenticated public channel with unlimited capacity is available to the source and destination for the secret sharing process. This situation is a special case of the “channel model with wiretapper” considered by Ahlswede and Csisz´ar. An extension of their result to continuous channel alphabets is employed to evaluate the key capacity of the fast-fading MIMO wiretap channel. The effects of spatial dimensionality provided by the use of multiple antennas at the source, destination, and eavesdropper are then investigated.
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HindawiPublishingCorporation EURASIPJournalonWirelessCommunicationsandNetworking Volume2009,ArticleID506973, 17 pages doi:10.1155/2009/506973ResearchArticle SecretSharingoverFast-FadingMIMOWiretapChannelsTanF.Wong,1MatthieuBloch,2,3andJohnM.Shea11WirelessInformationNetworkingGroup,UniversityofFlorida,Gainesvilles,FL32611-6130,USA2SchoolofElectricalandComputerEngineering,Geo rgiaInstituteofTechnology,Atlanta,GA30332,USA3GT-CNRSUMI2958,2-3rueMarconi,57070Metz,France CorrespondenceshouldbeaddressedtoTanF.Wong, twong@u.edu Received1December2008;Revised25June2009;Accepted14September2009 RecommendedbyShlomoShamai(Shitz) Secretsharingoverthefast-fadingMIMOwiretapchannelisconsidered.Asourceandadestinationtrytosharesecretinformation overafast-fadingMIMOchannelinthepresenceofaneavesdropperwhoalsomakeschannelobservationsthataredi erentfrom butcorrelatedtothosemadebythedestination.Aninteractive,authenticatedpublicchannelwithunlimitedcapacityisavailable tothesourceanddestinationforthesecretsharingprocess.Thissi tuationisaspecialcaseofthe"channelmodelwithwiretapper" consideredbyAhlswedeandCsisz ar.Anextensionoftheirresulttocontinuouschannelalphabetsisemployedtoevaluatethekey capacityofthefast-fadingMIMOwiretapchannel.Thee ectsofspatialdimensionalityprovidedbytheuseofmultipleantennas atthesource,destination,andeavesdropperaretheninvestigated. Copyright2009TanF.Wongetal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense, whichpermitsunrestricteduse,distribution,andreproductio ninanymedium,providedtheoriginalworkisproperlycited.1.IntroductionThewiretapchannelconsideredintheseminalpaper[ 1 ]is therstexamplethatdemonstratesthepossibilityofsecure communicationsatthephysicallayer.Itisshownin[ 1 ] thatasourcecantransmitamessageatapositive(secrecy) ratetoadestinationinsuchawaythataneavesdropper onlygathersinformationatanegligiblerate,whenthe source-to-eavesdropperchannelisadegradedversionofthe source-to-destinationchannel,thesource-to-eavesdropper andsource-to-destinationchannelswillhereafterbereferred toaseavesdropperanddestinationchannels,respectively.A similarresultfortheGaussianwiretapchannelisprovidedin [ 2 ].Theworkin[ 3 ]furtherremovesthedegradedwiretap channelrestrictionshowingthatpositivesecrecycapacityis possibleifthedestinationchannelis"morecapable"("less noisy"forafullextensionoftherateregionin[ 1 ])thanthe eavesdropper'schannel.Recently,therehasbeenaurryof interestinextendingtheseearlyresultstomoresophisticated channelmodels,includingfadingwiretapchannels,multiinputmulti-output(MIMO)wiretapchannels,multipleaccesswiretapchannels,broadcastwiretapchannels,and relaywiretapchannels.Wedonotattempttoprovidea comprehensivesummaryofallrecentdevelopmentsbut highlightonlythoseresultsthataremostrelevanttothe presentwork.Wereferinterestedreaderstotheintroduction andreferencelistof[ 4 ]foraconciseandextensiveoverview ofrecentworks. Whenthedestinationandeavesdropperchannelsexperienceindependentfading,thestrictrequirementofhavinga morecapabledestinationchannelforpositivesecrecycapacitycanbeloosened.Thisisduetothesimpleobservation thatthedestinationchannelmaybemorecapablethanthe eavesdropper'schannelundersomefadingrealizations,even ifthedestinationisnotmorecapablethantheeavesdropper onaverage.Hence,ifthechannelstateinformation(CSI)of boththedestinationandeavesdropperchannelsisavailable atthesource,itisshownin[ 4 5 ]thatapositivesecrecy capacitycanbeachievedbymeansofappropriatepower controlatthesource.Thekeyideaistoopportunistically transmitonlyduringthosefadingrealizationsforwhich thedestinationchannelismorecapable[ 6 ].Forblockergodicfading,itisalsoshownin[ 5 ](seealso[ 7 ])thata positivesecrecycapacitycanbeachievedwithavariable-rate transmissionschemewithoutanyeavesdropperCSIavailable atthesource. Whenthesource,destination,andeavesdropperhave multipleantennas,theresultingchannelisknownasa MIMOwiretapchannel(see[ 8 12 ]),whichmayalsohave positivesecrecycapacity.SincetheMIMOwiretapchannel

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2EURASIPJournalonWirelessCommunicationsandNetworking isnotdegraded,thecharacterizationofitssecrecycapacity isnotstraightforward.Forinstance,thesecrecycapacityof theMIMOwiretapchannelischaracterizedin[ 9 ]asthe saddlepointofaminimaxproblem,whileanalternative characterizationbasedonarecentresultformultiantenna broadcastchannelsisprovidedin[ 11 ].Interestinglyall characterizationspointtothefactthatthecapacityachieving schemeisonethattransmitsonlyinthedirectionsin whichthedestinationchannelismorecapablethanthe eavesdropper'schannel.Obviously,thisisonlypossiblewhen thedestinationandeavesdropperCSIisavailableatthe source.Itisshownin[ 9 ]thatiftheindividualchannels fromantennastoantennassu erfromindependentRayleigh fading,andtherespectiveratiosofthenumbersofsource anddestinationantennastothatofeavesdropperantennas arelargerthancertainxedvalues,thenthesecrecycapacity ispositivewithprobabilityonewhenthenumbersofsource, destination,andeavesdropperantennasbecomeverylarge. Asdiscussedabove,theavailabilityofdestination(and eavesdropper)CSIatthesourceisanimplicitrequirement forpositivesecrecycapacityinthefadingandMIMO wiretapchannels.Thus,anauthenticatedfeedbackchannel isneededtosendtheCSIfromthedestinationbackto thesource.In[ 5 7 ],thisfeedbackchannelisassumedto bepublic,andhencethedestinationCSIisalsoavailable totheeavesdropper.Inaddition,itisassumedthatthe eavesdropperknowsitsownCSI.Withtheavailabilityofa feedbackchannel,iftheobjectiveofhavingthesourcesend secretinformationtothedestinationisrelaxedtodistilling asecretkeysharedbetweenthesourceanddestination,itis shownin[ 13 ]thatapositivekeyrateisachievablewhenthe destinationandeavesdropperchannelsaretwoconditionally independent(giventhesourceinputsymbols)memoryless binarychannels,evenifthedestinationchannelisnotmore capablethantheeavesdropper'schannel.Thisnotionof secretsharingisformalizedin[ 14 ]basedontheconcept of commonrandomness betweenthesourceanddestination. Assumingtheavailabilityofaninteractive,authenticated publicchannelwithunlimitedcapacitybetweenthesource anddestination[ 14 ]suggeststwodi erentsystemmodels, calledthe"sourcemodelwithwiretapper"(SW)andthe "channelmodelwithwiretapper"(CW).TheCWmodelis similartothe(discretememoryless)wiretapchannelmodel thatwehavediscussedbefore.TheSWmodeldi ersinthat therandomsymbolsobservedatthesource,destination,and eavesdropperarerealizationsofadiscretememorylesssource withmultiplecomponents.BothSWandCWmodelshave beenextendedtothecaseofsecretsharingamongmultiple terminals,withthepossibilityofsometerminalsactingas helpers[ 15 17 ].Keycapacitieshavebeenobtainedforthe twospecialcasesinwhichtheeavesdropper'schannelisa degradedversionofthedestinationchannelandinwhich thedestinationandeavesdropperchannelsareconditionally independent[ 13 14 ].Similarresultshavebeenderivedfor multiterminalsecretsharing[ 16 17 ],withthetwospecial casesabovesubsumedbythemoregeneralconditionthat theterminalsymbolsformaMarkovchainonatree. Authenticationofthepublicchannelcanbeachievedby theuseofaninitialshortkeyandthenasmallportionof thesubsequentsharedsecretmessage[ 18 ].Adetailedstudy ofsecretsharingoveranunauthenticatedpublicchannelis givenin[ 19 21 ]. Otherapproachestoemployfeedbackhavealsobeen recentlyconsidered[ 22 24 ].Inparticular,itisshownin [ 22 ]thatpositivesecrecycapacitycanbeachievedforthe modulo-additivediscretememorylesswiretapchanneland themodulochannelifthedestinationisallowedtosend signalsbacktothesourceoverthesamewiretapchanneland bothterminalscanoperateinfull-duplexmanner.Infact, fortheformerchannel,thesecrecycapacityisthesameasthe capacityofsuchachannelintheabsenceoftheeavesdropper. Inthispaper,weconsidersecretsharingoverafast-fading MIMOwiretapchannel.Thus,weareinterestedintheCW modelof[ 14 ]withmemorylessconditionallyindependent destinationandeavesdropperchannelsandcontinuous channelalphabets.Weprovideanextensionofthekey capacityresultin[ 14 ]forthiscasetoincludecontinuous channelalphabets( Theorem1 ).Usingthisresult,weobtain thekeycapacityofthefast-fadingMIMOwiretapchannel ( Section3 ).Ourresultindicatesthatthekeycapacityis alwayspositive,nomatterhowlargethechannelgainof theeavesdropper'schannelis;inadditionthisholdseven ifthedestinationandeavesdropperCSIisavailableonlyat thedestinationandeavesdropper,respectively.Ofcourse,the availabilityofthepublicchannelimpliesthatthedestination CSIcouldbefedbacktothesource.However,duetothe restrictionsimposedonthesecret-sharingstrategies(see Section2 ),onlycausalfeedbackisallowed,andthusany destinationCSIavailableatsourceis"outdated."Thisdoes notturnouttobeaproblemsince,unliketheapproaches mentionedabove,thesourcedoesnotusetheCSItoavoid sendingsecretinformationwhenthedestinationisnotmore capablethantheeavesdropper'schannel.Asamatterof fact,thefadingprocessofthedestinationchannelprovides asignicantpartofthecommonrandomnessfromwhich thesourceandthedestinationdistillasecretkey.This factisreadilyobtainedfromthealternativeachievability proofgivenin Section4 .Wenotethat[ 25 26]consider theproblemkeygenerationfromcommonrandomnessover wiretapchannelsandexploitaWyner-Zivcodingscheme tolimittheamountofinformationconveyedfromthe sourcetothedestinationviathewiretapchannel.Unlike thesepreviousworks,weonlyemployWyner-Zivcoding toquantizethedestinationchanneloutputs.Ourcode constructionstillreliesonapublicchannelwithunlimited capacitytoachievethekeycapacity. Finally,wealsoinvestigatethelimitingvalueofthe keycapacityunderthreeasymptoticscenarios.Intherst scenario,thetransmissionpowerofthesourcebecomes asymptoticallyhigh( Corollary1 ).Inthesecondscenario, thedestinationandeavesdropperhavealargenumberof antennas( Corollary2 ).Inthethirdscenario,thegainadvantageoftheeavesdropper'schannelbecomesasymptotically large( Corollary3 ).Thesethreescenariosrevealtwodi erent e ectsofspatialdimensionalityuponkeycapacity.Intherst scenario,weshowthatthekeycapacitylevelso asthepower increasesiftheeavesdropperhasnofewerantennasthan thesource.Ontheotherhand,whenthesourcehasmore

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EURASIPJournalonWirelessCommunicationsandNetworking3 antennas,thekeycapacitycanincreasewithoutboundwith thesourcepower.Inthesecondscenario,weshowthatthe spatialdimensionalityadvantagethattheeavesdropperhas overthedestinationhasexactlythesamee ectasthechannel gainadvantageoftheeavesdropper.Inthethirdscenario, weshowthatthelimitingkeycapacityispositiveonlyifthe eavesdropperhasfewerantennasthanthesource.Theresults inthesescenariosconrmthatspatialdimensionalitycanbe usedtocombattheeavesdropper'sgainadvantage,whichwas alreadyobservedfortheMIMOwiretapchannel.Perhaps moresurprisingly,thisisachievedwithneitherthesource nordestinationneedinganyeavesdropperCSI.2.SecretSharingandKeyCapacityWeconsidertheCWmodelof[ 14 ],andwerecallitscharacteristicsforcompleteness.Weconsiderthreeterminals, namely,asource,adestination,andaneavesdropper.The sourcesendssymbolsfromanalphabet X .Thedestination andeavesdropperobservesymbolsbelongingtoalphabets Y and Z ,respectively.Unlikein[ 14 ], X Y ,and Z neednotto bediscrete.Infact,in Section3 wewillassumethattheyare multi-dimensionalvectorspacesoverthecomplexeld.The channelfromthesourcetothedestinationandeavesdropper isassumedmemoryless.Agenericsymbolsentbythesource isdenotedby X andthecorrespondingsymbolsobservedby thedestinationandeavesdropperaredenotedby Y and Z respectively.Fornotationalconvenience(andwithoutlossof generality),weassumethat( X Y Z )arejointlycontinuous, andthechannelisspeciedbytheconditionalprobability densityfunction(pdf) pY Z|X( y z|x ).Inaddition,we restrictourselvestocasesinwhich Y and Z areconditionally independentgiven X ,thatis, pY Z|X( y z|x )=pY|X( y|x ) pZ|X( z|x ),whichisareasonablemodelforsymbols broadcastinawirelessmedium.Hereafter,wedropthe subscriptsinpdfswhenevertheconcernedsymbolsarewell speciedbytheargumentsofthepdfs.Weassumethat aninteractive,authenticatedpublicchannelwithunlimited capacityisalsoavailableforcommunicationbetweenthe sourceanddestination.Here, interactive meansthatthe channelistwo-wayandcanbeusedmultipletimes, unlimited capacity meansthatitisnoiselessandhasinnitecapacity, and public and authenticated meanthattheeavesdroppercan perfectlyobserveallcommunicationsoverthischannelbut cannottamperwiththemessagestransmitted. Weconsidertheclassofpermissiblesecret-sharing strategiessuggestedin[ 14 ].Consider k timeinstantslabeled by1,2, ... k ,respectively.The( X Y Z )channelisused n timesduringthese k timeinstantsat i1 0,thereexistsapermissible secret-sharingstrategyoftheformdescribedabovesuchthat (1)Pr{K /=L}< (2)(1 /n ) I ( K ; Zn, k, k) < (3)(1 /n ) H ( K ) >RŠ (4)(1 /n )log|K|< (1 /n ) H ( K )+ forsu cientlylarge n .The keycapacity ofthechannel ( X Y Z )isthelargestachievablekeyratethroughthe channel.Weareinterestedinndingthekeycapacity.For thecaseofcontinuouschannelalphabetsconsideredhere, wealsoaddthefollowingpowerconstrainttothesymbol sequence Xnsentoutbythesource: 1 nnj=1 Xj 2P (1) withprobabilityone(w.p.1)forsu cientlylarge n Theorem1. ThekeycapacityofaCWmodel ( X Y Z ) with conditionalpdf p ( y z|x )=p ( y|x ) p ( z|x ) isgivenby maxX : E [|X|2]P[ I ( X ; Y )ŠI ( Y ; Z )] Proof. Thecasewithdiscretechannelalphabetsisestablished in[ 14 ,Corollary2ofTheorem2],whoseachievabilityproof (alsotheonesin[ 16 17 ])doesnotreadilyextendto continuouschannelalphabets.Neverthelessthesamesingle backwardmessagestrategysuggestedin[ 14 ]isstillapplicable forcontinuousalphabets.Thatstrategyuses k=n +1 timeinstantswith ij=j for j=1,2, ... n .Thatis,the sourcerstsends n symbolsthroughthe( X Y Z )channel; afterreceivingthese n symbols,thedestinationfeedsback asinglemessageatthelasttimeinstanttothesourceover thepublicchannel.AcarefullystructuredWyner-Zivcode canbeemployedtosupportthissecret-sharingstrategy. Thedetailedargumentsareprovidedinthealternative achievabilityproofin Section4 Hereweoutlineanachievabilityargumentbasedon theconsiderationofaconceptualwiretapchannelfromthe

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4EURASIPJournalonWirelessCommunicationsandNetworking destinationbacktothesourceandeavesdroppersuggestedin [ 13 ,Theorem3].First,assumethesourcesendsasequence ofi.i.d.symbols Xn,eachdistributedaccordingto p ( x ),over thewiretapchannel.Supposethat E [|X|2]P .Becauseof thelawoflargenumbers,wecanassumethat Xnsatises thepowerconstraint( 1 )withoutlossofgenerality.Let Ynand Znbetheobservationsofthethedestinationsand eavesdropper,respectively.Totransmitasequence Unof symbolsindependentof( Xn, Yn, Zn),thedestinationsends Un+ Ynbacktothesourceviathepublicchannel.This createsaconceptualmemorylesswiretapchannelfromthe destinationwithinputsymbol U tothesourceinthe presenceoftheeavesdropper,wherethesourceobserves ( U + Y X )whiletheeavesdropperobserves( U + Y Z ). Employingthecontinuousalphabetextensionofthewell knownresultin[ 3 ],thesecrecycapacityoftheconceptual wiretapchannel(andhencethekeycapacityoftheoriginal channel)islowerboundedby maxU[ I ( U ; U + Y X )ŠI ( U ; U + Y Z ) ] (2) Notethattheinputsymbol U hasnopowerconstraintsince thepublicchannelhasinnitecapacity.But I ( U ; U + Y X )ŠI ( U ; U + Y Z )=I ( U ; X ) + I ( U ; U + Y|X )Š[ I ( U ; Z ) + I ( U ; U + Y|Z ) ]=h ( U )Šh ( U|X ) + h ( U + Y|X )Šh ( U + Y|U X )Šh ( U ) + h ( U|Z )Šh ( U + Y|Z ) + h ( U + Y|U Z )=h ( Y|Z )Šh ( Y|X ) + [ h ( U + Y|X )Šh ( U|X ) ]Š[ h ( U + Y|Z )Šh ( U|Z ) ]h ( Y|Z )Šh ( Y|X )Š[ h ( U + Y|X )Šh ( U|X ) ]h ( Y|Z )Šh ( Y|X )Š[ h ( U + Y )Šh ( U ) ] (3) wherethethirdequalityresultsfrom h ( U + Y|U X )=h ( Y|U X )=h ( Y|X )duetotheindependenceof U and Y ,therstinequalityfollowsfromthefact h ( U + Y|Z )Šh ( U|Z )h ( U + Y|Z Y )Šh ( U|Z )=h ( U|Z Y )Šh ( U|Z )=0, (4) whichisagainduetoindependencebetween( Y Z )and U andtheinequalityonthelastlinefollowsfrom h ( U + Y|X )Šh ( U|X )=h ( U + Y|X )Šh ( U )h ( U + Y )Šh ( U ). Withoutlossofgeneralityandfornotationalsimplicity, assumethat Y and U arebothone-dimensionalrealrandom variables.Now,choose U tobeGaussiandistributedwith mean0andvariance 2 U.Then h ( U + Y )Šh ( U )1 2 log ( 2 e var ( U + Y ))Š1 2 log2 e2 U =1 2 log2 U+var ( Y ) 2 U, (5) wheretherstinequalityfollowsfrom[ 27 ,Theorem8.6.5], andthelastequalityisduetotheindependencebetween Y and U .Combining( 3 )and( 5 ),forevery > 0,wecanchoose 2 Ulargeenoughsuchthat I ( U ; U + Y X )ŠI ( U ; U + Y Z )h ( Y|Z )Šh ( Y|X )Š=I ( X ; Y )ŠI ( Y ; Z )Š. (6) Since isarbitrary,thekeycapacityislowerboundedby maxE [|X|2]P[ I ( X ; Y )ŠI ( Y ; Z )]. Theconverseproofin[ 14 ]isdirectlyapplicableto continuouschannelalphabets,providedthattheaverage powerconstraint( 1 )canbeincorporatedintothearguments in[ 14 ,pp.1129-1130].Thislatterrequirementissimplied bytheadditiveandsymmetricnatureoftheaveragepower constraint[ 28 ,Section3.6].Toavoidtoomuchrepetition,we outlinebelowonlythestepsoftheproofthatarenotdirectly availablein[ 14 ,pp.1129-1130]. Foreverypermissiblestrategywithachievablekeyrate R wehave 1 n I ( K ; L )=1 n H ( K )Š1 n H ( K|L )1 n H ( K )Š1 n1+Pr{K /=L}log|K| > 1 n H ( K )Š1 nŠ1 n H ( K ) + > ( 1Š )( RŠ )Š1 nŠ2, (7) wherethesecondlinefollowsfromFano'sinequality,the thirdlineresultsfromconditions( 1 )and( 7 )inthedenition ofachievablekeyrate,andthelastlineisduetocondition( 5 ). Thusitsu cestoupperbound I ( K ; L ).Fromcondition( 3 ) inthedenitionofachievablekeyrateandthechainrule,we have 1 n I ( K ; L ) < 1 n IK ; L|Zn, k, k+ 1 n IMX; MY, Yn|Zn, k, k+ (8) wherethesecondinequalityisduetothefactthat K=K ( MX, k)and L=L ( MY, Yn, k).Byrepeatedusesofthe chainrule,theconstructionofpermissiblestrategies,and

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EURASIPJournalonWirelessCommunicationsandNetworking5 thememorylessnatureofthe( X Y Z )channel,itisshown in[ 14 ,pp.1129-1130]that 1 n IMX; MY, Yn|Zn, k, k 1 nnj=1IXj; Yj|Zj. (9) Nowlet Q beauniformrandomvariablethattakesvalue from{1,2, ... n}andisindependentofallotherrandom quantities.Dene(X ,Y ,Z )=( Xj, Yj, Zj)if Q=j .Then itisobviousthat pY ,Z| X(y ,z| x )=pY Z|X(y ,z| x ),and( 9 ) canberewrittenas 1 n IMX; MY, Yn|Zn, k, k I X ;Y| Z Q I X ;Y| Z, (10) wherethesecondinequalityisduetothefactthat Q X(Y ,Z )formsaMarkovchain.Ontheotherhand,thepower constraint( 1 )impliesthat E X 2 =1 nnj=1E Xj 2 P. (11) Combining( 7 ),( 8 ),and( 10 ),weobtain R< 1 1ŠI X ;Y| Z+2 + 1 n. (12) Since canbearbitrarilysmallwhen n issu cientlylarge, ( 12 ),togetherwith( 11 ),gives RI X ;Y| Z maxX : E [|X|2]PI ( X ; Y|Z )=maxX : E [|X|2]P[ I ( X ; Y )ŠI ( Y ; Z ) ] (13) wherethelastlineisduetothefactthat p ( y z|x )=p ( y|x ) p ( z|x ). 3.KeyCapacityofFast-FadingMIMO WiretapChannelConsiderthatthesource,destination,andeavesdropperhave mS, mD,and mWantennas,respectively.Theantennasin eachnodeareseparatedbyatleastafewwavelengths,and hencethefadingprocessesofthechannelsacrossthetransmit andreceiveantennasareindependent.Usingthecomplex basebandrepresentationofthebandpasschannelmodel: YD=HDX + ND, YW=HWX + NW, (14) where (i) X isthe mS1complex-valuedtransmitsymbol vectorbythesource, (ii) YDisthe mD1complex-valuedreceivesymbol vectoratthedestination, (iii) YWisthe mW1complex-valuedreceivesymbol vectorattheeavesdropper, (iv) NDisthe mD1noisevectorwithindependent identicallydistributed(i.i.d.)zero-mean,circularsymmetriccomplexGaussian-distributedelements ofvariance 2 D(i.e.,therealandimaginarypartsof eachelementsareindependentzero-meanGaussian randomvariableswiththesamevariance), (v) NWisthe mW1noisevectorwithi.i.d. zero-mean,circular-symmetriccomplexGaussiandistributedelementsofvariance 2 W, (vi) HDisthe mDmSchannelmatrixfromthesourceto destinationwithi.i.d.zero-mean,circular-symmetric complexGaussian-distributedelementsofunitvariance, (vii) HWisthe mWmSchannelmatrixfromthesource toeavesdropperwithi.i.d.zero-mean,circularsymmetriccomplexGaussian-distributedelements ofunitvariance, (viii) > 0modelsthegainadvantageoftheeavesdropper overthedestination. Notethat HD, HW, ND,and NWareindependent.The wirelesschannelmodeledby( 14 )isused n timesasthe ( X Y Z )channeldescribedin Section2 with Y=[ YDHD] and Z=[ YWHW].Weassumethatthe n usesofthewireless channelin( 14 )arei.i.d.sothatthememorylessrequirement ofthe( X Y Z )channelissatised.Since HDand HWare includedintherespectivechannelsymbolsobservableby thedestinationandeavesdropper(i.e., Y and Z ,resp.), thismodelalsoimplicitlyassumesthatthedestinationand eavesdropperhaveperfectCSIoftheirrespectivechannels fromthesource.Inpractice,wecanseparateadjacentuses ofthewirelesschannelbymorethanthecoherencetimeof thechanneltoapproximatelyensurethei.i.d.channeluse assumption.Training(known)symbolscanbesentright beforeorafter(withinthechannelcoherenceperiod)bythe sourcesothatthedestinationcanacquiretherequiredCSI. Theeavesdroppermayalsousethesetrainingsymbolsto acquiretheCSIofitsownchannel.IftheCSIrequiredat thedestinationisobtainedinthewayjustdescribed,then aunitofchanneluseincludesthesymbol X togetherwith theassociatedtrainingsymbols.However,asin[ 29 ],wedo notcountthepowerrequiredtosendthetrainingsymbols (cf.( 1 )).Moreoverwenotethatthesource(andalsothe eavesdropper)maygetsomeinformationabouttheoutdated CSIofthedestinationchannel,becauseinformationabout thedestinationchannelCSI,uptotheprevioususe,maybe fedbacktothesourcefromthedestinationviathepublic channel.Morespecically,attimeinstant ij,thesource symbol Xjisafunctionofthefeedbackmessage ijŠ1, whichisinturnsomefunctionoftherealizationsof HDat time i1, i2, ... ijŠ1.Wealsonotethatneitherthesourcenor destinationhasanyeavesdropperCSI.Referringbackto( 14 ), thesetwofactsimplythat X isindependentof HD, HW, ND,

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6EURASIPJournalonWirelessCommunicationsandNetworking and NW;thatis,thecurrentsourcesymbol X isindependent ofthecurrentchannelstate. SincethefadingMIMOwiretapchannelmodelin( 14 )is aspecialcaseoftheCWmodelconsideredin Section2 ,the keycapacity CKisgivenby Theorem1 as CK=maxX : E [|X|2]P[ I ( X ; YD, HD)ŠI ( YD, HD; YW, HW) ] (15) Notethat I ( X ; YD, HD)ŠI ( YD, HD; YW, HW)=I ( X ; YD|HD)ŠI ( YD; YW|HD, HW)=h ( YD|YW, HD, HW)Šh ( YD|X HD)=h ( YD|YW, HD, HW)ŠmDloge2 D. (16) Substitutingthisbackinto( 15 ),weget CK=maxX : E [|X|2]Ph ( YD|YW, HD, HW)ŠmDloge2 D. (17) Asaresult,thekeycapacityofthefast-fadingwiretapchannel describedby( 14 )canbeobtainedbymaximizingtheconditionalentropy h ( YD|YW, HD, HW).Thismaximization problemissolvedbelow. Theorem2. Onehas CK=E log detImS+2P/mS2 WHWHW+P/mS2 DHDHD detImS+2P/mS2 WHWHW (18) wheredenotesconjugatetranspose. Proof. Todeterminethekeycapacity,weneedthefollowing upperboundontheconditionalentropy h ( U|V ). Lemma1. Let U and V betwojointlydistributedcomplex randomvectorsofdimensions mUand mV,respectively.Let KU, KV,and KUVbethecovarianceof U ,covarianceof V ,and cross-covarianceof U and V ,respectively.If KVisinvertible, then h ( U|V )logdetKUŠKUVKŠ1 VKVU+ mUlog ( e ) (19) Theupperboundisachievedwhen [ UTVT]TisacircularsymmetriccomplexGaussianrandomvector. Proof. Wecanassumethatboth U and V havezeromeans withoutlossofgenerality.Alsoassumetheexistenceofall unconditionalandconditionalcovariancesstatedbelow.For each v h ( U|V=v )log( e )mUdetKU|v, (20) where KU|visthecovarianceof U withrespecttothe conditionaldensity pU|V( u|v )[ 29 ,Lemma2].Thisimplies h ( U|V )EVlog( e )mUdetKU|V logdetEVKU|V+ mUlog ( e )logdetKUŠKUVKŠ1 VKVU+ mUlog ( e ) (21) Thesecondinequalityaboveisduetotheconcavityofthe functionlogdetoverthesetofpositivedenitesymmetric matrices[ 30 ,7.6.7],andtheJensen'sinequality.Togetthe thirdinequality,observethat EV[ KU|V]canbeinterpretedas thecovarianceoftheestimationerrorofestimating U bythe conditionalmeanestimator E [ U|V ].Ontheotherhand, KUŠKUVKŠ1 VKVUisthecovarianceoftheestimationerror ofusingthelinearminimummeansquarederrorestimator KUVKŠ1 VV instead.Theinequalityresultsfromthefactthat KUŠKUVKŠ1 VKVUEV[ KU|V](i.e.,[ KUŠKUVKŠ1 VKVU]ŠEV[ KU|V]ispositivesemidenite)[ 31 ]andtheinequalityof det( A )det( B )if A and B arepositivedenite,and AB [ 30 ,,7.7.4]. Supposethat[ UTVT]Tisacircular-symmetriccomplexGaussianrandomvector.Foreach v ,theconditional covarianceof U ,conditionedon V=v ,isthesameas the(unconditional)covarianceof UŠKUVKŠ1 VV .Since UŠKUVKŠ1 VV isacircular-symmetriccomplexGaussian randomvector[ 29 ,Lemma3],sois U conditionedon V=v Henceby[ 29 ,Lemma2],theupperboundin( 20 )isachieved with KU|v=KUŠKUVKŠ1 VKVU,whichalsogivestheupper boundin( 21 ). Toprovethetheorem,werstobtainanupperboundon CKandthenshowthattheupperboundisachievable.Using Lemma1 ,wehave h ( YD|YW, HD, HW)ŠmDloge2 D ElogdetKYDŠKYDYWKŠ1 YWKYWYD ŠmDlog 2 D, (22) where KYDand KYWare,respectively,theconditionalcovariancesof YDand YW,given HDand HW,and KYDYWand KYWYDarethecorrespondingconditionalcross-covariances. Substituting( 22 )into( 17 ),anupperboundon CKis maxX : E [|X|2]PElogdetKYDŠKYDYWKŠ1 YWKYWYD ŠmDlog 2 D. (23) Thusweneedtosolvethemaximizationproblem( 23 ).Todo so,let 1, 2, ... mSbethe(nonnegative)eigenvaluesof KX. Sinceboththedistributionsof HDand HWareinvariantto

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EURASIPJournalonWirelessCommunicationsandNetworking7 anyunitarytransformation[ 29 ,Lemma5],wecanwithout anyambiguitydene f1, 2, ... mS =ElogdetImD+ 1 2 DHDK1 / 2 X ImS+ 2 2 WK1 / 2 XHWHWK1 / 2 XŠ1K1 / 2 XHD (24) Thatis,wecanassume KX=diag( 1, 2, ... mS)withno lossofgenerality.Thenwehavethefollowinglemma,which suggeststhattheobjectivefunctionin( 23 )isaconcave functiondependingonlyontheeigenvaluesofthecovariance of X Lemma2. Supposethat X hasanarbitrarycovariance KX, whose(nonnegative)eigenvaluesare 1, 2, ... mS,then ElogdetKYDŠKYDYWKŠ1 YWKYWYD ŠmDlog 2 D=f1, 2, ... mS(25) isconcavein ={i0 for i=1,2, ... mS}. Proof. Firstwrite AD=HDK1 / 2 Xand AW=HWK1 / 2 X.Itis easytoseefrom( 14 )that KYD=ADAD+ 2 DImD, KYW=AWAW+ 2 WImW,and KYDYW=ADAW.Then KYDŠKYDYWKŠ1 YWKYWYD=2 DImD+ 1 2 DADImSŠAWAWAW+ 2 WImWŠ1AWAD =2 D ImD+ 1 2 DADImS+ 1 2 WAWAWŠ1AD (26) wherethelastequalityisduetothematrixinversionformula. Substitutingthisresultintotheleft-handsideof( 25 ),we obtaintheright-handsideof( 24 ),andhence( 25 ). Toshowconcavityof f ,itsu cestoconsideronlydiagonal KX=diag( 1, 2, ... mS)in .Notethatthemapping H : KX KY DKY D Y WKY W Y DKY Wislinearin .Alsothemapping F :KY DKY D Y WKY W Y DKY W KYDŠKYDYWKŠ1 YWKYWYDismatrix-concave in H ( )[ 32 ,Ex.3.58].Thusthecompositiontheorem[ 32 ] givesthatthemapping G : KXKYDŠKYDYWKŠ1 YWKYWYDismatrix-concavein ,since G=FH .Anotheruseof thecompositetheoremtogetherwiththeconcavityofthe functionlogdetasmentionedintheproofof Lemma1 shows thatlogdet G isconcavein .Thus( 25 )impliesthat f isalso concavein Henceitsu cestoconsideronlythose X withzeromeanin ( 23 ). Nowdenetheconstraintset P={i0for i=1,2, ... mSand!mSi=1iP}. Lemma2 impliesthat wecanndtheupperboundon CKbycalculating maxPf ( 1, 2, ... mS),whosevalueisgivenbythenext lemma. Lemma3. Onehas maxPf1, 2, ... mS =f"P mS, P mS, ... P mS#. (27) Proof. Sincetheelementsofboth HDand HWarei.i.d., f isinvarianttoanypermutationofitsarguments.This meansthat f isasymmetricfunction.By Lemma2 f is alsoconcavein P.ThusitisSchur-concave[ 33 ].Hence aSchur-minimalelement(anelementmajorizedbyany anotherelement)in Pmaximizes f .Itiseasytocheck that( P/mS, P/mS, ... P/mS)isSchur-minimalin P.Hence maxPf ( 1, 2, ... mS)=f ( P/mS, P/mS, ... P/mS). Combiningtheresultsin( 23 ),( 24 ),Lemmas 2 and 3 ,we obtaintheupperboundonthekeycapacityas CKE logdet$ %ImD+ P mS2 DHDImS+ 2P mS2 WHWHWŠ1 HD =E log detImS+2P/mS2 WHWHW+P/mS2 DHDHD detImS+2P/mS2 WHWHW (28) wheretheidentitydet( I + UVŠ1U)=det( V + UU ) / det( V ) forinvertible V [ 34 ,Theorem18.1.1]hasbeenused. Ontheotherhand,considerchoosing X tohave i.i.d.zero-mean,circular-symmetriccomplexGaussiandistributedelementsofvariance P/mS.Thenconditionedon HDand HW,[ YT DYT W]Tareacircular-symmetriccomplex Gaussianrandomvector,byapplying[ 29 ,Lemmas3and4] tothelinearmodelof( 14 ).Hence Lemma1 gives h ( YD|YW, HD, HW)=ElogdetKYDŠKYDYWKŠ1 YWKYWYD+ mDlog ( e ) (29) where KYD=( P/mS) HDHD+ 2 DImD, KYW=( 2P/ mS) HWHW+ 2 WImW,and KYDYW=( P/mS) HDHW.Substitutingthisbackinto( 16 )andusingthematrixinversion formulatosimplifytheresultingexpression,weobtainthe sameexpressionontherstlineof( 28 )for I ( X ; YD, HD)ŠI ( YD, HD; YW, HW).Thustheupperboundin( 28 )isachievablewiththischoiceof X ;henceitisinfactthekey capacity. In Figure1 ,thekeycapacitiesofseveralfast-fading MIMOchannelswithdi erentnumbersofsource,destination,andeavesdropperantennasareplottedagainstthe sourcesignal-to-noiseratio(SNR) P/2,where 2 D=2 W=2.Thechannelgainadvantageoftheeavesdropperisset

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8EURASIPJournalonWirelessCommunicationsandNetworking 0 5 1 1 5 2 2 5 3 3 5 4 4 5 5 5 5CK(bits/channelsymbol)02468101214161820 P/2(dB) mS=1, mD=1, mW=1 mS=2, mD=1, mW=1 mS=2, mD=2, mW=2 mS=1, mD=10, mW=10 Figure 1:Keycapacitiesoffast-fadingMIMOwiretapchannels withdi erentnumbersofsource,destination,eavesdropperantennas.Theeavesdropper'schannelgain 2=0dB,and 2 D=2 W=2.to 2=1.Weobservethatthekeycapacitylevelso as P/2increasesinthreeofthefourchannels,exceptthe caseof( mS, mD, mW)=(2,1,1),consideredin Figure1 .It appearsthattherelativeantennadimensionsdeterminethe asymptoticbehaviorofthekeycapacitywhentheSNRis large.Tomorepreciselystudythisbehavior,weevaluatethe limitingvalueof CKastheinputpower P ofthesource becomesverylarge.Tohighlightthedependenceof CKon P ,weusethenotation CK( P ). Corollary1. (1) If mWmS,then limPCK( P )=E log detHWHW+2 W/22 DHDHD detHWHW (30) (2) Supposethat mW
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EURASIPJournalonWirelessCommunicationsandNetworking9 columnsof V .Employingtheunitarypropertyof UWand VW,itisnothardtoverifythat&f ( P )=logdetImD+ P mS2 DHD&VW&VWHD+ HDVWW( P )VWHD, (37)&f( P )=logdetImD+ P mS2 DHD&VW&VWHD,(38) where W( P )=( 2 W/22 D)(( mS2 W/2P ) ImW+ S2 W)Š1.From ( 37 )and( 38 ),itisclearthat&f( P ) &f ( P ). Furtherlet t ( P )=tr( HDVWW( P )VWHD).Since t ( P ) ImDHDVWW( P )VWHD,&f ( P )logdet[ 1+ t ( P ) ] ImD+ P mS2 DHD&VW&VWHD =mDlog ( 1+ t ( P )) +logdetImD+ P mS2 D[ 1+ t ( P ) ] HD&VW&VWHD. (39) Let 1, 2, ... jbethepositiveeigenvaluesof HD&VW&VWHD. Notethat1jmin( mD, mSŠmW),becauseofthe factthattheelementsof HDarecontinuouslyi.i.d.andare independentoftheelementsof HW.Hence,from( 38 ),( 39 ), andthefactthat&f( P ) &f ( P ),wehave 0 &f ( P )Š &f( P )mDlog ( 1+ t ( P )) +log$ % 'j i=11+Pi/mS2 D( 1+ t ( P )) 'j i=11+Pi/mS2 D =mDlog ( 1+ t ( P )) +ji=1log( 1 / ( 1+ t ( P ))) +mS2 D/Pi 1+mS2 D/Pi (40) Nownotethat limPt ( P )=2 W 22 DtrHDVWSŠ2 WVWHD =2 W 22 Dtr" HŠ1 WHDHŠ1 WHD#, (41) where HŠ1 WdenotesthePenrose-Moorepseudoinverseof HW. Then( 40 )impliesthat 0liminfP &f ( P )Š &f( P ) limsupP &f ( P )Š &f( P ) mDŠjlog1+ 2 W 22 Dtr" HŠ1 WHDHŠ1 WHD# w p 1 (42) HencebyFatou'slemma,weget 0liminfP[ CK( P )ŠC( P ) ]limsupP[ CK( P )ŠC( P ) ]E mDŠjlog1+ 2 W 22 Dtr" HŠ1 WHDHŠ1 WHD# (43) From( 38 ),itisclearthat&f( P )increaseswithoutbound in P w.p.1;hence C( P )alsoincreaseswithoutbound. Combiningthisfactwith( 43 ),wearriveattheconclusion ofPart(2)ofthelemma. Part(1)ofthelemmaveriestheobservationsshownin Figure1 thatthekeycapacitylevelso astheSNRincreases ifthenumberofsourceantennasisnolargerthanthatof eavesdropperantennas.Whenthesourcehasmoreantennas, Part(2)ofthelemmasuggeststhatthekeycapacitycan growwithoutboundas P increasessimilarlytoaMIMO fadingchannelwithcapacity C( P ).Notethatthematrix ImSŠHW( HWHW)Š1HWintheexpressionthatdenes C( P ) isaprojectionmatrixtotheorthogonalcomplementof thecolumnspaceof HW.Thus C( P )hasthephysical interpretationthatthesecretinformationispassedacross thedimensionsnotobservablebytheeavesdropper.The mostinterestingaspectisthatthismodeofoperationcanbe achievedevenifneitherthesourcenorthedestinationknows thechannelmatrix HW. Wenotethattheasymptoticbehaviorofthekeycapacity inthehighSNRregimesummarizedin Corollary1 issimilar totheideaofsecrecydegreeoffreedomintroducedin[ 35 ]. Thesubtledi erencehereisthatnoup-to-dateCSIofthe destinationchannelisneededatthesource. Anotherinterestingobservationfrom Figure1 isthatfor thecaseof( mS, mD, mW)=(1,10,10),thesourcepower P seemstohavelittlee ectonthekeycapacity.Asmallamount ofsourcepowerisenoughtogetclosetothelevelingkey capacityofabout1bitperchanneluse.Thisobservation isgeneralizedbelowby Corollary2 ,whichcharacterizes thee ectofspatialdimensionalityofthedestinationand eavesdropperonthekeycapacitywhenthedestinationand eavesdropperbothhavealargenumberofantennas.

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10EURASIPJournalonWirelessCommunicationsandNetworking Corollary2. When mDand mWapproachinnityinsucha waythat limmD, mWmW/mD= CKŠmSlog1+ 1 22 D/2 W. (44) Proof. Thiscorollaryisadirectconsequenceofthefactthat (1 /mD) HDHDImSand(1 /mW) HWHWImSw.p.1, whichisinturnduetothestronglawoflargenumbers. Notethatwecaninterprettheratio asthespatial dimensionalityadvantageoftheeavesdropperoverthe destination.Theexpressionforthelimiting CKinthe corollaryclearlyindicatesthatthisspatialdimensionality advantagea ectsthekeycapacityinthesamewayasthe channelgainadvantage 2. In Figure2 ,thekeycapacitiesofseveralfast-fading MIMOchannelswithdi erentnumbersofsource,destination,andeavesdropperantennasareplottedagainstthe eavesdropper'schannelgainadvantage 2,with P/2=10dB.Theresultsin Figure2 showtheothere ectof spatialdimensionality.Weobservethatthekeycapacity decreasesalmostreciprocallywith 2inthechannelswith ( mS, mD, mW)=(1,1,1)and( mS, mD, mW)=(2,2,2),but staysalmostconstantforthechannelwith( mS, mD, mW)=(2,1,1).Itseemsthattherelativenumbersofsourceand eavesdropperantennasagainplaythemainroleindi erentiatingthesetwodi erentbehaviorsofthekeycapacity. Toverifythat,weevaluatethelimitingvalueof CKasthe gainadvantage 2oftheeavesdropperbecomesverylarge. Tohighlightthedependenceof CKon 2,weusethenotation CK( 2). Corollary3. Onehas limCK2 = 0, if mWmS, C( P ) if mWn ( R3Š8 ).Puttingtheseboundsback into( 71 ),wehave R3Š( 8 R3+8 ) Š1 n < 1 n H ( K )R3. (72) Nextwebound I ( K ; Zn, J ).Notethat I ( K ; Zn, J )=I ( L ; Zn, J ) + I ( K ; Zn, J|L )ŠI ( L ; Zn, J|K )I ( L ; Zn, J ) + I ( K ; Zn, J|L )I ( L ; Zn, J ) + H ( K|L )I ( L ; Zn, J ) +8 nR3+1, (73) wherethelastinequalityisobtainedfromPart(1)of Lemma6 andFano'sinequalitylikebefore.Inaddition,it holdsthat I ( L ; Zn, J )=H ( L )ŠH ( L|Zn, J )=H ( L )ŠH ( L J|Zn) + H ( J|Zn)=H ( L ) + H ( J|Zn)ŠH ( L J M|Zn) + H ( M|Zn, L J )H ( L ) + H ( J )ŠH ( M|Zn)ŠH ( L J|M Zn) + H ( M|Zn, L J )H ( L ) + H ( J ) + I ( M ; Zn)ŠH ( M ) +8 nR1 +1, (74) wherethesecondlastinequalityfollowsfrom H ( J|Zn)H ( J ),andthelastinequalityfollowsfrom H ( L J|M Zn)=0(bydenitionof J and L )and H ( M|Zn, L J )1+ 8 nR1 (byFano'sinequalityappliedtothectitiousreceiver). Byconstructionofthecode Cn,itholdsthat H ( L )nR2and H ( J )nR3.Inaddition,Part(3)of Lemma6 implies H ( M )n ( R1Š8 ).Finally,notethat I ( M ; Zn)I ( Yn; Zn)=nI ( Y ; Z )bythedata-processinginequalitysince M isadeterministicfunctionof Ynandthememoryless propertyofthechannelbetween Ynand Zn.Combining theseobservationsandsubstitutingthevaluesof R1, R2,and R3givenby( 46 )backinto( 73 ),weobtain 1 n I ( K ; Zn, J )R2+ R3ŠR1+ I ( Y ; Z ) + ( 8 R1+8 R3+8 ) + 2 nI ( Y ; Z )ŠI &Y ; Z+ ( 8 R1+8 R3+26 ) (75) when n issu cientlylarge.Withoutanyratelimitationon thepublicchannel,wecanchoosethetransitionprobability p (&y|y )suchthat I ( Y ; Z )ŠI (&Y ; Z ) ;therefore, 1 n I ( K ; Zn, J )( 8 R1+8 R3+27 ) (76) Since > 0canbechosenarbitrarily,Part(1)of Lemma6 ( 72 ),and( 76 ),establishtheachievabilityofthesecretkey rate I ( Y ; X )ŠI ( Y ; Z ).

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16EURASIPJournalonWirelessCommunicationsandNetworking5.ConclusionWeevaluatedthekeycapacityofthefast-fadingMIMOwiretapchannel.Wefoundthatspatialdimensionalityprovided bytheuseofmultipleantennasatthesourceanddestination canbeemployedtocombatachannel-gainadvantageof theeavesdropperoverthedestination.Inparticularifthe sourcehasmoreantennasthantheeavesdropper,then thechannelgainadvantageoftheeavesdroppercanbe completelyovercomeinthesensethatthekeycapacitydoes notvanishwhentheeavesdropperchannelgainadvantage becomesasymptoticallylarge.Thisisthemostinteresting observationofthispaper,asnoeavesdropperCSIisneeded atthesourceordestinationtoachievethenon-vanishingkey capacity.AcknowledgmentsTheworkofT.F.WongandJ.M.Sheawassupportedin partbytheNationalScienceFoundationunderGrantCNS0626863andbytheAirForceO ceofScienticResearch underGrantFA9550-07-10456.Theauthorswouldalsolike tothankDr.ShlomoShamaiandtheanonymousreviewers fortheirdetailedcommentsandthoughtfulsuggestions. Theyaregratefultothereviewerwhopointedouta signicantoversightintheproofof Theorem1 intheoriginal versionofthepaper.Theyarealsoindebtedtoanother reviewerwhosuggestedtheconcavityargumentintheproof of Lemma2 ,whichismuchmoreelegantthantheauthors' originalone.References[1]A.Wyner,"Thewire-tapchannel," BellSystemTechnical Journal ,vol.54,pp.13551387,1975. [2]S.K.Leung-Yan-CheongandM.Hellman,"TheGaussian wire-tapchannel," IEEETransactionsonInformationTheory vol.24,no.4,pp.451456,1978. [3]I.Csisz arandJ.Korner,"Broadcastchannelswithcondential messages," IEEETransactionsonInformationTheory ,vol.24, no.3,pp.339348,1978. [4]Y.Liang,H.Poor,andS.Shamai,"Securecommunicationover fadingchannels," IEEETransactionsonInformationTheory vol.54,no.6,pp.24702492,2008. [5]P.Gopala,L.Lai,andH.ElGamal,"Onthesecrecycapacity offadingchannels," IEEETransactionsonInformationTheory vol.54,no.10,pp.46874698,2008. 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