IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.54,NO.6,JUNE20082)Totransmitme - PDF document

IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.54,NO.6,JUNE20082)Totransmitmessage ,user ndsthetwocodewordscorrespondingtocomponents andalsouniformlychoosesacodeword from .User thenaddsallthesecodewordsandtransmitstheresultingcodeword ,sothatitactuallytransmitsoneof codewords.Let Notethatsinceallcodewordsarechosenuniformly,user essentiallytransmitsoneof codewordsatrandomforeachmessage ,anditsoverallrateoftransmission cally,wechoosetheratestosatisfy (26) withequalityif (27) whichwecanalsowriteas (29) withequalityif (30) Notethatif(31)iszeroforagroupofusers,wecannotachievesecrecyforthoseusers.When ,ifthesumca-pacityofthemainchannelislessthanthatoftheeavesdropperchannel,i.e., ,secrecyisnotpossibleforthesystem.Assumethisquantityispositive.Toensurethatwecanmutuallysatisfyboth(30)and(31),wecanreclassifysomeopenmes-sagesassecret.Clearly,ifwecanguaranteesecrecyforalargersetofmessages,secrecyisachievedfortheoriginalmessages.Fromtherstsetofconditionsin(25)andtheGMACcodingtheorem[44]withhighprobabilitythereceivercandecodethecodewordswithlowprobabilityoferror.Toshowthesecrecyconditionin(12),rstnotethatthecodingschemedescribedisequivalenttoeachuser selectingoneof messages,andsendingauniformlychosencodewordfromamong codewordsforeach.De ,andwehave (32) (33) (34) whereweused ,andthuswehave toget(35).Wewillconsiderthetwotermsindividually.First,wehavethetrivialboundduetochannelcapacity Nowwrite Sinceuser independentlysendsoneof codewordsequallylikelyforeachsecretmessage (38) (39) Wecanalsowrite (41)where as since,withhighprobability,theeavesdroppercandecode given dueto(30)andcodegeneration.Using(36),(37),(40),and(41)in(35),weget (42) Now,letusconsidertheTDMAregiongivenin(22).Thisre-gionisobtainedwhenuserswhocanachievesingle-usersecrecyuseasingle-userwiretapcodeasin[12]inaTDMAschedule,wherethetimeshareofeachuser isgivenby and .Atransmitter whocanachievesecrecy,i.e.,having ,transmitsfor portionofthetimewhenallotherusersaresilent,using power,satisfyingitsaveragepowerconstraintovertheTDMAtimeframe.Thisapproachwasusedin[40]toachievesecrecysumcapacityforindividualconstraints.Whenthechannelisdegraded,i.e., forall ,thenforcollectiveconstraintstheTDMAregionisseentobeasubsetofthesuperpositionregion.However,thisisnotnecessarilytrueforthegeneralcase,andbytimesharingbetweenthetwoschemes,wecangenerallyachievealargerachievableregion,givenin(25). Weremarkthatitispossibletofurtherdividethesagestogetmoresetsofprivatemessages,whicharealsoper-fectlysecret,i.e.,ifwelet , ,thenaslongasweimposethesamerestrictionson as ,wecanachieveperfectsecrecyof ,asin[12].However,thisdoesnotmeanthatwehaveperfectsecrecyatchannelcapacity,asthesecrecysubcodescarryinformationabouteachother.Observethatevenfor users,aratepointinthisregionisfourdimensional,andhencecannotbeaccuratelydrawn.Wecaninsteadfocusonthesecrecyrateregion,theregionofallachievable .Thesubregions and shownfordifferentchannelgainsinFig.3forxedtransmitpowers,and users.Fig.4representshowtheseregionschangewithdifferenttransmitpowerswhenthechannelgains TEKINANDYENER:GGMAC-WTANDGTW-WT:ACHIEVABLERATESANDCOOPERATIVEJAMMING2747single-userdecodable,andifithasenoughpowertomaketheotheruserintheneweffectivechannel.B.GTW-WTOnceagain,weproposetomaximizethesecrecysumrateusingcooperativejammingwhenuseful.Thisproblemisfor-mallystatedasfollows: wherewerecallthat isgivenby(71)and (102) Notethat sincethereareonlytwoterminals.AsimilarargumenttotheGGMAC-WTcasecaneasilybeusedtoestablishthatwecanassumeausertobeeithertransmittingorjamming,butnotboth.Sincethejamminguserisalsothereceiverthattheotheruseriscommunicatingwithandknowsthetransmittedsignal,thisschemeentailsnolossofcapacityasfarasthetransmittinguserisconcerned.Theoptimumpowerallocationsaregivenasfollows.Theorem6:Theachievablesecrecysumrateforthecollabo-rativeschemedescribedis (104)where isthesetoftransmittingusersandtheoptimumpowerallocationsaregivenby(105)shownatthebottomofthepage.Proof:SimilartotheGGMAC-WT,westartwiththesub-problemofndingtheoptimalpowerallocationgivenajam-mingset.TheLagrangianisgivenby Takingthederivativewehave if if sinceauser satises ,itmusthave Consideruser .Weagainarguethatif ,then andif ,then .Nowexamineauser .Itiseasytoseethatsincesuchauseronlyharmsthejammer,theoptimaljammingstrategyshouldhave ,i.e.,themaximumpower.Thiscanalsobeseenbynotingthat(107)forthiscasesimpliesto andhencewemusthave forall Thejammingsetwillbeoneof ,sincethereisnopointinjammingwhenthereisnotransmission.Also,ifanyofthetwousersisjamming,bytheargumentabove, , .Wecaneasilyseethatjammingbyauser offersanadvantageif ,i.e., iff for .Thus,when ,bothusersshouldbetransmittinginsteadofjamming.However,whenanyuser ,jammingalwaysdoesbetterthanthecasewhenbothusersaretransmitting.Inthiscase, someuser ,andtheobjectivefunctionin(101)isminimizedwhenthisuserisjamming,andtheotheroneistransmitting.If,however, ,thenitwillnottransmit,andweshouldnotbejamming.Consolidatingalloftheseresults,wecomeupwiththepowerallocationinTheorem6. Remark4:Asufcient,butnotnecessaryconditionfortheweakerusertobethejamminguserisif ;thiscasecorrespondstohavinghighersignal-to-noiseratio(SNR)attheeavesdropperfortheoriginal,nonstandardizedmodel.Thiscanbeinterpretedasjamwithmaximumpowerifitispossibletochangeuser1seffectivechannelgainsuchthatitisnolongersingle-userdecodable.Forthesimplecaseofequalpowercon- ,itiseasilyseenthatuser1shouldneverbejamming.Theoptimalpowerallocationinthatcasereduces bothtransmit if transmits jams if VI.NESULTSInthissection,wepresentnumericalresultstoillustratetheachievableratesobtained,aswellasthecooperativejammingschemeanditseffectonachievablesecrecysumrates.Asmentionedearlierinthispaper,examplesofachievablese-crecyrateregionsaregiveninFigs.4and6fortheGGMAC-WT andGTW-WT,respectively.ComparingFigs.4and6,weseethattheGTW-WTachievesalargersecrecyrateregion bothtransmit if transmits jams if transmits jams if , transmits jams if , otherwise.(105) TEKINANDYENER:GGMAC-WTANDGTW-WT:ACHIEVABLERATESANDCOOPERATIVEJAMMING2743IV.MAXIMIZATIONOFATETheachievableregionsgiveninTheorems1and2dependonthetransmitpowers.Weare,thus,naturallyinterestedinthepowerallocation thatwouldmaximizethetotalsecrecysumrate.Recallthatthestandardizedchannelgainforuser is ,andthatthehigher is,thebetterthecorre-spondingeavesdropperchannel.Withoutlossofgenerality,as-sumethatusersareorderedintermsofincreasingstandardizedeavesdropperchannelgains,i.e., .Notethatweonlyneedtoconcernourselveswiththecase sincewecancombineuserswiththesamechannelgainsintoonesuperuser.Wecanthensplittheresultingoptimumpowerallocationforasuperuseramongtheactualconstitutingusersinanywaywechoose,sincetheywouldallresultinthesamesumrate.Inaddition,fromaphysicalpointofview,assumingthatthechannelparametersaredrawnaccordingtoacontinuousdistributionandthenxed,theprobabilitythattwouserswouldhavethesameexactstandardizedchannelgainiszero.A.GGMAC-WTrstexaminethesuperpositionregiongivenin(20).ThesecrecysumrateachievablewithsuperpositioncodingfortheGGMAC-WTwasgiveninTheorem1as - andwewouldliketondthepowerallocationthatmaximizesthisquantity.Statedformally,weareinterestedinthetransmitpowersthatsolvethefollowingoptimizationproblem: (57) (58)where (59)and yields(58).Inobtaining(58),wesimplyusedthemonotonicityofthe function.Thesolutiontothisproblemisgivenbelow.Theorem3:Thesecrecysum-ratemaximizingpoweralloca-tionfor satises if and is where issomelimitingusersatisfying andwede and .Notethatthisallocationshowsthatonlyasubsetofthestrongusersmustbetransmitting.Proof:WestartwithwritingtheLagrangiantobemini- EquatingthederivativeoftheLagrangiantozero,weget wherewede foranyset Itiseasytoseethatif ,then ,andwehave .If ,thenwesimilarlyndthat .Finally,if ,thenwealsohave (64)and doesnotdependon ,sowecanset withnoeffectonthesecrecysumrate.Thus,wehave if ,and if .Then,theoptimalsetoftransmittersisoftheform sinceifauser istransmitting,alluserssuchthat mustalsobetransmitting.Wealsonotethat .Let bethelastusersatisfyingthisproperty,i.e., and .Notethat (65) Inotherwords,allsets for alsosat-isfythispropertyandareviablecandidatesfortheoptimalsetoftransmittingusers.Therefore,wecanclaimthat istheop-timumsetoftransmittingusers,sincefromabovewecanitera-tivelyseethat forall . Notethat,forthespecialcaseof users,theoptimumpowerallocationis if , if , WealsoneedtoconsidertheTDMAregion.Inthiscase,themaximumachievablesecrecysumrateis Thisisasimplecomplexoptimizationproblemthatcaneasilybesolvednumerically.Forthedegradedcase,wecanobtainaclosedformsolution: asin[40].Ingeneral,wecannotobtainsuchasolution.However,itistrivialtonotethatuserswith shouldnotbetransmittinginthisscheme.ThesecrecysumrateisthenthemaximumofthesolutionsgivenbythesuperpositionandTDMAregions. IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.54,NO.6,JUNE2008whenthewiretapcapacityiszero.AhslwedeandCsiszr[17],[18]examinedthesecretkeycapacityandcommonrandomnesscapacity,forseveralchannels.Theseresultsalsobenetfrom[14]toprovidesecretkeycapacities.Maureralsoex-aminedthecaseofactiveadversaries,wherethewiretapperhasread/writeaccesstothechannelin[19][21].Thesecretkeygenerationproblemwasinvestigatedfromamultipartypointofviewin[22]and[23].Notably,CsiszrandNarayanconsideredthecaseofmultipleterminalswhereanumberofterminalstrytodistillasecretkeyandasubsetoftheseterminalscanactashelperterminalstotherestin[24]and[25].Recently,severalnewmodelshaveemerged,examiningse-crecyforparallelchannels[26],[27],relaychannels[28],andfadingchannels[29],[30].Fadingandparallelchannelswereexaminedtogetherin[31]and[32].Broadcastandinterferencechannelswithcondentialmessageswereconsideredin[33].LiangandPoor[34]andLiuetal.[35]examinedthemultiple-accesschannelwithcondentialmessageswheretwotransmit-terstrytokeeptheirmessagessecretfromeachotherwhilecommunicatingwithacommonreceiver.In[34],anachiev-ableregionwasfoundingeneral,andthecapacityregionwasfoundforsomespecialcases.Multiple-input(MIMO)channelswereconsideredin[36]and[37].In[38][41],weinvestigatedmultiple-accesschannelswheretransmitterscommunicatewithanintendedreceiverinthepres-enceofanexternalwiretapperfromwhomthemessagesmustbekeptcondential.In[38][40],weconsideredthecasewherethewiretappergetsadegradedversionofaGMACsignal,andnedtwoseparatesecrecymeasuresextendingWynersmea-suretomultiuserchannelstoreecttheleveloftrustthenetworkmayhaveineachnode.Achievablerateregionswerefoundfordifferentsecrecyconstraints,anditwasshownthatthese-crecysumcapacitycanbeachievedusingGaussianinputsandstochasticencoders.Inaddition,time-divisionmultipleaccess(TDMA)wasshowntoalsoachievethesecrecysumcapacity.Gaussianandbinaryadditivetwo-waywiretapchannelswereexaminedin[42].Inthispaper,weconsiderthegeneralGaussianmultiple-ac-cesswiretapchannel(GGMAC-WT)andtheGaussiantwo-waywiretapchannel(GTW-WT),bothofwhichareofinterestinwirelesscommunicationsastheycorrespondtothecasewhereasinglephysicalchannelisutilizedbymultipletransmitters,suchasinanadhocnetwork.WeconsideranexternaldropperthatreceivesthetransmitterssignalsthroughageneralGaussianmultiple-accesschannel(GGMAC)inbothsystemmodels.Weutilizeasuitablesecrecyconstraintthatisthenor-malizedconditionalentropyofthetransmittedsecretmessagesgiventheeavesdropperssignal,correspondingtothetivesecrecyconstraintsusedin[40].Weshowthatsatisfyingthisconstraintimpliesthesecrecyofthemessagesforallusers.Inbothscenarios,transmittersareassumedtohaveonesecretandoneopenmessagetotransmit.Thisisdifferentfrom[40]inthatthesecrecyratesarenotconstrainedtobeatleastaxedportionoftheoverallrates.Wendanachievablesecrecyrateregion,whereuserscancommunicatewitharbitrarilysmallEventhoughwefaithfullyfollowWynersterminologyinnamingthechan-nels,admittedlyinwirelesssystemmodels,eavesdropperisamoreappropriatetermfortheadversary.probabilityoferrorwiththeintendedreceiverunderperfectse-crecyfromtheeavesdropper,whichcorrespondstotheresultof[40]forthedegradedcase.Wenotethat,inaccordancewiththerecentliterature,whenweusethetermperfectsecrecy,wearereferringtosecrecy,wheretherateofinformationleakedtotheadversaryislimited.Assuch,thiscanbethoughtofasalmostperfectsecrecy.Wealsondthesum-ratemax-imizingpowerallocationsforthegeneralcase,whichismoreinterestingfromapracticalpointofview.Itisseenthataslongastheusersarenotsingle-userdecodableattheeaves-dropper,asecrecy-ratetradeoffispossiblebetweentheusers.Next,weshowthatanontransmittingusercanhelpincreasethesecrecycapacityforatransmittinguserbyeffectivelytheeavesdropper,andevenenablesecretcommunica-tionsthatwouldnotbepossibleinasingle-userscenario.Wetermthisnewschemecooperativejamming.TheGTW-WTisshowntobeespeciallyusefulforsecretcommunications,asthemultiple-accessnatureofthechannelhurtstheeavesdropperwithoutaffectingthecommunicationrate.Thisisbecausethetransmittedmessagesofeachuseressentiallyhelphidetheotherssecretmessages,andreducetheextrarandomnessneededinwiretapchannelstoconfusetheeavesdropper.Therestofthispaperisorganizedasfollows.SectionIIde-scribesthesystemmodelfortheGGMAC-WTandGTW-WTandtheproblemstatement.SectionIIIdescribesthegeneralachievableratesfortheGGMAC-WTandtheGTW-WT.Sec-tionsIVandVgivethesecrecysumratemaximizingpowerallocations,andtheachievablerateswithcooperativejamming.SectionVIgivesournumericalresultsfollowedbyourconclu-sionsandfutureworkinSectionVII.II.SODELANDROBLEMTATEMENTWeconsider userscommunicatinginthepresenceofaneavesdropperwhohasthesamecapabilities.Eachtrans- hastwomessages, issecretand whichisopen,fromtwosetsofequallylikelymessages , .Let , , , ,and .Themessagesareencodedusing codesinto ,where .Theencodedmessages arethentransmitted.Weassumethechannelparametersareuniversallyknown,andthattheeavesdropperalsohasknowledgeofthecodebooksandthecodingscheme.Inotherwords,thereisnosharedsecret.Thetwochannelsweconsiderinthispaperaredescribednext.A.GGMAC-WTThisisascenariowheretheuserscommunicatewithacommonbasestationinthepresenceofaneavesdropper,wherebothchannelsaremodeledasGaussianmultiple-accesschannelsasshowninFig.1.Theintendedreceiverandthewiretapperreceive and ,respectively.Thereceiverdecodes togetanestimateofthetransmitted .WewouldliketocommunicatewiththeWewouldliketostressthatisnotthesameas,i.e.,wedonotimposeadecodabilityconstraintfortheopenmessagesattheeavesdropper. IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.54,NO.6,JUNE2008Theorem2:Therateregiongivenbelowisachievablefortheconvexclosureof Proof:TheproofisverysimilartotheproofofTheorem1.WeusethesamecodingschemeasTheorem1,butthemaindifferenceisthatwechoosetheratestosatisfy (48) withequalityif (49) orequivalently (51) withequalityif (52) assuming(53)ispositive.Thedecodabilityof from and comesfrom(51)andthecapacityregionoftheGaussiantwo-waychannel[5].Thisgivestherstsetoftermsintheachievableregion.Thekeyhereisthatsinceeachtransmitterknowsitsowncodeword,itcansubtractitsself-interferencefromthereceivedsignalandgetaclearchannel.Therefore,theGaussiantwo-waychanneldecomposesintotwoparallelchan-Thesecondgroupoftermsin(45),resultingfromthesecrecyconstraint,canbeshownthesamewayastheproofofTheorem1,since hasthesameformforbothchannels.Inotherwords,asfarastheeavesdropperisconcerned,thechannelisstillaGMACwith users.Assuch,weneedtosend extracodewordsintotal,whichneedtobesharedbythetwoterminalsprovidedtheyarenotsingle-userdecodable. Fordifferentchannelgains,theregionofall (45)isshowninFig.5.Sincewerequirefourdimensionsforanaccuratedepictionofthecompleterateregion,weonlyfocusonourmaininterest,i.e.,thesecrecyrateregion.Fig.6showstheachievablesecrecyrateregionasafunctionoftransmitpowers.Wenotethathigherpowersalwaysresultinalargerregion.Weindicatetheconstraintontheoverallrates,correspondingtothecapacityregionoftheGaussiantwo-waychannel,bythedottedline.NotethatthesecrecyregionhasastructuresimilartotheGGMAC-WTwith .Asfarastheeavesdropperisconcerned,thereisnodifferencebetweenthetwochannels.However,sincethemainchannelbetweenusersdecomposesintotwoparallelchannels,higherratescanbeachievedbetween Fig.5.GTW-WTachievableregionsfordifferentchannelparameters (P =2) Fig.6.GTW-WTachievablesecrecyregionwhen P h 3,andh thelegitimateterminals(users).Thus,ineffect,eachusertransmittedcodewordsactasasecretkeyfortheotherusertransmittedcodewords,requiringfewerextraneouscodewordsoveralltoconfusetheeavesdropper,andalargersecrecyre-gion.Wenotethatausermayeitherachievesecrecyornot,dependingonwhetheritissingle-userdecodable.Asaresult,TDMAdoesnotenlargetheregion,sinceeachusercanatleastachievetheirsingle-usersecrecyrates.Toseethis,notethattheconstraintonthesecrecysumratecanbewrittenas (54) sothattransmittinginthetwo-waychannelalwaysprovidesanadvantageoverthesingle-userchannels. ofthetransmittedmessagegiventhereceivedsignalatthewiretapperasthesecrecymeasure,hefoundtheregionofallpos- pairs,andtheexistenceofasecrecycapacity therateuptowhichitispossibletolimittherateofinformationtransmittedtothewiretappertoarbitrarilysmallvalues. TEKINANDYENER:GGMAC-WTANDGTW-WT:ACHIEVABLERATESANDCOOPERATIVEJAMMING2741 Fig.3.GGMAC-WTachievableregionsfordifferentchannelparameters (P =2) Fig.4.GGMAC-WTachievablesecrecyregionwhen P h ,and xed.Forthecaseshown,weneedtheconvexhullopera-tion,astheachievableregionisacombinationofdifferentsu-perpositionandTDMAregions.Notealsothatthemainextraconditionforthesuperpositionregionisonthetotalextraran-domnessadded.Asaresult,itispossibleforuserstousersbycontributingmoretothenecessaryextranumberofcodewords,whichisthesumcapacityoftheeaves-dropper.Suchaweakuseronlyhastomakesurethatitisnotsingle-userdecodable,providedthestrongerusersarewillingtocesomeoftheirownrateandgeneratemoresupercodewords.Inotherwords,weseethatusersinaset arefur-therprotectedfromtheeavesdropperbythefactthatusersin arealsoundecodable,comparedtothesingle-usercase.TheTDMAregion,ontheotherhand,doesnotallowuserstohelpeachotherthisway.Assuch,onlyuserswhosechannelgainsallowthemtoachievesecrecyontheirownareallowedtoForthespecialdegradedcaseof theperfectsecrecyrateregionfor becomestheregiongivenby[40,Th.1]for .Wealsoobservethateventhoughthereisalimitonthesecrecysumrateachievedbyourscheme,itispossibletosendopenmessagestotheintendedreceiveratratessuchthatthesumofthesecrecyrateandopenrateforallusersisinthecapacityregionoftheMACtotheintendedreceiver.Eventhoughwecannotsendatcapacitywithsecrecy,thecodewordsusedtoconfusetheeavesdroppermaybeusedtocommunicatemeaningfulinformationtotheintendedreceiver.B.GTW-WTInthissection,wepresentanachievableregionfortheGTW-WTusingasuperpositioncodingsimilartothatusedtoachievetheregion fortheGGMAC-WT.Wenethefollowing.nition5(GTW-WTSuperpositionRegion ):Let .Then,theGTW-WTsuperpositionregion isgivenby whichcanbewrittenas Remark2:Wecanalsowritethisregionmorecompactlyasthefollowing: (46) TEKINANDYENER:GGMAC-WTANDGTW-WT:ACHIEVABLERATESANDCOOPERATIVEJAMMING2737 Fig.1.StandardizedGMAC-WTsystemmodel.receiverwitharbitrarilylowprobabilityoferror,whilekeepingthewiretapper(eavesdropper)ignorantofthesecretmessages .Thesignalsattheintendedreceiverandthewiretapperaregivenby (1a) (1b)where and aretheadditivewhiteGaussiannoise(AWGN), isthetransmittedcodewordofuser ,and and arethechannelgainsofuser totheintendedreceiver ),andtheeavesdropper(wiretap ),respectively.Eachcomponentof and .Wealsoassumethefollowingtransmitpowerconstraints: Similartothescalingtransformationtoobtainthestandardformoftheinterferencechannel[43],wecanrepresentanyGMAC-WTbyanequivalentstandardform[40] (3a) where,foreach ,wehavethefollowing:thecodewordsarescaledtoget thenewpowerconstraintsare thewiretappersnewchannelgainsare thenoisesarenormalizedtoget and Wecanshowthattheeavesdroppergetsastochasticallyde-gradedversionofthereceiverssignalif .Weconsideredthisspecialcasein[39]and[40].B.GTW-WTInthisscenario,twotransmitter/receiverpairscommunicatewitheachotheroveracommonchannel.Eachreceiver gets andtheeavesdroppergets .Receiver decodes togetanestimateofthetransmittedmessagesoftheotheruser.Theuserswouldliketocommunicatetheopenandsecretmessageswitharbitrarilylowprobabilityoferror,whilemaintainingsecrecyofthesecretmessages.Thesignalsattheintendedreceiverandthewiretapperaregivenby (4a) (4b) (4c)where and .Wealsoas-sumethesamepowerconstraintsgivenin(2)(with ),andagainuseanequivalentstandardformasillustratedinFig.2 (5a) (5b) wherewehavethefollowing:thecodewords arescaledtoget and themaximumpowersarescaledtoget and thetransmittersnewchannelgainsaregivenby and thewiretappersnewchannelgainsaregivenby and ; TEKINANDYENER:GGMAC-WTANDGTW-WT:ACHIEVABLERATESANDCOOPERATIVEJAMMING2745wherewerecallthat isgivenby(59),suchthat (78) Toseethatausershouldnotbesplittingitspoweramongjammingandtransmitting,itissufcienttonotethatregardlessofhowausersplitsitspower, willbethesame,andtheuseronlyaffects .Assumetheoptimumsolutionissuchthatuser splitsitspower,so and .Then,itiseasytoseethatif ,thesumrateisincreasedwhenthatuserusesitsjammingpowertotransmit,andwhen ,thesumrateisincreasedwhentheuserusesitstransmitpowertojam.When ,thenregardlessofhowitspowerissplit,thesumrateisthesame,andwecanassumeuser eithertransmitsorjams.Notethatwemusthave tohaveanonzerosecrecysumrate,and tohaveanadvantageovernotjamming.Thisschemecanbeshowntoachievethefol-lowingsecrecysumrate.Theorem5:Thesecrecysumrateusingcooperativejamming (80)where isthesetoftransmittersandtheoptimumpowerallo-cationisoftheform with (81)and (82) (83) wheneverthepositiverealrootexists,and Proof:rstsolvethesubproblemofndingtheoptimalpowerallocationforasetofgiventransmitters .Thesolutiontothiswillalsogiveusinsightintothestructureoftheoptimalsetoftransmitters .WestartwithwritingtheLagrangian ThederivativeoftheLagrangiandependsontheuser if if sinceauser satises ,itmusthave Considerauser .Thesameargumentasinthesum-ratemaximizationproofleadsto if and if .Nowexamineauser .Wecanwrite(86)as (87)where (88)Let Then,wehave iff ,and iff .Thus,weagainndthatwemusthave forall .Also,if ,then .Onlyif ,canwehave .Now,since ,wemusthave .Thus,wendthat .Then,weknowthatforagivensetoftrans- ,thesolutionissuchthatallusers transmitwithpower if .Inthesetofjammers ,allusershave ,andwhenthisinequalityisnotsatisedwithequality,thejammersjamwithmaximumpower.Iftheequalityissatisedforsomeusers theirjammingpowerscanbefoundfromsolving Byrearrangingtermsin(88),wenotethattheoptimumpowerallocationforthisuser,callituser ,isfoundbysolvingthe thesolutionofwhichisgivenin(81).Notethat(90)denesan(upright)parabola.Iftherootgivenin(90)existsandispositive,then .Thiscomes TEKINANDYENER:GGMAC-WTANDGTW-WT:ACHIEVABLERATESANDCOOPERATIVEJAMMING2749 Fig.10.GGMAC-WTcooperativejammingexampledarkershadescorrespondtohighervalues.examinethetransmit/jampowersforthisareawhentheeaves-dropperisknowntobeat usingaxedpath-lossmodelforthechannelgains,andplotthetransmit/jampowersandtheachievedsecrecysumratesasafunctionoftheeavesdropperlo-cation.ItisreadilyseenthatwhentheeavesdropperisclosetotheBS,thesecrecysumratefallstozero.Also,whentheeaves-dropperisinthevicinityofatransmitter,thattransmittercannottransmitinsecrecy.However,inthiscase,thetransmittercanjamtheeavesdroppereffectivelyandallowtheothertransmittertotransmitand/orincreaseitssecrecyratewithlittlejammingpower.ThesituationfortheGTW-WTissimilarandisshowninFig.11.Inthiscase,jammingismoreusefulascomparedtotheGGMAC-WT,andweseethatitispossibletoprovidese-crecyforamuchlargerareawheretheeavesdropperislocated,asthejammingsignaldoesnothurttheintendedreceiver.VII.CONCLUSIONANDInthispaper,wehaveconsideredtheGaussianmultiple-ac-cessandtwo-waychannelsinthepresenceofanexternaleaves-dropperwhoreceivesthetransmittedsignalsthroughamul-tiple-accesschannel,andprovidedachievablesecrecyrates.Wehaveshownthatthemultiple-accessnatureofthechannelscon-sideredcanbeutilizedtoimprovethesecrecyofthesystem.Inparticular,wehaveshownthatthetotalextrarandomnessiswhatmattersmainlyconcerningtheeavesdropper,ratherthantheindividualrandomnessinthecodes.Assuch,itmaybepos-sibleforuserswhosesingle-userwiretapcapacityarezero,tocommunicatewithnonzerosecrecyrate,aslongasitispos-sibletoputtheeavesdropperatanoveralldisadvantage.Thisisevenclearerfortwo-waychannels,whereeventhoughtheeavesdropperschannelgainmaybebetterthanaterminals,theextraknowledgeofitsowncodewordbythatterminalenablescommunicationinperfectsecrecyaslongastheeavesdropperreceivedsignalisnotstrongenoughtoallowsingle-userde-WefoundachievablesecrecyrateregionsfortheGGMAC-WTandtheGTW-WT.WealsoshowedthatfortheGGMAC-WTthesecrecysumrateismaximizedwhenonlyuserswithchannelstotheintendedreceiverasopposedtotheeavesdroppertransmit,andtheydosousingalltheiravailablepower.FortheGTW-WT,thesumrateismaximizedwhenbothterminalstransmitwithmaximumpoweraslongastheeavesdropperschannelisnotgoodenoughtodecodethemusingsingle-userdecoding.Finally,weproposedaschemetermedcooperativejammingwhereadisadvantagedusermayhelpimprovethesecrecyratebyjammingtheeavesdropper.Wefoundtheoptimumpowerallocationsforthetransmittingandjammingusers,andweshowedthatsignicantrategainsmaybeachieved,especiallywhentheeavesdropperhasmuchhigherSNRthanthereceivers TEKINANDYENER:GGMAC-WTANDGTW-WT:ACHIEVABLERATESANDCOOPERATIVEJAMMING2739 .Ourachievableratescannotguaranteese-crecyforsuchagroupofusers.III.ACHIEVABLEATEA.GGMAC-WTInthissection,wepresentourmainresultsfortheGGMAC-WT.Werstdenetwoseparateregionsandthengiveanachievableregion.nition3(GGMAC-WTSuperpositionRegion): forall .Then,thesuperpositionregion isgivenby whichcanbewrittenas nition4(GGMAC-WTTDMARegion): besuch forall and .Let forall .Then,theTDMAregion isgivenby whichisequivalentto Remark1:ThesuperpositionandTDMAregionscanalsobewrittenasfollows: (23) inaccordancewiththedenitionsin(14)Theorem1:TherateregiongivenbelowisachievablefortheGGMAC-WT convexclosureof Proof:rstshowthatthesuperpositionencodingrateregiongivenin(20)foraxedpowerallocationisachievable.Considerthefollowingcodingschemeforrates forsome SuperpositionEncodingScheme:Foreachuser ,considerthefollowingscheme.1)Generatethreecodebooks , ,and . sistsof codewords,eachcomponentofwhichisdrawnfrom .Codebook has codewordswitheachcomponentrandomlydrawnfrom and has codewordswitheachcomponentrandomlydrawnfrom where isanarbitrarilysmallnumbertoensurethatthepowerconstraintsonthecodewordsaresatisedwithhighprob-abilityand .De and . IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.54,NO.6,JUNE2008B.GTW-WTNow,wewillexaminethepowerallocationthatmaximizesthesecrecysumrategiveninTheorem2as Thisproblemisformallystatedbelow (70)where (71)and yields(70).Theoptimumpowerallocationisstatedbelow.Theorem4:Thesecrecysum-ratemaximizingpoweralloca-tionfortheGTW-WTisgivenby if if Proof:TheLagrangianis EquatingthederivativeoftheLagrangiantozeroforuser weget (74)where AnargumentsimilartotheonefortheGGMAC-WTestab-lishesthatif ,orequiva-lently,if ,then .Whenequalityissatised,then regardlessof ,andassuch canbeseentonotdependon .Toconservepower,weagainset inthiscase.Ontheotherhand,if ,then Consideruser1.If ,and ,thisimpliesthat .Since ,wecannothave .Asaconsequenceofthiscontradiction,weseethat whenever .Assume ,andconsiderthetwoalternatives .Wewillhave if ;and if .Thesecasescorrespondto and ,respectively.Thus,wehave(72)asthesecrecysum-ratemaximizingpowerallocation. Remark3:ObservethatthesolutioninTheorem4hasastructuresimilartothatinTheorem3.Insummary,itisseenthataslongasauserisnotsingle-userdecodable,itshouldbetransmittingwithmaximumpower.Hence,whenbothuserscanbemadetobenonsingle-userdecodable,thenthemaximumpowerswillprovidethelargestsecrecysumrate.Ifthisisnotthecase,thentheuserwhoissingle-userdecodablecannottransmitwithnonzerosecrecyandwilljustmakethesecrecysum-rateconstrainttighterfortheremaininguserbytransmittingopenComparing(72)to(67),weseethatthesameformofsolu-tionsisfound,buttherangeofchannelgainswheretransmissionispossibleislarger,showingthatGTW-WTallowssecrecyevenwhentheeavesdropperschannelisnotveryweak.V.SHROUGHOOPERATIVEInSectionIV,wefoundthesecrecysum-ratemaxi-mizingpowerallocations.ForboththeGGMAC-WTandtheGTW-WT,iftheeavesdropperisnotdisadvantagedforsomeusers,thentheseuserstransmitpowersaresettozero.Wepositthatsuchausermaybeabletotransmittinguser,sinceitcancausemoreharmtotheeaves-dropperthantotheintendedreceiver.Weonlyconsiderthesuperpositionregion,sinceintheTDMAregionauserhasadedicatedtimeslot,andhencedoesnotaffecttheothers.Wewillnextshowthatthistypeofcooperativebehaviorisindeeduseful,notablyexploitingthefactthattheestablishedachievablesecrecysumrateisadifferenceofthesum-capacityexpressionsfortheintendedchannel(s)andtheeavesdropperchannel.Asaresult,reducingthelattermorethantheformeractuallyresultsinanincreaseintheachievablesecrecysumFormally,theschemeweareconsideringimpliespartitioningthesetofusers intoasetoftransmittingusers andasetofjammingusers .Ifauser isjamming,thenit insteadofcodewords.Inthiscase,wecanshowthatwecanachievehighersecrecyrateswhentheweakerusersarejamming.WealsoshowthattheGTW-WThasanadditionaladvantagecomparedtotheGGMAC-WT;thatisthefactthatthereceiveralreadyknowsthejammingsequence.Assuch,thisschemeonlyharmstheeavesdropperandnottheintendedreceivers,achievinganevenhighersecrecysumrate.Onceagain,withoutlossofgenerality,weconsider .Inaddition,wewillassumethatausercaneithertaketheactionoftransmittingitsinformationorjammingtheeaves-dropper,butnotboth.ItisreadilyshowninSectionV-Athatwedonotloseanygeneralitybydoingso,andthatsplittingthepowerofauserbetweenthetwoactionsissuboptimalfromthesecrecysum-ratemaximizationpointofview.A.GGMAC-WTTheproblemisformallypresentedbelow (76) (77) IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.54,NO.6,JUNE2008 Fig.7.GGMAC-WTcooperativejammingsecrecysumrateasafunctionof withdifferent forP =P h .Thecirclesindicateoptimumjammingpower.thantheGGMAC-WT,andoffersmoreprotectiontousers.Inaddition,TDMAdoesnotenlargetheachievablere-gionforGTW-WTsincesuperpositioncodingalwaysallowsuserstoachievetheirsingle-usersecrecyratesforanytransmitpower.Letushaveacloserlookatthesecrecyadvantageofthetwo-waychannelovertheMACwithtwousers.FortheGGMAC-WTwith ,theachievablemaximumsecrecysumrate islimitedbythechannelparameters.Itwasshownin[39]thatforthedegradedcase ,thesecrecysumcapacity isanincreasingfunctionofthetotalsumpower .However,itislimited as .Forthegeneralcase,where ,Theorem3impliesthatthesumrateismaximizedwhenonlyuser1transmits(assuming ),andisboundedsimilarlyby .Ontheotherhand,fortheGTW-WT,unliketheGGMAC-WT,itispossibletoincreasethesecrecycapacitybyincreasingthetransmitpowers.Thismainlystemsfromthefactthattheusersnowhavetheextraadvantageovertheeavesdropperthattheyknowtheirowntransmittedcodewords.Ineffect,eachuserhelpsencrypttheotheruserstransmission.Toseethismoreclearly,considerthesymmetriccasewhere and ,whichmakesallusersreceiveasimilarlynoisyversionofthesamesummessage.Theonlydisad-vantagetheeavesdropperhasisthathedoesnotknowanyofthecodewordswhereasuser knows .Inthiscase, isachievable,andthisrateapproaches as .Thus,itispossibletoachieveasecrecyrateincreaseatthesamerateastheincreaseinchannelcapacity.Next,weexaminethesecrecysum-ratemaximizingpowerallocationsandoptimumpowersforthecooperativejammingscheme.Figs.7and8showtheachievablesecrecyrateimprove-mentforthecooperativejammingschemeforvariouschannelparametersfortheGGMAC-WTwith .Theplotsarethesecrecyratesforuser1whenuser2isjammingwithagiven Fig.8.GGMAC-WTcooperativejammingsecrecysumrateasafunctionof withdifferent forP =P =100 .Thecirclesindicateoptimumjammingpower. Fig.9.GTW-WTcooperativejammingsecrecysumrateasafunctionof withdifferent forP =P h power,whichcorrespondtouser1ssingle-usersecrecy[12],sinceonlyoneuseristransmitting.When thesecrecycapacityisseentobezero,unlessuser2hasenoughpowertoconvertuser1srestandardizedchannelgaintoless .FortheGTW-WT,itisalwaysoptimalforuser2tojamaslongasitenablesuser1totransmit,asseeninFig.9.There-sultsshow,asexpected,thatsecrecyisachievableforbothuserssolongaswecankeeptheeavesdropperfromsingle-userde-codingthetransmittedcodewordsbytreatingtheremaininguserasnoise.Sincethecodingschemesconsideredhereassumeknowledgeofeavesdropperschannelgains,applicationsarelimited.Onepracticalapplicationcouldbesecuringofaphysicallyprotectedareasuchasinsideabuilding,whentheeavesdropperisknowntobeoutside.Insuchacase,wecandesignfortheworstcasescenario.AnexampleisgiveninFig.10fortheGGMAC-WT,whereweassumeasimplepath-lossmodelandxedlocationsfortwotransmitters andonereceiver atthecenter.We IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.54,NO.6,JUNE2008fromthefactthatif ,then forall ,andwemusthave .If,ontheotherhand,(90)givesacomplexornegativesolution,thentheparaboladoesnotintersectthe axis,andisalwayspositive.Hence, ,and doesnotbelongto ,i.e., Theformofthissolutionisintuitivelypleasing,sinceitmakesmoresenseforweakeruserstojamastheyharmtheeaves-droppermorethantheydotheintendedreceiver.Whatweseeisthatalltransmittingusers ,suchthat ,transmitwithmaximumpoweraslongastheirstandardizedchannelgain lessthansomelimit ,andalljammingusersmusthave Weclaimthatallusersin musthave allusersin have .Tomakethisargument,weneedtoshowthata suchthatthereexistssome with and suchthat cannotbetheop-timumset.Toseethis,let betheoptimumpowerallocationforaset .Consideranewpowerallocationandsetsuchthat ,i.e.,user isnowjamming,andlet , , and ,forsomesmall .Wethenhave (91) (92) (93) whichisalowervaluefortheobjectivefunction,provingthat isnotoptimum.Thisshowsthatallusers have forallusers .Sincethelastuserin has ,necessarily forall ,and forall Summarizing,theoptimumpowerallocationissuchthatthereisasetoftransmittingusers with for ,thereisasetofsilentusers andthereisasetofjammingusers with for and isfoundfrom ThisiswhatispresentedinthestatementinTheorem5.Notethatto and ,wecansimplydoanexhaustivesearchaswehavenarrowedthenumberofpossibleoptimalsets insteadof andfoundtheoptimalpowerallocationsforeach. Two-userGGMAC-WT:Forillustrationpurposes,letusconsiderthefamiliarcasewith transmitters.Inthiscase,weknowthateitheruser2jams,ornouserdoes.Thesolutioncanbefoundfromcomparingthetwocases.If,withoutjamming,user2cantransmit,thenitisoptimalforittocon-tinuetotransmit,andjammingwillnotimprovethesumrate.Otherwise,user2maybejammingtoimprovethesecrecyrateofuser1.Theoptimumpowerallocationforuser1isequivalent if and if Thepowerforuser2isfoundfrom(81).Fortwousers,wecansimplywrite(90)as (95)where (96) (97) (98)If ,weautomaticallyhave .Inaddition,wehave ,soweonlyneedtoconcernourselveswiththepossiblypositiveroot .Wendwhen .Weseethat forall if ,equivalenttohavingtwonegativeroots,or ,equivalenttohavingnorealrootsof .Nowexaminewhen Thisispossibleifandonlyif .Since ,thishappensonlywhen or .However,if ,wearebetterofftransmittingthanjamming.Thelastcasetoexamineiswhen .Thisimpliesthat anditissatisedwhen .Assume .Inthiscase,weareguaranteed .If ,thenwemusthave sincethesecrecyrateis .Wewouldliketondwhenwecanhave .Since ,wemusthave ,and .Thisimplies .Itiseasytosee if and Thus,for users,thesolutionsimpliesto(99)shownatthebottomofthepage,where Thissolutioncanbecheckedtobeinaccordancewiththesum-ratemaximizingpowerallocationofTheorem3.Wenotethatinthecaseunaccountedforin(99),i.e.,when and ,bothusersshouldbetransmitting.Ingeneral,thesolutionshowsthattheweakerusershouldjamifitisnot if , if if , if , (99) IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.54,NO.6,JUNE2008 Fig.2.StandardizedGTW-WTsystemmodel.thenoisesarenormalizedby , and C.PreliminaryDeÞnitionsInthissection,wepresentsomeusefulpreliminarydeincludingthesecrecyconstraintwewilluse.Inparticular,thesecrecyconstraintweusedisthecollectivesecrecyconstraintwedenedin[38]and[40],anditissuitableforthemultiaccessnatureofthesystemsofinterest.DeÞnition1(CollectiveSecrecyConstraint):Weusethenor-malizedjointconditionalentropyofthetransmittedmessagesgiventheeavesdroppersreceivedsignalasoursecrecycon-straint,i.e., foranyset ofusers.Forperfectsecrecyofalltransmittedsecretmessages,wewouldlike (7)Assume forsomearbitrarilysmall asrequired. (8) (9) (10) (11)where as .If ,thenwede .Thus,theperfectsecrecyofthesystemimpliestheperfectsecrecyofanygroupofusers,guaranteeingthatwhenthesystemissecure,soiseachindividualuser.DeÞnition2(AchievableRates): .Theratevector issaidtobeachievableifforanygiven thereexistsacodeofsufcientlength suchthat (12a) (12b)and sent istheaverageprobabilityoferror.Inaddition,weneed (12d)where denotesoursecrecyconstraintandisdenedin(7).Wewillcallthesetofallachievablerates,thesecrecy-capacityregion,anddenoteit fortheGGMAC-WT,and theGTW-WT,respectively.Beforewestateourresults,wealsodenethefollowingno-tation,whichwillbeusedextensivelyintherestofthispaper: (13) (14) (15) (16) (17) Last,weinformallycallthe thuserstrong ,and .Thisisawayofindicatingwhetherthein-tendedreceiverorthewiretapperisatamoreofanadvan-tageconcerningthatuser,andisequivalenttostatingwhetherthesingle-usersecrecycapacityofthatuserispositiveorzero.Welaterextendthisconcepttorefertouserswhocanachievepositivesecrecyratesandthosewhocannot.Inaddition,wewillsaythatauserissingle-userdecodableifitsrateissuchthatitcanbedecodedbytreatingtheotheruserasnoise.Ausergroup issingle-userdecodablebytheeavesdropperif IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.54,NO.6,JUNE2008TheGeneralGaussianMultiple-AccessandTwo-WayWiretapChannels:AchievableRatesandCooperativeJammingEnderTekin,StudentMember,IEEE,andAylinYener,Member,IEEEThegeneralGaussianmultiple-accesswiretapchannel(GGMAC-WT)andtheGaussiantwo-waywiretapchannel(GTW-WT)areconsidered.IntheGGMAC-WT,mul-tipleuserscommunicatewithanintendedreceiverinthepresenceofaneavesdropperwhoreceivestheirsignalsthroughanotherGMAC.IntheGTW-WT,twouserscommunicatewitheachotheroveracommonGaussianchannel,withaneavesdropper TEKINANDYENER:GGMAC-WTANDGTW-WT:ACHIEVABLERATESANDCOOPERATIVEJAMMING2751[13]I.CsiszrandJ.KBroadcastchannelswithcondentialmes-IEEETrans.Inf.Theory,vol.IT-24,no.3,pp.339348,May[14]U.MaurerandS.Wolf,Information-theoretickeyagreement:Fromweaktostrongsecrecyforfree,LectureNotesinComputer.Berlin,Germany:Springer-Verlag,2000,vol.1807,pp.[15]U.Maurer,SecretkeyagreementbypublicdiscussionfromcommonIEEETrans.Inf.Theory,vol.39,no.3,pp.733742,May[16]C.H.Bennett,G.Brassard,C.Crpeau,andU.Maurer,izedprivacyampliIEEETrans.Inf.Theory,vol.41,no.6,pp.19151923,Nov.1995.[17]R.AhslwedeandI.CsiszCommonrandomnessininformationtheoryandcryptographyPartI:Secretsharing,IEEETrans.Inf.,vol.39,no.4,pp.11211132,Jul.1993.[18]R.AhslwedeandI.CsiszCommonrandomnessininformationtheoryandcryptographyPartII:CRcapacity,IEEETrans.Inf.,vol.44,no.1,pp.225240,Jan.1998.[19]U.MaurerandS.Wolf,Secret-keyagreementoverunauthenticatedpublicchannelsPartI:Denitionsandacompletenessresult,Trans.Inf.Theory,vol.49,no.4,pp.822831,Apr.2003.[20]U.MaurerandS.Wolf,Secret-keyagreementoverunauthenticatedpublicchannelsPartII:Thesimulatabilitycondition,IEEETrans.Inform.Theory,vol.49,no.4,pp.832838,Apr.2003.[21]U.MaurerandS.Wolf,Secret-keyagreementoverunauthenticatedpublicchannelsPartIII:PrivacyampliIEEETrans.Inf.,vol.49,no.4,pp.839851,Apr.2003.[22]S.VenkatesanandV.Anantharam,Thecommonrandomnesscapacityofapairofindependentdiscretememorylesschannels,IEEETrans.Inf.Theory,vol.44,no.1,pp.215224,Jan.1998.[23]S.VenkatesanandV.Anantharam,Thecommonrandomnessca-pacityofanetworkofdiscretememorylesschannels,IEEETrans.Inf.Theory,vol.46,no.2,pp.367387,Mar.2000.[24]I.CsiszrandP.Narayan,Commonrandomnessandsecretkeygen-erationwithahelper,IEEETrans.Inf.Theory,vol.46,no.2,pp.366,Mar.2000.[25]I.CsiszrandP.Narayan,Secrecycapacitiesformultipleterminals,IEEETrans.Inf.Theory,vol.50,no.12,pp.30473061,Dec.2004.[26]H.Yamamoto,Onsecretsharingcommunicationsystemswithtwoorthreechannels,IEEETrans.Inf.Theory,vol.IT-32,no.3,pp.393,May1986.[27]H.Yamamoto,AcodingtheoremforsecretsharingcommunicationsystemswithtwoGaussianwiretapchannels,IEEETrans.Inf.Theoryvol.37,no.3,pp.634638,May1991.[28]Y.Oohama,Codingforrelaychannelswithcondentialmessages,Proc.IEEEInf.TheoryWorkshop,2001,pp.87[29]J.BarrosandM.R.D.Rodrigues,Secrecycapacityofwirelesschan-Proc.IEEEInt.Symp.Inf.Theory,Seattle,WA,Jul.92006,pp.356[30]M.Bloch,J.Barros,M.R.D.Rodrigues,andS.W.McLaughlin,Wirelessinformation-theoreticsecurityPartI:Theoreticalaspects,IEEETrans.Inf.Theory,acceptedforpublication.[31]Y.LiangandV.Poor,Securecommunicationoverfadingchannels,Proc.AllertonConf.Commun.ControlComput.,Monticello,IL,Sep.2729,2006.[32]L.Zang,R.Yates,andW.Trappe,Secrecycapacityofindependentparallelchannels,Proc.AllertonConf.Commun.ControlComput.Monticello,IL,Sep.2729,2006.[33]R.Liu,I.Maric,R.D.Yates,andP.Spasojevic,Discretememorylessinterferenceandbroadcastchannelswithcondentialmessages,Proc.AllertonConf.Commun.ControlComput.,Monticello,IL,Sep.29,2006.[34]Y.LiangandV.Poor,Generalizedmultipleaccesschannelswithcon-dentialmessages,IEEETrans.Inf.Theory,acceptedforpublication.[35]R.Liu,I.Maric,R.D.Yates,andP.Spasojevic,Thediscretemem-orylessmultipleaccesschannelwithcondentialmessages,Proc.IEEEInt.Symp.Inf.Theory,Seattle,WA,Jul.914,2006,pp.957[36]A.KhistiandG.W.Wornell,Securetransmissionwithmultiplean-tennas:TheMISOMEwiretapchannel,IEEETrans.Inf.Theory,sub-mittedforpublication.[37]S.Shaee,N.Liu,andS.Ulukus,TowardsthesecrecycapacityoftheGaussianMIMOwire-tapchannel:The2-2-1channel,IEEETrans.Inf.Theory,submittedforpublication.[38]E.Tekin,S.Serbetli,andA.Yener,OnsecuresignalingfortheGaussianmultipleaccesswire-tapchannel,Proc.AsilomarConf.SignalSyst.Comput.,Asilomar,CA,Nov.1,2005,pp.1747[39]E.TekinandA.Yener,TheGaussianmultiple-accesswire-tapchannelwithcollectivesecrecyconstraints,Proc.IEEEInt.Symp.Inf.Theory,Seattle,WA,Jul.914,2006,pp.1164[40]E.TekinandA.Yener,TheGaussianmultiple-accesswire-tapIEEETrans.Inf.Theory[Online].Available:http://arxiv.org/format/cs.IT/0605028,submittedforpublication[41]E.TekinandA.Yener,AchievableratesforthegeneralGaussianmul-tipleaccesswire-tapchannelwithcollectivesecrecy,Proc.AllertonConf.Commun.ControlComput.,Monticello,IL,Sep.2729,2006.[42]E.TekinandA.Yener,Achievableratesfortwo-waywire-tapchan-Proc.IEEEInt.Symp.Inf.Theory,Nice,France,Jun.24[43]A.B.Carleial,Interferencechannels,IEEETrans.Inf.Theory,vol.IT-24,no.1,pp.6070,Jan.1978.[44]T.M.CoverandJ.A.Thomas,ElementsofInformationTheory.NewYork:Wiley,1991.[45]E.TekinandA.Yener,Secrecysum-ratesforthemultiple-accesswire-tapchannelwithergodicblockfading,Proc.AllertonConf.Commun.ControlComput.,Monticello,IL,Sep.2628,2007.[46]A.Thangaraj,S.Dihidar,A.R.Calderbank,S.McLaughlin,andJ.-M.ApplicationsofLDPCcodestothewiretapchannel,Trans.Inf.Theory,vol.53,no.8,pp.29332945,Aug.2007.[47]M.Bloch,J.Barros,M.R.D.Rodrigues,andS.W.McLaughlin,Wirelessinformation-theoreticsecurityPartII:Practicalimple-IEEETrans.Inf.Theory,acceptedforpublication. IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.54,NO.6,JUNE2008 Fig.11.GTW-WTcooperativejammingexampledarkershadescorrespondtohighervalues.andnormalsecretcommunicationsisnotpossible.ThegainscanbesignicantforboththeGGMAC-WTandtheGTW-WT.Thiscooperativebehaviorisusefulwhenthemaximumsecrecysumrateisofinterest.Wehavealsocontrastedthesecrecyratesofthetwochannelsweconsidered,notingthebenetofthetwo-waychannelswherethefactthateachreceiverhasperfectknowledgeofitstransmittedsignalbringsanadvantagewitheachusereffectivelyencryptingthecommunicationsoftheotheruser.Inthispaper,weonlypresentedachievablesecrecyratesfortheGGMAC-WTandtheGTW-WT.Thesecrecycapacityregionforthesechannelsarestillopenproblems.In[45],wealsofoundanupperboundforthesecrecysumrateoftheGGMAC-WTandnotedthattheachievablesecrecysumrateandtheupperboundwefoundonlycoincideforthedegradedcase,sothatwehavethesecrecysumcapacityforthedegradedGMAC-WT.Eventhoughthereisagapbetweentheachievablesecrecysumratesandupperbounds,cooperativejammingwasshownin[45]togiveasecrecysumratethatisclosetotheupperboundingeneral.Finally,wenotethattheresultsprovidedareofmainlythe-oreticalinterest,sinceasofyettherearenocurrentlyknownpracticalcodesformultiple-accesswiretapchannelsunlikethesingle-usercasewhereinsomecasespracticalcodeshavebeenshowntobeusefulforthewiretapchannel[46],[47].Further-more,accurateestimatesoftheeavesdropperchannelparame-tersarerequiredforcodedesignforwiretapchannelswherethechannelmodelisquasi-static,asinourmodelsconsideredinthispaper.[1]R.Ahlswede,Multi-waycommunicationchannels,Proc.2nd.Int.Symp.Inf.Theory,Tsahkadsor,ArmenianS.S.R.,1971,pp.23[2]H.Liao,Multipleaccesschannels,Ph.D.dissertation,Dept.Electr.Engr.,Univ.Hawaii,Honolulu,HI,1972.[3]C.E.Shannon,Two-waycommunicationchannels,Proc.4thBerkeleySymp.Math.Statist.Probab.,1961,vol.1,pp.611[4]G.Dueck,Thecapacityregionofthetwo-waychannelcanexceedtheinnerbound,Inf.Control,no.40,pp.258266,1979.[5]T.S.Han,Ageneralcodingschemeforthetwo-waychannel,Trans.Inf.Theory,vol.IT-30,no.1,pp.3544,Jan.1984.[6]R.Ahlswede,Thecapacityregionofachannelwithtwosendersandtworeceivers,Ann.Probab.,vol.2,no.5,pp.805814,1974.[7]H.Sato,Two-usercommunicationchannels,IEEETrans.Inf.Theoryvol.IT-23,no.3,pp.295304,May1977.[8]A.ElGamalandT.Cover,Multipleuserinformationtheory,Proc.,vol.68,no.12,pp.14661483,Dec.1980.[9]C.E.Shannon,Communicationtheoryofsecrecysystems,BellSyst.Tech.J.,vol.28,pp.656715,1949.[10]A.Wyner,Thewire-tapchannel,BellSyst.Tech.J.,vol.54,pp.1387,1975.[11]A.B.CarleialandM.E.Hellman,AnoteonWynerswiretapIEEETrans.Inf.Theory,vol.IT-23,no.3,pp.387May1977.[12]S.K.Leung-Yan-CheongandM.E.Hellman,Gaussianwire-tapIEEETrans.Inf.Theory,vol.IT-24,no.4,pp.451456,Jul.