Research Article Securing OFDM over Wireless Time-Varying Channels Using Subcarrier Overloading with Joint Signal Constellations

Transcription

1 Hndaw Publshng Corporaton EURASIP Journal on Wreless Communcatons and Networkng Volume 2009, Artcle ID , 8 pages do:0.55/2009/ Research Artcle Securng OFDM over Wreless Tme-Varyng Channels Usng Subcarrer Overloadng wth Jont Sgnal Constellatons Gll R. Tsour and Dov Wulch (EURASIP Member) 2 Department of Electrcal Engneerng, Rochester Insttute of Technology, Rochester, NY 4623, USA 2 Department of Electrcal and Computer Engneerng, Ben-Guron Unversty of the Negev, Beer-sheva 8405, Israel Correspondence should be addressed to Gll R. Tsour, grteee@rt.edu Receved 29 November 2008; Revsed 29 May 2009; Accepted 30 July 2009 Recommended by Merouane Debbah A method of overloadng subcarrers by multple transmtters to secure OFDM n wreless tme-varyng channels s proposed and analyzed. The method s based on reverse plotng, superposton modulaton, and jont decodng. It makes use of channel randomness, recprocty, and fast decorrelaton n space to secure OFDM wth low overheads on encrypton, decrypton, and key dstrbuton. These propertes make t a good alternatve to tradtonal software-based nformaton securty algorthms n systems where the costs assocated wth such algorthms are an mplementaton obstacle. A necessary and suffcent condton for achevng nformaton theoretc securty n accordance wth channel and system parameters s derved. Securty by complexty s assessed for cases where the condton for nformaton theoretc securty s not satsfed. In addton, practcal means for mplementng the method are derved ncludng generatng robust jont constellatons, decodng data wth low complexty, and mtgatng the effects of mperfectons due to moblty, power control errors, and synchronzaton errors. Copyrght 2009 G. R. Tsour and D. Wulch. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.. Introducton Orthogonal Frequency Dvson Multplexng (OFDM) s a leadng choce for many current and future ar nterfaces. When usng OFDM over a wreless channel the broadcast nature of the channel exposes transmsson to eavesdroppng. Securng communcaton lnks from eavesdroppng s commonly done by mplementng encpherng and decpherng algorthms n software, and s usually detached from the physcal layer of communcaton. Promnent methods rely on publc keys cryptography such as RSA [], or symmetrc cryptography wth a common secret, such as the US Natonal Data Encrypton Standard (DES) [2]. There are some encrypton methods whch rely on the physcal layer of communcaton for ther mplementaton, such as spread spectrum Frequency Hoppng (FH) and Drect Sequence (DS) [3]. In FH and DS a key has to be generated and dstrbuted securely between the communcatng partes. The key s used to set the FH hoppng pattern or DS spreadng sequence. Promnent key dstrbuton methods rely on the Dffe- Hellman algorthm [4]. Encrypton, decrypton, and key dstrbuton mpose overheads on data throughput, energy consumpton, memory space, and computaton power. These overheads are a crucal mplementaton ssue for low complexty systems wth strct constrants on system resources [5], such as sensor and moble networks [6 8]. Secrecy capacty analyss of random, nosy, and fadng channels showed that n theory, a communcaton lnk can be perfectly secured from eavesdroppng for certan lmted nformaton rates [9 ]. Recent work provded specfc analyss of the secrecy capacty of wreless fadng channels [2 7]. Past work suggested practcal methods for usng randomness n the wreless channel to allevate the need for key dstrbuton. In [8 20], recprocal channel estmaton of a slow fadng wreless tme varyng channel was used as a common secret to generate and dstrbute encrypton keys to be used by tradtonal encrypton algorthms. In [2] a dfferental frequency modulaton technque coupled wth reverse plotng was used n a multtone channel to acheve secure transmsson for pont-to-pont systems. In [22] a practcal approach s depcted for key agreement n wreless channels based on multlevel and Low Densty Party Check

2 2 EURASIP Journal on Wreless Communcatons and Networkng (LDPC) codes. In [8 22] the wreless channel was assumed to be a recprocal slow flat fadng channel, whch decorrelates rapdly n space. The assumptons of recprocty and space decorrelaton were well establshed n prevous work [2] and are adopted n ths work as well. It s common practce n wreless communcatons to have the transmtter use asynchronous bursts of transmsson to access the channel. The burst starts wth a pror known plot sgnal followed by modulated data symbols. The plot s used by the recever to estmate the channel. The recever then uses the channel estmate to compensate for channel attenuaton and phase pror to decodng. Decodng of the receved data symbols s done based on the a pror known sgnal constellaton of the transmtter. Snce the wreless channel changes wth tme, a plot sgnal has to be sent every channel coherence tme. An eavesdropper can use the plot sgnal to estmate the channel from the transmtter to tself and decode the nformaton n exactly the same manner as the ntended recever. Ths means that sendng a plot sgnal from the transmtter to the recever compromses securty. In [23] the asynchronous burst transmsson approach was replaced wth synchronous transmsson to acheve securty. A reverse plotng protocol was proposed to secure transmsson bursts n a narrow-band sngle carrer pontto-pont system over slow flat fadng channels. Synchronous transmsson allows for the plot sgnal to be sent from the recever nstead of the transmtter. The transmtter can estmate the channel from the recever to tself usng the recever s plot sgnal and deduce the channel from tself to the recever based on channel recprocty. The transmtter can then send a burst of channel-compensated data symbols over the same frequency as the plot, and the recever would receve a readly decodable channel-compensated sgnal. The recever can use decson feedback to compensate for small changes n the channel, so that ts channel estmate remans accurate untl the orgnal channel s fully decorrelated n tme. After the channel decorrelates n tme the recever can send a new plot sgnal. Snce no plot sgnal s sent from the transmtter, the eavesdropper would be deprved of estmatng the channel from the transmtter to tself pror to recevng the data symbols, and would be forced to estmate the channel usng blnd estmaton. Even for a noseless channel, no decodng of the data would be possble untl blnd estmaton s completed because the eavesdropper wll not be able to map the receved symbols to decoded bts. In ths work, the reverse plotng protocol presented n [23] s elaborated to support a pluralty of transmtters n a superposton modulaton settng wth jont decodng at the recever. It s shown that the elaborated protocol can be practcally used to obtan nformaton theoretc securty and securty by complexty wth low mplementaton complexty, no memory requrements and no overhead on throughput and energy. In contradstncton to prevous work n lterature, the focus of ths work s on facltatng channel overloadng of multpletransmttersoversubcarrersnanofdmsystem. The purpose s to ncrease securty strength and decodng gan for transmtters n an OFDM system wth lmted emsson power, memory space, and onlne computaton power. Although we focus our attenton on OFDM, our analyss and results hold for securng narrowband sngle carrer transmsson as well. The novelty of ths work s n suggestng the use of reverse plotng for mplementng superposton modulaton wth jont decodng to acheve nformaton securty. The man contrbutons are n two categores: analyss of securty strength of the proposed method and practcal mplementaton of the proposed method. The analyss of securty strength results n a quanttatve condton for achevng nformaton theoretc securty, gven a pror known channel and system parameters and assessment of securty by complexty when the condton s not satsfed. Practcal mplementaton consderatons nclude generatng robust sgnal constellatons, low complexty Maxmum Lkelhood (ML) decodng, networkoptmzaton, evaluaton of the effects of moblty, power control errors and synchronzaton errors, and formulatng smple plotng rules for mtgatng ther effect. The rest of the paper s organzed as follows. In Secton 2 the method s presented usng a multple access protocol. In Secton 3 the mathematcal model used for analyss s defned. In Secton 4 securty strength s analyzed. In Secton 5 practcal mplementaton s consdered. Secton 6 depcts an llustratve scenaro of Raylegh fadng and three transmtters, and Secton 7 concludes the work. 2. Proposed Method In the proposed method the frequency and phase of subcarrers from multple transmtters are matched and synchronzed at the recever. All transmtters transmt together and ther start of transmsson s coordnated by the recever to have ther sgnals reach the recever smultaneously. All transmtters use the same subcarrer frequences. The recever s equpped wth a sngle Matched Flter (MF) matched to each subcarrer. Each transmtter s assgned (offlne) a set of arbtrary Base Band (BB) symbols to represent ts nformaton bts. Snce the transmt-channelreceve path s lnear, the recever s MF output s a sum of the BB symbols of all the transmtters, and represents the transmtted bts of all the transmtters at once. Ths s n fact superposton modulaton of multple transmtters on a sngle subcarrer whch s repeated ndvdually for multple subcarrers. The BB symbol sets are computed offlne to have all ther possble summatons create a sgnal constellaton wth hghest resstance to nose. The recever decodes the nformaton bts of all the transmtters at once from a sngle receved symbol as f orgnatng from a sngle transmtter. Snce the receved sgnal constellaton s jontly formed by all the transmtters BB symbols, the proposed method s heren termed Jont Constellaton Multple Access (JCMA). A multple access reverse plotng protocol over a sngle subcarrer s gven n what follows to mplement the method. () The recever obtans knowledge on sgnal propagaton tme from each transmtter to tself through some standard assocaton procedure. (2) The recever assgns ndex and delay parameter to each transmtter.

3 EURASIP Journal on Wreless Communcatons and Networkng 3 (3) Each transmtter uses ts ndex to access a preloaded table for retrevng a BB symbol set. (4) The recever sends a plot sgnal and starts sensng for an ncomng sgnal. (5) Each transmtter ndvdually estmates the recprocal channel s phase and ampltude usng the plot sgnal from the recever. (6) Each transmtter awats ts delay and sends a burst of nformaton symbols compensated for channel phase and ampltude. (7) The recever receves a burst of jont nformaton symbols, whch belong to ts predefned jont sgnal set. (8) The recever decodes all transmtters at once. (9) The recever uses decson feedback to compensate for slow channel decorrelaton n tme. (0) Steps 4 9 are repeated after a channel decorrelaton perod has passed. A prelmnary smpler verson of the proposed protocol n ths work was dsclosed n [24], wth the purpose of enhancng decodng but no nformaton securty consderatons. JCMA s superposton modulaton wth jont decodng. It should be dstngushed from the well-known Superposton Modulaton wth Successve Decodng (SM-SD) [25]. In SM-SD the transmtters transmt at the same tme and frequency and a jont sgnal s formed at the recever. Each transmtter s symbol s treated as nose to the other transmtters. The recever decodes the nformaton of each transmtter ndvdually n a successve manner. Frst, the transmtter wth the hghest receved sgnal energy s decoded. The decoded bts are used as feedback to remove the transmtter s sgnal from the receved jont sgnal. The next transmtter wth the hghest receved sgnal energy s decoded and so on. In SM-SD the transmtters send plot sgnals to facltate channel estmaton at the recever. As explaned before, ths compromses securty. In JCMA plots are sent only by the recever. It follows that knowledge of the channel s obtaned only by the transmtters. JCMA should also be dstngushed from multuser detecton [26] and rather recent advances n cooperatve transmt dversty [27]. In multuser detecton the structure of the nterferng sgnals from multple transmtters s used to reduce ther effect. The achevable codng gans are consderable, but the use of multple MFs s requred and the computatonal complexty grows exponentally wth the number of transmtters. In addton, no securty s ganed. In cooperatve transmt dversty, the transmtter sends ts own nformaton whle relayng the nformaton of another transmtter to the recever. The method offers some performance gans, but the lmted power of the relayng transmtter has to be dstrbuted between the data streams of the partcpatng nodes and no securty s ganed. Besde SM-SD and multuser detecton, lterature presents other approaches to channel overloadng of transmtters over the same frequency band. For example, the work n [28 30] uses symbol-synchronous superposton modulaton to create a jont rectangular lattce at the recever to support multple transmtters usng trells codes. There are also numerous works dealng wth the adder-channel for performng jont codng from multple transmtters through supermposed sgnals. In general, codebooks of ndvdual transmtters are optmzed under some crteron over the jont sgnal at the recever usually the focus s on codng gan. More relevant to the focus of ths work s the work n [3, 32], where the jont mnmal Eucldean dstance was used as the optmzng crteron of a symbolsynchronous supermposed sgnal. A comprehensve revew of other channel overloadng technques s provded n [39]. The work n [25, 27 32] addresses the ssue of desgnng jont sgnal constellatons to satsfy varous optmzaton crtera accordng to the problem explored. We are unaware of prevous work (ncludng [25, 27 32]) usng secure, low complexty, jont symbol by symbol decodng as the optmzaton crtera. As would be apparent n the followng sectons, the suggested method offers a low complexty soluton for securng OFDM over the tme-varyng wreless channel. The recever uses a sngle MF and performs decodng of multple transmtters wth the same complexty as decodng a sngle transmtter. The transmtters complexty s the same as that of a pont-to-pont scenaro wthout securty features. No memory space, computaton power, or transmtted energy s requred n order to secure transmsson. For comparson, consder the analyss of energy consumpton due to mplementng securty algorthms performed n [5]. In [5], t was shown that a typcal sensor-node usng asymmetrc key establshment coupled wth symmetrc encrypton per transmsson sesson losses 20% 80% of ts battery lfe due to encrypton, dependng on the sesson length. See [5] for energy consumpton of RSA, DES, and DH n specfc systems. Other analyss provded n [8] consdered the Central Processng Unt (CPU), memory and transmsson overheads requred for mplementng standard securty algorthms mplemented n specfc off-the-shelf nodes. The analyss n [8] concluded that DES s too resource demandng to be used n a sensor network and that a mnmum of 28 KB RAM and ablty to tolerate a consderable delay n data delvery are requred to mplement securty algorthms of lesser strength. See [8] for CPU processng and memory and transmsson overheads of encrypton algorthms DES, TEA, RC6, RC5, and SkpJack n specfc systems. 3. Mathematcal Model In JCMA a jont constellaton s constructed over each subcarrer separately. It follows that most of the analyss can be done usng a model for a sngle subcarrer. In what follows, a sngle subcarrer s consdered. The expanson of the analyss to an entre OFDM symbol s done when evaluatng securty of the entre OFDM transmsson. The BB model of the multple access scheme for a sngle subcarrer s depcted n Fgure, x ; =, 2,..., N represent the transmtted complex BB symbols. Subcarrers n OFDM experence flat fadng. It follows that the channel

4 4 EURASIP Journal on Wreless Communcatons and Networkng s s 2 TX TX2. x x 2 h h 2 h N Superposton due to channel lnearty y n r Recever ML decson devce transmtter symbols to be less thanp (peak power constrant on transmtted power). Fndng the best BB symbol sets s explctly formulated as follows. Gven the defnton: { } y = mn y j, (3) d mn def j fnd S ; =,..., N whch yeld max{d mn }, whle satsfyng the constrant: s N TXN x N Fgure : Baseband model of JCMA. over a sngle subcarrer s flat and s descrbed by a sngle complex number h. The transmtted symbols are summed va channel superposton and the MF output, wthout nose, s gven by y. Also, n s the BB addtve whte complex Gaussan nose. The receved sgnal r s fed to an ML decson devce, whch decodes all the transmtted bts at once, as f orgnatng from a sngle transmtter (the so-called superuser n [25]). Let us defne h = α exp ( jβ ) ; x = exp( jβ ) α s, () where s s an nformaton-carryng symbol wth unt energy whch belongs to a predefned set S. It s assumed that /α s small enough to allow the transmtter to adjust ts power wthn ts power constrant. If ths s not the case, communcaton would be severed, as would also happen n a standard pont to pont scenaro. For now, t s assumed that β, α are known wthout error at the th transmtter-perfectchannel State Informaton (CSI), and that h remans constant durng the decorrelaton tme. In later sectons, dervatons to the model are defned to analyze the effects of mperfect CSI and moblty on system performance. From ()andfgure t follows that r = y + n = s + n. (2) The jont sgnal constellaton at the recever s made up of all the permutatons n s, =, 2,..., N whch generate y. Note that S ; =,..., N are sets of complex numbers, where S = {s, s 2,..., s 2M }. In addton, M s the number of bts per symbol per transmtter, and y s a set of complex numbers made of all possble summatons of N numbers, where each number belongs to a dfferent set S. Moreover, y ={y, y 2,..., y 2 MN }, y = N j= l j ; l j S j where {l, l 2,..., l N } {lk, l2 k,..., ln k } for all k. The symbol sets are determned offlne. For addtve Gaussan nose, the mnmum Eucldan dstance n the jont constellaton represented by y should be maxmal whle constranng the average nstantaneous energy over the 2 M 2 M S j j= 2 P; =,..., N. (4) For addtve Gaussan nose, ML detecton translates to fndng the jont constellaton symbol g n y whch has the smallest Eucldean dstance from the receved symbol q,so { } ( ) g = arg mn q y. (5) y 4. Securty Analyss To facltate analyss, we assume that the eavesdropper uses a sngle MF for decodng the data. At frst glance, ths seems to be an unreasonable lmtaton on the eavesdropper resources. The justfcaton for ths constrant would be gven as part of the dscusson n ths secton. In JCMA, the jont constellaton at the recever s MF output s the result of a coherent sum of the transmtter BB symbols and s unque to the locaton of the transmtters and recever n space. Ths s acheved due to the ndvdual compensaton of delay, phase, and attenuaton of each transmtter. The recever s jont constellaton would always be the same and the bt mappng from the transmtters to the jont symbols would be known a pror to the recever. Snce an eavesdropper would naturally occupy a dfferent locaton n space, the transmtters BB symbols would create a dfferent jont constellaton than that of the recever. Ths s practcally always true even when the eavesdropper s close to the recever because the wreless channel decorrelates fast n space. A dstance of a few carrer wavelengths apart (a few centmeters for frequences of the order of GHz) decorrelates the channel almost completely [33]. It follows that the eavesdropper would have no a pror knowledge of the bt mappng from transmtters to jont symbols. Ths s the bass for achevng nformaton theoretc securty. The jont constellaton s constructed optmally at the recever s locaton n space. It follows that the jont constellaton at the eavesdropper locaton would have suboptmal structure and would change after every reverse plot sent due to dfferent channel compensaton performed at the transmtters. In addton, the sgnals from the transmtters would not reach the eavesdropper smultaneously. It follows that even after dscoverng the bt mappng from transmtters to jont symbols, the eavesdropper would have to decode the nformaton based on a deterorated jont sgnal constellaton and the decodng complexty would be hgher than that of the recever. Ths s the bass for securty by complexty.

5 EURASIP Journal on Wreless Communcatons and Networkng 5 [s, s 2, s N ] y M : Element wse multplcaton (.) : Element wse nverson Message source Encpherer x Encpherer 2 TX TX2 TXN s s 2. s N T K = M K Channel compensaton done at TX s x 2. x N T K2 = sum(t K (M) K ) Channel fadng and superposton at eavesdropper locaton E y h h 2 h N h h 2 h N K Key source K Dynamc fadng channel Fgure 2: JCMA as a Shannon secrecy model. For decodng data successfully the eavesdropper must frst perform blnd channel estmaton to map ts receved jont constellaton symbols to nformaton bts. Fndng the bt mappng s an act of decpherng, where the receved jont symbols are the cpher-text, the channel coeffcents are the encrypton key and the nformaton bts are the encrypted message. After decpherng, the eavesdropper must decode the data from a deterorated sgnal. 4.. Informaton Theoretc Securty. We start by descrbng the JCMA BB model as a Shannon secrecy model. Fgure 2 depcts the secrecy model for an eavesdropper. The plot from the recever nvokes key generaton and dstrbuton from the wreless medum to the transmtters. Ths key s the channel estmates at the transmtters. Each part of the key s known exclusvely at each transmtter. In accordance wth the Shannon secrecy model and ts notaton n [34], the message M as defned n [34] corresponds to the vector of transmtted symbols across the transmtters. Ths jont message s encrypted by the transmtters each transmtter transforms ts symbol by scalng and rotatng t wth the key. So K n [34] corresponds to the vector of channel estmates at the transmtters. For the recever s locaton n space, the channel tself performs a decpherng operaton by rotatng and scalng the encrypted message back to the orgnal y. For the eavesdropper locaton n space the channel performs another encpherng operaton wth another key K, beng the channels from the transmtters to the eavesdropper locaton. The eavesdropper constellaton pont y corresponds to the cryptogram E n [34]. The eavesdropper has to decpher M from E usng all possble pror knowledge, such as the offlne-determned transmtters BB symbol sets, the channel statstcs, and the pror probabltes of the messages M. A new key s generated every tme a reverse plot sgnal s sent by the recever. Due to the tme varyng nature of the fadng channel, a key uncorrelated wth prevous keys s nvoked after the channel decorrelates n tme. Ths s why the recever s requred to track the channel durng the decorrelaton perod. Note that encrypton and decrypton are done automatcally by the channel, so no overheads are requred for these operatons. The uncty dstance was defned n [34] as the amount of ntercepted cryptograms by the eavesdropper beyond whch the eavesdropper can deduce the key. Equvalently, the cryptogram s undecpherable when the message length s less than the uncty dstance [34]. So, to acheve nformaton theoretc securty the message should be encphered by a new key before the eavesdropper gathers enough cryptograms for decpherng. To secure an entre OFDM transmsson burst, the number of jont symbols wthn the channel decorrelaton perod must be less than the uncty dstance. In the pont-to-pont narrow-band scenaro (N = ) descrbed n [23] a sngle channel took part n encryptng a sngle message. It was shown that the channel acts as a shft cpher for cases where the phase of the complex Random Varable (RV) representng the channel s unformly dstrbuted. Unform dstrbuton of phase s a common assumpton for descrbng wreless tme-varyng channels. For the multpont to pont scenaro we assume that the phase of each channel s unformly dstrbuted over the range [0, 2π). We derve the uncty dstance for JCMA based

6 6 EURASIP Journal on Wreless Communcatons and Networkng on ths sngle assumpton. For clarty of presentaton we consder the case where a sngle bt s transmtted by each transmtter (M = ). The dervaton scales easly for general M. The receved jont constellaton at the eavesdropper s gven by (see Fgure 2) ( ) h y = s. (6) h Usng ampltude-phase representaton a sngle transmtter symbol set s gven by S = { s exp ( jθ ), s 2 exp ( jθ 2 )}. (7) The structure of any sgnal constellaton must be constraned to acheve an output sgnal wth zero mean, so that no power s wasted. Ths means that the two sgnals n the constellaton are antpodal: s = s 2 s ; θ = θ 2 + π. (8) Usng s = s exp(jθ s ), h = h exp(jθ ), h = h exp(jθ ), (7)and(8)n(6)gve ( y = δ exp ( j ( ))) N ϕ s, (9) where ϕ = θ s + θ θ and δ = s h / h. We fnd that s orgnatng from a sngle transmtter arrves at the eavesdropper wth the same ampltude δ for ether of the two possble bt values. The phase of s s the sum of three RVs. Two of the RVs (θ,θ ) are unformly and ndependently dstrbuted over the range [0, 2π) and the thrd (θ s ) assumes one of two possble values θ, θ2 correspondng to the two possble bt values. Snce the phase s unformly dstrbuted and cyclc over [0,2π), takng ts colnear values (multplyng the RV wth ) results n the same dstrbuton. It follows that the Probablty Densty Functon (PDF) of θ θ s equal to the PDF of θ + θ whch s the result of convoluton of two unform PDFs: [ ( )] [ ( )] z π z π f θ θ (z) = 2π Π 2π 2π Π, 2π (z π) (4π 2 ), π z<3π, = (z 3π) (2π) (4π 2, 3π z<5π, ) 0, otherwse. (0) Due to the cyclc property of the phase (modulo 2π), (0) s equvalent to f θ θ (z) ( ) ( ) (4π) + z + (4π) + z, 0 z<π, = ( ) (2π) (z π) + (z π), π z<2π, = (2π), 0 z<2π, 0, otherwse. () It follows that θ θ s unformly dstrbuted over the range [0, 2π). Addng θ s to θ θ results n a cyclc shft of phase values over the range [0, 2π) but does not change the PDF regardless the value of θ s. It follows that ϕ s always unformly dstrbuted over the range [0, 2π). Ths means that the mappngs from the two possble transmtter bt values to arecevedsymbol s at the eavesdropper are equally probable. Snce the channel decorrelates fast across space, the receved ndvdual symbols from the dfferent transmtters at the eavesdropper s, =,..., N are uncorrelated, each receved symbol from a transmtter has two equally probable bt mappngs. The result s 2 N equally probable bt mappngs for the jont sgnal constellaton at the eavesdropper. It follows that JCMA s analogous to a substtuton cpher wth 2 N equally probable keys. The uncty dstance of a substtuton cpher s gven by [34] U d = H(K) η, (2) where H(K) s the entropy of the key. Also, η s the effcency of the nformaton source defned as η def = D, whered s the redundancy of the nformaton source [34]. In our case, H(K) = H(2 N ) = N and the JCMA uncty dstance s gven by U d = N η. (3) The model n Fgure 2 depcts a sngle subcarrer. Neghborng subcarrers n OFDM experence smlar channel response due to channel correlaton across bandwdth. Ths means that neghborng subcarrers can be used by the eavesdropper to dscover ther smlar channel response (encrypton key). It s common to assume that the channel decorrelates n frequency every coherence bandwdth (B c ). To be on the safe sde, t s assumed that all subcarrers wthn B c experence exactly the same channel. Ths s a strct assumpton, snce subcarrers whch are less than B c apart are not fully correlated. Secrecy s requred for the entre data transmsson burst. Ths means that the uncty dstance must be larger than the number of receved jont symbols durng the channel decorrelaton perod (L c ) tmes the number of subcarrers

7 EURASIP Journal on Wreless Communcatons and Networkng 7 wthn the channel coherence bandwdth (N c ). It follows from (3) that for N c L c N η, (4) nformaton theoretc securty s acheved for the entre data burst. Note that L c and N c are gven by N c = B c T s, L c = T d T s, (5) where T s s the transmtted symbol tme. The channel temporal decorrelaton tme (T d ), s governed by the Doppler spread (B d ) and s defned as T d def = α B d, (6) where α s set to acheve suffcent channel decorrelaton n tme. However, B d s determned by the relatve movement of the transmtter wth respect to the recever descrbed by the Doppler shft, and by the movement of reflectors n ther path. Moreover, B d ncreases wth moblty and carrer frequency. In the context of encrypton strength a short channel decorrelaton tme (hgher moblty) s preferred. Ths results n a hgher key generatng rate. In ths sense, the worse-case scenaro s a statonary transmtter and recever, for whch B d has the smallest possble value. Usng (5), (6)n(4) results n ( ) η αbc B d. (7) N If the system and channel parameters satsfy (7) the data transmsson burst s secured from eavesdroppng. If (7) sdffcult to satsfy due to small B d,securty can be obtaned by shortenng the number of data symbols n the transmsson burst so that t equals the uncty dstance. Ths results n an equalty n (4) regardless of the channel decorrelaton perod. Ths approach would result n throughput reducton as the transmtters must wat n dle mode for the channel to decorrelate. Throughput loss can be avoded by havng multple JCMA groups accessng the channel n a TDMA fashon, so that one group uses the channel whle the others wat for t to decorrelate. Alternatvely, the protocol can be appled to only some of the subcarrers preferably spaced apart as much as possble across the bandwdth. We now justfy constranng the use of a sngle MF at the eavesdropper. An MF s the optmal demodulator for achevng a maxmal SNR from the receved sgnal [35]. Usng multple MFs connected to a sgnal antenna makes no sense because the jont constellaton s constructed over a sngle complex dmenson for any number of transmtters. It follows that the optmal choce for the eavesdropper s to use a sngle MF wth soft decodng. If the eavesdropper would try to use multple antennas for fndng the bt mappng faster (for decreasng the uncty dstance), the result would be multple equvalent decpherng problems and the eavesdropper would gan nothng as far as decpherng tme s concerned Securty by Complexty. In the prevous secton, nformaton theoretc securty of the entre data transmsson burst was found. Decpherng amounted to dscoverng the bt mappng from the transmtters to each jont symbol. Knowledge of the mappng s obtaned by the eavesdropper after the uncty dstance has passed only f the eavesdropper uses the best possble algorthm to decpher the data. After decpherng s accomplshed, the eavesdropper stll has to decode the data. When nformaton theoretc securty s not achevable, securty bols down to makng decodng much more dffcult for the eavesdropper compared to the ntended recever. We dentfy four such securty-bycomplexty features of JCMA. () The eavesdropper would have dffculty to know the expected jont symbol constellaton at ts MF output, snce t has no knowledge of the channel compensaton done at each transmtter, no mmedate knowledge of the CSI from the nodes to tself, and t receves nosy samples. (2) The sgnals from the group nodes would not reach the eavesdropper smultaneously, resultng n an overlap of past and present symbols. (3) The eavesdropper jont constellaton would change every decorrelaton perod. Ths s due to the changng of channel compensaton at the transmtters. Ths makes t mpossble to desgn a constant and computatonally effcent decodng algorthm, meanng that the eavesdropper would have to perform an exhaustve search for ML detecton of every receved symbol. At the same tme, the recever decodng algorthm would be constant because each channel nstance s compensated for. (4) The jont constellaton formed at the eavesdropper MF output s not optmal for decodng, snce t was made to be optmal ata dfferent and unque locaton n space that of the recever. Due to Factor, the eavesdropper must frst decde on ts jont sgnal constellaton. Ths precedes decpherng (mappng bts to jont symbols) and could prove to be a dffcult task, snce the jont samples are nosy and the jont constellaton changes wth every new plot from the recever. The deteroraton of the sgnal at the eavesdropper due to Factor 2 s substantal when the eavesdropper s far from the recever and s dffcult to evaluate as t depends greatly on the multpath propagaton of the transmtted sgnal. Factor 2 could be compromsed when the eavesdropper s close enough to the recever. Due to Factor 3, the asymptotc decodng complexty of the eavesdropper s O(2 N ), correspondng to an exhaustve search over all possble constellaton ponts. It would be

8 8 EURASIP Journal on Wreless Communcatons and Networkng shown n Secton 5 that the asymptotc decodng complexty of the recever s O(N). The dfference n complexty can become substantal for small N as well. For example, for N = 5 the recever would be requred to perform fve smple calculatons per symbol, whle the eavesdropper would have to calculate and compare 32 Eucldean dstances. Although the decodng complexty of the eavesdropper s expected to be hgh, t s prudent to assume that the eavesdropper mght have unlmted computatonal power, whch would allow t to perform ML detecton usng exhaustve search, so Factor 3 could be compromsed. Due to Factor 4, the eavesdropper has to perform decodng usng a suboptmal jont symbol constellaton and s expected to suffer a consderable penalty on Bt Error Rate (BER) compared to the recever. The exact decodng loss depends on the number of transmtters, and the characterstcs of the channel whch prevents a general analyss. Evaluaton of decodng loss for three transmtters over a Raylegh channel s gven n Secton 6. The eavesdropper could use soft decodng and recepton usng multple antennas to acheve decodng gans. However, at least some of the gans can be matched by the recever. The llustratve scenaro provded n Secton 6 demonstrates that the requred gan to match the recever s hard-decodng BER s 20 db for three transmtters n Raylegh fadng. 5. Practcal Implementaton JCMA s based on superposton modulaton wth jont decodng. Superposton modulaton schemes requre accurate symbol synchronzaton and power control, and are adversely affected by moblty. These lmtatons are commonly deemed prohbtve n practcal applcatons. Although superposton modulaton s assumed n many theoretcal works, practcal means of mplementaton are usually not addressed, see [28 32], for examples. To mplement JCMA, robust jont sgnal constellatons and the transmtters symbols sets that construct them must be found offlne. The jont constellatons must not result n performance loss. Effcent low complexty jont decodng must be formulated as well. The expected ncreased senstvty to synchronzaton and power control errors must be mtgated wthout ncreasng mplementaton complexty. These ssues are addressed n what follows. 5.. Jont Sgnal Constellatons 5... Decodng Gan for Power Lmted Transmtters. The overall sgnal energy collected by the recever s MF n JCMA grows wth the number of transmtters. However, ths does not necessarly mean that performance would be enhanced. The BER of ML detecton n Gaussan nose s governed by the Eucldean dstance between ponts n the constellaton [35]. A key requrement for enhancng performance wth JCMA s that the ncreased energy per jont symbol translates to an ncrease n the mnmal Eucldean dstance. In what follows, a probablstc approach s taken to prove that ths s ndeed the case. From (2), y N = s n, (8) n= where the ndex N s ntroduced to denote that y was created by N transmtters. Assumng equal probablty for each transmtted symbol per transmtter, Pr ( ) s n = s n l = 2 M, l =, 2,...,2 M. (9) Let μ n and σn 2 denote the mean value and the varance of the RV s n, respectvely. It s assumed that the transmtters transmt ndependent data and, therefore, {s n } N n= are mutually ndependent. For N suffcently large, y N s a complex normal RV wth mean μ N = N n= μ n and varance σn 2 = σn. 2 (20) n= The followng RV s defned Δ N def = y () N y (2) N, (2) where y () N and y (2) N are two receved constellaton ponts. Snce the probablty for recevng one jont constellaton pont s ndependent of the probablty for recevng any other constellaton pont, y () N and y (2) N are ndependent RVs. Also, y () N and y (2) N are characterzed by complex normal dstrbutons wth mean μ N and varance σn. 2 Another RV s defned as ρ N def = Δ N 2, (22) where ρ N has a ch-squared dstrbuton wth two degrees of freedom and ts PDF s gven by [36] f ρn (x) = exp( x/2σn 2 ), (23) 2σ 2 N where ρ N s the squared Eucldean dstance between two randomly chosen ponts from the jont constellaton. It follows that for Gaussan nose, ρ N governs system performance as t s drectly related to BER. To proceed, let us defne a d 2 N such that Pr ( ρ N >d 2 N ) = δ, (24) where δ s a small postve number. However, (24) means that the probablty that the Eucldan dstance between two arbtrarly chosen ponts from the jont constellaton would be greater than d N s very close to. Let us fnd dn 2 that satsfes (24). From (23), Pr ( ρ N >dn 2 ) = f ρn (x)dx = exp dn 2 Introducng (24) to(25) yelds ( d2 N 2σ 2 N ). (25) d 2 N = 2σ 2 N ln( δ) = σ 2 N ln [ ( δ) 2]. (26)

9 EURASIP Journal on Wreless Communcatons and Networkng 9 From (20), t follows that σ 2 N+ >σ 2 N, therefore, from (26), d 2 N+ >d 2 N. (27) The mnmum dstance ncreases (n a probablty sense) as N ncreases. In other words, the ncreased energy reachng the recever s translated to ncreased dstance between jont constellaton ponts at the recever. Now consder a Tme Dvson Multple Access (TDMA) system wth power lmted transmtters and the same data rate as a JCMA system. To mantan the same data rate as that of the JCMA system, the constellaton of each transmtter would have to become more crowded as N ncreases. Because the overall constellaton power remans constant, the mnmal dstance of the receved constellaton would decrease and performance would deterorate. dmn(jcma)/dmn(2 N -QAM TDMA) Number of transmtters N Fgure 3: Performance gan buldup usng random symbol search Suboptmal Symbol Search. To fnd the optmal jont constellaton, one must solve (3), (4) analytcally. The goal s to fnd BB symbol sets whch maxmze the mnmal Eucldean dstance n the resultng jont symbol constellaton. Ths s an optmzaton problem wth quadratc constrant. Whle a closed form soluton may be found under lmtng assumptons, ths approach s avoded and suboptmal search methods are used nstead. Ths alternatve approach s justfed because optmzaton s done only once for a set of transmtters and s performed offlne, so there s no need for a fast real-tme soluton. For smplcty of presentaton t s assumed that each transmtter has two symbols n ts symbol set, whch means that each transmtter has a sngle nformaton bt represented n the jont constellaton (or chp when forward error correcton s employed). To make sure no transmsson power s wasted, each BB symbol set s made to be a rotated and scaled verson ofbnary Phase Shft Keyng (BPSK). Ths nsures that the mean value of each BB symbol set and the mean value of the jont constellaton are zero. The dervatons that follow are easly applcable to more than one bt (chp) per symbol. A random search approach can be used, for whch symbol sets are found usng a Monte Carlo smulaton. A random set of BB symbols for each transmtter s randomly generated and s normalzed to meet the power constrant n (4). The resultng jont constellaton s derved and ts mnmal Eucldean dstance as defned n (3) s evaluated. Thsprocesssrepeatednnumeroustralsandthecollecton of BB sets whch result n the best jont constellaton s chosen. It s possble to add constrants on the BB symbol sets szes to comply wth Qualty of Servce (QoS) demands. In addton, the peak power constrants may vary across transmtters to address unequal channel fadng attenuatons whch could represent near-far scenaros common n multple access scenaros. Prelmnary results for such optmzaton scenaros were presented n [37]. The solutons found wth the random symbol search are asymptotcally optmal as the number of search trals approaches nfnty. A parametrc symbol search approach can be used as well. The phase and ampltude of each BB constellaton pont s quantzed wth some resoluton. The resultng jont constellaton s tested for each sample of phase and ampltude. The granularty of the parametrc search ncreases wth the quantzaton resoluton, and the result s asymptotcally optmal for nfntely small granularty. To explore the proposed approach the jont constellaton for N transmtters was optmzed usng a random symbol search. For comparson, a TDMA system usng a 2 N -QAM constellaton s used, whch results n the same overall bt rate. Fgure 3 dsplays the gan n d mn for the receved constellatons. It s clearly seen that JCMA results n better constellatons for all tested N. Ths result shows that the performance gan buldup proven before s achevable usng the random symbol search approach and that JCMA s energy effcent. An example of a jont constellaton obtaned varandomsymbolsearchsgvennsecton Maxmum Lkelhood Detecton. The receved jont sgnal sample at the MF output s to be decoded based on the expected jont constellaton usng ML detecton. ML detecton n the presence of Gaussan nose bols down to fndng the constellaton pont whch s closest to the receved sample (mnmal Eucldean dstance). Performng ML detecton usng exhaustve search over all possble constellaton ponts has algorthmc complexty of O(2 N ). Ths complexty grows exponentally wth the number of transmtters. For systems wth constraned resources, exhaustve search mght be mpractcal to mplement even for small values of N. For example, N = 3 requres calculatng and comparng eght Eucldean dstances for every receved symbol. Ths complexty problem exsts for any sngle transmtter user constellaton wth 2 N constellaton ponts. Tradtonally, n pont-to-pont systems the decodng complexty s reduced by usng suboptmal constellatons such as rectangular QAM. Suboptmal constellatons are desgned to exhbt structural symmetres whch are used for effcent decodng of the receved samples [38].

10 0 EURASIP Journal on Wreless Communcatons and Networkng The JCMA constellaton exhbts structural symmetres as well. Usng a smple manpulaton on the expresson for the receved jont symbol gves N y N = s + s n ; =,..., N. (28) n= n Symmetry lnes can be drawn between the BB symbols of s shfted to N n= s n. If the symmetry lnes are drawn for all n =,..., N symbol sets, the recever s observaton space s dvded to decson regons wth a constellaton pont at the center of each regon. The ML decodng problem n Gaussan nose reduces to fndng the decson regon n whch the receved sample s located. Assumng the angles of the symmetry lnes wth the real axs value n the complex plane are gven by φ ; =,..., N, the followng gudelnes for fndng a computatonally effcent ML decodng algorthm are defned. () Rotate the receved jont symbol sample r by an angle of ϕ : q = r exp( jφ ). (2) If q R = Re(q) s to the rght of the symmetry lne ntersecton wth the real axs, set L =, else set L = 0. (3) Repeat steps,2 for all. (4) Use L ; =,..., N to access a predefned Look Up Table (LUT) contanng the bt loadng of the constellaton pont n the decson regon of the receved sample. The algorthmc complexty of the suggested effcent ML decodng s O(N). Ths s also the complexty of tradtonal 2 N -QAM ML decodng of a sngle transmtter wth equal rate to a JCMA system wth N transmtters and bt per symbol per transmtter Effects of Imperfect Power Control, Synchronzaton Errors and Moblty. JCMA requres synchronzaton between the transmtters at the symbol level. Ths s a hgher synchronzaton demand than that of TDMA, for example. Methods wth equvalent synchronzaton demands as those of JCMA are symbol-synchronous Code Dvson Multple Access (CDMA) [26, 4], SM-SD [25], and the methods defned n [28 32]. In addton, JCMA demands a hgher accuracy n power control. In ths secton, the effects of synchronzaton errors, power control errors, and moblty on JCMA are dscussed. For coherency of presentaton, the rgorous analyss leadng to ths dscusson s gven n Appendx A. Lack of perfect synchronzaton s manfested by errors n the channel phase estmates, and naccurate power control s manfested by errors n the channel ampltude estmate. It follows that both are accurately modeled by an addtve complex RV representng an error n the channel estmate (CSI errors). The effects of CSI errors (phase and ampltude) on system performance are analyzed n a comparatve manner to TDMA. Snce the transmtters are requred to be smple n desgn, we assume that each transmtter obtans ts CSI by estmatng the channel coeffcent usng some lnear estmator, such as a lnear Mnmum Mean Square Error (MMSE) estmator. In Appendx A the requred energy of the JCMA recever sent plot sgnal s found wth respect to that of a TDMA system, so that the JCMA system performance loss would be the same as the TDMA system performance loss. Ths analyss results n a quanttatve estmate of the effects of synchronzaton and power control errors, and the requred plot energy to support the synchronzaton demands of JCMA. In Appendx A t s shown that n order to acheve the same synchronzaton/power control error n JCMA as that of TDMA whle mantanng the decodng gan, the plot energy used for channel estmaton must have N tmes more energy. Ths ncrease n energy can be acheved by usng longer plots resultng n decreased throughput or by ncreasng plot sgnal power. Alternatvely, the performance gan can be waved by reducng the transmtters power emsson resultng n no need for ncreasng plot energy. In JCMA a plot s sent every channel decorrelaton tme. In Appendx A the rate of plot transmsson for JCMA s calculated to acheve the same channel decorrelaton as for TDMA. Ths analyss results n a quanttatve estmate of the effects of moblty of transmtters and the requred system resources to support such moblty (throughput loss due to plot sgnal transmsson). It s also shown that the JCMA system effectvely shortens the channel decorrelaton tme. Ths results n a need for more frequent plot transmsson whch causes throughput reducton. As far as securty s concerned, the shortenng of decorrelaton tme s a beneft. Ths s because faster varaton n the channel allows for a hgher cryptographc key generatng rate. If the codng gan s waved by reducng power emsson, the decorrelaton tme s the same for JCMA and TDMA and no overhead on throughout s ncurred. Followng the analyss n Appendx A we defne the followng plotng rule: f the transmtters n a JCMA settng use all ther avalable power, decodng gan and faster key generatng rate are obtaned at the expense of hgher energy consumpton. To support these benefts wthout degradng decodng the recever s plot energy must be ncreased tmes the number of transmtters. Alternatvely, the decodng gan and fast channel decorrelaton can be waved by reducng the transmtters power emsson. Ths would result n same energy consumpton as that of a sngle transmtter and there would be no need to ncrease plot energy Network Management. JCMA s expected to operate n a multple access scenaro. Multple access scenaros requre network management protocols to resolve near-far problems, facltate QoS requrements, and optmze the overall data throughput. In the classcal CDMA near-far scenaro, a domnant transmtter deprves other transmtters from servce by maskng ther transmtted sgnal at the recever. The soluton s to use power-control by the recever to reduce the power of the domnant transmtter so that other transmtters can be detected as well. In JCMA each transmtter adjusts ts power

11 EURASIP Journal on Wreless Communcatons and Networkng emsson accordng to channel fadng and the symbol set t s assgned by the recever. The recever can solve near-far scenaros by assgnng symbol sets whch should be receved wth low power to transmtters wth hgher attenuatng channels. Transmtters wth lower attenuatng channels can be assgned a symbol set wth hgher power n the jont constellaton. Another possble soluton s to match transmtters nto JCMA groups accordng to the channel they experence over dfferent subcarrers. Ths s an extenson of OFDM-Access (OFDMA). In OFDMA subcarrers are allocated to dfferent transmtters accordng to the qualty of ther channel over the frequency selectve bandwdth. The basc dea s to allocate parts of the bandwdth to each transmtter n an effcent way so that the overall network throughput s ncreased. In JCMA transmtters should be grouped to facltate the constructon of the jont sgnal constellaton at the recever wth mnmal power-loss due to power reducton at the domnant transmtters. In addton, t s possble to perform offlne optmzaton of symbol sets to resolve specfc near-far scenaros. The resultng symbol sets would be kept at the transmtters symbol sets tables and used when the recever dms t approprate. Ths possblty does not exst n a classcal CDMA system. Offlne optmzaton of symbol sets can be defned wth dfferent szes of the transmtters symbols sets. A transmtter wth a larger symbol set sze would be represented at the jont constellaton wth more bts and so ts throughout would be hgher than that of the other transmtters. 6. Illustratve Scenaro Raylegh Fadng Performance n Raylegh fadng channels s evaluated n what follows. Frst, the channel condtons for achevng nformaton theoretc securty are depcted for varous N n accordance wth the condton n (7). Followng analyss of nformaton theoretc securty, a JCMA settng of three transmtters s depcted, ncludng generatng the jont constellaton and an algorthm for effcent ML decodng. The effects of securty factors, 3, and 4 descrbed before are evaluated and demonstrated as well. A scenaro wth three transmtters was also used n the prelmnary analyss gven n [40]. 6.. Informaton Theoretc Securty. Recall that α n (7) has to be set accordng to the temporal decorrelaton behavor of the channel. The normalzed temporal autocorrelaton functon for a Raylegh fadng channel s gven by [33] R(τ) = J 0 (2πB d τ), (29) where J 0 ( ) s the zero-order Bessel functon of the frst knd and τ s the tme dfference. J 0 (2.4) 0 for the shortest elapsed tme. Ths s equvalent to settng τ = 2.4/2πB d n (29), so t makes sense to set α = T d B d (2.4/2π) = Usng α = n (7) results n ( ) η 0.382Bc B d. (30) N Doppler spread (Hz) η = 0.9 η = 0.99 N =, 2, 3, 4, 5 N =, 2, 3, 4, Coherent bandwdth (KHz) Fgure 4: Mnmal Doppler spread for securty, Raylegh fadng; N =, 2, 3, 4, 5. Fgure 4 depcts the mnmum B d requred for achevng nformaton theoretc securty for the entre channel decorrelaton perod as a functon of B c for N =, 2, 3, 4, 5 and η = 0.9, The arrows drectons ndcate ncreasng values of N. It s evdent that ncreasng N lowers the requred B d dramatcally for a gven B c. The decrease s moderated as N ncreases. For η = 0.9, the requred B d s very hgh, makng t dffcult to use the method for securng the entre decorrelaton perod. However, for η = 0.99, the requred B d s reasonably low for practcal channels. For example, consder the IEEE 802.6e standard (a.k.a. moble WMax) [42]. In [42] the delay spread s assumed to be 0 μs, whch means that B c = 00 khz. For ths value of B c, the Raylegh fadng channel acheves securty for N =, 2, 3, 4, 5 when B d 382 Hz, 9 Hz, 27 Hz, 96 Hz, 76 Hz, respectvely. Assumng a carrer frequency of f c = 3.5 GHz, these values of B d correspond to relatve veloctes between transmtter and recever of v 60, 30, 20, 5, 2 [km/h], respectvely. These veloctes are expected for many moble scenaros Securty by Complexty Three Transmtters. In ths llustratve Raylegh fadng scenaro, subcarrers are overloaded by 3 transmtters n a JCMA settng. The symbol sets S n, n =, 2, 3 were found usng a random symbol search wth max{d mn } defned n (3) beng the optmzng crteron and (4) beng the optmzaton constrant. The followng symbol sets werefoundofflne and assgned arbtrarly to the transmtters: S = { exp ( j ) ;0.724 exp ( j )}, S 2 = { exp ( j.3720 ) ; exp ( j.7696 )}, S 3 = { exp ( j0.206 ) ; exp ( j )}. (3) The receved jont constellaton at the ntended recever along wth the correspondng bt mappng s depcted n