Multi-Antenna Gaussian Broadcast Channels with Confidential Messages

Multi-Antenna Gaussian Broadcast Channels with Confidential Messages Ruoheng Liu and H. V incent Poor Departmen t of E lectrical En gineerin g, Princeton Un iv ersity , Princeto n, NJ 08544 Email: { r liu,poo r } @prince ton.edu Abstract — In wireless data n etworks, communication is partic- ularly susceptible to ea vesdr opping due to its broadcast n ature. Security and privacy systems hav e become critical fo r wireless prov iders and enterprise networks. This paper consi ders the problem of secre t communication over a Gaussi an broadcast channel, wh ere a multi-antenn a transmitter sends indep endent confidential messages to two u sers with information-theoretic se- crecy . That is, each user would like to obtain its own confidential message i n a reliable and safe manner . This communication model is referr ed to as t he mult i-antenna Gaussian broadcast channel with confid ential messages (MGBC-CM). Under this communication scenario, a secret dirty-paper coding scheme and the corresponding achiev able secrecy rate region are first deve loped based on Gaussian codebooks. Ne xt, a computable Sato-type outer bound on th e secrecy capacity regio n is pro vided fo r the MGBC-CM. Furthermore, the S ato-type outer bound prov es to be consistent with the bound ary of t he secret d irty- paper coding achievable rate region, and hence, the secrecy capacity region of the MGBC-CM i s establi shed. Finally , a numerical example demonstrates that both users can achieve positive rates simultaneously u nder the information-theoretic secrecy requireme nt. I . I N T RO D U C T I O N The demand for efficient, r eliable, and secret da ta com- munication over wireless networks has becom e increasing ly critical in rece nt years. D ue to its b roadcast natu re, wireless commun ication is particular ly suscep tible to eavesdropping. The inh erent nature of wireless networks exposes not o nly vulnerab ilities that a malicio us user can exploit to severely compro mise the network, but also multip lies inform ation confidentiality concern s with resp ect to in -network term inals. Hence, security and pr iv acy systems have b ecome cr itical for wireless p roviders and enterpr ise networks. In this work , we con sider multiple anten na secret bro adcast in wir eless networks. This resear ch is inspired b y th e sem inal paper [ 1], in which W yner introdu ced the so-called wir etap channel and pro posed an in formation theoretic appro ach to secret commu nication sch emes. Un der the assump tion that the channel to the eavesdropper is a degrad ed version of that to the desired receiver , W yn er characterized th e ca pacity- secrecy trade off fo r the discrete memoryless wir etap ch annel and showed that secret commu nication is possible withou t sharing a secre t key . Later , th e result was extended by Csisz ´ ar This research was support ed by the Nati onal Science Found ation under Grants ANI-03-38807, CNS-06-25637 and CCF-07-28208. p( x |w 1 ,w 2 ) user 1 user 2 transmi tter (W 1, W 2 ) Y 1 Y 2 X stoc hastic encoder Z 1 Z 2 h g dec W 1 H(W 2 | Y 1 ) dec W 2 H(W 1 | Y 2 ) Fig. 1. Channel m odel of multiple-an tenna Gaussian broadcast channel w ith confident ial messages and K ¨ orner who determined the secrecy cap acity for the non - degraded br oadcast chan nel (BC) with a single confide ntial message in tended f or o ne o f th e users [2 ]. In mo re general wireless network scena rios, secret commu- nication may inv olve m ultiple users an d multiple a ntennas. Consequently , a sign ificant re cent r esearch effort h as been in vested in the study of the info rmation- theoretic limits o f secret comm unication in different wireless ne twork en viron- ments in cluding mu lti-user co mmunicatio n with confid ential messages [3] –[8], secret wireless co mmunic ation o n fading channels [9]–[ 11], an d th e Ga ussian multiple-inp ut single- output (MISO) and mu ltiple-inpu t multiple- output (MI MO) wiretap channels [1 2]–[1 6]. These issues and r esults motiv ate us to study th e multi- antenna Gaussian BC with confiden tial messages (MGBC- CM), in which indepe ndent confiden tial messages fr om a multi-anten na transmitter are to be communicated t o two users. The c orrespon ding broadcast comm unication mod el is shown in Fig. 1. Each u ser would like to obtain its own m essage reliably and co nfidentially . T o gi ve insight into this problem , we first conside r a single- antenna Gaussian BC. Note that this channel is degraded [1 7], which m eans that if a message ca n be successfu lly deco ded by the inferio r u ser , th en the sup erior user is also ensured of decod ing it. Henc e, the secrecy rate of the in ferior user is zero and this problem is reduced to the scalar Gaussian wiretap channel problem [ 18] wh ose secrecy cap acity is now th e m ax- imum rate achievable by the sup erior user . This analysis gives rise to th e qu estion: c an the transmitter, in fact, co mmunica te with both users confidentially at nonzero rate under some other condition s? Roughly speaking, the answer is in the affirmativ e. In par ticular, the tran smitter can com municate when e quipped with sufficiently separated multiple antenn as. W e here have two go als motiv ated directly by question s arising in practice. The first is to determine cond itions u nder which b oth users can obtain their own con fidential m essages in a reliable and safe man ner . This is equivalent to ev aluating the secrecy cap acity region fo r the MGBC-CM. The second is to show how the transm itter shou ld br oadcast co nfidentially , which is equiv alent to design ing an achievable secret coding scheme. T o this end, we first descr ibe a secr et dirty-pa per coding (DPC) scheme a nd derive the correspon ding achiev able rate region ba sed on Gaussian code books. The secr et DPC is based on double-binn ing [6] which enables both joint encoding and p reserving con fidentiality . Next, a comp utable Sato-type outer bound on the secrecy cap acity region is developed for the MGBC-CM. Furth ermore , the Sato-typ e outer bo und proves to be co nsistent with the bo undar y of the secret dirty- paper coding achiev able rate region, a nd hence, the secrecy capacity region o f the MGBC-CM is established. Finally , a n umerical example demo nstrates that both u sers can achieve positive rates simultaneou sly under the in formatio n-theor etic secrecy requirem ent. I I . S Y S T E M M O D E L A N D D E FI N I T I O N S A. Channel Mod el W e co nsider the commu nication of confidential message s to two users over a Gaussian BC via t ≥ 2 transmit-an tennas. Each user is equippe d with a single r eceiv e-antenn a. As shown in Fig. 1, the transmitter send s indepen dent confidential messages W 1 and W 2 in n cha nnel uses with nR 1 and nR 2 bits, respe ctiv ely . The m essage W 1 is destined for user 1 and eav esdropp ed upo n b y user 2, whereas the message W 2 is destined for u ser 2 and eavesdropped up on by user 1. This commun ication scenar io is referr ed to as the multi-anten na Gaussian BC with con fidentia l messages . The Gau ssian BC is an additive no ise ch annel and the received symb ols at user 1 and user 2 can be rep resented as fo llows: y 1 ,i = h H x i + z 1 ,i y 2 ,i = g H x i + z 2 ,i , i = 1 , . . . , n (1) where x i ∈ C t is a c omplex inp ut vector at time i , { z 1 ,i } and { z 2 ,i } corresp ond to two indepen dent, z ero-mea n, unit- variance, complex Gaussian n oise sequ ences, and h , g ∈ C t are fixed, comp lex ch annel a ttenuation vectors imposed on user 1 a nd user 2, respectively . The channe l input is con - strained b y tr( K X ) ≤ P , wh ere K X is the channel inp ut covariance matrix and P is the average total power limitatio n at the transmitter . W e also assume that bo th the tr ansmitter and users are aware of th e attenuation vectors. B. Important Cha nnel P arameters for the MGBC-CM For the MGBC-CM, we are interested in the following importan t pa rameters, wh ich a re r elated to the gener alized eigenv alue p roblem . L et λ 1 and e 1 denote the largest g eneral- ized eigenv alue and the corre spondin g normalized eigen vector of th e pencil ( I + P hh H , I + P gg H ) so th at e H 1 e 1 = 1 an d ( I + P hh H ) e 1 = λ 1 ( I + P gg H ) e 1 . ( 2) Similarly , we define λ 2 and e 2 as the la rgest g eneralized eigenv alue and th e corre spondin g norma lized eigenv ector o f the p encil ( I + P gg H , I + P hh H ) so th at e H 2 e 2 = 1 an d ( I + P gg H ) e 2 = λ 2 ( I + P hh H ) e 2 . ( 3) A useful proper ty of λ 1 and λ 2 is d escribed as follows. Lemma 1 : For any chan nel attenu ation vector pair h an d g , the largest gen eralized eigenv alues of the pencil ( I + P hh H , I + P gg H ) and the p encil ( I + P gg H , I + P hh H ) satisfy λ 1 ≥ 1 and λ 2 ≥ 1 . Moreover, if h and g ar e linear ly indepen dent, th en b oth λ 1 and λ 2 are strictly greate r than 1 . C. Defi nitions W e now defin e the secr et co deboo k, the pro bability of e rror, the secrecy level, an d the secrecy capacity region fo r the MGBC-CM as fo llows. An (2 nR 1 , 2 nR 2 , n ) secret c odeboo k for the MGBC-CM consists of th e fo llowing: 1) T wo message sets: W k = { 1 , . . . , 2 nR k } , f or k = 1 , 2 . 2) A stoch astic en coding functio n specified by a c ondi- tional proba bility den sity p ( x n | w 1 , w 2 ) , wher e x n = [ x 1 , . . . , x n ] ∈ C t × n , w k ∈ W k for k = 1 , 2 , and Z x n p ( x n | w 1 , w 2 ) = 1 . 3) Deco ding function s φ 1 and φ 2 . The d ecoding fun ction at user k is a deter ministic ma pping φ k : Y n k → W k . At the receiver ends, the error perfor mance and the secrecy lev el are ev aluated by the fo llowing p erform ance me asures. 1) The re liability is measu red by the maxim um error prob - ability P ( n ) e , max  P ( n ) e, 1 , P ( n ) e, 2  where P ( n ) e,k is th e error p robab ility fo r u ser k . 2) The secrecy levels with respect to confide ntial mes- sages W 1 and W 2 are measured , respectively , at u ser 2 and user 1 with respect to the eq uivocation rates 1 n H ( W 2 | Y n 1 ) and 1 n H ( W 1 | Y n 2 ) . A rate pa ir ( R 1 , R 2 ) is said to be achiev able for the MGBC- CM if, for any ǫ > 0 , th ere exists an (2 nR 1 , 2 nR 2 , n ) code that satisfies P ( n ) e ≤ ǫ , and the information -theoretic secrecy requirem ent nR 1 − H ( W 1 | Y n 2 ) ≤ nǫ and nR 2 − H ( W 2 | Y n 1 ) ≤ nǫ. (4) The secr ecy capa city re gio n C MG s of the MGBC-CM is the closure of the set of all achievable rate pairs ( R 1 , R 2 ) . I I I . M A I N R E S U LT The two-user Gaussian BC with multiple transmit-a ntennas is non-d egraded. For this ch annel, we have the following closed-fro m result on the secrecy ca pacity region under the informa tion-theo retic secrecy req uirement. Theor em 1: Consid er an MGBC-CM mo deled as in (1). Let γ 1 ( α ) = 1 + αP | h H e 1 | 2 1 + αP | g H e 1 | 2 , γ 2 ( α ) be the largest gen eralized eig en value of the pen cil  I + (1 − α ) P gg H 1 + αP | g H e 1 | 2 , I + (1 − α ) P hh H 1 + αP | h H e 1 | 2  , (5) and R MG ( α ) deno te the union of all ( R 1 , R 2 ) satisfy ing 0 ≤ R 1 ≤ log 2 γ 1 ( α ) and 0 ≤ R 2 ≤ log 2 γ 2 ( α ) . Then the secr ecy capac ity region of the MGBC-CM is C MG s = co    [ 0 ≤ α ≤ 1 R MG ( α )    where co {S } denotes the conve x h ull o f the set S . Pr oof: The achiev ability part of Th eorem 1 is b ased o n secret dirty- paper coding inner bou nd in Sec. IV. The co n verse part is b ased on Sato- type ou ter bo und in Sec. V. W e p rovide the comple te proof in [ 14]. Cor o llary 1: For the M GBC-CM, th e maximu m secrecy rate of user 1 is given by R 1 , max = max 0 ≤ α ≤ 1 log 2 γ 1 ( α ) = log 2 λ 1 where λ 1 is d efined in (2). Example: (MI SO wiretap chann els) A special case of the MGBC-CM m odel is th e Gau ssian MISO wir etap c hannel studied in [12], [19] , [20 ], wh ere the transmitter sends co n- fidential info rmation to only on e user and tr eats ano ther u ser as an eavesdropper . Let us con sider a Gaussian MISO wiretap channel m odeled in (1), where u ser 1 is the legitimate r eceiver and user 2 is the eav esdropp er . Corollary 1 implies th at the secrecy capacity of the Gaussian MISO wiretap chann el correspo nds to the corne r point of C MG s . He nce, th e secrecy capacity of the Gau ssian MISO wir etap ch annel is given by C MISO s = log 2 λ 1 , which coincides with th e result o f [19] . For the MGBC-CM, the actions of user 1 an d user 2 are symmetric to e ach other, i.e. , ea ch user dec odes its own message an d eavesdrops u pon the confide ntial in formatio n belongin g to the other u ser . Based on symmetry of th is two- user BC mo del, we can express the secrecy capacity r egion C MG s in an alternative way . Cor o llary 2: For an M GBC-CM modeled in as (1), the secrecy capac ity region can be written as C MG s = co    [ 0 ≤ β ≤ 1 R MG − 2 ( β )    where R MG − 2 ( β ) denotes th e u nion o f all ( R 1 , R 2 ) satisfying 0 ≤ R 1 ≤ log 2 ξ 1 ( β ) and 0 ≤ R 2 ≤ log 2 ξ 2 ( β ) , ξ 1 ( β ) is th e largest generalized eigenv alue o f the p encil  I + (1 − β ) P hh H 1 + β P | h H e 2 | 2 , I + (1 − β ) P gg H 1 + β P | g H e 2 | 2  and ξ 2 ( β ) = 1 + β P | g H e 2 | 2 1 + β P | h H e 2 | 2 . Remark 1 : Theorem 1 and Co rollary 2 imply tha t if α and β satisfy the imp licit fun ction γ 1 ( α ) = ξ 1 ( β ) , then R MG ( α ) = R MG − 2 ( β ) . For example, it is easy to check R MG (1) = R MG − 2 (0) . Now , by app lying Coro llary 2 a nd setting β = 1 , we can show that the rate pair (0 , log 2 λ 2 ) is the corn er po int correspo nding to the m aximum ach iev able rate of user 2 in the cap acity r egion C MG s . Cor o llary 3: For the MGBC-CM, the m aximum secrecy rate o f user 2 is given b y R 2 , max = log 2 λ 2 where λ 2 is defined in (3). Corollaries 1 an d 3 imply that f or the MGBC-CM, both users can ach iev e positive rates with inf ormation -theor etic secrecy if and o nly if λ 1 > 1 and λ 2 > 1 . Furth ermore , Lemma 1 illustrates that this condition can be ensure d whe n the atten uation vector s h and g are linearly indep enden t. I V . A C H I E V A B I L I T Y : S E C R E T D P C S C H E M E A. Double-Bin ning Inner b ound for the B C-CM An achievable rate r egion for the broadcast chann el with confidential messages (BC-CM) has b een established in [6] based on a d ouble- binning scheme that enab les both join t encodin g at the tran smitter by usin g Gel’fand-Pinsker binning and preservin g c onfidentiality by using rando m binnin g. Lemma 2 : ( [ 6, T heorem 3]) Let V 1 and V 2 be auxiliary random variables, Ω denote the class o f jo int probab ility densities p ( v 1 , v 2 , x , y 1 , y 2 ) th at factor as p ( v 1 , v 2 ) p ( x | v 1 , v 2 ) p ( y 1 , y 2 | x ) , and R I ( π ) den ote th e un ion of all ( R 1 , R 2 ) satisfying 0 ≤ R 1 ≤ I ( V 1 ; Y 1 ) − I ( V 1 ; Y 2 , V 2 ) and 0 ≤ R 2 ≤ I ( V 2 ; Y 2 ) − I ( V 2 ; Y 1 , V 1 ) for a given joint probability d ensity π ∈ Ω . For the BC-CM, any rate pair ( R 1 , R 2 ) ∈ co ( [ π ∈ Ω R I ( π ) ) (6) is ach iev able. B. Secr et DPC S cheme for the MGBC-CM The ach iev able strategy in Le mma 2 introd uces a double- binning cod ing schem e. Howe ver , when the rate r egion (6) is used as a constructive tech nique, it not clear how to choose the auxiliary random variables V 1 and V 2 to implement th e double- binning codeb ook, and hence, one has to “guess” the density of p ( v 1 , v 2 , x ) . Here, we employ the DPC techn ique with the double- binnin g code structu re to d ev elop the secret DPC ( S-DPC) achiev able rate region. For the M GBC-CM, we co nsider a secret d irty-pap er en- coder with Gaussian codebook s. Based on Lemma 2 , we obtain a S-DPC r ate region fo r the MGBC-CM as fo llows. Lemma 3 : [ S-DPC region] Let R S − DPC I ( K U 1 , K U 2 ) de- note the union of all ( R 1 , R 2 ) satisfy ing 0 ≤ R 1 ≤ log 2 1 + h H K U 1 h 1 + g H K U 1 g (7) and 0 ≤ R 2 ≤ log 2 1 + g H ( K U 1 + K U 2 ) g 1 + h H ( K U 1 + K U 2 ) h + log 2 1 + h H K U 1 h 1 + g H K U 1 g . (8) Then, any rate pair ( R 1 , R 2 ) ∈ co    [ tr( K U 1 + K U 2 ) ≤ P R S − DPC I ( K U 1 , K U 2 )    is ach iev able for the MGBC-CM. Pr oof: A d etail proo f can be f ound in [14] . The S- DPC ach iev able rate region requir es optimization of the covariance matrices K U 1 and K U 2 . In order to achieve the boun dary of C MG s , we choose K U 1 and K U 2 as f ollows: K U 1 = αP e 1 e H 1 and K U 2 = (1 − α ) P c 2 ( α ) c H 2 ( α ) , 0 ≤ α ≤ 1 (9) where e 1 is d efined in (2) and c 2 ( α ) is a norm alized eigenv ec- tor of the pen cil (5) corr espondin g to γ 2 ( α ) . Next, inserting (9) into ( 7) and (8), we o btain 1 + h H K U 1 h 1 + g H K U 1 g = γ 1 ( α ) and [1 + g H ( K U 1 + K U 2 ) g ][1 + h H K U 1 h ] [1 + h H ( K U 1 + K U 2 ) h ][1 + g H K U 1 g ] = γ 2 ( α ) . (10) Now , by substituting (1 0) into Lem ma 3, we o btain the desired achiev able result. V . C O N V E R S E : S ATO - T Y P E O U T E R B O U N D A. Sato-T yp e Outer Bo und W e consider an important p roperty for th e BC-CM in the following lemma. Lemma 4 : L et P d enote the set of chan nels p ˜ Y 1 , ˜ Y 2 | X whose marginal distributions satisfy p ˜ Y 1 | X ( y 1 | x ) = p Y 1 | X ( y 1 | x ) and p ˜ Y 2 | X ( y 2 | x ) = p Y 1 | X ( y 2 | x ) for all y 1 , y 2 and x . T he secrecy capacity region C MG s is the same fo r the channels p ˜ Y 1 , ˜ Y 2 | X ∈ P . W e note that P is the set of channe ls p ˜ Y 1 , ˜ Y 2 | X that have the same margin al distributions as the orig inal chann el tran sition density p Y 1 ,Y 2 | X . Lemma 4 implies that the secrecy cap acity region C MG s depend s only on margin al distributions. Theor em 2: Let R O  P ˜ Y 1 , ˜ Y 2 | X , P X  denote the union of all rate p airs ( R 1 , R 2 ) satisfying R 1 ≤ I ( X ; ˜ Y 1 | ˜ Y 2 ) and R 2 ≤ I ( X ; ˜ Y 2 | ˜ Y 1 ) for giv en distrib utions P X and P ˜ Y 1 , ˜ Y 2 | X . The secrecy cap acity region C MG s of th e BC-CM satisfies C MG s ⊆ \ P ˜ Y 1 , ˜ Y 2 | X ∈P ( [ P X R O  P ˜ Y 1 , ˜ Y 2 | X , P X  ) . ( 11) Pr oof: A d etail pr oof can be fou nd in [1 4]. Remark 2 : The outer boun d (11) f ollows by ev aluating the secrecy lev el at each r eceiv er end in an in dividual manner, while letting the users d ecode their messages in a cooperative manner . In this sense, we refer to this bound as “Sato -type” outer bo und. For examp le, w e consider the confiden tial message W 1 that is destined f or user 1 (corresp onding to ˜ Y 1 ) and eavesdropped upon by user 2 (correspond ing to ˜ Y 2 ). W e assume that a genie giv es user 1 th e signal ˜ Y 2 as side information for decoding W 1 . Note that the e av esdropp ed u pon signal ˜ Y 2 at user 2 is always a degraded version of the en tire received sign al ( ˜ Y 1 , ˜ Y 2 ) . Th is permits th e use o f the wiretap chan nel result of [1] . Remark 3 : Although Theo rem 2 is b ased on a degr aded argument, the ou ter bou nd (11) ca n be app lied to the general broadc ast channel with c onfidential m essages. B. Sato-T yp e Ou ter B ound fo r the MGBC-CM For the Gaussian BC, the family P is the set o f ch annels ˜ y 1 = h H x + ˜ z 1 ˜ y 2 = g H x + ˜ z 2 where ˜ z 1 and ˜ z 2 correspo nd to arbitrar ily corr elated, zero- mean, unit-variance, complex Gau ssian rando m variables. Let ρ denote th e covariance between ˜ Z 1 and ˜ Z 2 , i.e, Cov  ˜ Z 1 , ˜ Z 2  = ρ and | ρ | 2 ≤ 1 . Now , th e rate region R O  P ˜ Y 1 , ˜ Y 2 | X , P X  is a fu nction of th e noise covariance ρ and the input covariance matrix K X . W e consider a comp utable Sato -type o uter bo und for the MGBC- CM in the following lemma. Lemma 5 : L et R MG O ( ρ, K X ) deno te th e un ion of all rate pairs ( R 1 , R 2 ) satisfy ing 0 ≤ R 1 ≤ f 1 ( ρ, K X ) and 0 ≤ R 2 ≤ f 2 ( ρ, K X ) where f 1 ( ρ, K X ) = min ν ∈ C log 2 ( h − ν g ) H K X ( h − ν g ) + ψ 1 ( ν, ρ ) (1 − | ρ | 2 ) f 2 ( ρ, K X ) = min µ ∈ C log 2 ( g − µ h ) H K X ( g − µ h ) + ψ 2 ( µ, ρ ) (1 − | ρ | 2 ) ψ 1 ( ν, ρ ) = 1 + | ν | 2 − ν ∗ ρ − ρ ∗ ν and ψ 2 ( µ, ρ ) = 1 + | µ | 2 − µ ∗ ρ − ρ ∗ µ. For th e M GBC-CM, the secrecy capacity region C MG s satisfies C MG s ⊆ [ tr( K X ) ≤ P R O ( ρ, K X ) for any 0 ≤ | ρ | ≤ 1 . Remark 4 : Lemma 5 is based on the fact tha t Gaussian input d istributions maximize R O  P ˜ Y 1 , ˜ Y 2 | X , P X  for Gau ssian broadc ast channel. T o illustra te this point, we conside r I ( X ; ˜ Y 1 | ˜ Y 2 ) = h ( ˜ Y 1 | ˜ Y 2 ) − log 2 (2 π e )(1 − | ρ | 2 ) ≤ h ( ˜ Y 1 − ν ˜ Y 2 ) − log 2 (2 π e )(1 − | ρ | 2 ) . Moreover , the maximum- entropy the orem [ 17] implies that h ( ˜ Y 1 − ν ˜ Y 2 ) is maximized by Gaussian inpu t distributions. Finally , we prove that the Sato-type outer bound o f Lemma 5 c oincides with th e secrecy cap acity region C MG s by choosing the pa rameter ρ = ( g H e 1 ) / ( h H e 1 ) . A d etail p roof can b e foun d in [14]. V I . N U M E R I C A L E X A M P L E S In this section , we stud y a nu merical example to illu strate the secrecy capacity region of the MGBC-CM. For simp licity , we assume that the Gaussian BC has re al input and outp ut alphabets and the channe l a ttenuation vectors h and g are also real. Under th ese conditions, a ll calculated r ate values are d ivided b y 2 . 0 0.4 0.8 1.2 1.6 2 0 0.4 0.8 1.2 1.6 2 R 1 (bits) R 2 (bits) time−sharing capacity region Fig. 2. Secrecy capa city region vs. time-sha ring secrec y rate re gion for the exa mple MGBC-CM in (12) In p articular, we co nsider the f ollowing MGBC-CM  y 1 y 2  =  1 . 5 0 1 . 801 0 . 871   x 1 x 2  +  z 1 z 2  (12) where h = [1 . 5 , 0 ] T , g = [1 . 801 , 0 . 872] T , a nd the to tal power co nstraint is set to P = 10 . Fig. 2 illustrates the secrecy capacity r egion fo r the channe l (12). W e o bserve that ev en th ough e ach compo nent of the attenua tion vecto r h ( imposed o n u ser 1) is strictly less than th e cor respond ing compon ent of g (impo sed o n user 2), both user s ca n ach iev e positive r ates simu ltaneously u nder the informatio n-theo retic secrecy r equireme nt. Moreover, we co mpare the secrecy ca- pacity region with th e secrecy rate region achieved b y the time-sharing scheme (in dicated by the dash -dot line). Fig. 2 demonstra te that the time-sharing sch eme is s trictly suboptimal for providing the secrecy capacity region . 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