The Secrecy Capacity Region of the Degraded Vector Gaussian Broadcast Channel

The Secrec y Capacity Re gion of the De graded V ector Gaussian Broadcast Channel Ghadamali Bagherikaram, Abolfazl S. Motahari, Amir K. Khan dani Coding a nd Sign al T ransmission L aboratory , Department of Elec trical and Computer Engineering, Univ ersity of W aterloo, W aterloo, Ontario, N2L 3G1 Emails: { gba gheri,abolfazl,khan dani } @cst.uwaterloo.ca 1 Abstract — In this paper , we consider a scenario where a source node wishes to broadcast two confidential messages for two r esp ectiv e rec eiver s via a Gaussian MIMO broadcast channel. A wire-tapper also receiv es the transmitted signal via another MIMO channel. It is assumed th at the channels are degraded and the wire-tapper has the worst chann el. W e establish the capacity region of this scenario. Our achieva bility scheme is a combination of the superposition of Gaussian codes and randomization within the layers whi ch we w ill refer to as Secret Superposition Coding. For the outerbound , we use the notion of enhanced chann el to show th at the secret superposition of Gaussian codes is optimal. It is shown that we only n eed to en hance the chann els of t he legitimate receiv ers, and the channel of the eave sdropper r emains unchanged. I . I N T R O D U C T I O N Recently there has bee n significant research conduc ted in both theoretical and practical aspects of wireless co mmuni- cation systems with Multiple-In put Multiple-Output (MIMO) antennas. Most works have focused on the role of MIMO in enhancin g th e thr oughp ut and robustness. In this work, howe ver , we fo cus o n the role of such multiple antennas in enhancin g w ireless secu rity . The inform ation-theo retic single user secure com munication problem was first characterized by W yner in [1]. W y ner considered a scen ario in which a wire-tapper receives the transmitted sig nal over a degraded chann el with r espect to the legitimate recei ver’ s channel. He me asured the le vel of ign orance at the ea vesdr opper by its equi vocation and characterized th e cap acity-equivocation r egion. W yne r’ s work was then extend ed to the genera l broadcast chan nel with confidential m essages by Csiszar et al. [2 ]. They co nsidered transmitting confiden tial info rmation to th e legitimate r eceiv er while transm itting common info rmation to bo th the legitimate receiver an d the wire- tapper . They established a capacity- equiv ocation region of this channel. The secrecy capacity for the Gaussian wire-tap channel was cha racterized by Leu ng- Y an-Cheong in [3]. The Gaussian MIMO wir e-tap channel h as recently bee n considered by Khisti et al. in [4], [5]. Finding the o ptimal 1 Financi al support provided by Nortel and the correspondi ng matching funds by the Natural Sciences and Engineering Researc h Council of Canada (NSERC), and Ontario Centers of Excellen ce (OCE) are gratefull y ackno wl- edged. distribution, which maximiz es the secrecy capac ity for this channel is a nonco n vex proble m. Khisti et al., howe ver , followed an indir ect approach to ev aluate th e secrecy c apacity of Cs iszar et al. They used a genie-aided up per bo und and characterized the secrecy capacity as the saddle- value o f a min-max p roblem to show that Ga ussian d istribution is optimal. Motivated by the broadcast natu re of the wireless commun ication systems, we co nsidered the secure bro adcast channel in [6 ]. In this work, we characterize d the secre cy capacity region of the degra ded broadcast chan nel an d showed that th e secret sup erposition cod ing is optimal. The capa city region of the con vention al Gau ssian MIMO broadc ast chann el is studied in [7] by W ein garten et al. The notion of an enhanced b roadcast chann el is in troduced in this work and is used join tly with entr opy power ineq uality to characterize the capacity region of th e degraded vector Gaus- sian broadca st ch annel. They showed th at th e superp osition o f Gaussian codes is optima l for the degraded vector Gau ssian broadc ast cha nnel and that dirty-p aper co ding is optimal for the n ondegrad ed case. In this paper, we aim to ch aracterize the secrecy c apacity region of a secure d egraded vector Gaussian MIMO bro adcast channel. Ou r ac hiev ab ility scheme is a co mbination o f the superpo sition of Gaussian codes and r andomizatio n within the layers. T o prove th e co n verse, we use the notion o f enhan ced channel and show that th e secret superposition of Gaussian codes is op timal. W e ha ve extended the results of this paper to the gener al Gaussian MIMO bro adcast channel in [8] a nd showed that secre t dirty paper codin g of Gaussian codes is optimal. W e a cknowledge two oth er indepen dent an d concur rent works of [9], [10 ] wher e the authors con sidered the secrecy capacity regio n of the Gaussian MI MO br oadcast cha nnel. The rest of th e paper is organized as follows. In section II we intro duce some p reliminaries. In sectio n III , we establish the secrecy capacity region of the Gaussian vector broa dcast channel. In Section V , we co nclude th e pa per . I I . P R E L I M I N A R I E S Consider a Secu re Gaussian Mu ltiple-Inpu t Multiple-Output Broadcast Channel (SGMBC) as dep icted in Fig. 1. In this Encoder Decoder1 Decoder2 Eavesdropper H 3 V 3 ( W 1 , W 2 ) V 2 d W 1 d W 2 Z H 1 Y 1 V 1 Y 2 H 2 X Fig. 1. Secure Gaussian MIMO Broadcast Channel confidential setting , the transmitter wishes to send two inde- penden t m essages ( W 1 , W 2 ) to the respectiv e receivers in n uses of th e chan nel and p revent the eav esdropp er f rom having any in formation about the m essages. At a sp ecific time, the signals received by the destinations and the ea vesdropper are giv en by y 1 = H 1 x + n 1 , y 2 = H 2 x + n 2 , (1) z = H 3 x + n 3 , where • x is a r eal in put vector of size t × 1 under an input covariance con straint. W e require that E [ x T x ]  S for a positive semi-d efinite matrix S  0 . Here, ≺ ,  , ≻ , a nd  repr esent partial or dering between symmetric matr ices where B  A mea ns th at ( B − A ) is a po siti ve semi- definite matrix. • y 1 , y 2 , and z are rea l output vectors which are r eceiv ed by the destination s an d the eavesdropper respectively . These are vectors of size r 1 × 1 , r 2 × 1 , and r 3 × 1 , respectively . • H 1 , H 2 , an d H 3 are fixed, real ga in matrices which model th e ch annel g ains between the transmitter and the receivers. These are matrices of size r 1 × t , r 2 × t , and r 3 × t respectively . The channel state informatio n is assumed to be k nown p erfectly at the transmitter an d at all recei vers. • n 1 , n 2 and n 3 are real Ga ussian rand om vectors with zero means and cov ariance matrices N 1 = E [ n 1 n 1 T ] ≻ 0 , N 2 = E [ n 2 n 2 T ] ≻ 0 , and N 3 = E [ n 3 n 3 T ] ≻ 0 respectively . Let W 1 and W 2 denote the the me ssage indices of user 1 and user 2 , respectively . Fur thermore, let X , Y 1 , Y 2 , and Z denote the ran dom chann el inpu t and random ch annel o utputs matrices over a block o f n samp les. Let V 1 , V 2 , an d V 3 denote th e additiv e noises of th e ch annels. Thus, Y 1 = H 1 X + V 1 , Y 2 = H 2 X + V 2 , (2) Z = H 3 X + V 3 . Note that V i is an r i × n rando m matrix an d H i is an r i × t deterministic matr ix wh ere i = 1 , 2 , 3 . T he co lumns of V i are indep endent Gau ssian ran dom vecto rs with cov ar iance matrices N i for i = 1 , 2 , 3 . In addition V i is indepen dent of X , W 1 and W 2 . A ((2 nR 1 , 2 nR 2 ) , n ) code for the above channel consists of a stochastic en coder f : ( { 1 , 2 , ..., 2 nR 1 } × { 1 , 2 , ..., 2 nR 2 } ) → X , (3) and two d ecoders, g 1 : Y 1 → { 1 , 2 , ..., 2 nR 1 } , (4) and g 2 : Y 2 → { 1 , 2 , ..., 2 nR 2 } . (5) where a script letter with dou ble ov erline denotes the finite alphabet of a ran dom vector . The a verag e p robability of er ror is d efined as the probab ility that th e deco ded message s ar e not equal to the transmitted m essages; that is, P ( n ) e = P ( g 1 ( Y 1 ) 6 = W 1 ∪ g 2 ( Y 2 ) 6 = W 2 ) . (6) The secr ecy levels of confidential m essages W 1 and W 2 are measured at the e a vesdropp er in terms of eq uiv o cation rates, which are defined as follows. Definition 1 The equivocation rates R e 1 , R e 2 and R e 12 for the secu r e br o adcast chan nel are: R e 1 = 1 n H ( W 1 | Z ) , (7) R e 2 = 1 n H ( W 2 | Z ) , R e 12 = 1 n H ( W 1 , W 2 | Z ) . The p erfect secrecy rates R 1 and R 2 are the amount of informa tion that can b e sen t to the legitimate recei vers b oth reliably a nd confid entially . Definition 2 A secrecy rate pair ( R 1 , R 2 ) is said to be achievable if fo r an y ǫ > 0 , ǫ 1 > 0 , ǫ 2 > 0 , ǫ 3 > 0 , there exis ts a seq uence of ((2 nR 1 , 2 nR 2 ) , n ) codes, such that for sufficiently lar ge n , P ( n ) e ≤ ǫ, (8) R e 1 ≥ R 1 − ǫ 1 , (9) R e 2 ≥ R 2 − ǫ 2 , (10) R e 12 ≥ R 1 + R 2 − ǫ 3 . (11) In the above definition, the first condition conc erns the reli- ability , while the other conditions guarante e perfect secrecy for each individual message and both messages as well. The model presen ted in (1) is SGMBC. For lack o f space, the SGMBC cann ot be discu ssed within this paper, and we will only consider a subclass of this ch annel h ere. The special subclass that we will con sider is the Secure Aligne d Degrad ed MIMO Broadcast Channel (SADBC). The MIMO broadcast channel of (1) is said to b e align ed if the n umber of transmit antennas is equal to the nu mber of recei ve a ntennas at each of the u sers and the ea vesdr opper ( t = r 1 = r 2 = r 3 ) an d th e gain ma trices ar e all id entity matr ices ( H 1 = H 2 = H 3 = I ) . Furthermo re, if the ad ditive noise vectors’ covariance matrices are ordered such th at 0 ≺ N 1  N 2  N 3 , then the ch annel is SADBC. I I I . T H E C A P AC I T Y R E G I O N O F T H E S A D B C In this section, we character ize th e ca pacity region of the SADBC. In [6], we co nsidered the degrade d broadcast chann el with confiden tial messages an d establish its secr ecy capacity region. Theorem 1 The c apacity r e gion for transmitting indep endent secr et messages over the degr aded b r oad cast channe l is the conve x hull of the closur e of all ( R 1 , R 2 ) satisfying R 1 ≤ I ( X ; Y 1 | U ) − I ( X ; Z | U ) , (12) R 2 ≤ I ( U ; Y 2 ) − I ( U ; Z ) . (13) for some jo int distribution P ( u ) P ( x | u ) P ( y 1 , y 2 , z | x ) . Pr oof: Our achiev able co ding sch eme is ba sed on Cover’ s superpo sition scheme and random binnin g. W e re fer to this scheme as the Secre t Sup erposition Sch eme. I n this sch eme, random ization in the first layer increases the secrecy rate of the second layer . Our co n verse proo f is based on a combination of the co n verse proof of the co n ventional degraded b roadcast channel and Csiszar Lemma. Please see [6] for d etails. Note th at find ing optimal d istribution w hich char acterizes the bound ary points of (12) and for the Gaussian channe ls in- volves solving a fun ctional, n onconve x o ptimization proble m. Usually non trivial techniqu es and stro ng inequa lities ar e used to solve optimization prob lems of this type. I ndeed, for the single antenna case, we successfully ev aluated the capacity expression of this scheme in [11]. Liu et al. in [12] evaluated the capacity expression of MIMO wire- tap channel by using the ch annel en hancemen t method. In the f ollowing section, we state and prove o ur r esult fo r the capacity region o f SADBC. First, we defin e the achiev able rate region due to Gaussian codebo ok under a cov arian ce matrix constraint S  0 . The achiev ability scheme o f T heorem 1 is the secret superpo sition of Gaussian codes and successive d ecoding at the first receiv er . According to the above theo rem, for any cov ar iance matrix input constra int S an d two semi-definite matric es B 1  0 an d B 2  0 such th at B 1 + B 2  S , it is possible to ach iev e the following rates, R G 1 ( B 1 , 2 , N 1 , 2 , 3 ) = 1 2 log | N − 1 1 ( B 1 + N 1 ) | − 1 2 log | N − 1 3 ( B 1 + N 3 ) | , R G 2 ( B 1 , 2 , N 1 , 2 , 3 ) = 1 2 log | B 1 + B 2 + N 2 | | B 1 + N 2 | − 1 2 log | B 1 + B 2 + N 3 | | B 1 + N 3 | . Definition 3 Let S b e a positive semi-definite ma trix. Then , the Gaussian rate r egion of SADBC under a covarian ce matrix constraint S is given by R G ( S , N 1 , 2 , 3 ) =   R G 1 ( B 1 , 2 , N 1 , 2 , 3 ) , R G 2 ( B 1 , 2 , N 1 , 2 , 3 )  | s.t S − ( B 1 + B 2 )  0 , B k  0 , k = 1 , 2  . (1 4) W e will sho w that R G ( S , N 1 , 2 , 3 ) is the capacity region of the SADBC. Before that, certain p reliminaries need to be addressed. Definition 4 The rate vector R ∗ = ( R 1 , R 2 ) is said to be an optimal Gaussian r ate vector under the covariance matrix S , if R ∗ ∈ R G ( S , N 1 , 2 , 3 ) and if there is no other rate vector R ′ ∗ = ( R ′ 1 , R ′ 2 ) ∈ R G ( S , N 1 , 2 , 3 ) such that R ′ 1 ≥ R 1 and R ′ 2 ≥ R 2 wher e at least on e th e inequ alities is strict. The set of positive semi-definite matrices ( B ∗ 1 , B ∗ 2 ) such that B ∗ 1 + B ∗ 2  S is said to b e r ealizing matrices of an optima l Gau ssian rate vector if the rate vector  R G 1 ( B ∗ 1 , 2 , N 1 , 2 , 3 ) , R G 2 ( B ∗ 1 , 2 , N 1 , 2 , 3 )  is an optimal Gaussian rate vector . Definition 5 A SADBC with n oise covariance matrices of ( N ′ 1 , N ′ 2 , N ′ 3 ) is an enhan ced ver sion of an other SADB C with noise c ovariance matrices ( N 1 , N 2 , N 3 ) if N ′ 1  N 1 , N ′ 2  N 2 , N ′ 3 = N 3 , N ′ 1  N ′ 2 . (15) Obviously , the capacity region of the enhanc ed version con- tains the capacity region of the orig inal chann el. Note that in characterizin g the capacity region of the c on ventional Gaussian MIMO broad cast chann el, all ch annels must be enhanc ed by reducin g the noise cov ariance matr ices. In o ur schem e, howe ver , we only en hance th e chann els for the legitimate receivers and the chann el of the eavesdropper remains un - changed . Th is is due to the fact that the cap acity region of th e enhanced channel must contain the or iginal cap acity region. Reducing the noise covariance ma trix of the ea vesdropper ’ s channel, howe ver , may red uce the secr ecy cap acity region. The following theorem conn ects th e definition s o f the optim al Gaussian r ate vector and the enhan ced chan nel. Theorem 2 Consider a SADBC with positive defi nite n oise covariance matrices ( N 1 , N 2 , N 3 ) . Let B ∗ 1 and B ∗ 2 be r e- alizing ma trices o f an optima l Gaussian rate vector unde r a transmit covariance matrix co nstraint S ≻ 0 . Th er e then exis ts an enh anced SADBC with noise covariance matrices ( N ′ 1 , N ′ 2 , N ′ 3 ) that th e follo wing p r operties hold. 1) Enhancement: N ′ 1  N 1 , N ′ 2  N 2 , N ′ 3 = N 3 , N ′ 1  N ′ 2 , 2) Pr o portiona lity: Ther e exists an α ≥ 0 and a ma trix A such th at ( I − A )( B ∗ 1 + N ′ 1 ) = α A ( B ∗ 1 + N ′ 3 ) , 3) Rate and optimality p r eservatio n: R G k ( B ∗ 1 , 2 , N 1 , 2 , 3 ) = R G k ( B ∗ 1 , 2 , N ′ 1 , 2 , 3 ) ∀ k = 1 , 2 , furthermor e, B ∗ 1 and B ∗ 2 ar e realizing matrices of a n optimal Gaussian rate vector in the e nhanced channel. Theorem 2 states that if there exists the realizing matrices of the bou ndary of R G ( S , N 1 , 2 , 3 ) , then the secre t superpo sition coding with Gaussian c odeboo k is the op timal choice fo r the cap acity r egion of a SADBC. Note that this The orem provides a sufficient condition to ev alu ate th e cap acity r egion of SADBC. Pr oof: The realizing matrices B ∗ 1 and B ∗ 2 are the solution of the fo llowing o ptimization pro blem: max ( B 1 , B 2 ) R G 1 ( B 1 , 2 , N 1 , 2 , 3 ) + µR G 2 ( B 1 , 2 , N 1 , 2 , 3 ) (16) s.t B 1  0 , B 2  0 , B 1 + B 2  S , where µ ≥ 1 . Using the Lagran ge Multip lier m ethod, the above constraint optimization problem is eq uiv ale nt to the following uncon ditional optimiz ation problem : max ( B 1 , B 2 ) R G 1 ( B 1 , 2 , N 1 , 2 , 3 ) + µR G 2 ( B 1 , 2 , N 1 , 2 , 3 ) + T r { B 1 O 1 } + T r { B 2 O 2 } + T r { ( S − B 1 − B 2 ) O 3 } , where O 1 , O 2 , a nd O 3 are positi ve semi-definite t × t matrices such that T r { B ∗ 1 O 1 } = 0 , T r { B ∗ 2 O 2 } = 0 , and T r { ( S − B ∗ 1 − B ∗ 2 ) O 3 } = 0 . As all B ∗ k , k = 1 , 2 , O i , i = 1 , 2 , 3 , and S − B ∗ 1 − B ∗ 2 are positive semi-definite matrices, the n we mu st have B ∗ k O k = 0 , k = 1 , 2 and ( S − B ∗ 1 − B ∗ 2 ) O 3 = 0 . A ccording to the n ecessary KKT condition s, and after som e man ipulations we have: ( B ∗ 1 + N 1 ) − 1 + ( µ − 1)( B ∗ 1 + N 3 ) − 1 + O 1 = µ ( B ∗ 1 + N 2 ) − 1 + O 2 , (17) µ ( B ∗ 1 + B ∗ 2 + N 2 ) − 1 + O 2 = µ ( B ∗ 1 + B ∗ 2 + N 3 ) − 1 + O 3 . (18) W e choo se the no ise covariance matrices of the enhan ced SADBC as the following: N ′ 1 =  N 1 − 1 + O 1  − 1 , (19) N ′ 2 =  ( B ∗ 1 + N 2 ) − 1 + 1 µ O 2  − 1 − B ∗ 1 , N ′ 3 = N 3 . As O 1  0 an d O 2  0 , th en the above choice has the en - hancemen t property . Th e expression  ( B ∗ 1 + N 1 ) − 1 + O 1  − 1 can be written as:  ( B ∗ 1 + N 1 ) − 1 + O 1  − 1 =  ( B ∗ 1 + N 1 ) − 1 ( I + ( B ∗ 1 + N 1 ) O 1 )  − 1 ( a ) = ( I + N 1 O 1 ) − 1 ( B ∗ 1 + N 1 ) − B ∗ 1 + B ∗ 1 = ( I + N 1 O 1 ) − 1  ( B ∗ 1 + N 1 ) − ( I + N 1 O 1 ) B ∗ 1  + B ∗ 1 ( b ) = ( I + N 1 O 1 ) − 1 N 1 + B ∗ 1 =  N 1  N − 1 1 + O 1  − 1 N 1 + B ∗ 1 =  N − 1 1 + O 1  − 1 + B ∗ 1 = B ∗ 1 + N ′ 1 , where ( a ) and ( b ) follows from the fact that B ∗ 1 O 1 = 0 . Similarly , it can be shown that µ ( B ∗ 1 + N 2 ) − 1 + O 2 = µ ( B ∗ 1 + N ′ 2 ) − 1 , Therefo re, accordin g to (17) th e following p roperty ho lds fo r the en hanced channel. ( B ∗ 1 + N ′ 1 ) − 1 + ( µ − 1)( B ∗ 1 + N ′ 3 ) − 1 = µ ( B ∗ 1 + N ′ 2 ) − 1 . Since N ′ 1  N ′ 2  N ′ 3 then, there exists a matrix A suc h that N ′ 2 = ( I − A ) N ′ 1 + AN ′ 3 where A = ( N ′ 2 − N ′ 1 )( N ′ 3 − N ′ 1 ) − 1 . Th erefore, the above equ ation can be written as. ( B ∗ 1 + N ′ 1 ) − 1 + ( µ − 1)( B ∗ 1 + N ′ 3 ) − 1 = µ h ( I − A )( B ∗ 1 + N ′ 1 ) + A ( B ∗ 1 + N ′ 3 ) i − 1 . Let ( I − A )( B ∗ 1 + N ′ 1 ) = α A ( B ∗ 1 + N ′ 3 ) then after some manipulatio ns, th e above equation becomes 1 α I + ( µ − 1 − 1 α ) A = µ α + 1 I . (20) The above eq uation is satisfied by α = 1 µ − 1 which completes the p ropor tionality p roperty . W e can no w prove the r ate conservation pro perty . The expression | B ∗ 1 + N ′ 1 | | N ′ 1 | can be written as follow . | B ∗ 1 + N ′ 1 | | N ′ 1 | = | I | | N ′ 1  B ∗ 1 + N ′ 1  − 1 | (21) = | I | |  B ∗ 1 + N ′ 1 − B ∗ 1   B ∗ 1 + N ′ 1  − 1 | = | I | | I − B ∗ 1  B ∗ 1 + N ′ 1  − 1 | = | I | | I − B ∗ 1 (( B ∗ 1 + N 1 ) − 1 + O 1 ) | ( a ) = | I | | I − B ∗ 1 ( B ∗ 1 + N 1 ) − 1 | = | B ∗ 1 + N 1 | | N 1 | , where ( a ) on ce again follo ws from the f act that B ∗ 1 O 1 = 0 . T o complete the proof of rate conservation, consider the follo wing equalities. | B ∗ 1 + B ∗ 2 + N ′ 2 | | B ∗ 1 + N ′ 2 | = | B ∗ 2  B ∗ 1 + N ′ 2  − 1 + I | | I | (22) = | B ∗ 2  ( B ∗ 1 + N 2 ) − 1 + 1 µ O 2  + I | | I | ( a ) = | B ∗ 1 + B ∗ 2 + N 2 | | B ∗ 1 + N 2 | , where ( a ) fo llows from the fact B ∗ 2 O 2 = 0 . Therefo re, accordin g to (21), (22), and the fact th at N ′ 3 = N 3 , the rate preservation pr operty holds f or the enhanced channel. T o prove the optimality preservation, we need to sho w that ( B ∗ 1 , B ∗ 2 ) are also realizin g matrice s of an op timal Gaussian r ate vector in the enhanced channel. F or that purpo se, n ote that the necessary KKT conditio ns fo r the enhanced channel coincid es with the KKT conditions o f the original channel. W e c an now u se T heorem 2 to prove that R G ( S , N 1 , 2 , 3 ) is the capacity region o f the SADBC. W e follo w Bergmans’ ap- proach [1 3] to prove a co ntradiction . Note that since th e o rig- inal cha nnel is no t prop ortional, we cann ot apply Bergmans’ proof on the o riginal ch annel directly . Here we a pply his pr oof on the en hanced chan nel instead. Theorem 3 Consider a SADBC with positive defi nite n oise covariance matrices ( N 1 , N 2 , N 3 ) . Let C ( S , N 1 , 2 , 3 ) de note the ca pacity r e gion of the S ADBC under a covariance matrix constraint S ≻ 0 . Then, C ( S , N 1 , 2 , 3 ) = R G ( S , N 1 , 2 , 3 ) . Pr oof: The achievability scheme is secre t superposition coding with Gau ssian code book. For the converse proof , we use a contrad iction argument and assume that ther e exists a n achiev able rate vector ( R 1 , R 2 ) whic h is not in the Gaussian region. W e can apply the steps o f Bergmans’ proof of [1 0] on the en hanced ch annel to show that this assumption is impossible. According to the Theor em 1, R 1 is bound ed as follows. R 1 ≤ h ( y 1 | u ) − h ( z | u ) − ( h ( y 1 | x , u ) − h ( z | x , u )) = h ( y 1 | u ) − h ( z | u ) − 1 2  log | N ′ 1 | − log | N ′ 3 | )  Since R 1 > R G 1 ( B 1 , 2 , N ′ 1 , 2 , 3 ) , the above in equality mean s that h ( y 1 | u ) − h ( z | u ) > 1 2  log | B ∗ 1 + N ′ 1 | − lo g | B ∗ 1 + N ′ 3 | )  By the definition of matrix A and since y 1 → y 2 → z forms a Morkov chain, th e re ceiv ed signals z and y 2 can be wr itten as z = y 1 + e n and y 2 = y 1 + A 1 2 e n whe re e n is an in depende nt Ga ussian no ise with cov ariance matrix e N = N ′ 3 − N ′ 1 . According to Costa’ s Entropy Power I nequality and the previous ineq uality , we have h ( y 2 | u ) − h ( z | u ) ≥ t 2 log  | I − A | 1 t 2 2 t ( h ( y 1 | u ) − h ( z | u )) + | A | 1 t )  > t 2 log | I − A | 1 t | B ∗ 1 + N ′ 1 | 1 t | B ∗ 1 + N ′ 3 | 1 t + | A | 1 t ) ! ( a ) = 1 2 log( B ∗ 1 + N ′ 2 ) − 1 2 log( B ∗ 1 + N ′ 3 ) (23) where (a) is due to the p ropor tionality prop erty . The rate R 2 is b ounde d as fo llows R 2 ≤ h ( y 2 ) − h ( z ) − ( h ( y 2 | u ) − h ( z | u )) Using (23) and the fact that R 2 > R G 2 ( B 1 , 2 , N ′ 1 , 2 , 3 ) , the above inequality means that h ( y 2 ) − h ( z ) ≥ R 2 + h ( y 2 | u ) − h ( z | u ) > 1 2 log( B ∗ 1 + B ∗ 2 + N ′ 2 ) − 1 2 log( B ∗ 1 + B ∗ 2 + N ′ 3 ) which is a con tradiction with the fact that Gaussian distribution maximizes h ( x + n 2 ) − h ( x + n 3 ) [14]. I V . C O N C L U S I O N A scen ario where a source nod e wishes to bro adcast two con fidential messages for two respectiv e receivers via a Gaussian MIMO broadc ast ch annel, while a wir e-tapper also receives the transmitted signal v ia ano ther MIMO chann el is considere d. W e considered the secur e vector Gaussian degraded broadcast channel and established its capacity re- gion. Our achievability schem e is the secr et sup erposition of Gaussian codes. I nstead of solving a nonconve x pr oblem, we used the notion o f an en hanced ch annel to show that secret superpo sition of Gau ssian codes is optimal. T o char acterize the secrecy capacity r egion of th e vector G aussian degrad ed broadc ast chan nel, we only en hanced the ch annels for the legitimate re ceiv ers, and the cha nnel o f the ea vesdrop per remains unchanged . R E F E R E N C E S [1] A. W yner , “The Wi re-tap Channel”, Bell System T echnical J ournal , vol. 54, pp. 1355-1387, 1975 [2] I. Csiszar and J. Korner , “Broadcast Channels with Confidential Mes- sages”, IEE E T rans. Info rm. Theory , vol. 24, no. 3, pp. 339-348, May 1978. [3] S. K. L eung-Y an-Ch eong and M. E. Hellman, “Gaussian Wire tap Chan- nel”, IEEE T rans. Inform. Theory , v ol. 24, no. 4, pp. 451-456, July 1978. [4] A. Khisti, G. W ornell, A. W iesel, and Y . Eldar , “On the Gaussian MIMO W iretap Channel” , in P r oc. IEEE Int. Symp. Information Theory (ISIT) , Nice, France, Jun. 2007. [5] A. Khisti and G. W ornell, “Secure Transmission with Multiple Antennas: The M ISOME W iretap Channel” , av ailable at http://arxiv.o rg/PS_cache/arxi v/pdf/0708/0708.4219v1.pdf . [6] G. Ba gherikara m, A. S. Mota hari and A. K. Khandani, “Secure Broadc ast- ing: The Secrec y Rate Regio n”, Allerton Confer ence on Communications, Contr ol and Computin g, , September 2008. [7] H. W eingarten, Y . Steinbe rg, S. Shamai(Shitz), “The Capacit y Region of the Gaussian Multiple-Input Multiple-Out put Broadcast Channel”, IEEE T rans. Inform. Theory , vol. 52, no. 9, pp. 3936-3964, September 2006. [8] G. Bagherikara m, A. S. Motahari and A. K. Khandani, “The Secrecy Capaci ty Regi on of the Gaussian MIMO Broadcast Channel ”, Submitted to IEEE T rans. Inform. Theory , March 200 9, ava ilable at http://arxiv.o rg/PS_cache/arxi v/pdf/0903/0903.3261v1.pdf . [9] E. Ekrem and S. Ulukus, “The Secrec y Capacity Region of the Gaussian MIMO Multi -Recei ver W iretap Channel”, Submitted to IEEE T rans. Inform. Theory , March 200 9, av ailabl e at http://arxiv.o rg/PS_cache/arxi v/pdf/0903/0903.3096v1.pdf . [10] R. L iu, T . L iu, H. V . Poor , and S. Shamai (Shitz),“Mul tiple Input Mul- tiple Output Gaussian Broadcast Channels with Confidentia l Messages”, Submitted to IEEE T rans. Inform. Theory , March 200 9, ava ilable at http://arxiv.o rg/PS_cache/arxi v/pdf/0903/0903.3786v1.pdf . [11] G. Bagheri karam, A. S. Motahari and A. K. Khandani , “Secrec y Capaci ty Region of Gaussian Broadcast Channel ”, presented at the 43rd annual Confer ence on Informat ion Sciences and Systems (CISS 2009 ), , March 2009. [12] T . Liu, S. Shamai(Shitz), “ A Note on t he Secrecy Capacity of the Multi-ant enna Wire tap Channel”, February 2008. av ailabl e at http://arxiv.o rg/PS_cache/arxi v/pdf/0710/0710.4105v1.pdf .. [13] P . P . Bergmans, “ A Simple Con verse for Broadca st Channels with Additi ve White Gaussian Noise”, IEEE T rans. Inform. Theory , vol. IT -20, no. 2, pp. 279-280, March 1974. [14] T . L iu, P . V iswana th, “ An Extremal Inequality Motiv ated by Multiter - minal Information Theoreti c Problems”, IEEE T rans. on Inf . Theory , vol. 53, no. 5, pp. 1839-1851, May 2007.