Capacity-Equivocation Region of the Gaussian MIMO Wiretap Channel

Capacit y-Equiv o cation Region of the Gaussian MIMO Wiretap Channel ∗ Ersen Ekrem Senn ur Ulukus Departmen t of Electrical and Computer Engineering Univ ersit y of Maryland, College P ark, MD 20 7 42 ersen@umd.e du ulukus@umd.e du No vem ber 13, 20 21 Abstract W e study the Gauss ian m ultiple-input m ultiple-output (MIMO) wiretap channel, whic h consists of a tr an s mitter, a legitimate user, and an ea v esdropp er. In this c hannel, the transmitter sends a common message to b oth the legitimat e user and the ea v es- dropp er . In addition to this common message, the legitimat e user receiv es a priv a te message, wh ic h is desired to b e kept hid den as muc h as p ossib le from the eav esdrop- p er. W e obtain the entire capacit y-equiv o cation region of the Gaussian MIMO wir etap c hannel. This region con tains all ac hiev a ble common message, priv ate message, and priv ate message’s equiv o cation (secrecy) rates. In p articular, we show the sufficiency of join tly Gaussian a uxiliary random v ariables and c hannel input to ev aluate the existing single-lette r description of the capacit y-equivocation r egion du e to C siszar-Korner. ∗ This work was supp orted by NSF Grants CCF 04-4 7613, CCF 05 -1484 6, CNS 0 7-163 1 1 a nd CCF 07 - 29127 . 1 1 In tr o duct i on W e consider the G aussian m ultiple-input m ultiple-output (MIMO) wiretap channel, whic h consists of a transmitter, a legitimate user, and an ea v esdropp er. In this channe l, the trans- mitter sends a common message to b oth the legitimate user a nd the eav esdropp er in addition to a priv ate message whic h is directed to only the legitimate user. There is a secrecy concern regarding this priv ate message in the sense that the priv ate message needs to b e k ept secret as m uc h as p ossible fro m the eav esdropp er. The secrecy of the priv ate message is measured b y its equiv o cation at the eav esdropp er. Here, w e obtain the capacity-equiv o cation region of the Gaussian MIMO wiretap channel. This region contains all ac hiev able rate triples ( R 0 , R 1 , R e ), where R 0 denotes the common message ra te, R 1 denotes the priv ate message rate, and R e denotes the priv ate message’s equiv o cation (secrecy) rate. I n fact, this region is kno wn in a single-letter form due t o [1]. In this w ork, w e show that jointly G aussian auxiliary ra ndo m v ariables and channe l input are sufficien t to ev aluate this single-letter description for the capacit y-equiv o cation region of the Gaussian MIMO wiretap c hannel. W e pro v e the sufficiency of the join tly Gaussian auxiliary random v ariables and c hannel input by using c ha nnel enhancemen t [2] and an extremal inequality from [3]. In our pro of, we also use the equiv alence b et w een the G aussian MIMO wiretap c hannel and the Gaussian MIMO wiretap c hannel with public messages [4, Problem 33- c], [5]. In the la t ter channel mo del, the tra nsmitter has three messages, a common, a confiden tial, and a public mess age. The common m essage is sen t to b oth the legitimate user and t he eav esdropp er, while the confidential and public messages are directed to only the legitimate user. Here, the confiden tia l message needs to b e transmitted in p erfect secrecy , whereas there is no secrecy constrain t on t he public message. Since the Gaussian MIMO wiretap c hannel and the Gaussian MIMO wiretap c hannel with public messages are equiv alen t, i.e., there is a one-to-o ne corresp ondence b et w een the capacit y regions of these t w o mo dels, in our pro of, we obtain the capacity region of the Gaussian MIMO wiretap c ha nnel with public messages, whic h, in turn, giv es us t he capacit y-equiv o cation region of the G a ussian MIMO wiretap c hannel. Our result subsumes the f ollo wing previous findings about the capacit y-equiv o cation region of the Gaussian MIMO wiretap channel: i) The secrecy capacit y of this c ha nnel, i.e., ma x R 1 when R 0 = 0 , R e = R 1 , is obtained in [6, 7] fo r the general case, a nd in [8] for the 2-2-1 case. ii) The common and confiden tial rate region under p erfect secrecy , i.e., ( R 0 , R 1 ) region with R e = R 1 , is obtained in [9]. iii) The c apacity -equiv o cation region without a common message, i.e., ( R 1 , R e ) region with R 0 = 0, is obtained in [5]. iv) The capacit y region of the Gaussian MIMO broadcast channel with degraded message sets without a secrecy concern, i.e., ( R 0 , R 1 ) r egion with no consideration on R e , is obtained in [10]. Here, w e obtain the entire ( R 0 , R 1 , R e ) region. 2 2 Discret e Memor yl e ss Wiretap C h annels The discrete memoryless wiretap c hannel consists of a tra nsmitter, a legitimate user and an ea v esdropp er. The c hannel tra nsition pro ba bility is denoted b y p ( y , z | x ), where x ∈ X is the c ha nnel input, y ∈ Y is the legitimat e user’s observ ation, and z ∈ Z is the eav esdropp er’s observ ation. W e consider the following scenario for the discrete memoryless wiretap channel: The transmitter sends a common message to b oth the legitimate user and the eav esdropp er, and a priv ate message t o t he legitimate user whic h is desired to b e k ept hidden a s m uc h as p ossible from the ea v esdropp er. An ( n, 2 nR 0 , 2 nR 1 ) co de for this channel consists of tw o message sets W 0 = { 1 , . . . , 2 nR 0 } , W 1 = { 1 , . . . , 2 nR 1 } , one enco der at the transmitter f : W 0 × W 1 → X n , one deco der a t the legitimate user g u : Y n → W 0 × W 1 , and one deco der at the eav esdropp er g e : Z n → W 0 . The probabilit y of error is defined as P n e = max { P n e,u , P n e,e } , where P n e,u = Pr[ g u ( Y n ) 6 = ( W 0 , W 1 )] , P n e,e = P r[ g e ( Z n ) 6 = W 0 ], and W j is a uniformly distributed random v ariable in W j , j = 0 , 1. The secrecy o f t he legitimate user’s priv ate message is measured b y its equiv o cation at the ea v esdropp er [1, 11], i.e., 1 n H ( W 1 | W 0 , Z n ) (1) A ra te triple ( R 0 , R 1 , R e ) is said to b e achie v able if there exists an ( n, 2 nR 0 , 2 nR 1 ) co de suc h that lim n →∞ P n e = 0, and R e = lim n →∞ 1 n H ( W 1 | W 0 , Z n ) (2) The capacity-eq uiv o cation region of the discrete memoryless wiretap channel is defined as the conv ex closure of all ac hiev able rat e triples ( R 0 , R 1 , R e ), and denoted b y C . The capacit y- equiv o cation region of the discrete memoryless wiretap channel, whic h is obtained in [1], is stated in the follow ing theorem. Theorem 1 ( [1, Theorem 1]) The c ap acity-e quivo c ation r e g ion of the discr ete memoryless wir etap chan n el C is given by the union of r ate triples ( R 0 , R 1 , R e ) sa tisfying 0 ≤ R e ≤ R 1 (3) R e ≤ I ( V ; Y | U ) − I ( V ; Z | U ) (4) R 0 + R 1 ≤ I ( V ; Y | U ) + min { I ( U ; Y ) , I ( U ; Z ) } (5) R 0 ≤ min { I ( U ; Y ) , I ( U ; Z ) } (6) for so me U, V , X such that U → V → X → ( Y , Z ) (7) 3 W e next pro vide an alternat ive description for C . This alternat ive description will arise as the capacity region of a differen t, ho w ev er related, comm unication scenario for the discrete memoryless wiretap c hannel. In this comm unication scenario, the transmitter has three messages, W 0 , W p , W s , where W 0 is sen t to b oth the legitimate user and the ea v esdropp er, and W s , W p are sen t only to the legitimate user. In this scenario, W s needs to b e transmitted in p erfect secrecy , i.e., it needs to satisfy lim n →∞ 1 n I ( W s ; Z n , W 0 ) = 0 (8) and there is no secrecy constrain t on the public message W p . T o distinguish this comm uni- cation scenario from the previous one, w e call the c hannel mo del a rising from this scenario the discrete memoryless wiretap c hannel with public messages. W e note that this alternativ e description f or wiretap c hannels has b een previously conside red in [4, Problem 33-c], [5]. An ( n, 2 nR 0 , 2 nR p , 2 nR s ) co de for this scenario consists of three message sets W 0 = { 1 , . . . , 2 nR 0 } , W p = { 1 , . . . , 2 nR p } , W s = { 1 , . . . , 2 nR s } , one enco der at the transmitter f : W 0 × W p × W s → X n , one deco der at the legitimate us er g u : Y n → W 0 × W p × W s , and one deco der at the eav esdropp er g e : Z n → W 0 . The probabilit y of error is defined as P n e = max { P n e,u , P n e,e } , w here P n e,u = Pr[ g u ( Y n ) 6 = ( W 0 , W p , W s )] and P n e,e = Pr[ g e ( Z n ) 6 = W 0 ]. A rate triple ( R 0 , R p , R s ) is said to b e achiev able if there exists an ( n, 2 nR 0 , 2 nR p , 2 nR s ) co de suc h that lim n →∞ P n e = 0 and ( 8 ) is satisfied. The capacit y region C p of the discrete memoryless wiretap ch annel with public messages is defined as the con v ex closure of all ach iev able rate triples ( R 0 , R p , R s ). The follo wing lemma establishes the equiv alence b et w een C and C p . Lemma 1 ( R 0 , R p , R s ) ∈ C p iff ( R 0 , R s + R p , R s ) ∈ C . The pro of of this lemma is giv en in App endix A. Using Lemma 1 and Theorem 1, w e can express C p as stated in the follo wing theorem. Theorem 2 Th e c ap acity r e gion of the discr ete memoryless wir etap cha nnel with public messages C p is gi ven by the union of r ate triples ( R 0 , R p , R s ) satisfying 0 ≤ R s ≤ I ( V ; Y | U ) − I ( V ; Z | U ) (9) R 0 + R p + R s ≤ I ( V ; Y | U ) + min { I ( U ; Y ) , I ( U ; Z ) } (10) R 0 ≤ min { I ( U ; Y ) , I ( U ; Z ) } (11) for so me ( U, V , X ) such that U → V → X → ( Y , Z ) (12) 4 3 Gaussian MIMO Wiretap Channel The G aussian MIMO wiretap channel is defined b y Y = H Y X + N Y (13) Z = H Z X + N Z (14) where t he c hannel input X is a t × 1 v ector, Y is an r Y × 1 c olumn v ector denoting the legitimate user’s observ ation, Z is an r Z × 1 column v ector denoting the eav esdropp er’s observ ation, H Y , H Z are the c hannel ga in matrices of sizes r Y × t, r Z × t , r esp ectiv ely , and N Y , N Z are Gaussian random v ectors with cov ariance matrices Σ Y , Σ Z 1 , resp ectiv ely , whic h are a ssumed to b e strictly p ositiv e-definite, i.e., Σ Y ≻ 0 , Σ Z ≻ 0 . W e consider a co v ariance constrain t o n the c hannel input as follo ws E  XX ⊤   S (15 ) where S  0 . The capacit y-equiv o cation region of the Gaussian MIMO wiretap channel is denoted b y C ( S ), and is stated in the following theorem. Theorem 3 Th e c ap acity-e quivo c ation r e gi o n of the Gaussian MIMO wir etap cha nnel C ( S ) is gi ven by the union of r ate triples ( R 0 , R 1 , R e ) sa tisfying 0 ≤ R e ≤ 1 2 log | H Y KH ⊤ Y + Σ Y | | Σ Y | − 1 2 log | H Z KH ⊤ Z + Σ Z | | Σ Z | (16) R 0 + R 1 ≤ 1 2 log | H Y KH ⊤ Y + Σ Y | | Σ Y | + min  1 2 log | H Y SH ⊤ Y + Σ Y | | H Y KH ⊤ Y + Σ Y | , 1 2 log | H Z SH ⊤ Z + Σ Z | | H Z KH ⊤ Z + Σ Z |  (17) R 0 ≤ min  1 2 log | H Y SH ⊤ Y + Σ Y | | H Y KH ⊤ Y + Σ Y | , 1 2 log | H Z SH ⊤ Z + Σ Z | | H Z KH ⊤ Z + Σ Z |  (18) for so me p ositive semi-definite matrix K such that 0  K  S . Similar to what we did in the previous section, w e can establish a n alternativ e statemen t for Theorem 3 b y considering the Gaussian MIMO wiretap channe l with public messages, where the legitimate user’s priv ate message is divided in to tw o parts suc h that one part (confidential message) needs to b e transmitted in p erfect secrecy a nd there is no secrecy constrain t on the other pa rt (public message). The capacit y region for this a lternativ e sce nario is denoted b y C p ( S ), and can b e obtained by using Lemma 1 and Theorem 3 as stated in the next theorem. 1 Without lo ss o f g enerality , we can se t Σ Y = Σ Z = I . How ever, we let Σ Y , Σ Z be arbitrar y for ease of presentation. 5 Theorem 4 Th e c ap acity r e g ion of the Gaussian MIMO wir etap channe l with public mes- sages C p ( S ) is given by the union of r ate triples ( R 0 , R p , R s ) sa tisfying 0 ≤ R s ≤ 1 2 log | H Y KH ⊤ Y + Σ Y | | Σ Y | − 1 2 log | H Z KH ⊤ Z + Σ Z | | Σ Z | (19) R 0 + R p + R s ≤ 1 2 log | H Y KH ⊤ Y + Σ Y | | Σ Y | + min  1 2 log | H Y SH ⊤ Y + Σ Y | | H Y KH ⊤ Y + Σ Y | , 1 2 log | H Z SH ⊤ Z + Σ Z | | H Z KH ⊤ Z + Σ Z |  (20) R 0 ≤ min  1 2 log | H Y SH ⊤ Y + Σ Y | | H Y KH ⊤ Y + Σ Y | , 1 2 log | H Z SH ⊤ Z + Σ Z | | H Z KH ⊤ Z + Σ Z |  (21) for so me p ositive semi-definite matrix K such that 0  K  S . W e next define a sub-class of Gaussian MIMO wiretap c hannels called the aligned Gaus- sian MIMO wiretap c hannel, whic h can b e obtained from (1 3)-( 1 4) by setting H Y = H Z = I , Y = X + N Y (22) Z = X + N Z (23) In this w ork, w e first prov e Theorems 3 and 4 for the aligned Gaussian MIMO wiretap c ha nnel. Then, w e establish the capacit y region for the general c hannel mo del in (13)-(14) b y f ollo wing the analysis in Section V.B of [2 ] and Section 7.1 of [12] in conjunction with the capacit y result we obtain for the aligned c hannel. 3.1 Capacit y Region under a P o w er Constrain t W e note tha t the cov aria nce constrain t on the c hannel input in (1 5 ) is a rather general constrain t t hat subsumes the av erage p ow er constraint E  X ⊤ X  = tr  E  XX ⊤  ≤ P (24) as a sp ecial case, see Lemma 1 and Corollary 1 of [2]. Therefore, using Theorem 3, the capacit y-equiv o catio n region arising from the av erage p o w er constraint in (24), C ( P ), can b e found as follows . Corollary 1 The c ap acity-e quivo c ation r e gion of the Gaussian MIMO w ir etap c hannel sub- je ct to an ave r age p ower c onstr aint P , C ( P ) , is gi v en by the union of r ate triples ( R 0 , R 1 , R e ) 6 satisfying 0 ≤ R e ≤ 1 2 log | H Y K 1 H ⊤ Y + Σ Y | | Σ Y | − 1 2 log | H Z K 1 H ⊤ Z + Σ Z | | Σ Z | (25) R 0 + R 1 ≤ 1 2 log | H Y K 1 H ⊤ Y + Σ Y | | Σ Y | + min  1 2 log | H Y ( K 1 + K 2 ) H ⊤ Y + Σ Y | | H Y K 1 H ⊤ Y + Σ Y | , 1 2 log | H Z ( K 1 + K 2 ) H ⊤ Z + Σ Z | | H Z K 1 H ⊤ Z + Σ Z |  (26) R 0 ≤ min  1 2 log | H Y ( K 1 + K 2 ) H ⊤ Y + Σ Y | | H Y K 1 H ⊤ Y + Σ Y | , 1 2 log | H Z ( K 1 + K 2 ) H ⊤ Z + Σ Z | | H Z K 1 H ⊤ Z + Σ Z |  (27) for so me p ositive semi-definite matric es K 1 , K 2 such that tr( K 1 + K 2 ) ≤ P . 4 Pro o f of Theorem 3 for t he Align e d Cas e Instead of pro ving Theorem 3, here we prov e Theorem 4, whic h implies Theorem 3 due to Lemma 1. Achiev ability of the region given in Theorem 4 can b e sho wn b y setting V = X in Theorem 2, and using jointly Gaussian ( U, X = U + T ), where U, T are indep enden t Gaussian random vec tors with cov ar ia nce matrices S − K , K , resp ectiv ely . In the rest of this section, w e provide the conv erse pro of. T o this end, we note tha t since C p ( S ) is con v ex b y definition, it can b e c haracterized b y solving the following optimization problem 2 f ( R ∗ 0 ) = max ( R ∗ 0 ,R p ,R s ) ∈C p ( S ) µ p R p + µ s R s (29) for all µ p ∈ [0 , ∞ ) , µ s ∈ [0 , ∞ ), and all p ossible common message ra t es R ∗ 0 , whic h is b ounded as fo llo ws 0 ≤ R ∗ 0 ≤ min { C Y ( S ) , C Z ( S ) } (30) where C Y ( S ) , C Z ( S ) are the single-user capa cities for the legitimate user and the eav esdrop- p er channels , resp ectiv ely , i.e., C Y ( S ) = 1 2 log | S + Σ Y | | Σ Y | (31) C Z ( S ) = 1 2 log | S + Σ Z | | Σ Z | (32) 2 Although characterizing C p ( S ) by solving the following optimization proble m max ( R 0 ,R p ,R s ) ∈C p ( S ) µ 0 R 0 + µ p R p + µ s R s (28) for all µ 0 , µ p , µ s seems to b e more natural, we find working with (29 ) more conv enient. Here , we characterize C p ( S ) by solving (29 ) for a ll µ p , µ s , for all fixed feas ible R ∗ 0 . 7 W e note that the optimization problem in (29) can b e expressed in the following more explicit form f ( R ∗ 0 ) = max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S µ p R p + µ s R s (33) s . t .      0 ≤ R s ≤ I ( V ; Y | U ) − I ( V ; Z | U ) R ∗ 0 + R p + R s ≤ I ( V ; Y | U ) + min { I ( U ; Y ) , I ( U ; Z ) } R ∗ 0 ≤ m in { I ( U ; Y ) , I ( U ; Z ) } (34) W e also consider the Gaussian rate region R G ( S ) which is defined a s R G ( S ) =          ( R 0 , R p , R s ) : 0 ≤ R s ≤ R s ( K ) R 0 + R p + R s ≤ R s ( K ) + R p ( K ) + min { R 0 Y ( K ) , R 0 Z ( K ) } R 0 ≤ m in { R 0 Y ( K ) , R 0 Z ( K ) } for some 0  K  S          (35) where R s ( K ) , R p ( K ) , R 0 Y ( K ) , R 0 Z ( K ) are give n as follo ws R s ( K ) = 1 2 log | K + Σ Y | | Σ Y | − 1 2 log | K + Σ Z | | Σ Z | (36) R p ( K ) = 1 2 log | K + Σ Z | | Σ Z | (37) R 0 Y ( K ) = 1 2 log | S + Σ Y | | K + Σ Y | (38) R 0 Z ( K ) = 1 2 log | S + Σ Z | | K + Σ Z | (39) T o pro vide the con v erse pro of , i.e., to prov e the optimalit y of jointly Gaussian ( U, V = X ) for the optimization problem in (33)-(34), we will sho w that f ( R ∗ 0 ) = g ( R ∗ 0 ) , 0 ≤ R ∗ 0 ≤ min { C Y ( S ) , C Z ( S ) } (40) where g ( R ∗ 0 ) is defined as g ( R ∗ 0 ) = max ( R ∗ 0 ,R p ,R s ) ∈R G ( S ) µ p R p + µ s R s (41) W e show (40) in tw o parts: • µ s ≤ µ p • µ p < µ s 8 4.1 µ s ≤ µ p In this case, f ( R ∗ 0 ) can b e written as f ( R ∗ 0 ) = max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S µ p ( R p + R s ) (42) s . t . ( R ∗ 0 + R p + R s ≤ I ( X ; Y | U ) + min { I ( U ; Y ) , I ( U ; Z ) } R ∗ 0 ≤ m in { I ( U ; Y ) , I ( U ; Z ) } (43) where we use the fact that µ s ≤ µ p , and the secret message rate R s can b e give n up in fa v or of the priv ate message rate R p . T his optimization problem giv es us the capacit y r egion of the t w o-user G a ussian MIMO broadcast c hannel with degraded message sets, where a common message is sen t to b ot h users, and a priv ate message, on which there is no secrecy constrain t, is sent to one of the tw o users [13]. The optimization pro blem for this case given in (42)-(43) is solv ed in [10] b y sho wing the optimality of join tly G a ussian ( U, X ), i.e., f ( R ∗ 0 ) = g ( R ∗ 0 ). This completes the con v erse pro of for the case µ s ≤ µ p . 4.2 µ p < µ s In this case, w e first study the optimization problem in (41 ). W e rewrite g ( R ∗ 0 ) as follow s g ( R ∗ 0 ) = max 0  K  S R p µ p R p + µ s R s ( K ) (44) s . t . ( R ∗ 0 + R p ≤ R p ( K ) + min { R 0 Y ( K ) , R 0 Z ( K ) } R ∗ 0 ≤ m in { R 0 Y ( K ) , R 0 Z ( K ) } (45) Let ( K ∗ , R ∗ p ) b e the maximizer for this optimization problem. The necess ary KKT conditions that ( K ∗ , R ∗ p ) needs to satisfy ar e give n in the fo llo wing lemma. Lemma 2 K ∗ ne e ds to satisfy ( µ s − µ p λ − β Y )( K ∗ + Σ Y ) − 1 + M = ( µ s − µ p λ + β Z )( K ∗ + Σ Z ) − 1 + M S (46) for so me p ositive semi-definite matric es M , M S such that K ∗ M = MK ∗ = 0 (4 7) ( S − K ∗ ) M S = M S ( S − K ∗ ) = 0 (48) 9 and for some λ = 1 − ¯ λ such that it satisfies 0 ≤ λ ≤ 1 and λ      = 0 if R 0 Y ( K ∗ ) > R 0 Z ( K ∗ ) = 1 if R 0 Y ( K ∗ ) < R 0 Z ( K ∗ ) 6 = 0 , 1 if R 0 Y ( K ∗ ) = R 0 Z ( K ∗ ) (49) and ( β Y , β Z ) ar e gi v en as fol low s ( β Y , β Z ) =          (0 , 0) if R ∗ 0 < min { R 0 Y ( K ∗ ) , R 0 Z ( K ∗ ) } (0 , > 0) if R ∗ 0 = R 0 Z ( K ∗ ) < R 0 Y ( K ∗ ) ( > 0 , 0) if R ∗ 0 = R 0 Y ( K ∗ ) < R 0 Z ( K ∗ ) ( > 0 , > 0) if R ∗ 0 = R 0 Y ( K ∗ ) = R 0 Z ( K ∗ ) (50) R ∗ p ne e ds to satify R ∗ p = R p ( K ∗ ) + min { R 0 Y ( K ∗ ) , R 0 Z ( K ∗ ) } − R ∗ 0 (51) The pr o of of Lemma 2 is giv en in App endix B. W e treat three cases separately: • R ∗ 0 < min { R 0 Y ( K ∗ ) , R 0 Z ( K ∗ ) } • R ∗ 0 = R 0 Y ( K ∗ ) ≤ R 0 Z ( K ∗ ) • R ∗ 0 = R 0 Z ( K ∗ ) < R 0 Y ( K ∗ ) 4.2.1 R ∗ 0 < min { R 0 Y ( K ∗ ) , R 0 Z ( K ∗ ) } In this case, w e hav e β Y = β Z = 0, see (50) . Thus , the KKT condition in (46) reduces to ( µ s − µ p λ )( K ∗ + Σ Y ) − 1 + M = ( µ s − µ p λ )( K ∗ + Σ Z ) − 1 + M S (52) W e first note that K ∗ satisfying (52) ac hiev es the secrec y capacit y of this Gaussian MIMO wiretap channel [14], i.e., R ∗ s = R s ( K ∗ ) (53) = C S ( S ) (54) = max 0  K  S 1 2 log | K + Σ Y | | Σ Y | − 1 2 log | K + Σ Z | | Σ Z | (55) Next, w e define a new cov aria nce matrix ˜ Σ Z as fo llo ws ( µ s − µ p λ )( K ∗ + ˜ Σ Z ) − 1 = ( µ s − µ p λ )( K ∗ + Σ Z ) − 1 + M S (56) 10 This new co v ariance matrix ˜ Σ Z has some useful prop erties whic h a r e listed in the following lemma. Lemma 3 We have the fol lowing facts. • ˜ Σ Z  Σ Z • ˜ Σ Z  Σ Y • ( K ∗ + ˜ Σ Z ) − 1 ( S + ˜ Σ Z ) = ( K ∗ + Σ Z ) − 1 ( S + Σ Z ) The pr o of of Lemma 3 is giv en in App endix C. Thu s, w e ha v e R 0 Z ( K ∗ ) = 1 2 log | S + Σ Z | | K ∗ + Σ Z | (57) = 1 2 log | S + ˜ Σ Z | | K ∗ + ˜ Σ Z | (58) ≥ 1 2 log | S + Σ Y | | K ∗ + Σ Y | (59) = R 0 Y ( K ∗ ) (60) where (5 8) comes from t he third part of Lemma 3, (59) is due to the fact that | A + B + ∆ | | B + ∆ | ≤ | A + B | | B | (61) for A  0 , ∆  0 , B ≻ 0 by noting the second part of Lemma 3. Therefore, w e ha v e R 0 Z ( K ∗ ) ≥ R 0 Y ( K ∗ ) (62) where K ∗ satisfies (52). Using (62) in (51), we find R ∗ p as fo llo ws R ∗ p = R p ( K ∗ ) + R 0 Y ( K ∗ ) − R ∗ 0 (63) W e also note the fo llo wing R ∗ 0 + R ∗ p + R ∗ s = R 0 Y ( K ∗ ) + R p ( K ∗ ) + R s ( K ∗ ) (64) = 1 2 log | S + Σ Y | | Σ Y | (65) = C Y ( S ) (66) No w, we sho w that g ( R ∗ 0 ) = f ( R ∗ 0 ) (67) 11 T o this end, w e assume that g ( R ∗ 0 ) < f ( R ∗ 0 ) (68) whic h implies that there exists a rate triple ( R ∗ 0 , R o p , R o s ) ∈ C p ( S ) such that µ p R ∗ p + µ s R ∗ s < µ p R o p + µ s R o s (69) T o pro v e (67 ), i.e., that (68) is not p ossible, w e note t he following b ounds R o s ≤ C S ( S ) = R ∗ s (70) R o p + R o s ≤ C Y ( S ) − R ∗ 0 = R ∗ p + R ∗ s (71) where (70) comes from (55) a nd the fact that the rate of the confiden tia l mess age, i.e., R s , cannot excee d the sec recy capacity , and (71) is due to (66) and the fact that the sum ra t e R 0 + R p + R s cannot exceed the legitimate user’s single-user capacit y . Th us, in view of µ s > µ p , (7 0)-(71) imply µ p R o p + µ s R o s ≤ µ p R ∗ p + µ s R ∗ s (72) whic h contradicts with (69); pro ving (67). This completes the conv erse pro of for this case. Before starting the proof s of the other t w o cases, w e no w r ecap our pro of for the case R ∗ 0 < min { R 0 Y ( K ∗ ) , R 0 Z ( K ∗ ) } . W e note t ha t w e did no t sho w the optimalit y of Ga ussian signalling directly , instead, w e prov e it indirectly b y showin g t he following g ( R ∗ 0 ) = f ( R ∗ 0 ) (73) First, w e sho w that for the giv en common message rate R ∗ 0 , w e can ac hiev e the secrecy capacit y , i.e., R ∗ s = C S ( S ), see (53)-(55). In other w ords, w e show tha t ( R ∗ 0 , 0 , R ∗ s ) is on the b oundary of the capacit y region C p ( S ). Secondly , w e show that fo r the give n common message ra te R ∗ 0 , ( R ∗ p , R ∗ s ) achie v e the sum capacit y of the public and confiden tial messages, i.e., R ∗ s + R ∗ p is sum rate optimal for the giv en common message ra t e R ∗ 0 , see (64)-(66) and (71). These t w o findings lead to the inequalities in (70)-(71). Finally , we use a time-sharing argumen t for these tw o inequalities in ( 70)-(71) to obtain (73), whic h completes the pro of. 4.2.2 R ∗ 0 = R 0 Y ( K ∗ ) ≤ R 0 Z ( K ∗ ) W e first rewrite the KKT condition in (46) as fo llo ws ( µ s − µ p λ − µ 0 β )( K ∗ + Σ Y ) − 1 + M = ( µ s − µ p λ + µ 0 ¯ β )( K ∗ + Σ Z ) − 1 + M S (74) 12 b y defining µ 0 = β Y + β Z , µ 0 β = β Y , and µ 0 ¯ β = β Z . W e note tha t if R 0 Y ( K ∗ ) < R 0 Z ( K ∗ ), w e ha v e β = λ = 1, if R 0 Y ( K ∗ ) = R 0 Z ( K ∗ ), w e hav e 0 < λ < 1 , 0 < β < 1. The pro of of these t w o cases are v ery similar, a nd w e consider only the case 0 < λ < 1 , 0 < β < 1 , i.e., w e assume R 0 Y ( K ∗ ) = R 0 Z ( K ∗ ). The other case can b e prov ed similarly . Similar t o Section 4.2.1, here also, we prov e the desired iden tity g ( R ∗ 0 ) = f ( R ∗ 0 ) (75) b y contradiction. W e first a ssume that g ( R ∗ 0 ) < f ( R ∗ 0 ) (76) whic h implies that there exists a rate triple ( R ∗ 0 , R o p , R o s ) ∈ C p ( S ) such that µ p R ∗ p + µ s R ∗ s < µ p R o p + µ s R o s (77) where w e define R ∗ s = R s ( K ∗ ). Since the sum rate R 0 + R p + R s needs to be smaller than the legitimate user’s single user capacity , we hav e R ∗ 0 + R o p + R o s ≤ C Y ( S ) (78) On t he other hand, we hav e the f ollo wing R ∗ 0 + R ∗ p + R ∗ s = min { R 0 Y ( K ∗ ) , R 0 Z ( K ∗ ) } + R p ( K ∗ ) + R s ( K ∗ ) (79) = R 0 Y ( K ∗ ) + R p ( K ∗ ) + R s ( K ∗ ) (80) = C Y ( S ) (81) where (79) comes from (51), and (80) is due to our assumption that R ∗ 0 = R 0 Y ( K ∗ ) = R 0 Z ( K ∗ ). Equations (78) a nd (81 ) imply that R o p + R o s ≤ R ∗ p + R ∗ s (82) In the r est of this section, w e prov e that w e hav e R o s ≤ R ∗ s for t he given common message rate R ∗ 0 , whic h, in conjunction with (82), will yield a con tradiction with (7 7); prov ing (75 ). T o this end, w e first define a new cov ar ia nce matr ix ˜ Σ Y as fo llo ws ( µ s − µ p λ )( K ∗ + ˜ Σ Y ) − 1 = ( µ s − µ p λ )( K ∗ + Σ Y ) − 1 + M (83) This new cov a r iance matrix ˜ Σ Y has some useful prop erties whic h are listed in the follow ing lemma. Lemma 4 We have the fol lowing facts. 13 • ˜ Σ Y  Σ Y • ˜ Σ Y  Σ Z • ( K ∗ + ˜ Σ Y ) − 1 ˜ Σ Y = ( K ∗ + Σ Y ) − 1 Σ Y The pro of of this lemma is giv en in App endix D. Us ing this new co v ariance matr ix, w e define a random v ector ˜ Y a s ˜ Y = X + ˜ N Y (84) where ˜ N Y is a Gaussian random ve ctor w ith c ov ariance ma t r ix ˜ Σ Y . Due t o the first and second statements of Lemma 4, w e hav e the follo wing Marko v c hains U → V → X → ˜ Y → Y (85) U → V → X → ˜ Y → Z (86) W e next study the f ollo wing o ptimization problem max ( R 0 ,R p ,R s ) ∈C p ( S ) µ 0 R 0 + ( µ s − µ p λ ) R s = max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S µ 0 min { I ( U ; Y ) , I ( U ; Z ) } + ( µ s − µ p λ ) [ I ( V ; Y | U ) − I ( V ; Z | U )] (87) Since we assume ( R ∗ 0 , R o p , R o s ) ∈ C p ( S ), we hav e the f ollo wing low er b ound for (87) µ 0 R ∗ 0 + ( µ s − µ p λ ) R o s ≤ max ( R 0 ,R p ,R s ) ∈C p ( S ) µ 0 R 0 + ( µ s − µ p λ ) R s (88) 14 No w we solv e the optimization problem in (87) as follow s max ( R 0 ,R p ,R s ) ∈C p ( S ) µ 0 R 0 + ( µ s − µ p λ ) R s = max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S µ 0 min { I ( U ; Y ) , I ( U ; Z ) } + ( µ s − µ p λ ) [ I ( V ; Y | U ) − I ( V ; Z | U )] (89) ≤ max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S µ 0 ¯ β I ( U ; Z ) + µ 0 β I ( U ; Y ) + ( µ s − µ p λ ) [ I ( V ; Y | U ) − I ( V ; Z | U )] (90) ≤ max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S µ 0 ¯ β I ( U ; Z ) + µ 0 β I ( U ; Y ) + ( µ s − µ p λ ) h I ( V ; ˜ Y | U ) − I ( V ; Z | U ) i (91) ≤ max U → X → ( Y , Z ) E [ XX ⊤ ]  S µ 0 ¯ β I ( U ; Z ) + µ 0 β I ( U ; Y ) + ( µ s − µ p λ ) h I ( X ; ˜ Y | U ) − I ( X ; Z | U ) i (92) ≤ µ 0 ¯ β 2 log | S + Σ Z | | K ∗ + Σ Z | + µ 0 β 2 log | S + Σ Y | | K ∗ + Σ Y | + µ s − µ p λ 2 " log | K ∗ + ˜ Σ Y | | ˜ Σ Y | − log | K ∗ + Σ Z | | Σ Z | # (93) = µ 0 ¯ β R 0 Z ( K ∗ ) + µ 0 β R 0 Y ( K ∗ ) + µ s − µ p λ 2 " log | K ∗ + ˜ Σ Y | | ˜ Σ Y | − log | K ∗ + Σ Z | | Σ Z | # (94) = µ 0 ¯ β R 0 Z ( K ∗ ) + µ 0 β R 0 Y ( K ∗ ) + µ s − µ p λ 2  log | K ∗ + Σ Y | | Σ Y | − log | K ∗ + Σ Z | | Σ Z |  (95) = µ 0 ¯ β R 0 Z ( K ∗ ) + µ 0 β R 0 Y ( K ∗ ) + ( µ s − µ p λ ) R s ( K ∗ ) (96) = µ 0 R ∗ 0 + ( µ s − µ p λ ) R ∗ s (97) where (90) comes fr om the fact that 0 < β = 1 − ¯ β < 1, (91)-(92) are due to the Marko v c ha ins in (85)- (86), resp ectiv ely , (9 3) can b e obtained b y using the ana lysis in [9, eqns (30)- (32)], (95) comes from the thir d part of Lemma 4, and (97) is due to our assumption that R ∗ 0 = R 0 Y ( K ∗ ) = R 0 Z ( K ∗ ). Th us, (9 7) implies max ( R 0 ,R p ,R s ) ∈C p ( S ) µ 0 R 0 + ( µ s − µ p λ ) R s ≤ µ 0 R ∗ 0 + ( µ s − µ p λ ) R ∗ s (98) Comparing ( 88) and (98) yields R o s ≤ R ∗ s (99) Using (82 ) and (99) and noting µ s > µ p , we can g et µ p R o p + µ s R o s ≤ µ p R ∗ p + µ s R ∗ s (100) whic h contradicts with (77); pro ving (75). This completes the conv erse pro of for this case. Before providing the pro of for the last case, w e recap our pro of for the case R ∗ 0 = 15 R 0 Y ( K ∗ ) ≤ R 0 Z ( K ∗ ). Sim ilar to Section 4.2.1, here also, w e prov e the optimalit y of Gaussian signalling indirectly , i.e., w e show the desired identit y g ( R ∗ 0 ) = f ( R ∗ 0 ) (101) indirectly . First, we sho w that for the giv en common message rate R ∗ 0 , R ∗ s + R ∗ p is sum rate optimal, i.e., ( R ∗ p , R ∗ s ) ac hiev e the sum capacit y of the public and confiden tial messages , by obtaining (82). Secondly , w e sho w that ( R ∗ 0 , 0 , R ∗ s ) is also on the b oundary of the capacit y region C p ( S ) by obta ining (98). These tw o findings giv e us t he inequalities in (82) and (99). Finally , w e use a time-sharing argumen t for these tw o inequalities in (82) and (99) to establish (1 01), whic h completes the pro of. 4.2.3 R ∗ 0 = R 0 Z ( K ∗ ) < R 0 Y ( K ∗ ) In this case, w e ha ve λ = β Y = 0, see (4 9 )-(50). Hence, the KKT condition in (4 6) reduces to µ s ( K ∗ + Σ Y ) − 1 + M = ( µ s + β Z )( K ∗ + Σ Z ) − 1 + M S (102) W e ag ain prov e the desired iden tity g ( R ∗ 0 ) = f ( R ∗ 0 ) (103) b y contradiction. W e first a ssume that g ( R ∗ 0 ) < f ( R ∗ 0 ) (104) whic h implies that there exists a rate triple ( R ∗ 0 , R o p , R o s ) ∈ C p ( S ) such that µ p R ∗ p + µ s R ∗ s < µ p R o p + µ s R o s (105) In the rest of the section, w e sho w that µ p R ∗ p + µ s R ∗ s ≥ µ p R o p + µ s R o s (106) to reac h a contradiction, and hence, pro v e (103). T o this end, w e define a new cov ariance matrix ˜ Σ Y as fo llo ws µ s ( K ∗ + ˜ Σ Y ) − 1 = µ s ( K ∗ + Σ Y ) − 1 + M (107) This new co v ariance matrix ˜ Σ Y has some useful prop erties listed in the following lemma. Lemma 5 We have the fol lowing facts. 16 • ˜ Σ Y  Σ Y • ˜ Σ Y  Σ Z • ( K ∗ + ˜ Σ Y ) − 1 ˜ Σ Y = ( K ∗ + Σ Y ) − 1 Σ Y The pro of of this lemma is v ery similar to the pro of Lemma 4 , and hence is omitted. Using this new cov ariance matrix ˜ Σ Y , we define a random v ector ˜ Y a s ˜ Y = X + ˜ N Y (108) where ˜ N Y is a Gaussian random ve ctor w ith c ov ariance ma t r ix ˜ Σ Y . Due t o the first and second statements of Lemma 5, w e hav e the follo wing Marko v c hains U → V → X → ˜ Y → Y (109) U → V → X → ˜ Y → Z (110) Next, w e study the follow ing optimization problem max ( R 0 ,R p ,R s ) ∈C p ( S ) ( µ p + β Z ) R 0 + µ p R p + µ s R s (111) W e no t e that since ( R ∗ 0 , R o p , R o s ) ∈ C p ( S ), w e ha ve the follo wing lo w er bound for the opti- mization pro blem in (111) ( µ p + β Z ) R ∗ 0 + µ p R o p + µ s R o s ≤ max ( R 0 ,R p ,R s ) ∈C p ( S ) ( µ p + β Z ) R 0 + µ p R p + µ s R s (112) W e next obtain the max im um for (111 ) . T o this end, we intro duce the follo wing lemma whic h provides an explicit form fo r this optimization problem. Lemma 6 F or µ s > µ p , we have max ( R 0 ,R p ,R s ) ∈C p ( S ) ( µ p + β Z ) R 0 + µ p R p + µ s R s = max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S ( µ p + β Z ) min { I ( U ; Y ) , I ( U ; Z ) } + µ p I ( V ; Z | U ) + µ s [ I ( V ; Y | U ) − I ( V ; Z | U )] (113) The pr o of of this lemma is giv en in App endix E. Next we in tro duce the fo llowing extremal inequalit y from [3], whic h will b e used subse- quen tly in the solution of (1 1 3). Lemma 7 ( [3, Corollary 4]) L et ( U, X ) b e an arbitr arily c orr el a te d r andom v e ctor, wher e X ha s a c ovarianc e c onstr aint E  XX ⊤   S and S ≻ 0 . L et N 1 , N 2 b e Gaussia n r and o m 17 ve ctors with c ovarianc e matric es Σ 1 , Σ 2 , r esp e ctively. They ar e indep enden t of ( U, X ) . F ur- thermor e, Σ 1 , Σ 2 satisfy Σ 1  Σ 2 . Assume that ther e exists a c ovarianc e matrix K ∗ such that K ∗  S and ν ( K ∗ + Σ 1 ) − 1 = γ ( K ∗ + Σ 2 ) − 1 + M S (114) wher e ν ≥ 0 , γ ≥ 0 and M S is p ositive semi-definite m a trix such that ( S − K ∗ ) M S = 0 . Then, for any ( U, X ) , we have ν h ( X + N 1 | U ) − γ h ( X + N 2 | U ) ≤ ν 2 log | (2 π e )( K ∗ + Σ 1 ) | − γ 2 log | ( 2 π e )( K ∗ + Σ 2 ) | (115) No w we use Lemma 7. T o this end, w e note that using (107) in (102), w e get µ s ( K ∗ + ˜ Σ Y ) − 1 = ( µ s + β Z )( K ∗ + Σ Z ) − 1 + M S (116) In view of (116) and the fact that ˜ Σ Y  Σ Z , Lemma 7 implies µ s h ( ˜ Y | U ) − ( µ s + β Z ) h ( Z | U ) ≤ µ s 2 log | (2 π e )( K ∗ + ˜ Σ Y ) | − µ s + β Z 2 log | (2 π e )( K ∗ + Σ Z ) | (117) 18 W e now consider the maximization in (113) as follows max ( R 0 ,R p ,R s ) ∈C p ( S ) ( µ p + β Z ) R 0 + µ p R p + µ s R s = max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S ( µ p + β Z ) min { I ( U ; Y ) , I ( U ; Z ) } + µ p I ( V ; Z | U ) + µ s [ I ( V ; Y | U ) − I ( V ; Z | U )] (118) ≤ max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S ( µ p + β Z ) I ( U ; Z ) + µ p I ( V ; Z | U ) + µ s [ I ( V ; Y | U ) − I ( V ; Z | U )] (1 19) ≤ max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S ( µ p + β Z ) I ( U ; Z ) + µ p I ( X ; Z | U ) + µ s [ I ( V ; Y | U ) − I ( V ; Z | U )] (120) ≤ max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S ( µ p + β Z ) I ( U ; Z ) + µ p I ( X ; Z | U ) + µ s h I ( V ; ˜ Y | U ) − I ( V ; Z | U ) i (121) ≤ max U → X → ( Y , Z ) E [ XX ⊤ ]  S ( µ p + β Z ) I ( U ; Z ) + µ p I ( X ; Z | U ) + µ s h I ( X ; ˜ Y | U ) − I ( X ; Z | U ) i (122) = max U → X → ( Y , Z ) E [ XX ⊤ ]  S ( µ p + β Z ) h ( Z ) + µ s h ( ˜ Y | U ) − ( µ s + β Z ) h ( Z | U ) − µ s 2 log | ( 2 π e ) ˜ Σ Y | + µ s − µ p 2 log | (2 π e ) Σ Z | (123) ≤ µ p + β Z 2 log | ( 2 π e )( S + Σ Z ) | + max U → X → ( Y , Z ) E [ XX ⊤ ]  S µ s h ( ˜ Y | U ) − ( µ s + β Z ) h ( Z | U ) − µ s 2 log | (2 π e ) ˜ Σ Y | + µ s − µ p 2 log | ( 2 π e ) Σ Z | (124) ≤ µ p + β Z 2 log | ( 2 π e )( S + Σ Z ) | + µ s 2 log | (2 π e )( K ∗ + ˜ Σ Y ) | − µ s + β Z 2 log | ( 2 π e )( K ∗ + Σ Z ) | − µ s 2 log | (2 π e ) ˜ Σ Y | + µ s − µ p 2 log | ( 2 π e ) Σ Z | (125) = µ p + β Z 2 log | S + Σ Z | | K ∗ + Σ Z | + µ p 2 log | K ∗ + Σ Z | | Σ Z | + µ s 2 " log | K ∗ + ˜ Σ Y | | ˜ Σ Y | − log | K ∗ + Σ Z | | Σ Z | # (126) = ( µ p + β Z ) R 0 Z ( K ∗ ) + µ p R p ( K ∗ ) + µ s 2 " log | K ∗ + ˜ Σ Y | | ˜ Σ Y | − log | K ∗ + Σ Z | | Σ Z | # (127) = ( µ p + β Z ) R 0 Z ( K ∗ ) + µ p R p ( K ∗ ) + µ s 2  log | K ∗ + Σ Y | | Σ Y | − log | K ∗ + Σ Z | | Σ Z |  (128) = ( µ p + β Z ) R 0 Z ( K ∗ ) + µ p R p ( K ∗ ) + µ s R s ( K ∗ ) (129) = ( µ p + β Z ) R ∗ 0 + µ p R ∗ p + µ s R ∗ s (130) 19 where (1 19) is due to min { a, b } ≤ a , (12 0) is due to the Mark o v ch ain in (110), (121)-(1 22) come from the Mark o v c hains in (109)-(110), resp ectiv ely , (124) is due to t he maxim um en tr o p y theorem [15], (12 5) comes from (117), and (128) is due to the third part of Lemma 5. Comparing ( 130) and (112) yields µ p R o p + µ s R o s ≤ µ p R ∗ p + µ s R ∗ s (131) whic h contradicts with o ur assumption in (105); implying (1 03). T his completes the con ve rse pro of for this case. W e no t e that con trary to Sections 4.2 .1 and 4.2.2, here w e prov e the optimalit y of G aus- sian signalling, i.e., g ( R ∗ 0 ) = f ( R ∗ 0 ) (132) directly . In other w ords, to sho w (1 32), w e did not find any o ther po ints on the b oundary of the capacit y r egio n C p ( S ) and did not hav e to use a time-sharing argument b et w een these p oin ts to reac h (132). (This w as our strategy in Sections 4.2 .1 and 4.2.2.) Instead, w e define a new optimization problem giv en in (113) whose solution yields (132). 5 Pro o f of Theorem 3 for t he Gener al Case The ac hiev ability of the region giv en in Theorem 3 can b e sho wn b y computing the region in Theorem 1 with the follo wing se lection of ( U, V , X ): V = X , X = U + T where T , U are indep enden t Ga ussian ra ndom ve ctors with co v ariance matrices K , S − K , respectiv ely , U = U . In the rest of this section, w e consider the con v erse pro of. W e first note that follo wing the approac hes in Section V.B of [2] and Section 7.1 o f [12], it can b e shown that a new Ga ussian MIMO wiretap channe l can b e constructed fro m an y Gaussian MIMO wiretap c hannel de scrib ed b y (13)- (14) such that the new channel has the same capacity - equiv o cation region with the original one and in the new c hannel, b oth the legitimate user and the eav esdropp er hav e the same num b er of ante nnas as the transmitter, i.e., r Y = r Z = t . Th us, without loss of generalit y , w e assume that r Y = r Z = t . W e next apply singular-v alue decomp osition to the c hannel gain matrices H Y , H Z as fo llo ws H Y = U Y Λ Y V ⊤ Y (133) H Z = U Z Λ Z V ⊤ Z (134) 20 where U Y , U Z , V Y , V Z are t × t orthogo nal matrices, and Λ Y , Λ Z are diagonal matrices. W e no w define a new Gaussian MIMO wiretap ch annel as follo ws Y = H Y X + N Y (135) Z = H Z X + N Z (136) where H Y , H Z are defined as H Y = U Y ( Λ Y + α I ) V ⊤ Y (137) H Z = U Z ( Λ Z + α I ) V ⊤ Z (138) for some α > 0. W e denote the capa city-equiv o cation region of the Gaussian MIMO wire- tap ch annel defined in (1 35)-(136) b y C α ( S ). Since H Y , H Z are inv ertible, the capacit y- equiv o cation region of the c hannel in (135)-(13 6) is equal to the capacit y-equiv o cation r egio n of the following aligned c hannel Y = X + H − 1 Y N Y (139) Z = X + H − 1 Z N Z (140) Th us, using the capacit y result f or the alig ned case, whic h w as prov ed in the previous section, w e obtain C α ( S ) a s the union of rate triples ( R 0 , R 1 , R e ) satisfying 0 ≤ R e ≤ 1 2 log    H Y KH ⊤ Y + Σ Y    | Σ Y | − 1 2 log    H Z KH ⊤ Z + Σ Z    | Σ Z | (141) R 0 + R 1 ≤ 1 2 log    H Y KH ⊤ Y + Σ Y    | Σ Y | + min    1 2 log    H Y SH ⊤ Y + Σ Y       H Y KH ⊤ Y + Σ Y    , 1 2 log    H Z SH ⊤ Z + Σ Z       H Z KH ⊤ Z + Σ Z       (142) R 0 ≤ min    1 2 log    H Y SH ⊤ Y + Σ Y       H Y KH ⊤ Y + Σ Y    , 1 2 log    H Z SH ⊤ Z + Σ Z       H Z KH ⊤ Z + Σ Z       (143) for some p ositiv e semi-definite matrix K such that 0  K  S . W e next obtain an outer b ound for the capacit y-equiv o cat io n r egio n of the original Gaus- sian MIMO wiretap channe l in (13)-(14) in terms of C α ( S ). T o this end, we first note the follo wing Mark o v c hains X → Y → Y (144) X → Z → Z (145) 21 whic h imply that if the messages ( W 0 , W 1 ) with r ates ( R 0 , R 1 ) are transmitted with a v an- ishingly small probability of error in the original G aussian MIMO wiretap channel giv en b y (13)-(14), they will b e transmitted with a v anishingly small probabilit y of error in the new Gaussian MIMO wiretap channel giv en by (135)-(1 36) as w ell. How ev er, as opp osed to the rates R 0 , R 1 , we cannot immediately conclude that if an equiv o cation rate R e is ac hiev able in the original G aussian MIMO wiretap c hannel giv en in (1 3)-(14), it is also a c hiev able in the new Gaussian MIMO wiretap c hannel in (135 )-(136). The reason for this is that b oth the legitimate user’s and the ea v esdropp er’s c hannel gain matr ices are enhanced in the new chan- nel giv en by (135)-(136), see (137)-(138) and/or (144)-(145), and consequen tly , it is not clear what the o v erall effect of these t w o enhancemen ts o n the equiv o cation rate will b e. Ho wev er, in the sequel, w e sho w that if ( R 0 , R 1 , R e ) ∈ C ( S ), then w e ha v e ( R 0 , R 1 , R e − γ ) ∈ C α ( S ). This will let us write do wn an outer bound for C ( S ) in t erms of C α ( S ). T o this end, we note that if ( R 0 , R 1 , R e ) ∈ C ( S ), w e need to hav e a rando m v ector ( U, V , X ) suc h that the inequalities giv en in Theorem 1 hold. Assume tha t w e use the same r andom v ector ( U, V , X ) for the new Gaussian MIMO wiretap channel in (135)-(136), a nd achiev e the rate triple ( R 0 , R 1 , R e ). Due to the Mark ov chains in (144)-(145), w e already ha v e R 1 ≤ R 1 , R 0 ≤ R 0 . F urthermore, fo llowing the analysis in Section 4 of [9], we can b ound the g ap b etw een R e and R e , i.e., γ , as fo llo ws γ = R e − R e ≤ 1 2 log    H Z SH ⊤ Z + Σ Z    | Σ Z | − 1 2 log | H Z SH ⊤ Z + Σ Z | | Σ Z | (146) Th us, we hav e C ( S ) ⊆ C α ( S ) + G ( S ) (147) where G ( S ) is G ( S ) =    (0 , 0 , R e ) : 0 ≤ R e ≤ 1 2 log    H Z SH ⊤ Z + Σ Z    | Σ Z | − 1 2 log | H Z SH ⊤ Z + Σ Z | | Σ Z |    (148) T aking α → 0 in (147), w e get C ( S ) ⊆ lim α → 0 C α ( S ) (149) where we use t he fact that lim α → 0 1 2 log    H Z SH ⊤ Z + Σ Z    | Σ Z | − 1 2 log | H Z SH ⊤ Z + Σ Z | | Σ Z | = 0 (150) whic h fo llows from the con tinu ity of log | · | in p ositiv e semi-definite matrices, and the fact 22 that lim α → 0 H Z = H Z . Finally , w e note t hat lim α → 0 C α ( S ) (151) con v erges to the regio n giv en in Theorem 3 due to the contin uity of log | · | in p ositiv e semi-definite matrices and lim α → 0 H Y = H Y , lim α → 0 H Z = H Z ; completing the pro of. 6 Conclus ions W e study the Gaussian MIMO wiretap channe l in whic h a common message is sent to b oth the legitimate use r and the ea ves dropp er in addition to the priv ate message sen t o nly to the legitimate user. W e first establish a n equiv alence betw een this o r ig inal definition of the wiretap channel and the wiretap c hannel with public messages, in whic h the priv ate message is divided in to tw o parts as the confiden tial message, whic h needs to b e transmitted in p erfect secrecy , and public message, on whic h there is no secrecy constrain t . W e next obta in capacit y regions fo r b oth cases. W e sho w that it is sufficien t to consider join tly Ga ussian auxiliary random v ariables and c hannel input to ev aluat e the single-letter description of the capacit y-equiv o catio n region due to [1]. W e prov e this by using c hannel enhancemen t [2 ] and a n extremal inequalit y from [3]. A Pro of of Lemma 1 The pro of of this lemma fo r R 0 = 0 is outlined in [4 , Pro blem 33-c], [5]. W e extend their pro of to t he g eneral case of interes t here. W e first note the inclusion C p ⊆ C , whic h follows from the fa ct that if ( R 0 , R p , R s ) ∈ C p , w e can attain the rate triple ( R 0 , R 1 = R s + R p , R e = R s ), i.e., ( R 0 , R s + R p , R s ) ∈ C . T o show the rev erse inclusion, we use the a c hiev ability pro of for Theorem 1 giv en in [1]. According to this a c hiev able sche me, W 1 can b e divided into t w o parts a s W 1 = ( W p , W s ) with rates ( R 1 − R e , R e ), resp ectiv ely , and w e hav e H ( W 1 | W 0 , Z n ) = H ( W p , W s | Z n , W 0 ) (152) ≥ H ( W s | Z n , W 0 ) (153) ≥ H ( W s ) − nγ n (154) for some γ n whic h satisfies lim n →∞ γ n = 0. Hence, using t his capacit y achie ving sc heme for C , w e can attain the rate tr iple ( R 0 , R p = R 1 − R e , R s = R e ) ∈ C p . This implies C ⊆ C p ; completing the pro of of the lemma. 23 B Pro o f of Lemma 2 Since t he pr o gram in (44)-( 45) is not necessarily con v ex, the KKT conditions a r e necessary but not sufficien t. The Lagrangia n for t his optimization problem is giv en b y L = µ s R s ( K ) + µ p R p + λ Y [ R p ( K ) + R 0 Y ( K ) − R p − R ∗ 0 ] + λ Z [ R p ( K ) + R 0 Z ( K ) − R p − R ∗ 0 ] + β Y [ R 0 Y ( K ) − R ∗ 0 ] + β Z [ R 0 Z ( K ) − R ∗ 0 ] + tr( KM ) + tr(( S − K ) M S ) (155) where M , M S are p ositiv e semi-definite ma t r ices, and λ Y ≥ 0 , λ Z ≥ 0, β Y ≥ 0 , β Z ≥ 0. The necessary KKT conditions that they need to satisfy a re give n as follo ws ∂ L ∂ R p | R p = R ∗ p = 0 (156) ∇ K L | K = K ∗ = 0 (157) tr( K ∗ M ) = 0 (158) tr(( S − K ∗ ) M S ) = 0 (159) λ Y  R p ( K ∗ ) + R 0 Y ( K ∗ ) − R ∗ p − R ∗ 0  = 0 (160) λ Z  R p ( K ∗ ) + R 0 Z ( K ∗ ) − R ∗ p − R ∗ 0  = 0 (161) β Y ( R 0 Y ( K ∗ ) − R ∗ 0 ) = 0 (162) β Z ( R 0 Z ( K ∗ ) − R ∗ 0 ) = 0 (163) The first K K T condition in (156) implies λ Y + λ Z = µ p . W e define λ Y = µ p λ, λ Z = µ p ¯ λ and consequen tly , w e ha v e 0 ≤ ¯ λ = 1 − λ ≤ 1. The second KKT conditio n in (157 ) implies (46). Since tr( AB ) = tr( BA ) and tr( AB ) ≥ 0 for A  0 , B  0 , (158)-(159) imply (47)-(48). T he KKT conditions in (1 60)-(161) imply (51). F urthermore, t he KKT conditions in (160)-(161) state the conditions that if R 0 Y ( K ∗ ) > R 0 Z ( K ∗ ), λ = 0, if R 0 Y ( K ∗ ) < R 0 Z ( K ∗ ), λ = 1, and if R 0 Y ( K ∗ ) = R 0 Z ( K ∗ ), λ is arbitrary , i.e., 0 < λ < 1. Similarly , the KKT c onditions in (162)-(163) imply (50 ). C Pro o f of Lemm a 3 W e note the following iden tit ies ( µ s − µ p λ )( K ∗ + ˜ Σ Z ) − 1 = ( µ s − µ p λ )( K ∗ + Σ Z ) − 1 + M S (164) ( µ s − µ p λ )( K ∗ + ˜ Σ Z ) − 1 = ( µ s − µ p λ )( K ∗ + Σ Y ) − 1 + M (165) 24 where (164) is due to (56), and (16 5) is obtained by plugging (164) into (52). Since M  0 , M S  0 , (1 64)-(165) implies ( µ s − µ p λ )( K ∗ + ˜ Σ Z ) − 1  ( µ s − µ p λ )( K ∗ + Σ Z ) − 1 (166) ( µ s − µ p λ )( K ∗ + ˜ Σ Z ) − 1  ( µ s − µ p λ )( K ∗ + Σ Y ) − 1 (167) Using the fact t ha t for A ≻ 0 , B ≻ 0 , if A  B , then A − 1  B − 1 in (166)-(1 6 7), w e can get the first and second parts of Lemma 3 . W e next sho w the t hir d part o f Lemma 3 as follows ( K ∗ + ˜ Σ Z ) − 1 ( S + ˜ Σ Z ) = I + ( K ∗ + ˜ Σ Z ) − 1 ( S − K ∗ ) (168) = I +  ( K ∗ + Σ Z ) − 1 + 1 µ s − µ p λ M S  ( S − K ∗ ) (169) = I + ( K ∗ + Σ Z ) − 1 ( S − K ∗ ) (170) = ( K ∗ + Σ Z ) − 1 ( S + Σ Z ) (171) where (1 69) is due to (164), and (170) comes from (48). The pro of is complete. D Pro o f of Lemma 4 W e note the following ( µ s − µ p λ )( K ∗ + ˜ Σ Y ) − 1 = ( µ s − µ p λ )( K ∗ + Σ Y ) − 1 + M (172) ( µ s − µ p λ )( K ∗ + ˜ Σ Y ) − 1 = ( µ s − µ p λ + µ 0 ¯ β )( K ∗ + Σ Z ) − 1 + µ 0 β ( K ∗ + Σ Y ) − 1 + M S (173) where (172) is ( 83), and (1 7 3) comes from plugging (172) into (74). Since M  0 , (172) implies ( µ s − µ p λ )( K ∗ + ˜ Σ Y ) − 1  ( µ s − µ p λ )( K ∗ + Σ Y ) − 1 (174) Using the fact that for A ≻ 0 , B ≻ 0 , if A  B , then A − 1  B − 1 in (174) yields the first statemen t of the lemma. Since 0 ≤ β = 1 − ¯ β ≤ 1 and M S  0 , (173) implies ( µ s − µ p λ )( K ∗ + ˜ Σ Y ) − 1  ( µ s − µ p λ )( K ∗ + Σ Z ) − 1 (175) 25 Using the fact tha t for A ≻ 0 , B ≻ 0 , if A  B , then A − 1  B − 1 in (175) yields the second statemen t of the lemma. W e next consider the third statemen t of this lemma as follows ( K ∗ + ˜ Σ Y ) − 1 ˜ Σ Y = I − ( K ∗ + ˜ Σ Y ) − 1 K ∗ (176) = I −  ( K ∗ + Σ Y ) − 1 + 1 µ s − µ p λ M  K ∗ (177) = I − ( K ∗ + Σ Y ) − 1 K ∗ (178) = ( K ∗ + Σ Y ) − 1 Σ Y (179) where (1 77) is due to (172) and (1 7 8) comes from ( 4 7). E Pro of of Lemma 6 The o ptimization pro blem in (113) can b e written as max U → V → X → ( Y , Z ) E [ XX ⊤ ]  S µ s R s + µ p R p + ( µ p + β Z ) R 0 (180) s . t .      0 ≤ R s ≤ I ( V ; Y | U ) − I ( V ; Z | U ) R s + R p + R 0 ≤ I ( V ; Y | U ) + min { I ( U ; Y ) , I ( U ; Z ) } R 0 ≤ min { I ( U ; Y ) , I ( U ; Z ) } (181) F or a giv en ( U, V , X ), we can rewrite the cost function in (180 ) as follow s µ s R s + µ p R p + ( µ p + β Z ) R 0 ≤ µ s R s + µ p [ I ( V ; Y | U ) + min { I ( U ; Y ) , I ( U ; Z ) } − R s − R 0 ] + ( µ p + β Z ) R 0 (182) = ( µ s − µ p ) R s + µ p [ I ( V ; Y | U ) + min { I ( U ; Y ) , I ( U ; Z ) } ] + β Z R 0 (183) ≤ ( µ s − µ p )[ I ( V ; Y | U ) − I ( V ; Z | U )] + µ p [ I ( V ; Y | U ) + min { I ( U ; Y ) , I ( U ; Z ) } ] + β Z R 0 (184) = µ s [ I ( V ; Y | U ) − I ( V ; Z | U )] + µ p [ I ( V ; Z | U ) + min { I ( U ; Y ) , I ( U ; Z ) } ] + β Z R 0 (185) ≤ µ s [ I ( V ; Y | U ) − I ( V ; Z | U )] + µ p [ I ( V ; Z | U ) + min { I ( U ; Y ) , I ( U ; Z ) } ] + β Z min { I ( U ; Y ) , I ( U ; Z ) } (186) = µ s [ I ( V ; Y | U ) − I ( V ; Z | U )] + µ p I ( V ; Z | U ) + ( µ p + β Z ) min { I ( U ; Y ) , I ( U ; Z ) } (187) where (182) comes from the sec ond constrain t in (181), (18 4) is due to the first constrain t in (181) and the assumption µ s > µ p , and (1 86) comes from the third constrain t in (1 81). 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