Table Of ContentMIMO Gaussian Broadcast Channels with
Confidential and Common Messages
Ruoheng Liu, Tie Liu, H. Vincent Poor, and Shlomo Shamai (Shitz)
Abstract—This paper considers the problem of secret com- H1 Z1n
munication over a two-receiver multiple-input multiple-output × + Y1n Receiver 1 (W^0,W^1)
(MIMO) Gaussian broadcast channel. The transmitter has two 1n_I(W2;Y1n )→0
independent,confidentialmessagesandacommonmessage.Each Xn
of the confidential messages is intended for one of the receivers (W0,W1,W2) Transmitter H2 Z2n
10 but needs to be kept perfectly secret from the other, and the × + Y2n Receiver 2 (W^0,W^2)
0 common message is intended for both receivers. It is shown 1n_I(W1;Y2n )→0
that a natural scheme that combines secret dirty-paper coding
2
withGaussiansuperpositioncodingachievesthesecrecycapacity Fig.1. Channelmodel
n region.Toprovethisresult,achannel-enhancementapproachand
a an extremal entropy inequality of Weingarten et al. are used.
J
6 As shown in Fig. 1, we consider the communication sce-
I. INTRODUCTION
1 nario in which there is a common message W and two
0
In this paper, we study the problem of secret communi- independent, confidential messages W1 and W2 at the trans-
]
T cation over a multiple-input multiple-output (MIMO) Gaus- mitter. Message W0 is intended for both receivers. Message
I sian broadcast channel with two receivers. The transmitter is W1 is intendedforreceiver1 butneedsto bekeptsecretfrom
s. equipped with t transmit antennas, and receiver k, k = 1,2, receiver 2, and message W2 is intended for receiver 1 but
c is equipped with r receive antennas. A discrete-time sample needs to be kept secret from receiver 2. The confidentiality
[ k
of the channel at time m can be written as ofthe messagesatthe unintendedreceiversis measuredusing
1 the normalized information-theoreticcriteria [1]
v Yk[m]=HkX[m]+Zk[m], k =1,2 (1) 1 1
6 I(W ;Yn)→0 and I(W ;Yn)→0 (3)
0 where Hk is the (real) channel matrix of size rk × t, and n 1 2 n 2 1
28 {adZdki[tmive]}mvecistoarnGinaduespsieanndennotiasendpirdoecnetsiscawllyithdizsterriobumteeda(ni.ia.dn.d) warheetraekeYnknas:=the(Ybklo[1ck],.l.e.n,gYthkn[n]→), k∞=. T1h,e2,goaanldisthteolcimhaitrs-
.
1 identity covariance matrix. The channel input {X[m]}m is acterize the entire secrecy rate region Cs[SBC](H1,H2,S) =
0 subject to the matrix power constraint: {(R ,R ,R )} that can be achieved by any coding scheme,
0 0 1 2
n whereR ,R andR arethecommunicationratescorrespond-
1 1 0 1 2
v: n (X[m]X⊺[m])(cid:22)S (2) ing to the common message W0, the confidential message
m=1 W destined for receiver 1 and the confidential message W
i X 1 2
X where S is a positive semidefinite matrix, and “(cid:22)” denotes destined for receiver 2, respectively.
r “less than or equal to” in the positive semidefinite partial In recent years, MIMO secret communication has been an
a
ordering between real symmetric matrices. Note that (2) is active area of research. Several special cases of the commu-
a rather general power constraint that subsumes many other nication problem that we consider here have been studied in
important power constraints including the average total and the literature. Specifically,
per-antenna power constraints as special cases.
• With only one confidential message (W1 or W2), the
problem was studied as the MIMO Gaussian wiretap
This research was supported bytheUnited States National Science Foun-
channel. The secrecy capacity of the MIMO Gaussian
dation under Grant CNS-09-05398, CCF-08-45848 and CCF-09-16867, by
the Air Force Office of Scientific Research under Grant FA9550-08-1-0480, wiretap channel was characterized in [2] and [3] under
by the European Commission in the framework of the FP7 Network of the matrix power constraint (2) and in [4] and [5] under
Excellence in Wireless Communications NEWCOM++, and by the Israel
an average total power constraint.
Science Foundation.
Ruoheng Liu and H. Vincent Poor are with the Department of Electri- • With bothconfidentialmessagesW1 and W2 butwithout
cal Engineering, Princeton University, Princeton, NJ 08544, USA (e-mail: the common message W , the problem was studied in
0
{rliu,poor}@princeton.edu).
[6] for the multiple-inputsingle-output(MISO) case and
Tie Liu is with the Department of Electrical and Computer Engineer-
ing, Texas A&M University, College Station, TX 77843, USA (e-mail: in [7] for general MIMO case. Rather surprisingly, it
[email protected]). was shown in [7] that, underthe matrix powerconstraint
Shlomo Shamai (Shitz) is with the Department of Electrical Engineering,
(2), both confidential messages can be simultaneously
Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel
(e-mail: [email protected]). communicatedattheirrespectedmaximumsecrecyrates.
• With only one confidential message (W1 or W2) and Theorem 2 (Aligned MIMO Gaussian broadcast channel):
the common message W , the secrecy capacity region ThesecrecycapacityregionC[SBC](N ,N ,S)of the aligned
0 s 1 2
of the channel was characterized in [8] using a channel- MIMO Gaussian broadcast channel (5) with a common
enhancement approach [9] and an extremal entropy in- messageW and confidentialmessagesW andW underthe
0 1 2
equality of Weingarten et al. [10]. matrix powerconstraint(2) is givenby the set of nonnegative
The main contribution of this paper is to provide a precise rate triples (R ,R ,R ) such that
0 1 2
characterization of the secrecy capacity region of the MIMO
1 S+N 1 S+N
Gaussian broadcast channel with a more complete message R ≤min log 1 , log 2
0 2 (S−B )+N 2 (S−B )+N
setthatincludesacommonmessageW andtwoindependent, (cid:26) (cid:12) 0 1(cid:12) (cid:12) 0 2(cid:12)(cid:27)
0 1 B +(cid:12)N 1 B(cid:12) +N (cid:12) (cid:12)
ecnohnafindceenmtiaelnmt aersgsuagmeesnWt o1fa[n8d].W2bygeneralizingthechannel- R1 ≤ 2log(cid:12) 1N(cid:12)(cid:12)1 1(cid:12)− 2log(cid:12) 1(cid:12)(cid:12)N2 2(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)
1 (cid:12)(S−B )(cid:12)+N (cid:12)1 (S−(cid:12) B )+N
II. MAIN RESULT R ≤ log(cid:12) 0 (cid:12) 2 −(cid:12) log (cid:12) 0 1
2 2 (cid:12) B +N(cid:12) (cid:12)2 (cid:12)B +N
Themainresultofthepaperissummarizedinthefollowing (cid:12) 1 2 (cid:12) (cid:12) 1 1 (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (6)
theorem. (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Theorem 1 (General MIMO Gaussian broadcast channel): for some B0 (cid:23)0, B1 (cid:23)0 and B0+B1 (cid:22)S.
Thesecrecycapacityregionofthe MIMOGaussianbroadcast Next, we prove Theorem 2 by generalizing the channel-
channel (1) with a common message W (intended for enhancementargumentof[8].Extensionfromthealignedcase
0
both receivers) and confidential messages W (intended for (6) to the general case (4) follows from the standard limiting
1
receiver 1 but needing to be kept secret from receiver 2) and argument[9]; the details are deferred to the extended version
W (intended for receiver 2 but needing to be kept secret of this work [11].
2
from receiver 1) under the matrix power constraint (2) is
III. PROOF OFTHEOREM2
given by the set of nonnegativerate triples (R ,R ,R ) such
0 1 2
that A. Achievability
1 H SH⊺+I The problem of a two-receiver discrete memoryless broad-
R ≤min log 1 1 r1 ,
0 2 H (S−B )H⊺+I cast channel with a common message and two confidential
1(cid:26) (cid:12)(cid:12) H1 SH⊺+0I 1 r1(cid:12)(cid:12) common messages was studied in [12], where an achievable
log (cid:12) 2 2 r2 (cid:12) secrecy rate region was given by the set of rate triples
2 H(cid:12)(S−B )H⊺+I (cid:12)
(cid:12) 2 0 2 r2(cid:12)(cid:27) (R ,R ,R ) such that
1 (cid:12) (cid:12) 0 1 2
R ≤ log|I (cid:12)+H B H⊺| (cid:12)
1 2 r1(cid:12) 1 1 1 (cid:12) R0 ≤min[I(U0;Y1), I(U0,Y2)]
1
− 2log|Ir2 +H2B1H⊺2| R1 ≤I(V1;Y1|U0)−I(V1;V2,Y2|U0)
1 I +H (S−B )H⊺ and R2 ≤I(V2;Y2|U0)−I(V2;V1,Y1|U0) (7)
and R ≤ log r2 2 0 2
2 2 I +H B H⊺ where U , V and V are auxiliary random variables such
(cid:12) r2 2 1 2 (cid:12) 0 1 2
− (cid:12)(cid:12)1log Ir1 +H1(S−B0(cid:12)(cid:12))H⊺1 (4) that (U0,V1,V2)→X→(Y1,Y2) forms a Markov chain.
(cid:12)2 I +H B H(cid:12)⊺ The scheme to achieve this secrecy rate region is a natural
(cid:12) r1 1 1 1 (cid:12) combination of secret dirty-paper coding and superposition
(cid:12) (cid:12)
for some B0 (cid:23) 0, B1 (cid:23) 0 (cid:12)(cid:12)and B0 + B1 (cid:22) S. He(cid:12)(cid:12)re, Irk coding. Thus, the achievability of the secrecy rate region (6)
denotes the identity matrix of size r ×r for k =1,2.
k k follows from that of (7) by setting
Remark 1: NotethatforanygivenB ,theupperboundson
0
R and R can be simultaneously maximized by a same B . V =U +FU
1 2 1 1 1 2
In fact, the upper bounds on R1 and R2 are fully symmetric V2 =U2
withrespectto H andH , eventhoughitisnotimmediately
1 2 and X=U +U +U
evident from the expressions themselves. 0 1 2
To prove Theorem 1, we shall follow [9] and first consider whereU ,U andU arethreeindependentGaussianvectors
0 1 2
the canonical aligned case. In an aligned MIMO Gaussian withzeromeansandcovariancematricesB ,B andS−B −
0 1 0
broadcast channel [9], the channel matrices H and H are B , respectively, and
1 2 1
square and invertible. Multiplying both sides of (1) by H−1,
k F:=BH⊺(I +H BH⊺)−1H .
the channel can be equivalently written as 1 r1 1 1 1
The details of the proof are deferred to the extended version
Y [m]=X [m]+Z [m], k =1,2 (5)
k k k
of this work [11].
where {Z [m]} is an i.i.d. additive vector Gaussian noise
k m
B. The converse
process with zero mean and covariance matrix N =
k
H−1H−⊺, k = 1,2. The secrecy capacity region of the Next, we prove the converse part of Theorem 2 assuming
k k
aligned MIMO Gaussian broadcast channel is summarized in that S ≻ 0. The case where S (cid:23) 0, |S| = 0 can be found in
the following theorem. the extended version of this work [11].
Let Hence, we have
f0(B0)=min 12log (S−SB+N)+1 N , (β1+β2)R0†+λ1R1†+λ2R2†
(cid:26) (cid:12) 0 1(cid:12) β1 S+N1 β2 S+N2
1 S(cid:12) +N (cid:12) = log + log
2log (S−(cid:12)(cid:12)B )+2 N (cid:12)(cid:12) 2 (cid:12)(S−B⋆0)+N1(cid:12) 2 (cid:12)(S−B⋆0)+N2(cid:12)
f1(B1)= 12log (cid:12)(cid:12)(cid:12)(cid:12)B1N+N01 − 212lo(cid:12)(cid:12)(cid:12)(cid:12)g(cid:27) B1N+N2 +λ1(cid:18)(cid:12)(cid:12)(cid:12)21log(cid:12)B⋆1N+1N1(cid:12)(cid:12)(cid:12)(cid:12)− 21log(cid:12)B(cid:12)(cid:12)(cid:12)⋆1N+2N2(cid:12)(cid:19) (cid:12)(cid:12)(cid:12)
and f2(B0,B1)= 12log(cid:12)(cid:12)(cid:12)(cid:12)(S−B1B+0)N(cid:12)(cid:12)(cid:12)(cid:12)+N2 (cid:12)(cid:12)(cid:12)(cid:12) 2 (cid:12)(cid:12)(cid:12)(cid:12) +λ2(cid:18)21log(cid:12)(cid:12)(cid:12)(cid:12)(S−B⋆1B+⋆0)N(cid:12)(cid:12)(cid:12)+2N2(cid:12) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)
− 21l(cid:12)(cid:12)(cid:12)(cid:12)og (S1−BB+02)N+N(cid:12)(cid:12)(cid:12)(cid:12) 1 . (8) −21log(cid:12)(S(cid:12)(cid:12)(cid:12)−B⋆1B+⋆0)N+1N1(cid:12)(cid:19)(cid:12)(cid:12)(cid:12)+ρ. (15)
(cid:12) 1 1 (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Then, the secrecy rate region (6)(cid:12)can be rewritten(cid:12)as (cid:12) (cid:12)
(cid:12) (cid:12)
Next, we shall find a contradictionto (15) by followingthe
R := (R ,R ,R )
in 0 1 2 following three steps.
B0(cid:23)0,B1(cid:23)[0,B0+B1(cid:22)S(cid:8) (cid:12)
R ≤f (B ), R ≤f (B ), R ≤f (B(cid:12) ,B ) . (9)
0 0 0 1 1 1 2 2 0 1
To show that Rin is indeed the secrecy capacity re(cid:9)gion of 1) Step 1–Split Each Receiver into Two Virtual Receivers:
the aligned MIMO Gaussian broadcast channel (5), we will Consider the following aligned MIMO Gaussian broadcast
consider proof by contradiction. Assume that (R†,R†,R†) is channel with four receivers:
0 1 2
an achievable secrecy rate triple that lies outside the region
Rin. Since (R0†,R1†,R2†) is achievable, we can bound R0† by Y1a[m]=X[m]+Z1a[m]
Y [m]=X[m]+Z [m]
1 S+N 1 S+N 1b 1b
R0† ≤min 2log N 1 , 2log N 2 =R0max. Y2a[m]=X[m]+Z2a[m]
(cid:18) (cid:12) 1 (cid:12) (cid:12) 2 (cid:12)(cid:19)
Moreover, if R† = (cid:12)(cid:12)R† = 0,(cid:12)(cid:12)then Rm(cid:12)(cid:12)ax can be(cid:12)(cid:12) achieved by and Y2b[m]=X[m]+Z2b[m] (16)
1 (cid:12) 2 (cid:12) 0(cid:12) (cid:12)
setting B0 =S and B1 =0 in (6). Thus, we can find λ1 ≥0 where{Z1a[m]},{Z1b[m]},{Z2a[m]}and{Z2b[m]}arei.i.d.
and λ ≥0 such that additive vector Gaussian noise processes with zero means
2
and covariance matrices N , N , N and N , respectively.
λ R†+λ R† =λ R⋆+λ R⋆+ρ (10) 1 1 2 2
1 1 2 2 1 1 2 2 Suppose that the transmitter has three independent messages
for some ρ>0, where λ R⋆+λ R⋆ is given by W0, W1 and W2, where W0 is intended for both receivers 1b
1 1 2 2
and 2b, W is intended for receiver 1a but needs to be kept
1
(Bm0,aBx1) λ1f1(B1)+λ2f2(B0,B1) secretfromreceiver2b,andW2 isintendedforreceiver2abut
needstobekeptsecretfromreceiver1b.Notethatthechannel
subject to f (B )≥R†
0 0 0 (16) has the same secrecy capacity region as the channel (5)
B (cid:23)0 under the same power constraints.
0
B (cid:23)0
1
B +B (cid:22)S. (11)
0 1
2) Step 2–Construct an Enhanced Channel: Let N be a
Let(B⋆,B⋆)beanoptimalsolutiontotheaboveoptimization
0 1 real symmetric matrix satisfying
program (11). Then, (B⋆,B⋆) must satisfy the following
Karush-Kuhn-Tucker(KK0T) c1onditions: 1 −1 e
N:= N−1+ M . (17)
(β1+λ2)[(S−B⋆0)+N1]−1+β2[(S−B⋆0)+N2]−1+M0 (cid:18) 1 λ1+λ2 1(cid:19)
=λ [(S−B⋆)+N ]−1+M (12) Note that the eabove definition implies that N (cid:22) N . Since
2 0 2 2 1
(λ1+λ2)(B⋆1+N1)−1+M1 M1B⋆1 =0, following [9, Lemma 11], we have
e
=(λ +λ )(B⋆+N )−1+M (13) (λ +λ )(B⋆+N)−1 =(λ +λ )(B⋆+N )−1+M
1 2 1 2 2 1 2 1 1 2 1 1 1
M B⋆ =0, M B⋆ =0, and M (S−B⋆−B⋆)=0 (14)
0 0 1 1 2 0 1 and
e
whereM0,M1andM2arepositivesemidefinitematrices,and |B⋆+N||N |=|B⋆+N ||N|. (18)
β , k = 1,2, is a nonnegative real scalar such that β > 0 if 1 1 1 1
k k
and only if Thus, by (13), we obtain
e e
1log S+Nk =R†. (λ1+λ2)(B⋆1+N)−1 =(λ1+λ2)(B⋆1+N2)−1+M2.
2 (S−B⋆)+N 0 (19)
(cid:12) 0 k(cid:12)
(cid:12) (cid:12) e
(cid:12) (cid:12)
(cid:12) (cid:12)
This implies that N (cid:22) N2. Consider the following enhanced capacity region of this channel satisfies Cs[DMC] ⊆Ro, where
aligned MIMO Gaussian broadcast channel R denotes the set of nonnegative rate triples (R ,R ,R )
o 0 1 2
e such that
Y [m]=X[m]+Z [m]
1a 1a
Y1b[m]=X[m]+Z1b[m] R0 ≤min[I(U;Y1b), I(U,Y2b)]
e e
Y2a[m]=X[m]+Z2a[m] R1 ≤I(X;Y1a|U)−I(X;Y2b|U)
and Ye2b[m]=X[m]+Ze2b[m] (20) and R2 ≤I(X;Ye2a|U)−I(X;Y1b|U) (25)
where{Z1a[m]},{Z1b[m]},{Z2a[m]}and{Z2b[m]}arei.i.d. for some p(u,x)=p(u)p(x|eu).
additivevectorGaussiannoise processeswith zeromeansand Theproofof Theorem3can be foundin the extendedversion
covarianece matrices N, N , Ne and N , respectively. Since of this work [11].
1 2
N(cid:22){N ,N }, we concludethatthe secrecy capacity region Now, we may combine Steps 1, 2 and 3 and consider an
1 2
of the channel (20) iseat least aes large as the secrecy capacity upper bound on the weighted secrecy sum-capacity of the
reegion of the channel (16) under the same power constraints. channel (5). By Theorem 3, for any achievable secrecy rate
triple (R ,R ,R ) for the channel (5) we have
Furthermore, based on (19), we have 0 1 2
[(S−B⋆)+N](B⋆+N)−1 (β1+β2)R0+λ1R1+λ2R2
0 1
β β
=[(S−B⋆0)+N2](B⋆1+N2)−1 (21) ≤ 1 log|2πe(S+N1)|+ 2 log|2πe(S+N2)|
e e 2 2
and hence, λ N λ N
+ 1 log 2 + 2 log 1 +η(λ ,λ ) (26)
1 2
(S−B⋆0)+N = (S−B⋆0)+N2 . (22) 2 (cid:12)(cid:12) N (cid:12)(cid:12) 2 (cid:12)(cid:12) N (cid:12)(cid:12)
Combining(cid:12)(cid:12)(cid:12)(cid:12)(12)Ba⋆1n+d (N19)e, w(cid:12)(cid:12)(cid:12)(cid:12)e m(cid:12)(cid:12)(cid:12)ay oBb⋆1ta+inN2 (cid:12)(cid:12)(cid:12) wηh(λer1e,λ2):=λ1h(X(cid:12)(cid:12) +eZ(cid:12)(cid:12)1a|U)+λ(cid:12)(cid:12)2he(X(cid:12)(cid:12) +Z2a|U)
(cid:12) e (cid:12) (cid:12) (cid:12)
−(λ +β )h(X+Z |U)−(λ +β )h(X+Z |U).
(λ1+λ2)[(S−B⋆0)+N]−1 2 1 e 1b 1 2e 2b
=(λ2+β1)[(S−B⋆0)+N1]−1 Note that 0≺N(cid:22){N1,N2}, 0≺B⋆0 (cid:22)S and B⋆0M0 =0.
+(λ +β )[(S−eB⋆)+N ]−1+M . (23) Using [10, Corollary 4] and (23), we may obtain
1 2 0 2 0
e
Substituting (18) and (22) into (15), we have η(λ1,λ2)≤(λ1+λ2)log 2πe(S−B⋆0)+N
(β1+β2)R0†+λ1R1†+λ2R2† −(λ2+β1)lo(cid:12)(cid:12)(cid:12)g|2πe(S−B⋆0)+eN(cid:12)(cid:12)(cid:12) 1|
β1 S+N1 β2 S+N2 −(λ1+β2)log|2πe(S−B⋆0)+N2|. (27)
= log + log
2 (cid:12)(S−B⋆0)+N1(cid:12) 2 (cid:12)(S−B⋆0)+N2(cid:12) Combining(26)and(27),foranyachievablesecrecyratetriple
+λ (cid:12)(cid:12)(cid:12)1log (S−B⋆0)+(cid:12)(cid:12)(cid:12) N − 1l(cid:12)(cid:12)(cid:12)og (S−B⋆0)+N(cid:12)(cid:12)(cid:12) 2 (R0,R1,R2) for the channel (5) we have
fol3l++o)wλρSi.21tneg p,1223w–leoOgush(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)te(arSllB−coouBnNNnees⋆0di)dfe+orrNaeeth(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)feo−uErn122-rhelaocngeci(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ve(edSrC−dhisaBcNNnr⋆0ne21)teel+:mNIne(m12t(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)h4o!!e)- (β≤1+++β21λλβl12o) gR(cid:12)(cid:12)(cid:12)(cid:12)1120(S+lloogg−λS(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)1B((R+SS⋆01N)−−++1BBλNeN⋆0⋆02))R1++(cid:12)(cid:12)(cid:12)(cid:12)2+NNeβ(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)22−−lo112gll(cid:12)(cid:12)(cid:12)(cid:12)oo(ggS(cid:12)(cid:12)(cid:12)(cid:12)−((SSSB+−−⋆0N)BBN+2⋆0⋆02))N++2NN(cid:12)(cid:12)(cid:12)(cid:12) 21(cid:12)(cid:12)(cid:12)(cid:12)!
rcoyonlnetfishsdeebsnertoicaarledcmcyaescstsaacpghaeacsnitnyaenrldegwpiiortonhv.iadecoamsimngolne-lmetetsesraoguetearnbdotuwnod <(β12+ β22)R0†(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)+λ1R1N†e+λ2Re2†(cid:12)(cid:12)(cid:12)(cid:12)(cid:12). 2 (cid:12)(cid:12)(cid:12)(cid:12) N1 (28(cid:12)(cid:12)(cid:12)(cid:12)!)
Theorem 3 (Discrete memoryless broadcast channel): Clearly, this contradicts the assumption that the rate triple
Consider a discrete memoryless broadcast channel with (R†,R†,R†) is achievable. Therefore, we have proved the
0 1 2
transition probability p(y ,y ,y ,y |x) and messages desired converse result for Theorem 2.
1a 1b 2a 2b
W (intended for both receivers 1b and 2b), W (intended for
0 1
receiver 1a but needing teo be kepteconfidential from receiver IV. NUMERICAL EXAMPLE
2b) and W2 (intended for receiver 2a but needing to be kept Inthissection,weprovideanumericalexampletoillustrate
confidential from receiver 1b). If both the secrecy capacity region of the MIMO Gaussian broadcast
channel with a common message and two confidential mes-
X→Y →(Y ,Y ) and X→Y →(Y ,Y )
1a 1b 2b 2a 1b 2b sages. In this example, we assume that both the transmitter
formMarkovchainsintheirrespectiveorder,thenthesecrecy andeachofthereceiversareequippedwithtwoantennas.The
e e
2.5
R=0
0
5 2
R=1.60
0
4
1.5
3 R0=2.40
R0 R2
2
1
R=3.45
1 0
0 0.5
0
0
1
R 1 0
2 2 2 R1 0 0.5 1 R1 1.5 2 2.5 3
3 3
Fig.3. Secrecyrateregions {(R1,R2)}forsomegivenR0
Fig.2. Secrecycapacity region{(R0,R1,R2)}
[5] F.OggierandB.Hassibi, “Thesecrecy capacity oftheMIMOwiretap
channelmatricesandthematrixpowerconstraintaregivenby
channel,”inProc.IEEEInt.Symp.InformationTheory,Toronto,Canada,
July2008,pp.524–528.
1.8 2.0 3.3 1.3
H = , H = [6] R. Liu and H. V. Poor, “Secrecy capacity region of a multi-antenna
1 1.0 3.0 2 2.0 −1.5 Gaussian broadcast channel with confidential messages,” IEEE Trans.
(cid:18) (cid:19) (cid:18) (cid:19)
Inf.Theory,vol.55,no.3,pp.1235–1249, Mar.2009.
5.0 1.25
and S= . [7] R. Liu, T. Liu, H. V. Poor, and S. Shamai (Shitz), “Multiple-
1.25 10.0 input multiple-output gaussian broadcast channels with confidential
(cid:18) (cid:19)
messages,” IEEE Trans. Inf. Theory, submitted March 2009. [Online].
which yield a nondegraded MIMO Gaussian broadcast Available: http://arxiv.org/abs/0903.3786.
channel. The boundary of the secrecy capacity region [8] H.D.Ly,T.Liu,andY.Liang,“Multiple-inputmultiple-outputGaussian
C[SBC](H ,H ,S) is plotted in Fig. 2. broadcast channels with common and confidential messages,” IEEE
s 1 2 Trans.Inf.Theory,submittedJuly2009.
InFig.3,wehavealsoplottedtheboundariesofthesecrecy [9] H.Weingarten,Y.Steinberg,andS.Shamai(Shitz),“Thecapacityregion
capacity region (R ,R ) for some given common rate R . It oftheGaussianmultiple-inputmultiple-outputbroadcastchannel,”IEEE
1 2 0
Trans.Inf.Theory,vol.52,no.9,pp.3936–3964, Sep.2006.
isparticularlyworthmentioningthatwithR =0,thesecrecy
0 [10] H.Weingarten,T.Liu,S.Shamai(Shitz),Y.Steinberg,andP.Viswanath,
capacity region{(R1,R2)} is rectangular,which implies that “The capacity region of the degraded multiple-input multiple-output
underthe matrix powerconstraint,bothconfidentialmessages compoundbroadcastchannel,”IEEETrans.Inf.Theory,vol.55,no.11,
pp.5011–5023, Nov.2009.
W andW canbesimultaneouslytransmittedattheirrespec-
1 2 [11] R. Liu, T. Liu, H. V. Poor, and S. Shamai (Shitz), “MIMO Gaussian
tive maximum secrecy rates. The readers are referred to [7] broadcast channels with confidential and common messages,” IEEE
for further discussion of this phenomenon. Trans.Inf.Theory,submittedforpublication.
[12] J. Xu, Y. Cao, and B. Chen, “Capacity bounds for broadcast channels
V. CONCLUSION with confidential messages,” IEEE Trans. Inf. Theory, vol. 55, no. 10,
pp.4529–4542, Oct.2009.
Inthispaper,wehaveconsideredtheproblemofcommuni-
catingtwoconfidentialmessagesandacommonmessageover
a two-receiver MIMO Gaussian broadcast channel. We have
shown that a natural scheme that combines secret dirty-paper
coding and Gaussian superposition coding achieves the entire
secrecycapacityregion.Toprovetheconverseresult,wehave
applied a channel-enhancement argument and an extremal
entropy inequalityof Weingarten et al., which generalizesthe
argumentof[8]forthecasewithacommonmessageandonly
one confidential message.
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