Gaussian MIMO broadcast channels with common and confidential messages

Abstract

We study the two-user Gaussian multiple-input multiple-output (MIMO) broadcast channel with common and confidential messages. In this channel, the transmitter sends a common message to both users, and a confidential message to each user which is kept perfectly secret from the other user. We obtain the entire capacity region of this channel. We also explore the connections between the capacity region we obtained for the Gaussian MIMO broadcast channel with common and confidential messages and the capacity region of its non-confidential counterpart, i.e., the Gaussian MIMO broadcast channel with common and private messages, which is not known completely.

Full text

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arXiv:1001.3297v1 [cs.IT] 19 Jan 2010
Gaussian MIMO Broadcast Channels with Common
and Confidential Messages
Ersen EkremSennur Ulukus
Department of Electrical and Computer Engineering
University of Maryland, College Park, MD 20742
ersen@umd.eduulukus@umd.edu
Abstract—We study the two-user Gaussian multiple-input
multiple-output (MIMO) broadcast channel with common and
confidential messages. In this channel, the transmitter sends
a common message to both users, and a confidential message
to each user which is kept perfectly secret from the other
user. We obtain the entire capacity region of this channel.
We also explore the connections between the capacity region
we obtained for the Gaussian MIMO broadcast channel with
common and confidential messages and the capacity region of its
non-confidential counterpart, i.e., the Gaussian MIMO broadcast
channel with common and private messages, which is not known
completely.
I. INTRODUCTION
We study the two-user Gaussian multiple-input multiple-
output (MIMO) broadcast channel for the following scenario:
The transmitter sends a common message to both users, and
a confidential message to each user which needs to be kept
perfectly secret from the other user. We call the channel model
arising from this scenario the Gaussian MIMO broadcast
channel with common and confidential messages.
The Gaussian MIMO broadcast channel with common and
confidential messages subsumes many other channel models
as special cases. The first one is the Gaussian MIMO wiretap
channel, where the transmitter has only one confidential mes-
sage for one (legitimate) user, which is kept perfectly secret
from the other user (eavesdropper). The secrecy capacity of
the Gaussian MIMO wiretap channel is obtained in [1], [2]
for the general case, in [3] for the 2 − 2 − 1 case. The
second channel model that the Gaussian MIMO broadcast
channel with common and confidential messages subsumes
is the Gaussian MIMO wiretap channel with common mes-
sage [4], in which the transmitter sends a common message to
both legitimate user and the eavesdropper, and a confidential
message to the legitimate user that is kept perfectly secret from
the eavesdropper. The capacity region of the Gaussian MIMO
wiretap channel with common message is obtained in [4]. The
last channel model that the Gaussian MIMO broadcast channel
with common and confidential messages encompasses is the
the Gaussian MIMO broadcast channel with confidential mes-
sages [5], where the transmitter sends a confidential message
to each user which is kept perfectly secret from the other user.
The capacity region of the Gaussian MIMO broadcast channel
with confidential messages is established in [5].
This work was supported by NSF Grants CCF 04-47613, CCF 05-14846,
CNS 07-16311 and CCF 07-29127.
Here, we obtain the capacity region of the Gaussian MIMO
broadcast channel with common and confidential messages.
In particular, we show that a variant of the secret dirty-
paper coding (S-DPC) scheme proposed in [5] is capacity-
achieving. Since the S-DPC scheme proposed in [5] is for the
transmission of only two confidential messages, it is modified
here to incorporate the transmission of a common message
as well. Similar to [5], we also notice an invariance property
of this achievable scheme with respect to the encoding order
used in the S-DPC scheme. In other words, two achievable
rate regions arising from two possible encoding orders used
in the S-DPC scheme are identical, and equal to the capacity
region. We provide the proof of this statement as well as the
converse proof for the capacity region of the Gaussian MIMO
broadcast channel with common and confidential messages
by using channel enhancement technique [6] and an extremal
inequality from [7].
We also explore the connections between the Gaussian
MIMO broadcast channel with common and confidential mes-
sages and its non-confidential counterpart, i.e., the (two-user)
Gaussian MIMO broadcast channel with common and private
messages. In the Gaussian MIMO broadcast channel with
common and private messages, the transmitter again sends a
common message to both users, and a private message to each
user, for which there is no secrecy constraint now, i.e., private
message of each user does not need to be kept secret from
the other user. Thus, the Gaussian MIMO broadcast channel
with common and confidential messages we study here can
be viewed as a constrained version of the Gaussian MIMO
broadcast channel with common and private messages, where
the constraints come through forcing the private messages to
be confidential. We note that although there are partial results
for the capacity region of the Gaussian MIMO broadcast
channel with common and private messages [8], [9], it is not
known completely. However, here, we are able to obtain the
entire capacity region for a constrained version of the Gaussian
MIMO broadcast channel with common and private messages,
i.e., for the Gaussian MIMO broadcast channel with common
and confidential messages. We provide an intuitive explanation
of this at-first-sight surprising point as well as the invariance
property of the achievable rate regions with respect to the
encoding orders that can be used in the S-DPC scheme by
using a result from [9] for the Gaussian MIMO broadcast
channel with common and private messages.

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II. CHANNEL MODEL AND MAIN RESULT
We study the two-user Gaussian MIMO broadcast channel
which is defined by
Y1= H1X + N1
Y2= H2X + N2
(1)
(2)
where the channel input X is a t×1 vector, Hjis the channel
gain matrix of size rj× t, the channel output of the jth user
Yjis a rj×1 vector, and the Gaussian random vector Njis
of size rj×1 with a covariance matrix Σjwhich is assumed
to be strictly positive-definite, i.e., Σj ≻ 0. We consider a
covariance constraint on the channel input as follows
E?XX⊤?? S
(3)
where S ? 0.
We study the following scenario for the Gaussian MIMO
broadcast channel: There are three independent messages
(W0,W1,W2) with rates (R0,R1,R2), respectively, where
W0is the common message that needs to be delivered to both
users, W1is the confidential message of the first user which
needs to be kept perfectly secret from the second user, and
similarly, W2 is the confidential message of the second user
which needs to be kept perfectly secret from the first user.
The secrecy of the confidential messages is measured by the
normalized equivocation rates [10], [11], i.e, we require
1
nI(W1;W0,W2,Yn
2) → 0 and
1
nI(W2;W0,W1,Yn
1) → 0
(4)
as n → ∞, where n denotes the number of channel uses. The
closure of all achievable rate triples (R0,R1,R2) is defined
to be the capacity region, and will be denoted by C(S). We
next define the following shorthand notations
R0j(K1,K2) =1
2log
|HjSH⊤
j+ Σj|
|Hj(K1+ K2)H⊤
j+ Σj|,j = 1,2
(5)
R1(K1,K2) =1
2log|H1(K1+ K2)H⊤
|H1K2H⊤
−1
|H2K2H⊤
2log|H2K2H⊤
|Σ2|
1+ Σ1|
1+ Σ1|
2+ Σ2|
2+ Σ2|
−1
2log|H2(K1+ K2)H⊤
(6)
R2(K2) =1
2+ Σ2|
2log|H1K2H⊤
1+ Σ1|
|Σ1|
(7)
using which, our main result can be stated as follows.
Theorem 1 The capacity region of the Gaussian MIMO
broadcast channel with common and confidential messages
C(S) is given by
C(S) = RS−DPC
12
(S) = RS−DPC
21
(S)
(8)
where RS−DPC
(R0,R1,R2) satisfying
12
(S) is given by the union of the rate triples
R0≤ min{R01(K1,K2),R02(K1,K2)}
(9)
R1≤ R1(K1,K2)
R2≤ R2(K2)
(10)
(11)
for some positive semi-definite matrices K1,K2 such that
K1 + K2 ? S, and RS−DPC
21
RS−DPC
12
(S) by swapping the subscripts 1 and 2.
Theorem 1 states that the common message, for which a
covariance matrix S−K1−K2is allotted, should be encoded
by using a standard Gaussian codebook, and the confidential
messages, for which covariance matrices K1,K2are allotted,
need to be encoded by using the S-DPC scheme proposed
in [5]. The receivers first decode the common message by
treating the confidential messages as noise, and then each
receiver decodes the confidential message intended to itself.
Depending on the encoding order used in S-DPC, one of the
users gets a clean link for the transmission of its confidential
message, where there is no interference originating from the
other user’s confidential message. Although one might expect
that two achievable regions arising from two possible encoding
orders that can be used in S-DPC could be different, i.e.,
RS−DPC
12
(S) ?= RS−DPC
21
(S), and taking a convex closure of
these two regions would yield a larger achievable rate region,
Theorem 1 states that RS−DPC
12
achievable rate region is invariant with respect to the encoding
order used in S-DPC.
We conclude this section by defining a sub-class of Gaussian
MIMO broadcast channels called the aligned Gaussian MIMO
broadcast channel, which can be obtained from (1)-(2) by
setting H1= H2= I, i.e.,
(S) can be obtained from
(S) = RS−DPC
21
(S), i.e., the
Y1= X + N1
Y2= X + N2
(12)
(13)
Here, we prove Theorem 1 for the aligned channel. The proof
for the general case in (1)-(2) can be carried out by using the
capacity result for the aligned case and the analysis in [12].
III. PROOF OF THEOREM 1 FOR THE ALIGNED CASE
Due to space limitations here, we omit the achievability
proof for Theorem 1. We provide the converse proof. Since
the capacity region C(S) is convex due to time-sharing, it can
be characterized by the solution of
max
(R0,R1,R2)∈C(S)R0+ µ1R1+ µ2R2
(14)
for µj ∈ [0,∞), j = 1,2. To this end, we first character-
ize the boundary of RS−DPC
12
(S) by studying the following
optimization problem
max
(R0,R1,R2)∈RS−DPC
12
(S)R0+ µ1R1+ µ2R2
(15)
which can be written as
max
0?Kj, j=1,2
K1+K2?S
min{R01(K1,K2),R02(K1,K2)}
+ µ1R1(K1,K2) + µ2R2(K2)
(16)

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Let K∗
the following KKT conditions.
1,K∗
2be the maximizer of (16), which needs to satisfy
Lemma 1 K∗
1,K∗
2need to satisfy
(µ1+ µ2)(K∗
1+ K∗
= (λ + µ2)(K∗
+ (¯λ + µ1)(K∗
2+ Σ2)−1+ M2
= (µ1+ µ2)(K∗
2+ Σ1)−1+ M1
1+ K∗
1+ K∗
2+ Σ1)−1
2+ Σ2)−1+ MS
(17)
(µ1+ µ2)(K∗
2+ Σ1)−1+ M1
(18)
for some positive semi-definite matrices M1,M2,MS such
that K∗
some λ = 1 −¯λ such that it satisfies 0 ≤ λ ≤ 1 and
1M1 = K∗
2M2 = (S − K∗
1− K∗
2)MS = 0 and for
λ



=
=
?=
0
1
0,1
if
if
if
R01(K∗
R01(K∗
R01(K∗
1,K∗
1,K∗
1,K∗
2) > R02(K∗
2) < R02(K∗
2) = R02(K∗
1,K∗
1,K∗
1,K∗
2)
2)
2)
(19)
Due to space limitations here, the proof of this lemma as well
as the proofs of the upcoming lemmas are omitted. We now
use channel enhancement [6] to define a new noise covariance
matrix˜Σ as follows
(µ1+ µ2)(K∗
2+˜Σ)−1= (µ1+ µ2)(K∗
2+ Σ2)−1+ M2
(20)
These new noise covariance matrix˜Σ has some useful prop-
erties which are listed in the following lemma.
Lemma 2 We have the following facts.
•˜Σ ? Σ1,˜Σ ? Σ2.
• (µ1+ µ2)(K∗
1+ K∗
2+˜Σ)−1=
(µ1+ µ2)(K∗
2+ Σ2)−1Σ2
2+˜Σ)−1(K∗
(K∗
1+ K∗
2+ Σ1)−1+ M1
• (K∗
• (K∗
2+˜Σ)−1˜Σ = (K∗
1+ K∗
2+˜Σ) =
1+ K∗
2+ Σ1)−1(K∗
2+ Σ1)
We now construct an enhanced channel using the new covari-
ance matrix˜Σ as follows
˜Y1=˜Y2=˜Y = X +˜N
Y1= X + N1
Y2= X + N2
(21)
(22)
(23)
where˜N is a Gaussian random vector with the covariance
matrix˜Σ. In the channel given by (21)-(23), the enhanced
first and second users have the same observation˜Y. For the
enhanced channel in (21)-(23), we consider the scenario that a
common message W0with rate R0is directed to the first and
second users, i.e., the users with observations Y1 and Y2,
respectively, W1 (resp. W2) with rate R1 (resp. R2) is the
confidential message of the enhanced first (resp. second) user
which is kept perfectly hidden from the second (resp. first)
user. We denote the capacity region by˜C(S). Since in the
enhanced channel, the receivers to which only the common
message is sent are identical to the receivers in the original
channel in (12)-(13), and the receivers to which confidential
messages are sent have better observations with respect to the
receivers in the original channel in (12)-(13), we have C(S) ⊆
˜C(S). We next introduce an outer bound for˜C(S).
Lemma 3 The capacity region of the enhanced channel in
(21)-(23)˜C(S) is contained in the union of the rate triples
(R0,R1,R2) satisfying
R0≤ min{I(U;Y1),I(U;Y2)}
R1≤ I(X;˜Y|U) − I(X;Y2|U)
R2≤ I(X;˜Y|U) − I(X;Y1|U)
(24)
(25)
(26)
for some (U,X) such that U → X →˜Y → (Y1,Y2) and
E?XX⊤?? S.
We also introduce the following extremal inequality from [7]:
Lemma 4 ([7], Corollary 4) (U,X) is an arbitrary random
vector, where E?XX⊤?? S and S ≻ 0. Let˜N,N1,N2be
Gaussian with covariance matrices˜Σ,Σ1,Σ2, respectively.
They are independent of (U,X). Moreover,˜Σ,Σ1,Σ2satisfy
˜Σ ? Σj, j = 1,2. Assume that there exists a covariance
matrix K∗such that K∗? S and
β(K∗+˜Σ)−1=
2
?
j=1
γj(K∗+ Σj)−1+ MS
(27)
where β ≥ 0,γj ≥ 0, j = 1,2 and MS is positive semi-
definite matrix such that (S − K∗)MS = 0. Then, for any
(U,X), we have
βh(X +˜N|U) −
2
?
j=1
γjh(X + Nj|U) ≤
β
2log|(2πe)(K∗+˜Σ)| −
2
?
j=1
γj
2log|(2πe)(K∗+ Σj)| (28)
We now use this lemma. For that purpose, we note that using
the second statement of Lemma 2 in (17) yields
(µ1+ µ2)(K∗
1+ K∗
2+˜Σ)−1= (λ + µ2)(K∗
+ (¯λ + µ1)(K∗
1+ K∗
2+ Σ1)−1
1+ K∗
2+ Σ2)−1+ MS (29)
using which in conjunction with Lemma 4, we get
(µ1+ µ2)h(˜Y|U) − (λ + µ2)h(Y1|U) − (¯λ + µ1)h(Y2|U)
≤µ1+ µ2
2
−λ + µ2
2
¯λ + µ1
2
which will be used subsequently. We are now ready to com-
plete the converse proof as follows
log|(2πe)(K∗
1+ K∗
2+˜Σ)|
log|(2πe)(K∗
1+ K∗
2+ Σ1)|
−
log|(2πe)(K∗
1+ K∗
2+ Σ2)|
(30)
max
(R0,R1,R2)∈C(S)R0+ µ1R1+ µ2R2
≤max
(R0,R1,R2)∈˜C(S)R0+ µ1R1+ µ2R2
≤ max min{I(U;Y1),I(U;Y2)} + (µ1+ µ2)I(X;˜Y|U)
− µ1I(X;Y2|U) − µ2I(X;Y1|U)
(31)
(32)
(33)

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≤ max λI(U;Y1) +¯λI(U;Y2) + (µ1+ µ2)I(X;˜Y|U)
− µ1I(X;Y2|U) − µ2I(X;Y1|U)
= max λh(Y1) +¯λh(Y2) + (µ1+ µ2)h(˜Y|U)
− (λ + µ2)h(Y1|U) − (¯λ + µ1)h(Y2|U)
2log|˜Σ|
≤λ
2log|(2πe)(S + Σ1)| +
+ max
?
−µ1
(34)
−µ1
|Σ2|−µ2
2log|˜Σ|
¯λ
2log|(2πe)(S + Σ1)|
|Σ1|
(35)
(µ1+ µ2)h(˜Y|U) − (λ + µ2)h(Y1|U)
− (¯λ + µ1)h(Y2|U)
?
2log|˜Σ|
¯λ
2log|(2πe)(S + Σ1)|
|Σ2|−µ2
2log|˜Σ|
|Σ1|
(36)
≤λ
2log|(2πe)(S + Σ1)| +
+(µ1+ µ2)
2
−(λ + µ2)
2
¯λ + µ1
2
−µ1
log|(2πe)(K∗
1+ K∗
2+˜Σ)|
log|(2πe)(K∗
1+ K∗
2+ Σ1)|
−
log|(2πe)(K∗
1+ K∗
2+ Σ2)|
2log|˜Σ|
|Σ2|−µ2
1,K∗
1+ K∗
|(K∗
2log|˜Σ|
2),R02(K∗
2+˜Σ)Σ2|
1+ K∗
1+ K∗
|(K∗
1,K∗
+ µ2R2(K∗
|Σ1|
(37)
= min{R01(K∗
2log|(K∗
1,K∗
2)}
+µ1
2+ Σ2)˜Σ|
2+˜Σ)Σ1|
2+ Σ1)˜Σ|
2),R02(K∗
+µ2
2log|(K∗
1+ K∗
(38)
= min{R01(K∗
1,K∗
2)} + µ1R1(K∗
1,K∗
2)
2)
(39)
where (32) comes from the fact that C(S) ⊆˜C(S), (33) is due
to Lemma 3, (34) results from the fact that 0 ≤ λ = 1−¯λ ≤ 1,
(36) is due to the maximum entropy theorem, (37) comes from
(30), and (39) will be shown next. We first note the following
R1(K∗
1,K∗
2) =1
2log|(K∗
1+ K∗
|(K∗
2+ Σ1)(K∗
2+ Σ1)(K∗
1+ K∗
2+ Σ2)−1|
2+ Σ2)−1|
(40)
=1
2log|(K∗
1+ K∗
|(K∗
2+˜Σ)(K∗
2+˜Σ)(K∗
1+ K∗
2+ Σ2)−1|
2+ Σ2)−1|
(41)
=1
2log|(K∗
1+ K∗
1+ K∗
2+˜Σ)Σ2|
2+ Σ2)˜Σ||(K∗
(42)
where (41) is due to the fourth statement of Lemma 2 and
(42) comes from the third statement of Lemma 2. We next
note the following identity
R2(K∗
2) =1
2log|(K∗
2log|(K∗
2+ Σ2)(K∗
|Σ2Σ−1
2+˜Σ)(K∗
|˜ΣΣ−1
2+ Σ1)−1|
1|
2+ Σ1)−1|
1|
(43)
=1
(44)
=1
2log|(K∗
1+ K∗
1+ K∗
2+˜Σ)Σ1|
2+ Σ1)˜Σ||(K∗
(45)
where (44) is due to the third statement of Lemma 2, and (2)
comes from the fourth statement of Lemma 2. Identities in
(42) and (45) give (39). Thus, in the view of (39), we have
shown that C(S) = RS−DPC
12
(S). Similarly, one can also show
C(S) = RS−DPC
21
(S); completing the proof of Theorem 1.
IV. CONNECTIONS TO THE GAUSSIAN MIMO BROADCAST
CHANNEL WITH COMMON AND PRIVATE MESSAGES
Here, we provide an intuitive explanation for the two
facts that Theorem 1 reveals: i) The achievable rate region
does not depend on the encoding order used in S-DPC, i.e.,
RS−DPC
12
(S) = RS−DPC
21
(S), ii) The capacity region of the
Gaussian MIMO broadcast channel with common and confi-
dential messages can be completely characterized, althoughthe
capacity region of the its non-confidential counterpart, i.e., the
Gaussian MIMO broadcast channel with common and private
messages, is not known completely.
In the Gaussian MIMO broadcast channel with common and
private messages, there are again three messages W0,W1,W2
with rates R0,R1,R2, respectively, such that W0 is again
sent to both users, W1 (resp. W2) is again directed to only
the first (resp. second) user, however, there are no secrecy
constraints on W1,W2. The capacity region of the Gaussian
MIMO broadcast channel with common and private messages
will be denoted by CNS(S). The achievable rate region for
the Gaussian MIMO broadcast channel with common and
private messages that can be obtained by using DPC will be
denoted by RNS−DPC
12
(S),RNS−DPC
21
encoding order), where RNS−DPC
12
triples (R0,R1,R2) satisfying
(S) (depending on the
(S) is given by the rate
R0≤ min{RNS
R1≤ RNS
R2≤ RNS
01(K1,K2),RNS
02(K1,K2)}
(46)
(47)
(48)
1(K1,K2)
2(K2)
for
such that K1 + K2
RNS
somepositivesemi-definite
?
S, and {RNS
2(K2) are defined as
|S + Σj|
|K1+ K2+ Σj|,
1(K1,K2) =1
matrices
0j(K1,K2)}2
K1,K2
j=1,
1(K1,K2),RNS
RNS
0j(K1,K2) =1
2log
2log|K1+ K2+ Σ1|
|K2+ Σ1|
2log|K2+ Σ2|
|Σ2|
j = 1,2
(49)
RNS
(50)
RNS
2(K2) =1
(51)
Moreover, RNS−DPC
by swapping the subscripts 2 and 1. This achievable rate region
was proposed in [13].
We now state a result of [9] on the capacity region of the
Gaussian MIMO broadcast channel with common and private
messages, which is that for a given common message rate
R0, the private messages sum rate capacity, i.e., R1+ R2, is
achieved by both RNS
21
(S) can be obtained from RNS−DPC
12
(S)
12(S) and RNS
21(S). This result can also

Page 5

be stated as follows
max
(R0,R1,R2)∈CNS(S)µ′
=
(R0,R1,R2)∈RNS−DPC
0R0+ µ′
1R1+ µ′
2R2
max
12
(S)µ′
0R0+ µ′
1R1+ µ′
2R2
(52)
=max
(R0,R1,R2)∈RNS−DPC
21
(S)µ′
0R0+ µ′
1R1+ µ′
2R2
(53)
for µ′
aforementioned two facts suggested by Theorem 1, which will
be explained next using (52)-(53).
1= µ′
2= µ′. This result is crucial to understand the
In the proof of Theorem 1, first, we characterize the
boundary of RS−DPC
12
(S) by finding the properties of the
covariance matrices that achieve the boundary of RS−DPC
see Lemma 1. According to Lemma 1, the boundary of
RS−DPC
12
(S) can be achieved by using the covariance matrices
K∗
1,K∗
covariance matrices, we can also achieve the boundary points
of RNS−DPC
12
(S), which are actually on the boundary of the
capacity region CNS(S) as well, and are the private message
sum rate capacity points for a given common message rate.
To see this point, we define µ′= µ1+ µ2,µ′
and γ =
conditions in (17)-(18) can be written as
12
(S),
2satisfying (17)-(18). On the other hand, using these
0= 1 + µ1+ µ2
¯λ+µ1
1+µ1+µ2. Thus, the
λ+µ2
1+µ1+µ2, i.e., ¯ γ = 1 − γ =
µ′(K∗
1+ K∗
2+ Σ1)−1+ M1= µ′
+ µ′
µ′(K∗
0γ(K∗
2+ Σ2)−1+ MS
2+ Σ1)−1+ M1
1+ K∗
2+ Σ1)−1
0¯ γ(K∗
1+ K∗
(54)
(55)
2+ Σ2)−1+ M2= µ′(K∗
which are the necessary conditions that the following problem
needs to satisfy
max
(R0,R1,R2)∈RNS−DPC
12
(S)µ′
0R0+ µ′(R1+ R2)
(56)
On the other hand, due to (52)-(53), we know that the solution
of (56) gives us the private message sum rate capacity for a
given common message rate, i.e., the points that achieve the
maximum in (56) are on the boundary of the capacity region
CNS(S). Furthermore, the maximum value in (56) can also be
achieved by using the other possible encoding order, i.e.,
max
(R0,R1,R2)∈RNS−DPC
12
(S)µ′
0R0+ µ′(R1+ R2)
=max
(R0,R1,R2)∈RNS−DPC
21
(S)µ′
0R0+ µ′(R1+ R2)
(57)
Thus, this discussion reveals that there is a one-to-one cor-
respondence between any rate triple on the boundary of
RS−DPC
12
(S) and the private messages sum rate capacity points
on CNS(S). Hence, the boundary of RS−DPC
RS−DPC
21
(S), can be constructed by considering the private
messages sum rate capacity points on CNS(S). This connection
between the private messages sum rate capacity points and the
boundaries of RS−DPC
12
(S), RS−DPC
the two facts suggested by Theorem 1: i) The achievable
rate region for the Gaussian MIMO broadcast channel with
common and confidential messages is invariant with respect to
the encoding order, i.e., RS−DPC
12
12
(S), similarly
21
(S) intuitively explains
(S) = RS−DPC
21
(S) because
the boundaries of these two regions correspond to those points
on the DPC region for the Gaussian MIMO broadcast channel
with common and private messages, for which encoding order
does not matter either. ii) We can obtain the entire capacity
region of the Gaussian MIMO broadcast channel with common
and confidential messages, although the capacity region of
its non-confidential counterpart is not known completely. The
reason is that the boundary of the capacity region of the Gaus-
sian MIMO broadcast channel with common and confidential
messages comes from those points on the boundary of the DPC
region of its non-confidential counterpart, which are known to
be tight, i.e., on the boundary of the capacity region of the
Gaussian MIMO broadcast channel with common and private
messages.
V. CONCLUSIONS
We study the Gaussian MIMO broadcast channel with
common and confidential messages, and obtain the entire
capacity region. We show that a variant of the S-DPC scheme
proposed in [5] is capacity-achieving. We provide the converse
proof by using channel enhancement [6] and an extremal in-
equality from [7]. We also investigate the connections between
the Gaussian MIMO broadcast channel with common and
confidential messages and its non-confidential counterpart to
provide further insight into capacity result we obtained.
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