# Multi-Antenna Gaussian Broadcast Channels with Confidential Messages

Ruoheng Liu and H. Vincent Poor This research was supported by the National
Science Foundation under Grants ANI-03-38807, CNS-06-25637 and CCF-07-28208.
Department of Electrical Engineering, Princeton University, Princeton, NJ
08544
Email<EMAIL_ADDRESS>
###### Abstract

In wireless data networks, communication is particularly susceptible to
eavesdropping due to its broadcast nature. Security and privacy systems have
become critical for wireless providers and enterprise networks. This paper
considers the problem of secret communication over a Gaussian broadcast
channel, where a multi-antenna transmitter sends independent confidential
messages to two users with _information-theoretic secrecy_. That is, each user
would like to obtain its own confidential message in a reliable and safe
manner. This communication model is referred to as the multi-antenna Gaussian
broadcast channel with confidential messages (MGBC-CM). Under this
communication scenario, a secret dirty-paper coding scheme and the
corresponding achievable secrecy rate region are first developed based on
Gaussian codebooks. Next, a computable Sato-type outer bound on the secrecy
capacity region is provided for the MGBC-CM. Furthermore, the Sato-type outer
bound proves to be consistent with the boundary of the secret dirty-paper
coding achievable rate region, and hence, the secrecy capacity region of the
MGBC-CM is established. Finally, a numerical example demonstrates that both
users can achieve positive rates simultaneously under the information-
theoretic secrecy requirement.

##  I Introduction

The demand for efficient, reliable, and secret data communication over
wireless networks has become increasingly critical in recent years. Due to its
broadcast nature, wireless communication is particularly susceptible to
eavesdropping. The inherent nature of wireless networks exposes not only
vulnerabilities that a malicious user can exploit to severely compromise the
network, but also multiplies information confidentiality concerns with respect
to in-network terminals. Hence, security and privacy systems have become
critical for wireless providers and enterprise networks.

In this work, we consider multiple antenna secret broadcast in wireless
networks. This research is inspired by the seminal paper [1], in which Wyner
introduced the so-called wiretap channel and proposed an information theoretic
approach to secret communication schemes. Under the assumption that the
channel to the eavesdropper is a degraded version of that to the desired
receiver, Wyner characterized the capacity-secrecy tradeoff for the discrete
memoryless wiretap channel and showed that secret communication is possible
without sharing a secret key. Later, the result was extended by Csiszár and
Körner who determined the secrecy capacity for the non-degraded broadcast
channel (BC) with a single confidential message intended for one of the users
[2].

Figure 1:  Channel model of multiple-antenna Gaussian broadcast channel with
confidential messages

In more general wireless network scenarios, secret communication may involve
multiple users and multiple antennas. Consequently, a significant recent
research effort has been invested in the study of the information-theoretic
limits of secret communication in different wireless network environments
including multi-user communication with confidential messages [3, 4, 5, 6, 7,
8], secret wireless communication on fading channels [9, 10, 11], and the
Gaussian multiple-input single-output (MISO) and multiple-input multiple-
output (MIMO) wiretap channels [12, 13, 14, 15, 16].

These issues and results motivate us to study the multi-antenna Gaussian BC
with confidential messages (MGBC-CM), in which independent confidential
messages from a multi-antenna transmitter are to be communicated to two users.
The corresponding broadcast communication model is shown in Fig. 1. Each user
would like to obtain its own message reliably and confidentially.

To give insight into this problem, we first consider a single-antenna Gaussian
BC. Note that this channel is degraded [17], which means that if a message can
be successfully decoded by the inferior user, then the superior user is also
ensured of decoding it. Hence, the secrecy rate of the inferior user is zero
and this problem is reduced to the scalar Gaussian wiretap channel problem
[18] whose secrecy capacity is now the maximum rate achievable by the superior
user. This analysis gives rise to the question: can the transmitter, in fact,
communicate with both users confidentially at nonzero rate under some other
conditions? Roughly speaking, the answer is in the affirmative. In particular,
the transmitter can communicate when equipped with sufficiently separated
multiple antennas.

We here have two goals motivated directly by questions arising in practice.
The first is to determine conditions under which both users can obtain their
own confidential messages in a reliable and safe manner. This is equivalent to
evaluating the secrecy capacity region for the MGBC-CM. The second is to show
_how_ the transmitter should broadcast confidentially, which is equivalent to
designing an achievable secret coding scheme. To this end, we first describe a
secret dirty-paper coding (DPC) scheme and derive the corresponding achievable
rate region based on Gaussian codebooks. The secret DPC is based on double-
binning [6] which enables both joint encoding and preserving confidentiality.
Next, a computable Sato-type outer bound on the secrecy capacity region is
developed for the MGBC-CM. Furthermore, the Sato-type outer bound proves to be
consistent with the boundary of the secret dirty-paper coding achievable rate
region, and hence, the secrecy capacity region of the MGBC-CM is established.
Finally, a numerical example demonstrates that both users can achieve positive
rates simultaneously under the information-theoretic secrecy requirement.

##  II System Model and Definitions

###  II-A Channel Model

We consider the communication of confidential messages to two users over a
Gaussian BC via $t\geq 2$ transmit-antennas. Each user is equipped with a
single receive-antenna. As shown in Fig. 1, the transmitter sends independent
confidential messages $W_{1}$ and $W_{2}$ in $n$ channel uses with $nR_{1}$
and $nR_{2}$ bits, respectively. The message $W_{1}$ is destined for user 1
and eavesdropped upon by user 2, whereas the message $W_{2}$ is destined for
user 2 and eavesdropped upon by user 1. This communication scenario is
referred to as the multi-antenna Gaussian BC with confidential messages. The
Gaussian BC is an additive noise channel and the received symbols at user 1
and user 2 can be represented as follows:

$\displaystyle y_{1,i}$ $\displaystyle=\mathbf{h}^{H}\mathbf{x}_{i}+z_{1,i}$
$\displaystyle y_{2,i}$
$\displaystyle=\mathbf{g}^{H}\mathbf{x}_{i}+z_{2,i},\qquad i=1,\dots,n$ (1)

where $\mathbf{x}_{i}\in\mathbb{C}^{t}$ is a complex input vector at time $i$,
$\\{z_{1,i}\\}$ and $\\{z_{2,i}\\}$ correspond to two independent, zero-mean,
unit-variance, complex Gaussian noise sequences, and
$\mathbf{h},\mathbf{g}\in\mathbb{C}^{t}$ are fixed, complex channel
attenuation vectors imposed on user 1 and user 2, respectively. The channel
input is constrained by ${\rm tr}(K_{\mathbf{X}})\leq P$, where
$K_{\mathbf{X}}$ is the channel input covariance matrix and $P$ is the average
total power limitation at the transmitter. We also assume that both the
transmitter and users are aware of the attenuation vectors.

###  II-B Important Channel Parameters for the MGBC-CM

For the MGBC-CM, we are interested in the following important parameters,
which are related to the generalized eigenvalue problem. Let $\lambda_{1}$ and
$\mathbf{e}_{1}$ denote the largest generalized eigenvalue and the
corresponding normalized eigenvector of the pencil
$(I+P\mathbf{h}\mathbf{h}^{H},I+P\mathbf{g}\mathbf{g}^{H})$ so that
$\mathbf{e}_{1}^{H}\mathbf{e}_{1}=1$ and

$\displaystyle(I+P\mathbf{h}\mathbf{h}^{H})\mathbf{e}_{1}=\lambda_{1}(I+P\mathbf{g}\mathbf{g}^{H})\mathbf{e}_{1}.$
(2)

Similarly, we define $\lambda_{2}$ and $\mathbf{e}_{2}$ as the largest
generalized eigenvalue and the corresponding normalized eigenvector of the
pencil $(I+P\mathbf{g}\mathbf{g}^{H},I+P\mathbf{h}\mathbf{h}^{H})$ so that
$\mathbf{e}_{2}^{H}\mathbf{e}_{2}=1$ and

$\displaystyle(I+P\mathbf{g}\mathbf{g}^{H})\mathbf{e}_{2}=\lambda_{2}(I+P\mathbf{h}\mathbf{h}^{H})\mathbf{e}_{2}.$
(3)

A useful property of $\lambda_{1}$ and $\lambda_{2}$ is described as follows.

###### Lemma 1

For any channel attenuation vector pair $\mathbf{h}$ and $\mathbf{g}$, the
largest generalized eigenvalues of the pencil
$(I+P\mathbf{h}\mathbf{h}^{H},I+P\mathbf{g}\mathbf{g}^{H})$ and the pencil
$(I+P\mathbf{g}\mathbf{g}^{H},I+P\mathbf{h}\mathbf{h}^{H})$ satisfy
$\lambda_{1}\geq 1$ and $\lambda_{2}\geq 1.$ Moreover, if $\mathbf{h}$ and
$\mathbf{g}$ are linearly independent, then both $\lambda_{1}$ and
$\lambda_{2}$ are strictly greater than $1$.

###  II-C Definitions

We now define the secret codebook, the probability of error, the secrecy
level, and the secrecy capacity region for the MGBC-CM as follows.

An $(2^{nR_{1}},2^{nR_{2}},n)$ secret codebook for the MGBC-CM consists of the
following:

  1. 1.

Two message sets: ${\mathcal{W}}_{k}=\\{1,\ldots,2^{nR_{k}}\\}$, for $k=1,2.$

  2. 2.

A stochastic encoding function specified by a conditional probability density
$p(\mathbf{x}^{n}|w_{1},w_{2})$, where
$\mathbf{x}^{n}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{n}]\in\mathbb{C}^{t\times
n}$, $w_{k}\in{\mathcal{W}}_{k}$ for $k=1,2$, and

$\displaystyle\int_{\mathbf{x}^{n}}p(\mathbf{x}^{n}|w_{1},w_{2})=1.$

  3. 3.

Decoding functions $\phi_{1}$ and $\phi_{2}$. The decoding function at user
$k$ is a deterministic mapping
$\phi_{k}:{\mathcal{Y}}_{k}^{n}\rightarrow{\mathcal{W}}_{k}.$

At the receiver ends, the error performance and the secrecy level are
evaluated by the following performance measures.

  1. 1.

The reliability is measured by the maximum error probability

$P_{e}^{(n)}\triangleq\max\bigl{\\{}P_{e,1}^{(n)},P_{e,2}^{(n)}\bigr{\\}}$

where $P_{e,k}^{(n)}$ is the error probability for user $k$.

  2. 2.

The secrecy levels with respect to confidential messages $W_{1}$ and $W_{2}$
are measured, respectively, at user 2 and user 1 with respect to the
equivocation rates $\frac{1}{n}H(W_{2}|Y_{1}^{n})$ and
$\frac{1}{n}H(W_{1}|Y_{2}^{n})$.

A rate pair $(R_{1},R_{2})$ is said to be achievable for the MGBC-CM if, for
any $\epsilon>0$, there exists an $(2^{nR_{1}},2^{nR_{2}},n)$ code that
satisfies $P_{e}^{(n)}\leq\epsilon$, and the information-theoretic secrecy
requirement

$\displaystyle nR_{1}-H(W_{1}|Y_{2}^{n})$ $\displaystyle\leq
n\epsilon~{}\text{and}~{}nR_{2}-H(W_{2}|Y_{1}^{n})\leq n\epsilon.$ (4)

The secrecy capacity region ${\mathcal{C}}^{\rm MG}_{s}$ of the MGBC-CM is the
closure of the set of all achievable rate pairs $(R_{1},R_{2})$.

##  III Main Result

The two-user Gaussian BC with multiple transmit-antennas is non-degraded. For
this channel, we have the following closed-from result on the secrecy capacity
region under the information-theoretic secrecy requirement.

###### Theorem 1

Consider an MGBC-CM modeled as in (1). Let

$\displaystyle\gamma_{1}(\alpha)=\frac{1+\alpha
P|\mathbf{h}^{H}\mathbf{e}_{1}|^{2}}{1+\alpha
P|\mathbf{g}^{H}\mathbf{e}_{1}|^{2}},$

$\gamma_{2}(\alpha)$ be the largest generalized eigenvalue of the pencil

$\displaystyle\left(I+\frac{(1-\alpha)P\mathbf{g}\mathbf{g}^{H}}{1+\alpha
P|\mathbf{g}^{H}\mathbf{e}_{1}|^{2}},\,I+\frac{(1-\alpha)P\mathbf{h}\mathbf{h}^{H}}{1+\alpha
P|\mathbf{h}^{H}\mathbf{e}_{1}|^{2}}\right),$ (5)

and ${{\mathcal{R}}}^{\rm MG}(\alpha)$ denote the union of all $(R_{1},R_{2})$
satisfying

$\displaystyle 0$ $\displaystyle\leq R_{1}\leq\log_{2}\gamma_{1}(\alpha)$ and
$\displaystyle 0$ $\displaystyle\leq R_{2}\leq\log_{2}\gamma_{2}(\alpha).$

Then the secrecy capacity region of the MGBC-CM is

$\displaystyle{\mathcal{C}}^{\rm MG}_{s}={\rm
co}\left\\{\bigcup_{0\leq\alpha\leq 1}{\mathcal{R}}^{\rm MG}(\alpha)\right\\}$

where ${\rm co}\\{{\mathcal{S}}\\}$ denotes the convex hull of the set
${\mathcal{S}}$.

###### Proof:

The achievability part of Theorem 1 is based on secret dirty-paper coding
inner bound in Sec. IV. The converse part is based on Sato-type outer bound in
Sec. V. We provide the complete proof in [14]. ∎

###### Corollary 1

For the MGBC-CM, the maximum secrecy rate of user 1 is given by

$\displaystyle R_{1,\max}=\max_{0\leq\alpha\leq
1}\log_{2}\gamma_{1}(\alpha)=\log_{2}\lambda_{1}$

where $\lambda_{1}$ is defined in (2).

Example: (MISO wiretap channels) A special case of the MGBC-CM model is the
Gaussian MISO wiretap channel studied in [12, 19, 20], where the transmitter
sends confidential information to only one user and treats another user as an
eavesdropper. Let us consider a Gaussian MISO wiretap channel modeled in (1),
where user 1 is the legitimate receiver and user 2 is the eavesdropper.
Corollary 1 implies that the secrecy capacity of the Gaussian MISO wiretap
channel corresponds to the corner point of ${\mathcal{C}}^{\rm MG}_{s}$.
Hence, the secrecy capacity of the Gaussian MISO wiretap channel is given by

$\displaystyle C^{\rm MISO}_{s}=\log_{2}\lambda_{1},$

which coincides with the result of [19].

For the MGBC-CM, the actions of user 1 and user 2 are symmetric to each other,
i.e., each user decodes its own message and eavesdrops upon the confidential
information belonging to the other user. Based on symmetry of this two-user BC
model, we can express the secrecy capacity region ${\mathcal{C}}^{\rm MG}_{s}$
in an alternative way.

###### Corollary 2

For an MGBC-CM modeled in as (1), the secrecy capacity region can be written
as

$\displaystyle{\mathcal{C}}^{\rm MG}_{s}={\rm
co}\left\\{\bigcup_{0\leq\beta\leq 1}{\mathcal{R}}^{\rm MG-2}(\beta)\right\\}$

where ${{\mathcal{R}}}^{\rm MG-2}(\beta)$ denotes the union of all
$(R_{1},R_{2})$ satisfying

$\displaystyle 0$ $\displaystyle\leq R_{1}\leq\log_{2}\xi_{1}(\beta)$ and
$\displaystyle 0$ $\displaystyle\leq R_{2}\leq\log_{2}\xi_{2}(\beta),$

$\xi_{1}(\beta)$ is the largest generalized eigenvalue of the pencil

$\displaystyle\left(I+\frac{(1-\beta)P\mathbf{h}\mathbf{h}^{H}}{1+\beta
P|\mathbf{h}^{H}\mathbf{e}_{2}|^{2}},\,I+\frac{(1-\beta)P\mathbf{g}\mathbf{g}^{H}}{1+\beta
P|\mathbf{g}^{H}\mathbf{e}_{2}|^{2}}\right)$

and

$\displaystyle\xi_{2}(\beta)=\frac{1+\beta
P|\mathbf{g}^{H}\mathbf{e}_{2}|^{2}}{1+\beta
P|\mathbf{h}^{H}\mathbf{e}_{2}|^{2}}.$

###### Remark 1

Theorem 1 and Corollary 2 imply that if $\alpha$ and $\beta$ satisfy the
implicit function $\gamma_{1}(\alpha)=\xi_{1}(\beta)$, then

${{\mathcal{R}}}^{\rm MG}(\alpha)={{\mathcal{R}}}^{\rm MG-2}(\beta).$

For example, it is easy to check ${{\mathcal{R}}}^{\rm
MG}(1)={{\mathcal{R}}}^{\rm MG-2}(0)$.

Now, by applying Corollary 2 and setting $\beta=1$, we can show that the rate
pair $(0,\log_{2}\lambda_{2})$ is the corner point corresponding to the
maximum achievable rate of user 2 in the capacity region ${\mathcal{C}}^{\rm
MG}_{s}$.

###### Corollary 3

For the MGBC-CM, the maximum secrecy rate of user 2 is given by

$R_{2,\max}=\log_{2}\lambda_{2}$

where $\lambda_{2}$ is defined in (3).

Corollaries 1 and 3 imply that for the MGBC-CM, both users can achieve
positive rates with information-theoretic secrecy if and only if
$\lambda_{1}>1$ and $\lambda_{2}>1$. Furthermore, Lemma 1 illustrates that
this condition can be ensured when the attenuation vectors $\mathbf{h}$ and
$\mathbf{g}$ are linearly independent.

##  IV Achievability: Secret DPC Scheme

###  IV-A Double-Binning Inner bound for the BC-CM

An achievable rate region for the broadcast channel with confidential messages
(BC-CM) has been established in [6] based on a double-binning scheme that
enables both joint encoding at the transmitter by using Gel’fand-Pinsker
binning and preserving confidentiality by using random binning.

###### Lemma 2

([6, Theorem 3]) Let $\mathbf{V}_{1}$ and $\mathbf{V}_{2}$ be auxiliary random
variables, $\Omega$ denote the class of joint probability densities
$p(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{x},y_{1},y_{2})$ that factor as

$\displaystyle
p(\mathbf{v}_{1},\mathbf{v}_{2})p(\mathbf{x}|\mathbf{v}_{1},\mathbf{v}_{2})p(y_{1},y_{2}|\mathbf{x}),$

and ${{\mathcal{R}}}_{\rm I}(\pi)$ denote the union of all $(R_{1},R_{2})$
satisfying

$\displaystyle 0$ $\displaystyle\leq R_{1}\leq
I(\mathbf{V}_{1};Y_{1})-I(\mathbf{V}_{1};Y_{2},\mathbf{V}_{2})$ and
$\displaystyle 0$ $\displaystyle\leq R_{2}\leq
I(\mathbf{V}_{2};Y_{2})-I(\mathbf{V}_{2};Y_{1},\mathbf{V}_{1})$

for a given joint probability density $\pi\in\Omega$. For the BC-CM, any rate
pair

$\displaystyle(R_{1},R_{2})\in{\rm
co}\left\\{\bigcup_{\pi\in\Omega}{\mathcal{R}}_{\rm I}(\pi)\right\\}$ (6)

is achievable.

###  IV-B Secret DPC Scheme for the MGBC-CM

The achievable strategy in Lemma 2 introduces a double-binning coding scheme.
However, when the rate region (6) is used as a constructive technique, it not
clear how to choose the auxiliary random variables $\mathbf{V}_{1}$ and
$\mathbf{V}_{2}$ to implement the double-binning codebook, and hence, one has
to “guess” the density of $p(\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{x})$. Here,
we employ the DPC technique with the double-binning code structure to develop
the secret DPC (S-DPC) achievable rate region.

For the MGBC-CM, we consider a secret dirty-paper encoder with Gaussian
codebooks. Based on Lemma 2, we obtain a S-DPC rate region for the MGBC-CM as
follows.

###### Lemma 3

[S-DPC region] Let ${{\mathcal{R}}}_{\rm I}^{\rm
S-DPC}(K_{\mathbf{U}_{1}},K_{\mathbf{U}_{2}})$ denote the union of all
$(R_{1},R_{2})$ satisfying

$\displaystyle 0$ $\displaystyle\leq
R_{1}\leq\log_{2}\frac{1+\mathbf{h}^{H}K_{\mathbf{U}_{1}}\mathbf{h}}{1+\mathbf{g}^{H}K_{\mathbf{U}_{1}}\mathbf{g}}$
(7) and $\displaystyle 0$ $\displaystyle\leq
R_{2}\leq\log_{2}\frac{1+\mathbf{g}^{H}(K_{\mathbf{U}_{1}}+K_{\mathbf{U}_{2}})\mathbf{g}}{1+\mathbf{h}^{H}(K_{\mathbf{U}_{1}}+K_{\mathbf{U}_{2}})\mathbf{h}}+$
$\displaystyle\qquad~{}~{}\quad\log_{2}\frac{1+\mathbf{h}^{H}K_{\mathbf{U}_{1}}\mathbf{h}}{1+\mathbf{g}^{H}K_{\mathbf{U}_{1}}\mathbf{g}}.$
(8)

Then, any rate pair

$\displaystyle(R_{1},R_{2})\in{\rm co}\left\\{\bigcup_{{\rm
tr}(K_{\mathbf{U}_{1}}+K_{\mathbf{U}_{2}})\leq P}{\mathcal{R}}_{\rm I}^{\rm
S-DPC}(K_{\mathbf{U}_{1}},K_{\mathbf{U}_{2}})\right\\}$

is achievable for the MGBC-CM.

###### Proof:

A detail proof can be found in [14]. ∎

The S-DPC achievable rate region requires optimization of the covariance
matrices $K_{\mathbf{U}_{1}}$ and $K_{\mathbf{U}_{2}}$. In order to achieve
the boundary of ${\mathcal{C}}^{\rm MG}_{s}$, we choose $K_{\mathbf{U}_{1}}$
and $K_{\mathbf{U}_{2}}$ as follows:

$\displaystyle K_{\mathbf{U}_{1}}$ $\displaystyle=\alpha
P\mathbf{e}_{1}\mathbf{e}_{1}^{H}$ and $\displaystyle K_{\mathbf{U}_{2}}$
$\displaystyle=(1-\alpha)P\mathbf{c}_{2}(\alpha)\mathbf{c}_{2}^{H}(\alpha),\quad
0\leq\alpha\leq 1$ (9)

where $\mathbf{e}_{1}$ is defined in (2) and $\mathbf{c}_{2}(\alpha)$ is a
normalized eigenvector of the pencil (5) corresponding to
$\gamma_{2}(\alpha)$. Next, inserting (9) into (7) and (8), we obtain

$\displaystyle\frac{1+\mathbf{h}^{H}K_{\mathbf{U}_{1}}\mathbf{h}}{1+\mathbf{g}^{H}K_{\mathbf{U}_{1}}\mathbf{g}}=\gamma_{1}(\alpha)\qquad\qquad\text{and}$
$\displaystyle\frac{[1+\mathbf{g}^{H}(K_{\mathbf{U}_{1}}+K_{\mathbf{U}_{2}})\mathbf{g}][1+\mathbf{h}^{H}K_{\mathbf{U}_{1}}\mathbf{h}]}{[1+\mathbf{h}^{H}(K_{\mathbf{U}_{1}}+K_{\mathbf{U}_{2}})\mathbf{h}][1+\mathbf{g}^{H}K_{\mathbf{U}_{1}}\mathbf{g}]}=\gamma_{2}(\alpha).$
(10)

Now, by substituting (10) into Lemma 3, we obtain the desired achievable
result.

##  V Converse: Sato-Type Outer Bound

###  V-A Sato-Type Outer Bound

We consider an important property for the BC-CM in the following lemma.

###### Lemma 4

Let ${\mathcal{P}}$ denote the set of channels
$p_{\tilde{Y}_{1},\tilde{Y}_{2}|\mathbf{X}}$ whose marginal distributions
satisfy

$\displaystyle p_{\tilde{Y}_{1}|\mathbf{X}}(y_{1}|\mathbf{x})$
$\displaystyle=p_{Y_{1}|\mathbf{X}}(y_{1}|\mathbf{x})$ and $\displaystyle
p_{\tilde{Y}_{2}|\mathbf{X}}(y_{2}|\mathbf{x})$
$\displaystyle=p_{Y_{1}|\mathbf{X}}(y_{2}|\mathbf{x})$

for all $y_{1}$, $y_{2}$ and $\mathbf{x}$. The secrecy capacity region
${\mathcal{C}}^{\rm MG}_{s}$ is the same for the channels
$p_{\tilde{Y}_{1},\tilde{Y}_{2}|\mathbf{X}}\in{\mathcal{P}}$.

We note that ${\mathcal{P}}$ is the set of channels
$p_{\tilde{Y}_{1},\tilde{Y}_{2}|\mathbf{X}}$ that have the same marginal
distributions as the original channel transition density
$p_{Y_{1},Y_{2}|\mathbf{X}}$. Lemma 4 implies that the secrecy capacity region
${\mathcal{C}}^{\rm MG}_{s}$ depends only on marginal distributions.

###### Theorem 2

Let ${\mathcal{R}}_{\rm
O}\bigl{(}P_{\tilde{Y}_{1},\tilde{Y}_{2}|\mathbf{X}},P_{\mathbf{X}}\bigr{)}$
denote the union of all rate pairs $(R_{1},R_{2})$ satisfying

$\displaystyle R_{1}$ $\displaystyle\leq
I(\mathbf{X};\tilde{Y}_{1}|\tilde{Y}_{2})$ and $\displaystyle R_{2}$
$\displaystyle\leq I(\mathbf{X};\tilde{Y}_{2}|\tilde{Y}_{1})$

for given distributions $P_{\mathbf{X}}$ and
$P_{\tilde{Y}_{1},\tilde{Y}_{2}|\mathbf{X}}$. The secrecy capacity region
${\mathcal{C}}^{\rm MG}_{s}$ of the BC-CM satisfies

$\displaystyle{\mathcal{C}}^{\rm
MG}_{s}\subseteq\bigcap_{P_{\tilde{Y}_{1},\tilde{Y}_{2}|\mathbf{X}}\in{\mathcal{P}}}\left\\{\bigcup_{P_{\mathbf{X}}}{\mathcal{R}}_{\rm
O}\bigl{(}P_{\tilde{Y}_{1},\tilde{Y}_{2}|\mathbf{X}},P_{\mathbf{X}}\bigr{)}\right\\}.$
(11)

###### Proof:

A detail proof can be found in [14]. ∎

###### Remark 2

The outer bound (11) follows by evaluating the secrecy level at each receiver
end in an individual manner, while letting the users decode their messages in
a _cooperative_ manner. In this sense, we refer to this bound as “Sato-type”
outer bound.

For example, we consider the confidential message $W_{1}$ that is destined for
user 1 (corresponding to $\tilde{Y}_{1}$) and eavesdropped upon by user 2
(corresponding to $\tilde{Y}_{2}$). We assume that a genie gives user 1 the
signal $\tilde{Y}_{2}$ as side information for decoding $W_{1}$. Note that the
eavesdropped upon signal $\tilde{Y}_{2}$ at user 2 is always a degraded
version of the entire received signal $(\tilde{Y}_{1},\tilde{Y}_{2})$. This
permits the use of the wiretap channel result of [1].

###### Remark 3

Although Theorem 2 is based on a _degraded_ argument, the outer bound (11) can
be applied to the _general_ broadcast channel with confidential messages.

###  V-B Sato-Type Outer Bound for the MGBC-CM

For the Gaussian BC, the family ${\mathcal{P}}$ is the set of channels

$\displaystyle\tilde{y}_{1}$
$\displaystyle=\mathbf{h}^{H}\mathbf{x}+\tilde{z}_{1}$
$\displaystyle\tilde{y}_{2}$
$\displaystyle=\mathbf{g}^{H}\mathbf{x}+\tilde{z}_{2}$

where $\tilde{z}_{1}$ and $\tilde{z}_{2}$ correspond to arbitrarily
correlated, zero-mean, unit-variance, complex Gaussian random variables. Let
$\rho$ denote the covariance between $\tilde{Z}_{1}$ and $\tilde{Z}_{2}$, i.e,

${\rm
Cov}\bigl{(}\tilde{Z}_{1},\tilde{Z}_{2}\bigr{)}=\rho\quad\text{and}\quad|\rho|^{2}\leq
1.$

Now, the rate region ${{\mathcal{R}}}_{\rm
O}\bigl{(}P_{\tilde{Y}_{1},\tilde{Y}_{2}|\mathbf{X}},P_{\mathbf{X}}\bigr{)}$
is a function of the noise covariance $\rho$ and the input covariance matrix
$K_{\mathbf{X}}$. We consider a computable Sato-type outer bound for the MGBC-
CM in the following lemma.

###### Lemma 5

Let ${{\mathcal{R}}}_{\rm O}^{\rm MG}(\rho,K_{\mathbf{X}})$ denote the union
of all rate pairs $(R_{1},R_{2})$ satisfying

$\displaystyle 0\leq R_{1}$ $\displaystyle\leq f_{1}(\rho,K_{\mathbf{X}})$ and
$\displaystyle 0\leq R_{2}$ $\displaystyle\leq f_{2}(\rho,K_{\mathbf{X}})$

where

$\displaystyle f_{1}(\rho,K_{\mathbf{X}})=\min_{\nu\in\mathbb{C}}$
$\displaystyle\log_{2}\frac{(\mathbf{h}-\nu\mathbf{g})^{H}K_{\mathbf{X}}(\mathbf{h}-\nu\mathbf{g})+\psi_{1}(\nu,\rho)}{(1-|\rho|^{2})}$
$\displaystyle f_{2}(\rho,K_{\mathbf{X}})=\min_{\mu\in\mathbb{C}}$
$\displaystyle\log_{2}\frac{(\mathbf{g}-\mu\mathbf{h})^{H}K_{\mathbf{X}}(\mathbf{g}-\mu\mathbf{h})+\psi_{2}(\mu,\rho)}{(1-|\rho|^{2})}$
$\displaystyle\psi_{1}(\nu,\rho)$
$\displaystyle=1+|\nu|^{2}-\nu^{*}\rho-\rho^{*}\nu$
$\displaystyle\text{and}\quad\psi_{2}(\mu,\rho)$
$\displaystyle=1+|\mu|^{2}-\mu^{*}\rho-\rho^{*}\mu.$

For the MGBC-CM, the secrecy capacity region ${\mathcal{C}}^{\rm MG}_{s}$
satisfies

$\displaystyle{\mathcal{C}}^{\rm MG}_{s}\subseteq\bigcup_{{\rm
tr}(K_{\mathbf{X}})\leq P}{{\mathcal{R}}}_{\rm O}(\rho,K_{\mathbf{X}})$

for any $0\leq|\rho|\leq 1$.

###### Remark 4

Lemma 5 is based on the fact that Gaussian input distributions maximize
${{\mathcal{R}}}_{\rm
O}\bigl{(}P_{\tilde{Y}_{1},\tilde{Y}_{2}|\mathbf{X}},P_{\mathbf{X}}\bigr{)}$
for Gaussian broadcast channel. To illustrate this point, we consider

$\displaystyle I(\mathbf{X};\tilde{Y}_{1}|\tilde{Y}_{2})$
$\displaystyle=h(\tilde{Y}_{1}|\tilde{Y}_{2})-\log_{2}(2\pi e)(1-|\rho|^{2})$
$\displaystyle\leq h(\tilde{Y}_{1}-\nu\tilde{Y}_{2})-\log_{2}(2\pi
e)(1-|\rho|^{2}).$

Moreover, the maximum-entropy theorem [17] implies that
$h(\tilde{Y}_{1}-\nu\tilde{Y}_{2})$ is maximized by Gaussian input
distributions.

Finally, we prove that the Sato-type outer bound of Lemma 5 coincides with the
secrecy capacity region ${\mathcal{C}}^{\rm MG}_{s}$ by choosing the parameter
$\rho=(\mathbf{g}^{H}\mathbf{e}_{1})/(\mathbf{h}^{H}\mathbf{e}_{1}).$ A detail
proof can be found in [14].

##  VI Numerical Examples

In this section, we study a numerical example to illustrate the secrecy
capacity region of the MGBC-CM. For simplicity, we assume that the Gaussian BC
has real input and output alphabets and the channel attenuation vectors
$\mathbf{h}$ and $\mathbf{g}$ are also real. Under these conditions, all
calculated rate values are divided by $2$.

Figure 2: Secrecy capacity region vs. time-sharing secrecy rate region for the
example MGBC-CM in (12)

In particular, we consider the following MGBC-CM

$\displaystyle\left[\begin{matrix}y_{1}\\\ y_{2}\end{matrix}\right]$
$\displaystyle=\left[\begin{matrix}1.5&0\\\
1.801&0.871\end{matrix}\right]\left[\begin{matrix}x_{1}\\\
x_{2}\end{matrix}\right]+\left[\begin{matrix}z_{1}\\\
z_{2}\end{matrix}\right]$ (12)

where $\mathbf{h}=[1.5,0]^{T}$, $\mathbf{g}=[1.801,0.872]^{T}$, and the total
power constraint is set to $P=10$. Fig. 2 illustrates the secrecy capacity
region for the channel (12). We observe that even though each component of the
attenuation vector $\mathbf{h}$ (imposed on user 1) is strictly less than the
corresponding component of $\mathbf{g}$ (imposed on user 2), both users can
achieve positive rates simultaneously under the information-theoretic secrecy
requirement. Moreover, we compare the secrecy capacity region with the secrecy
rate region achieved by the time-sharing scheme (indicated by the dash-dot
line). Fig. 2 demonstrate that the time-sharing scheme is strictly suboptimal
for providing the secrecy capacity region.

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