Discrete Memoryless Interference and Broadcast Channels With Confidential Messages: Secrecy Rate Regions

Abstract

We study information-theoretic security for discrete memoryless interference and broadcast channels with independent confidential messages sent to two receivers. Confidential messages are transmitted to their respective receivers while ensuring mutual information-theoretic secrecy. That is, each receiver is kept in total ignorance with respect to the message intended for the other receiver. The secrecy level is measured by the equivocation rate at the eavesdropping receiver. In this paper, we present inner and outer bounds on secrecy capacity regions for these two communication systems. The derived outer bounds have an identical mutual information expression that applies to both channel models. The difference is in the input distributions over which the expression is optimized. The inner bound rate regions are achieved by random binning techniques. For the broadcast channel, a double-binning coding scheme allows for both joint encoding and preserving of confidentiality. Furthermore, we show that, for a special case of the interference channel, referred to as the switch channel, derived bounds meet. Finally, we describe several transmission schemes for Gaussian interference channels and derive their achievable rate regions while ensuring mutual information-theoretic secrecy. An encoding scheme in which transmitters dedicate some of their power to create artificial noise is proposed and shown to outperform both time-sharing and simple multiplexed transmission of the confidential messages.

Full text

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 6, JUNE 20081
Discrete Memoryless Interference and Broadcast
Channels with Confidential Messages:
Secrecy Rate Regions
Ruoheng Liu, Member, IEEE, Ivana Mari´ c, Member, IEEE, Predrag Spasojevi´ c, Member, IEEE, and
Roy D. Yates Member, IEEE
Abstract—We study information-theoretic security for discrete
memoryless interference and broadcast channels with independent
confidential messages sent to two receivers. Confidential messages
are transmitted to their respective receivers while ensuring
mutual information-theoretic secrecy. That is, each receiver is
kept in total ignorance with respect to the message intended
for the other receiver. The secrecy level is measured by the
equivocation rate at the eavesdropping receiver. In this paper, we
present inner and outer bounds on secrecy capacity regions for
these two communication systems. The derived outer bounds have
an identical mutual information expression that applies to both
channel models. The difference is in the input distributions over
which the expression is optimized. The inner bound rate regions
are achieved by random binning techniques. For the broadcast
channel, a double-binning coding scheme allows for both joint
encoding and preserving of confidentiality. Furthermore, we show
that, for a special case of the interference channel, referred to
as the switch channel, derived bounds meet. Finally, we describe
several transmission schemes for Gaussian interference channels
and derive their achievable rate regions while ensuring mutual
information-theoretic secrecy. An encoding scheme in which
transmitters dedicate some of their power to create artificial noise
is proposed and shown to outperform both time-sharing and
simple multiplexed transmission of the confidential messages.
Index Terms—Broadcast channels, interference channels,
channel capacity, secrecy capacity region, equivocation rate,
information-theoretic secrecy, telecommunication security.
I. INTRODUCTION
The broadcast nature of a wireless medium allows for the
transmitted signal to be received by all users within the com-
munication range. Hence, wireless communication sessions are
very susceptible to eavesdropping. The information-theoretic
single user secure communication problem was first character-
ized using the wiretap channel model proposed by Wyner [1].
In this model, a single source-destination communication link
is eavesdropped upon by a wiretapper via a degraded channel.
Manuscript received February 16, 2007; revised July 30, 2007 and October
15, 2007. This work was supported by NSF Grant ANI 0338805. The material
in this paper was presented in part at the 44th Allerton Conference on
Communication, Control, and Computing, Urbana, IL, USA, September, 2006
and Information Theory and Application Workshop (ITA), San Diego, CA,
USA Jan-Feb, 2007.
R. Liu is with Department of Electrical Engineering, Princeton University,
Princeton, NJ 08544 USA (email: rliu@princeton.edu).
P. Spasojevi´ c and R. D. Yates are with WINLAB, Electrical and Com-
puter Engineering, Rutgers University, North Brunswick, NJ 08902 USA
(email: {spasojev,ryates}@winlab.rutgers.edu).
I. Mari´ c is with WSL, Department of Electrical Engineering, Stanford
University, Stanford, CA 94305 USA (email: ivanam@wsl.stanford.edu).
The secrecy level is measured by the equivocation rate at
the wiretapper. Wyner showed that secure communication is
possible without sharing a secret key between legitimate users,
and determined the tradeoff between the transmission rate
and the secrecy level [1]. This result was generalized by
Csisz´ ar and K¨ orner who determined the capacity region of
the broadcast channel with confidential messages [2] in which
a message intended for one of the receivers is confidential.
Following the work of Wyner [1] and Csisz´ ar and K¨ orner
[2], the more recent information-theoretic research on secure
communication focuses at implementing security on the phys-
ical layer. Based on independent efforts, the authors of [3]
and [4] described achievable secure rate regions and outer
bounds for a two-user discrete memoryless multiple access
channel with confidential messages. This model generalizes
the multiple access channel (MAC) [5, Sec. 14.3] by allowing
each user (or one of the users) to receive noisy channel outputs
and, hence, to eavesdrop the confidential information sent
by the other user. In addition, the Gaussian MAC wiretap
channel has been analyzed in [6]–[10]. The relay channel
with confidential messages where the relay node acts as both
a helper and a wiretapper has been considered in [11]. The
relay-eavesdropper channel has been proposed in [12]. More
recently, the cognitive interference channel with confidential
messages has been addressed in [13]. The effects of fading
on secure wireless communication have been studied in [14]–
[18].
In this paper, we study two distinct but related multi-
terminal secure communication problems following the
information-theoretic approach. We focus on discrete mem-
oryless interference and broadcast channels with indepen-
dent confidential messages sent to two receivers. Confidential
messages are transmitted to their respective receivers while
ensuring mutual information-theoretic secrecy. That is, each
receiver is kept in total ignorance with respect to the message
intended for the other receiver. We first derive outer bounds on
capacity regions for these two communication systems. These
bounds have an identical mutual information expression. The
expression is optimized over different input distributions, i.e.,
for the interference channel, the two senders offer independent
inputs to the channel and, for the broadcast channel, the sender
jointly encodes both messages. We also derive achievable rate
regions for the two channel models. Here, we only consider
sending confidential messages and, hence, no common mes-
sage in the sense of Marton [19] is conveyed to the receivers

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2IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 6, JUNE 2008
in the case of the broadcast channel. The inner bounds are
achieved using random binning techniques. For the broadcast
channel, a double-binning coding scheme which allows for
both joint precoding as in the classical broadcast channel
[19], and preserving of confidentiality. Similarly, ensuring
of confidential messages precludes partial decoding of the
message intended for the other receiver in the case of the
interference channel. Hence, rate-splitting encoding used by
Carleial [20] and Han and Kobayashi [21] employed with
the classical interference channel is precluded. Instead, the
encoders will use only stochastic encoders. We show that
for the special case of the interference channel, referred
to as the switch channel, derived bounds meet. Finally, we
describe several transmission schemes for general Gaussian
interference channels and derive their achievable rate regions
while still ensuring information-theoretic secrecy. An encoding
scheme in which transmitters dedicate some of their power to
create artificial noise is proposed and shown to outperform
both time-sharing and simple multiplexed transmission of the
confidential messages.
The remainder of this paper is organized as follows. The no-
tation and the channel model are given in Sec. II. We state the
main results in Sec. III. Outer bounds are derived in Sec. IV.
Inner bounds associated with the achievable coding scheme
for the interference and broadcast channels with confidential
messages are established in Sec. V. Finally, the results are
summarized in Sec. VI.
II. DEFINITIONS AND NOTATIONS
A. Notations
Throughout the paper, a random variable is denoted with an
upper case letter (e.g., X), its realization is denoted with the
corresponding lower case letter (e.g., x), the finite alphabet
of the random variable is denoted with the corresponding
calligraphic letter (e.g., X), and its probability distribution is
denoted with PX(x). For example, the random variable X with
the probability distribution P(x) = PX(x) takes on values in
the finite alphabet X. A boldface symbol denotes a sequence
with the following conventions
X = [X1,...,Xn],
˜Xi= [Xi,...,Xn].
Xi= [X1,...,Xi],
and
Finally, we use A(n)
typical sequences x with respect to P(x) (see [5] for more
details).
? (PX) to denote the set of (weakly) jointly
B. The Interference Channel with Confidential Messages
Consider a discrete memoryless interference channel with fi-
nite input alphabets X1, X2, finite output alphabets Y1, Y2, and
the channel transition probability distribution P(y1,y2|x1,x2).
Two transmitters wish to send independent, confidential mes-
sages to their respective receivers. We refer to such a channel
as the interference channel with confidential messages (IC-
CM). This communication model is shown in Fig. 1. Symbols
(x1,x2) ∈ (X1× X2) are the channel inputs at transmitters 1
Y1
encW2
X2
enc
Channel
p(y1,y2|x1,x2)
W1
W1
dec
Transmitter 1
Receiver 1
H(W2|Y1)
Transmitter 2
X1
Y2
W2
dec
Receiver 2
H(W1|Y2)
Fig. 1.Interference Channel with Confidential Messages.
and 2, and (y1,y2) ∈ (Y1× Y2) are the channel outputs at
receivers 1 and 2, respectively.
Transmitter t, t = 1,2, intends to send an independent
message Wt ∈ {1,...,Mt} to the desired receiver t in n
channel uses while ensuring information-theoretic secrecy. The
channel is memoryless in the sense that
P(y1,y2|x1,x2) =
n
?
i=1
P(y1,i,y2,i|x1,i,x2,i).
A stochastic encoder for transmitter t is described by a matrix
of conditional probabilities ft(xt|wt), where xt∈ Xn
Wt, and
?
Decoding functions are mappings ψt : Yt → Wt. Secrecy
levels at receivers 1 and 2 are measured with respect to the
equivocation rates
1
nH(W2|Y1)
An (M1,M2,n,P(n)
e
) code for the interference channel
consists of two encoding functions f1, f2, two decoding
functions ψ1, ψ2, and the corresponding maximum average
error probability
? max?P(n)
where, for t = 1,2,
?
A rate pair (R1, R2) is said to be achievable for the interfer-
ence channel with confidential messages if, for any ?0> 0,
there exists a (M1,M2,n,P(n)
e
t, wt∈
xt∈Xn
t
ft(xt|wt) = 1.
and
1
nH(W1|Y2).
(1)
P(n)
e
e,1, P(n)
e,2
?
(2)
P(n)
e,t=
w1,w2
1
M1M2P?ψt(Yt) ?= wt|(w1,w2) sent?.
) code such that
Mt≥ 2nRt
for t = 1,2
(3)
and the reliability requirement
P(n)
e
≤ ?0
(4)
and the security constraints
nR1− H(W1|Y2) ≤ n?0
nR2− H(W2|Y1) ≤ n?0
(5a)
(5b)
are satisfied. This definition corresponds to the so-called weak
secrecy-key rate [22]. A stronger measurement of the secrecy

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DISCRETE MEMORYLESS INTERFERENCE AND BROADCAST CHANNELS WITH CONFIDENTIAL MESSAGES: SECRECY RATE REGIONS3
Y1
W2
Channel
p(y1,y2|x)
W1
W1
dec
Transmitter
Receiver 1
H(W2|Y1)
X
Y2
W2
dec
Receiver 2
H(W1|Y2)
enc
Fig. 2.Broadcast Channel with Confidential Messages.
level has been defined by Maurer and Wolf in terms of the
absolute equivocation [22], where the authors have shown that
the former definition could be replaced by the latter without
any rate penalty for the wiretap channel.
The capacity region of the IC-CM is the closure of the set
of all achievable rate pairs (R1,R2), denoted by CIC.
C. The Broadcast Channel
We also consider a broadcast channel with confidential
messages (BC-CM) in which secret messages from a single
transmitter are to be communicated to two receivers, as shown
in Fig. 2. A discrete memoryless BC-CM is described using
finite sets X, Y1, Y2, and a conditional probability distri-
bution P(y1,y2|x). Symbols x ∈ X are channel inputs and
(y1,y2) ∈ (Y1×Y2) are channel outputs at receivers 1 and 2,
respectively. The transmitter intends to send an independent
message Wt∈ {1,...,Mt} ? Wtto the respective receiver
t ∈ {1, 2} in n channel uses while ensuring information-
theoretic secrecy as given by (5). The channel is memoryless
in the sense that
?
A stochastic encoder is specified by a matrix of conditional
probabilities f(x|w1,w2), where x ∈ Xn, wt∈ Wt, and
?
Note that f(x|w1,w2) is the probability that the pair of
messages (w1,w2) are encoded as the channel input x. The de-
coding function at the receiver t is a mapping φt : Yt→ Wt.
The secrecy levels of confidential messages W2 and W1
are measured, respectively, at receivers 1 and 2 in terms of
the equivocation rates (1). An (M1,M2,n,P(n)
the broadcast channel consists of the encoding function f,
decoding functions φ1, φ2, and the maximum error probability
P(n)
e
in (2), where, for t = 1,2,
?
A rate pair (R1, R2) is said to be achievable for the broadcast
channel with confidential messages if, for any ?0> 0, there
exists a (M1,M2,n,P(n)
e
) code which satisfies (3)–(5).
The capacity region of the BC-CM is the closure of the set
of all achievable rate pairs (R1,R2), denoted by CBC.
P(y1,y2|x) =
n
i=1
P(y1,i,y2,i|xi).
x∈Xn
f(x|w1,w2) = 1.
e
) code for
P(n)
e,t=
w1,w2
1
M1M2P?φt(Yt) ?= wt|(w1,w2) sent?.
(6)
III. MAIN RESULTS
In this section, we state our main results. We first describe
the outer and inner bounds on the capacity regions of both in-
terference and broadcast channels with confidential messages.
We then show that the derived bounds meet for a special
case of the interference channel, called the switch channel.
Finally, we propose several transmission schemes for Gaussian
interference channels and derive their achievable rate regions
under information-theoretic secrecy.
A. Interference Channel with Confidential Messages
Let U, V1, and V2 be auxiliary random variables. We
consider the following two classes of joint distributions for the
interference channel. Let πIC−Obe the class of distributions
P(u,v1,v2,x1,x2,y1,y2) that factor as
P(u)P(v1,v2|u)P(x1|v1)P(x2|v2)P(y1,y2|x1,x2),
and πIC−Ibe the class of distributions that factor as
P(u)P(v1|u)P(v2|u)P(x1|v1)P(x2|v2)P(y1,y2|x1,x2).
Theorem 1: (outer bound for IC-CM) Let RO(πIC−O)
denote the union of all (R1,R2) satisfying
?
0 ≤ R2≤ min
I(V2;Y2|V1,U) − I(V2;Y1|V1,U)
over all distributions P(u,v1,v2,x1,x2,y1,y2) in πIC−O. For
the interference channel (X1×X2,P(y1,y2|x1,x2),Y1×Y2)
with confidential messages, the capacity region
(7)
(8)
0 ≤ R1≤ min
I(V1;Y1|U) − I(V1;Y2|U),
I(V1;Y1|V2,U) − I(V1;Y2|V2,U)
I(V2;Y2|U) − I(V2;Y1|U),
?
?
(9a)
?
(9b)
CIC⊆ RO(πIC−O).
Proof: We provide a proof of Theorem 1 in Sec. IV.
Theorem 2: (inner bound for IC-CM) Let RIC(πIC−I)
denote the union of all (R1,R2) satisfying
0 ≤ R1≤ I(V1;Y1|U) − I(V1;Y2|V2,U)
0 ≤ R2≤ I(V2;Y2|U) − I(V2;Y1|V1,U)
over all distributions P(u,v1,v2,x1,x2,y1,y2) in πIC−I. Any
rate pair
(R1,R2) ∈ RIC(πIC−I)
is achievable for the interference channel with confidential
messages.
Proof: We provide a proof in Sec. V-A.
To derive the achievable rate region for the IC-CM, we
employ an auxiliary random variable U in the sense of Han-
Kobayashi [21]. For a given U, we consider two independent
stochastic encoders, that is, the pre-coding auxiliary random
variables V1 and V2 will be independent for a given U, as
given by (8). To ensure information-theoretic secrecy, the
achievable rate R1 includes a penalty term I(V1;Y2|V2,U),
which is a conditional mutual information of the receiver 2’s
eavesdropper channel while assuming the receiver 2 can first
decode its own information.
(10a)
(10b)

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4IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 6, JUNE 2008
B. Broadcast Channel with Confidential Messages
For the broadcast channel, we focus on the class of distri-
butions P(u,v1,v2,x,y1,y2) that factor as
P(u)P(v1,v2|u)P(x|v1,v2)P(y1,y2|x).
We refer to this class as πBC.
Theorem 3: (outer bound for BC-CM) Let RO(πBC) de-
note the union of all (R1,R2) satisfying
(11)
R1≥ 0, R2≥ 0
R1≤ min
?
?
I(V1;Y1|U) − I(V1;Y2|U),
I(V1;Y1|V2,U) − I(V1;Y2|V2,U)
I(V2;Y2|U) − I(V2;Y1|U),
I(V2;Y2|V1,U) − I(V2;Y1|V1,U)
over all distributions P(u,v1,v2,x,y1,y2) in πBCand auxil-
iary random variables U, V1, and V2satisfying
?
?
(12a)
R2≤ min
(12b)
U → V1→ X
and
U → V2→ X.
(13)
For the broadcast channel (X,P(y1,y2|x),Y1× Y2) with
confidential messages, the capacity region
CBC⊆ RO(πBC).
Proof: We provide a proof of Theorem 3 in Sec. IV.
Remark 1: Outer bounds for the BC-CM and the IC-CM
have a same mutual information expression RO(·), but, they
are optimized over different input distributions πBC and
πIC−O, respectively.
Theorem 4: (inner bound for BC-CM) Let RBC(πBC)
denote the union of all (R1,R2) satisfying
R1≥ 0, R2≥ 0
R1≤ I(V1;Y1|U) − I(V1;V2|U) − I(V1;Y2|V2,U)
R2≤ I(V2;Y2|U) − I(V1;V2|U) − I(V2;Y1|V1,U)
over all distributions P(u,v1,v2,x,y1,y2) in πBC. Any rate
pair
(R1,R2) ∈ RBC(πBC)
is achievable for the broadcast channel with confidential
messages.
Proof: We provide a proof in Sec. V-B.
We note that, for the broadcast channel, we can employ
joint encoding at the transmitter. Hence, the achievable coding
scheme for the BC-CM is based on the double-binning scheme
which combines the Gel’fand-Pinsker binning [23] and the
random binning. To preserve confidentiality, the achievabil-
ity bounds on R1 and R2 each include the penalty term
I(V1;V2|U). Without the confidentiality constraint, Marton’s
inner bound [19] on the broadcast channel illustrates only that
the sum rate has the penalty term I(V1;V2|U). To ensure
information-theoretic secrecy, the proposed coding scheme
pays “double” when jointly encoding at the transmitter.
Example 1: (less noisy broadcast channel) Consider a
special class of broadcast channels in which the channel
X → Y1is less noisy than the channel X → Y2, i.e.,
I(V ;Y1) ≥ I(V ;Y2)
(14a)
(14b)
(15)
for every V → X → (Y1,Y2) [2]. We first consider the outer
bound of the less noisy BC-CM. Based on the Markov chains
in (13) and the definition (15), we have
I(V1;Y1|U = u) ≥ I(V1;Y2|U = u)
I(V2;Y1|U = u) ≥ I(V2;Y2|U = u),
which implies that
I(V1;Y1|U) ≥ I(V1;Y2|U)
I(V2;Y1|U) ≥ I(V2;Y2|U).
Hence the outer bound can be rewritten as the union of all
(R1,R2) satisfying
R1≤ max
R2= 0
P(X)[I(X;Y1) − I(X;Y2)]
(16a)
(16b)
where (16a) follows from [2, Theorem 3]. Next, by applying
Theorem 4 and setting V2 = U = const, we obtain the
identical rate region as (16). This result implies that only
the “better” user can get the non-zero secrecy rate for the
less noisy BC-CM. Note that, the single-antenna Gaussian
broadcast channel is a special case of the less noisy broadcast
channel.
In the following, we consider a sufficient condition under
which both R1and R2can be strictly positive for the BC-CM.
Corollary 1: For a broadcast channel, if there exist a dis-
tribution P(u,v1,v2,x,y1,y2) ∈ πBCfor which
I(V1;Y1|U) > I(V1;Y2,V2|U)
and
I(V2;Y2|U) > I(V2;Y1,V1|U),
then both receivers can achieve strictly positive rates while
ensuring information-theoretic secrecy.
Proof: This result is obtained by applying Theorem 4 and
by setting R1> 0 and R2> 0.
More recently, motivated by this work, the multiple-antenna
Gaussian broadcast channel with confidential messages was
studied in [24]. It was shown that with multiple antennas at
transmitters, strictly positive rates to both receivers can be
achieved while ensuring information-theoretic secrecy.
(17a)
(17b)
C. Switch Channel
In this subsection, we obtain the secrecy capacity region
for a special case of the interference channel referred to as
the switch channel (SC). As shown in Fig. 3, receivers in the
SC cannot listen to both transmissions (from encoders 1 and
2) at the same time. For example, each encoder may transmit
at a different frequency, while each receiver may listen only
to one frequency during each symbol time i. We assume that
each receiver t ∈ {1,2} has a random switch st ∈ {1,2},
which chooses between t and¯t independently at each symbol
time i with probabilities
P(St,i= t) = τt
P(St,i=¯t) = 1 − τt,i = 1,...,n
where¯t is the complement of t. Therefore, receiver t listens
to its own information xt,ifrom encoder t whenever St,i= t,

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DISCRETE MEMORYLESS INTERFERENCE AND BROADCAST CHANNELS WITH CONFIDENTIAL MESSAGES: SECRECY RATE REGIONS5
X1
X2
Y1
Y2
S2
1:(τ1)
S1
2: (1-τ1)
1: (1-τ2)
2: (τ2)
Fig. 3.Switch channel model
while it eavesdrops upon the signal x¯ t,iwhen St,i =¯t. By
assuming that the switch state information is available at the
receiver, we have that
P(yt,i|x1,i,x2,i,st,i) = P(yt,i|x1,i)1(st,i= 1)
+ P(yt,i|x2,i)1(st,i= 2)
= P(yt,i|xst,i,i)
(18)
where 1(·) is the indicator function.
The switch state information {St,i}n
known at receiver t. Hence, we can consider st,ias a part of
the channel output, i.e., we set
i=1is an i.i.d. process
yt,i? {zt,i, st,i}
(19)
where zt,i represents the received signal value at receiver t.
Under this setting, we have the following theorem on the
secrecy capacity region CSCof SC-CM.
Theorem 5: For the switch channel with confidential mes-
sages, the capacity region CSC is the union of all (R1,R2)
satisfying
0 ≤ R1≤ I(V1;Y1|U) − I(V1;Y2|V2,U)
0 ≤ R2≤ I(V2;Y2|U) − I(V2;Y1|V1,U)
over all distributions P(u,v1,v2,x1,x2,y1,y2) in πIC−I.
Proof: We provide a proof in the Appendix.
Remark 2: In SC-CM, receiver t listens to the desired
information during time fraction τt, and intercepts the other
message during the time fraction (1−τt). When τ1= τ2= 1,
both receivers only listen to their own messages and thus SC-
CM reduces to two independent parallel channels without the
secrecy constraints. When τ1= 1 and τ2= 0, receiver 2 acts
as an eavesdropper only and both Y1and Y2are independent
with respect to the message W2. Hence, in this case, SC-CM
reduces to the wiretap channel [1].
Example 2: (noiseless memoryless switch channel) We
assume that the channel is discrete memoryless and that the
input-output relationship at each time instant satisfies
?
where P(St,i= t) = τtand τ1+ τ2≥ 1. Theorem 5 implies
that the secrecy capacity region of this channel is:
?
(20a)
(20b)
Yt,i=
X1,i,
X2,i,
St,i= 1
St,i= 2
for
i = 1,...,n
(21)
(R1,R2) :
R1≤ (τ1+ τ2− 1)H(X1)
R2≤ (τ1+ τ2− 1)H(X2)
?
.
(22)
receiver 1
receiver 2
transmitter 2
(W1)
transmitter 1
α1
α2
X1
X2
N1
N2
Y1
Y2
(W2)
W1
^
W2
W1
W2
^
Fig. 4.Gaussian interference channel with confidential messages
We note that here τ1+ τ2− 1 equals τ1− (1 − τ2), the time
that user 1 sends without user 2 listening and also equals τ2−
(1 − τ1), the time that user 2 sends without user 1 listening.
D. Gaussian Interference Channel with Confidential Messages
We next consider a Gaussian interference channel (GIC)
with confidential messages (GIC-CM) where each node em-
ploys a single antenna as shown in Fig. 4. We have proposed
this problem originally in [25].
We assume the channel input and output symbols to be from
an alphabet of real numbers. Following the standard form GIC
[20], the received symbols are
Y1= X1+ α1X2+ N1
Y2= α2X1+ X2+ N2
(23a)
(23b)
where α1and α2are normalized crossover channel gains, X1
and X2are transmitted symbols from encoders 1 and 2 with
the average power constraint
?
and N1 and N2 correspond to two independent, zero-mean,
unit-variance, Gaussian noise variables. In the following, we
focus on the weak interference channel, i.e., 0 ≤ α2
0 ≤ α2
achievable rate regions under the requirement of information-
theoretic secrecy.
1) Time-Sharing: The transmission period is divided into
two non-overlapping slots with time fractions ρ1and ρ2, where
ρ1 ≥ 0, ρ2 ≥ 0, and ρ1+ ρ2 = 1. Transmitter t sends
confidential message Wt in slot t with time fraction ρt and
power Pt/ρt, t = 1,2. We refer to this technique as the time-
sharing scheme. We note that, in each slot, the channel reduces
to a Gaussian wiretap channel [26]. Let R[T]
of (R1,R2) satisfying
?
0 ≤ R2≤ρ2
2
ρ2
over all time fractions (ρ1,ρ2) pairs. Following [26], we can
show that any rate pair
n
i=1
E[X2
t,i]
n
≤ Pt,
for t = 1,2,
1< 1 and
2< 1. We describe three transmission schemes and their
GICdenote the set
0 ≤ R1≤ρ1
2
log
?
?
1 +P1
ρ1
?
?
− log
?
?
1 + α2
2
P1
ρ1
P2
ρ2
??
???
log1 +P2
− log 1 + α2
1
(R1,R2) ∈ R[T]
GIC
is achievable for GIC-CM.