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Securing Visible Light Communications via Friendly
Jamming
Ayman Mostafa and Lutz Lampe
Department of Electrical and Computer Engineering
The University of British Columbia, Vancouver, BC, V6T 1Z4, Canada
Email: {amostafa,lampe}@ece.ubc.ca
Abstract—Despite offering higher security than radio fre-
quency (RF) channels, the broadcast nature of the visible light
communication (VLC) channel makes VLC links inherently
susceptible to eavesdropping by unauthorized users. In this work,
we consider the physical-layer security of VLC links aided by
friendly jamming. The jammer has multiple light sources, but
does not have access to the data transmitted. The eavesdropper’s
reception is degraded by a jamming signal that causes no
interference to the legitimate receiver. Due to the limited dynamic
range of typical light-emitting diodes (LEDs), both the data and
jamming signals are subject to amplitude constraints. Therefore,
we begin with deriving a closed-form secrecy rate expression for
the corresponding wiretap channel, and adopt secrecy rate as the
performance measure. Then, we formulate a linear programming
problem to maximize the secrecy rate when the eavesdropper’s
channel is accurately known to the jammer. Finally, we consider
robust beamforming to maximize the worst-case secrecy rate
when information about the eavesdropper’s channel is uncertain
due to location uncertainty. The robust scheme makes use of
simple linear programming, making real-time implementation
feasible in a variety of real-world scenarios.
I. INTRODUCTION
Visible light communication (VLC) refers to the wire-
less transmission of information by the means of intensity-
modulating light sources. VLC systems utilize solid-state illu-
mination devices, typically inexpensive, high-brightness light-
emitting diodes (LEDs), to establish high-speed, short-range
wireless communication links. Such VLC links benefit from
the license-free light spectrum and immunity to radio fre-
quency (RF) interference, making VLC a potential technology
to complement Wi-Fi networks and mitigate the degradation
in quality-of-service caused by the congested industrial, scien-
tific, and medical (ISM) radio bands.
The broadcast nature of the wireless medium gives rise
to concerns about the privacy and confidentiality of wireless
networks. Compared to RF channels, VLC links offer better
signal confinement and reduced probability of interception
due to line-of-sight (LoS) propagation and confinement of
light waves by opaque surfaces. Nevertheless, the security of
VLC links is still a concern, especially when the physical
area illuminated by the transmitter is accessible to, or shared
by, multiple users and, consequently, potential eavesdroppers.
Examples include classrooms, meeting rooms, public libraries,
and airplanes to name a few.
During the last few years, physical-layer security has
evolved as a promising research area to enhance the confi-
This work was supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC).
dentiality of wireless networks and complement encryption
techniques typically applied at higher layers of the network
stack. The framework of physical-layer security was pioneered
by Wyner in his celebrated paper [1] when he proposed the
wiretap channel model and secrecy capacity as an information-
theoretic security measure. Since then, physical-layer security
has attracted much interest from researchers in information
theory and signal processing. The secrecy capacity of the Gaus-
sian single-input, single-output (SISO) channel was considered
in [2], the single-input, multiple-output (SIMO) channel in [3],
the multiple-input, single-output (MISO) channel in [4], and
the multiple-input, multiple-output (MIMO) channel in [5], [6].
With the absence of multiple antennas at the transmitter, or the
lack of accurate information about the eavesdropper’s channel,
the use of jamming signals, also known as artificial noise, was
proposed in [7], [8] to degrade the eavesdropper’s reception
and improve the security of the source-destination link.
In this paper, we consider secure transmission over the
VLC channel with the help of a friendly jammer. The sender
transmits data via a single light source, typically a group
of LEDs, while the receiver and eavesdropper have a single
photodetector, each. The jammer is equipped with multiple
light sources, without having access to the transmitted data. A
jamming signal is transmitted to increase the interference seen
by the eavesdropper, resulting in higher secrecy rates over the
legitimate channel.
Typical LEDs have a limited dynamic range, which im-
poses an amplitude constraint on the data and jamming sig-
nals. Hence, we derive a closed-form secrecy rate expression
for the corresponding wiretap channel, subject to amplitude
constraints, and adopt secrecy rate as our performance metric.
Given accurate eavesdropper’s channel state information (CSI),
we formulate a simple linear programming problem to design
the optimal jamming beamformer that maximizes the achiev-
able secrecy rate. In addition, we adopt a bounded uncertainty
model about the eavesdropper’s channel, and consider robust
beamforming, via linear semi-infinite programming, to maxi-
mize the worst-case secrecy rate over all the admissible chan-
nels. We use grid-discretization to approximate the problem
into an ordinary linear program, and characterize worst-case
secrecy rates. The robust scheme, aided by implementation
simplicity, has the potential to mitigate performance degrada-
tion caused by inaccurate CSI in practical scenarios.
We use the following notation throughout the paper. The
set of n-dimensional real-valued numbers is denoted by R
n
,
and the set of n-dimensional non-negative real-valued numbers
is denoted by R
n
+
. Bold characters denote column vectors, and
vector transposition is denoted by the superscript {·}
T
. The all-
ones column vector is denoted by , and its dimension will
be clear from the context. The curled inequality symbol
between two vectors denotes componentwise inequality. We
use |·| around a vector to denote componentwise absolute
value, and use the same symbol around a set to denote the
cardinality of the set. The logarithm of base 2 is denoted
by log(·). A lower-case character x denotes one realization
of the random variable X. Differential entropy and mutual
information are denoted by (·) and I(·; ·), respectively.
The remainder of the paper is organized as follows. The
channel and system models are presented in Section II. In
Section III, we derive the closed-form secrecy rate expres-
sion, and consider optimal and robust jamming beamforming.
Simulation and numerical results are presented in Section IV.
Finally, we conclude the paper in Section V.
II. CHANNEL AND SYSTEM MODELS
A. VLC Channel Model
We consider a DC-biased pulse-amplitude modulation
(PAM) scheme. The transmit element is an illumination LED
driven by a fixed bias I
DC
. A zero-mean current signal x ∈ R is
superimposed on I
DC
, and imperceptibly modulates the output
intensity via an appropriate driver circuit. Typical LEDs have
a limited dynamic range, and, therefore, x must satisfy an
amplitude constraint expressed by |x| ≤ A, A = αI
DC
,
where α ∈ [0, 1] is the modulation index. At the receiver,
a photodiode (PD) converts the incident optical power into a
proportional current. Then, the DC bias is removed, and the
signal is amplified via a transimpedance amplifier to produce
a voltage signal y ∈ R.
Such a VLC channel is conventionally modelled by
y = hx + w (1)
where h ∈ R
+
is the channel gain and w is a zero-mean,
additive white Gaussian noise (AWGN) term. Assuming an
LED with Lambertian emission pattern, the channel gain is
given by [9], [10]
h =
1
2π
(m + 1) cos
m
(φ)
A
RX
d
2
cos(ψ)R |ψ| ≤ ψ
FoV
0 |ψ| > ψ
FoV
(2)
where m = −1/log
cos φ
1
2
is the order of Lambertian
emission with half irradiance at φ
1
2
(measured from the optical
axis of the LED), φ is the angle of irradiance, A
RX
is the
receiver collection area, d is the LoS distance between the
LED and PD, ψ is the angle of incidence (measured from the
axis normal to the receiver surface), R is the PD responsivity,
and ψ
FoV
is the receiver field-of-view (FoV). The receiver
collection area is given by [10]
A
RX
=
n
2
sin
2
(ψ
FoV
)
A
PD
(3)
where n is the refractive index of the optical concentrator and
A
PD
is the PD area.
h
JB
h
JE
Eve
Bob
Jammer
Jammer
JammerJammer Alice
h
AB
h
AE
Jamming
Data
Fig. 1. A VLC network with one transmitter (Alice), one receiver (Bob), one
eavesdropper (Eve), and one jammer equipped with multiple light sources.
B. System Model
We consider the VLC scenario illustrated in Fig. 1. The
room is illuminated by N
J
+ 1 identical light fixtures. Each
fixture consists of a group of LEDs modulated by the same
current signal. Alice, the transmitter, sends her data via a single
fixture. On the other hand, the jammer is equipped with N
J
fixtures but does not have access to the data transmitted by
Alice. The legitimate receiver, Bob, and the eavesdropper, Eve,
are equipped with a single PD, each.
Our goal is to secure the communication link between Alice
and Bob, by hiding information from Eve, without reliance
on encryption techniques probably applied at upper layers.
Without help from the jammer, securing the connection be-
tween Alice and Bob, both having a single transmit or receive
element, will not be possible unless Bob is closer to Alice than
Eve. On the other hand, a jammer equipped with multiple light
sources, and without having access to the transmitted data,
can help secure the connection by transmitting an intelligently-
designed jamming signal which increases the interference seen
by Eve and degrades her signal-to-interference-plus-noise ratio
(SINR) while causing minimum or no interference to Bob.
Utilizing the VLC channel model in (1), the signals re-
ceived by Bob and Eve, respectively, are given by
y = h
AB
x + h
T
JB
s + w
B
(4a)
z = h
AE
x + h
T
JE
s + w
E
(4b)
where h
AB
, h
AE
∈ R
+
are the channel gains from Alice to
Bob and Eve, respectively, h
JB
, h
JE
∈ R
N
J
+
are the channel
gain vectors form the jammer to Bob and Eve, respectively,
x ∈ R is the data signal, s ∈ R
N
J
is the jamming signal,
and w
B
, w
E
are zero-mean AWGN samples with variance σ
2
.
Similar to x, each entry of s assumes a zero-mean distribution
and, therefore, the illumination level remains unaffected. The
system of equations in (4) describes a SISO wiretap channel.
Due to the limited dynamic range of the LEDs, both the
data and jamming signals must satisfy an amplitude constraint,
as expressed by
|x| ≤ A, (5a)
|s| A. (5b)
To simplify the wiretap channel model in (4), we impose
the following restrictions. First, the jamming signal should
cause no interference to Bob, i.e., s is restricted to be in the
W
B
~ (0,σ
2
)
(AWGN)
W
E
~ (0,σ
2
)
(AWGN)
J ~
( A,A)
(Jamming signal)
X Y
Z
Alice Bob
Eve
V
h
AB
h
AE
h
T
JE
w
X ~
( A,A)
(Data signal)
Fig. 2. The wiretap channel corresponding to (6).
nullspace of h
T
JB
. Such a restriction is not necessarily optimum
as it might be possible that allowing some interference at
Bob would cause higher interference at Eve and probably
higher secrecy rate. The problem is further simplified when
the jammer adopts a potentially sub-optimal beamforming
strategy, i.e., the jammer chooses s = wj, where w ∈ R
N
J
,
|w| , is a deterministic vector, i.e., a beamformer, while
j ∈ R, |j| ≤ A, is a zero-mean random jamming symbol.
Beamforming is preferred as it permits simple implementation,
however it might be an inappropriate jamming strategy if
multiple eavesdroppers are existent with probably orthogonal
or near-orthogonal channels. Finally, to simplify deriving a
closed-form secrecy rate expression, we assume that both x
and j are uniformly distributed over the interval [−A, A].
Therefore, (4) simplifies to
y = h
AB
x + w
B
, (6a)
z = h
AE
x + h
T
JE
wj + w
E
, (6b)
and a block diagram illustrating (6) is shown in Fig. 2.
III. SECRECY RATE ANALYSIS
A. Secrecy Rate Expression
Theorem 1: An achievable secrecy rate, in (bits/sec/Hz),
for the wiretap channel in (6) is max(R
s
, 0), where R
s
is
given by
R
s
=
1
2
log
1 +
2h
2
AB
A
2
πeσ
2
−
log
h
AE
|
h
T
JE
w
|
+
|
h
T
JE
w
|
h
AE
log
√
e
|
h
T
JE
w
|
h
AE
≤ 1
h
AE
|
h
T
JE
w
|
log
√
e
|
h
T
JE
w
|
h
AE
> 1
=
1
2
log
1 +
2h
2
AB
A
2
πeσ
2
− min
log
h
AE
h
T
JE
w
+
h
T
JE
w
h
AE
log
√
e,
h
AE
h
T
JE
w
log
√
e
!
.
(7)
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
3.5
4
h
T
JE
w × 10
4
Secrecy rate (bits/sec/Hz)
h
AE
= 0.1000 × 10
−4
h
AE
= 0.2154 × 10
−4
h
AE
= 0.4642 × 10
−4
h
AE
= 1.0000 × 10
−4
h
AE
= 2.1544 × 10
−4
h
AE
= 4.6416 × 10
−4
h
AE
= 10.000 × 10
−4
h
AB
= 10
−4
A = 0.1
σ
2
= 10
−13
Fig. 3. The secrecy rate of Theorem 1.
Proof: See Appendix A.
Figure 3 depicts R
s
as a function of h
T
JE
w for dif-
ferent values of h
AE
. Notice that R
s
is upper-bounded by
1
2
log
1 +
2h
2
AB
A
2
πeσ
2
, which is an achievable rate over Alice-
Bob channel subject to |x| ≤ A [11, Theorem 5]. Notice also
that R
s
is a monotonically-increasing function of h
T
JE
w over
the open interval (0, ∞), while it is a monotonically-decreasing
function of h
AE
over the same interval.
B. Eve’s Channel Perfectly Known to the Jammer
In general, it is unreasonable to assume that Eve’s CSI is
accurately known to Alice or the jammer as Eve is typically a
passive attacker or a malicious user, and will not feed back
her CSI. Nevertheless, there exist certain scenarios where
such an assumption is justifiable. For example, Eve might
be a registered user in the network and her CSI is known
to the jammer, however, in a certain transmission session,
confidential messages shall be delivered from Alice to Bob
and kept hidden from Eve. Alternatively, the jammer might
be able to accurately determine Eve’s location and map the
location information into CSI using (2). Nevertheless, even
in practical scenarios, the assumption of accurate Eve’s CSI
is worth investigation as it provides an upper bound on the
performance in more realistic scenarios.
If h
JE
is perfectly known to the jammer, then the opti-
mal jamming beamformer w
∗
that maximizes the achievable
secrecy rate is given by
w
∗
= arg max
w
R
s
= arg max
w
h
T
JE
w (8a)
s.t.
h
T
JB
w = 0,
|w| .
(8b)
The maximization problem in (8) is a linear program which
can be efficiently solved.
C. Bounded Uncertainty about Eve’s Channel
In this scenario, we consider a bounded uncertainty model
about Eve’s channel. Such a model would naturally arise
if the jammer is aware of Eve’s physical location, with
some uncertainty, and uses the location information to get
an estimate of Eve’s channel. Alternatively, Eve might be
expected or permitted to exist within a certain bounded area
known to the jammer. For simplicity, we consider only two-
dimensional location uncertainty, and assume that Eve’s height
is fixed and accurately known. Extension to three-dimensional
location uncertainty should be straightforward. Let A
E
denote
the area bounding Eve’s location, and let H
A
E
be the set of all
admissible channel realizations for Eve within A
E
. Our goal is
to design the beamformer w so as to maximize the worst-case
secrecy rate over all the channel realizations in H
A
E
. That is
max
w
min
h
h
AE
h
JE
i
∈H
A
E
R
s
(9a)
s.t.
h
T
JB
w = 0,
|w| .
(9b)
Since R
s
is a monotonically-increasing function of h
T
JE
w/h
AE
over the open interval (0, ∞), the max-min problem can be
written as
max
w
min
h
h
AE
h
JE
i
∈H
A
E
h
T
JE
w
h
AE
(10)
subject to the constraints in (9b).
By introducing the auxiliary variable t, the problem can be
reformulated as
max
w,t
t (11a)
s.t.
h
AE
t − h
T
JE
w ≤ 0 ∀
h
AE
h
JE
∈ H
A
E
,
h
T
JB
w = 0,
|w| .
(11b)
Since the uncertainty set H
A
E
is continuous, i.e., an infinite set,
the problem in (11) is a linear semi-infinite program. One ap-
proach to tackle such a problem is to consider the approximate
problem obtained by grid-discretization [12], [13]. Hence, we
approximate the area A
E
by a two-dimensional grid A
E
⊂ A
E
,
A
E
= K < ∞, to impose a finite uncertainty set H
A
E
with
cardinality
H
A
E
= K.
Thus, the approximate problem can be written as
max
w,t
t (12a)
s.t.
h
AE
i
t − h
T
JE
i
w ≤ 0 1 ≤ i ≤ K,
h
T
JB
w = 0,
|w| .
(12b)
The maximization in (12) is an ordinary, i.e., finite, linear pro-
gramming problem and can be efficiently solved. Obviously,
the accuracy of such an approximate problem depends on the
discretization resolution.
IV. SIMULATION RESULTS
In this section, we provide numerical results by simulating
a typical indoor VLC scenario. The problem geometry is
876
54
321
x
y
5 m
1⅔ m
Alice
5 m
1⅔ m
Fig. 4. Problem geometry
TABLE I. SIMULATION PARAMETERS.
Problem geometry
Room dimensions (W × L × H) (5 × 5 × 3) m
3
Light fixtures height (Alice and the jammer) 3 m
Receivers height (Bob and Eve) 0.85 m
Total number of light fixtures N
J
+ 1 9
Transmitter characteristics
Number of LEDs per fixture 7
Average optical power per LED 1 W
Modulation index α 10%
LED half luminous intensity semi-angle φ
1
2
60
◦
Receiver characteristics
Receiver FoV ψ
FoV
70
◦
Lens refractive index n 1.5
PD responsivity R 0.54 (A/W)
PD geometrical area A
PD
1 cm
2
Average noise power σ
2
−98.39 dBm
illustrated in Fig. 4, and the simulation parameters are provided
in Table I. The room has a size of 5 × 5 × 3 m
3
, and it
is illuminated by nine identical light fixtures. Every fixture
consists of seven LEDs, and every LED radiates one Watt of
optical power. The fixture in the center is modulated by Alice
for data transmission, while the remaining eight fixtures are
exploited for jamming. The modulation index for all the LEDs
is 10%. Bob and Eve are located at height 0.85 m above the
floor level, and their receivers have a 70
◦
FoV and a single PD,
each. We use a two-dimensional coordinate system (x, y) to
identify the receivers locations. The origin (0, 0) corresponds
to the room center at the receivers level. Noise power is
calculated using [10, Eqn. (6) and Table I] with a receiver
bandwidth of 70 MHz, and the result is averaged over the
entire room area. The average noise power is −98.39 dBm.
In Fig. 5, we plot the secrecy rate of Theorem 1
as a function of Eve’s location when Bob is located at
(−0.70 m, −0.90 m), while in Fig. 6, we fix Eve’s location
at (0.30 m, −1.50 m) and plot the secrecy rate as a function
of Bob’s location. In both figures, Bob’s and Eve’s channels
are perfectly known to the jammer, and the beamformer in
(8) is applied. When Eve is located in the vicinity of Bob,
jamming is restrained by the nullspace of Bob, resulting in
considerably reduced secrecy rates. On the other hand, when
Bob and Eve are sufficiently separated, the jammer is able to
significantly degrade Eve’s reception, and the resulting secrecy
rate is almost independent of Eve’s channel and is upper-
bounded by the achievable rate between Alice and Bob.
In Fig. 7, we consider the robust beamforming problem.
−2
−1
0
1
2
−2
−1
0
1
2
0
0.5
1
1.5
2
2.5
x (m)
y (m)
Secrecy rate (bits/sec/Hz)
0
0.5
1
1.5
2
Fig. 5. Secrecy rate achieved via the jamming beamformer in (8) as a function
of Eve’s location. Bob is located at (−0.70 m, −0.90 m).
−2
−1
0
1
2
−2
−1
0
1
2
0
0.5
1
1.5
2
2.5
3
x (m)
y (m)
Secrecy rate (bits/sec/Hz)
0
0.5
1
1.5
2
2.5
Fig. 6. Secrecy rate achieved via the jamming beamformer in (8) as a function
of Bob’s location. Eve is located at (0.30 m, −1.50 m).
Bob is located at (−0.70 m, −0.90 m) and his channel is
perfectly known to the jammer. The room area is divided into
16 squares, each of area 1.25 × 1.25 m
2
. The area of each
square represents uncertainty about Eve’s location. The jammer
knows the index of the square surrounding Eve, but does not
know her exact location, i.e., Eve’s location information is
quantized into 16 squares. The problem is to design the optimal
jamming beamformer for each square in order to maximize
the worst-case secrecy rate. Every square is discretized into
a 26 × 26 grid, i.e., with a resolution of 5 × 5 cm
2
, and the
beamformer obtained by (12) is applied. The resulting secrecy
rate is shown as a function of Eve’s actual location within
every square, and worst-case secrecy rates are also shown.
V. CONCLUSION
In this paper, we proposed securing VLC links with the
help of friendly jammers. This approach is particularly useful
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
2.251
2.182
2.183
2.267
2.003
1.625
1.769
2.193
x (m)
1.477
1.754
2.193
Bob
1.981
1.931
2.071
2.257
y (m)
Secrecy rate (bits/sec/Hz)
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
Fig. 7. Secrecy rate as a function of Eve’s location when the robust beam-
former obtained via (12) is applied. Bob is located at (−0.70 m, −0.90 m).
Eve’s location information is quantized into 16 squares, each of area 1.25 ×
1.25 m
2
. The jammer is aware which square bounds Eve’s location. Worst-
case secrecy rates (bits/sec/Hz) are shown within each square.
when multiple antennas are not available to the data transmit-
ter, e.g., due to hardware or wiring limitations. We derived a
closed-form secrecy rate expression for the wiretap channel
to serve as the design equation. We formulated simple linear
programming problems to maximize achievable secrecy rates
with perfect and imperfect eavesdropper’s CSI. Of particular
interest is the robust scheme as it directly addresses a major
drawback of physical-layer security approaches, namely, per-
formance sensitivity to CSI assumptions.
APPENDIX A
PROOF OF THEOREM 1
A. The Trapezoidal Distribution
Let a and b be two positive real numbers, and X and
Y be two independent random variables uniformly distributed
over the intervals [−a, a] and [−b, b], respectively. Then, the
density function of the sum Z = X + Y is obtained by
the convolution f
Z
(z) = (f
X
∗ f
Y
)(z), giving rise to the
trapezoidal distribution
f
Z
(z) =
z+a+b
4ab
−a − b ≤ z ≤ −|a − b|,
min
1
2a
,
1
2b
−|a − b| ≤ z ≤ |a − b|,
−z+a+b
4ab
|a − b| ≤ z ≤ a + b,
0 otherwise,
(13)
as illustrated in Fig. 8 for b ≤ a. Thus, the differential entropy
of Z (in bits) is given by
(Z) = −
Z
f
Z
(z) log f
Z
(z)dz
=
log 2a +
b
a
log
√
e b ≤ a
log 2b +
a
b
log
√
e b > a
= min
log 2a +
b
a
log
√
e , log 2b +
a
b
log
√
e
.
(14)
A plot of (14) as a function of b, for a = 2, is shown in Fig.
9. Notice that (Z) is differentiable ∀b > 0, including b = a.
a a
1/2a
f
X
(x)
x
b b
1/2b
f
Y
(y)
y
(a+b) (a b) a b a+b
1/2a
f
Z
(z) = (f
X
*f
Y
)(z)
z
Fig. 8. The trapezoidal distribution (b ≤ a).
0 1 2 3 4 5 6 7 8 9 10
2
3
4
5
6
7
8
b
Differential entropy (bits)
log 2a +
b
a
log
√
e
log 2b +
a
b
log
√
e
min(log 2a +
b
a
log
√
e , log 2b +
a
b
log
√
e)
a = 2
Fig. 9. Differential entropy of the trapezoidal distribution (a = 2).
B. Derivation of the Secrecy Rate Expression in (7)
Without loss of generality, in the following we assume that
h
T
JE
w is non-negative. For the case h
T
JE
w < 0, w is simply
replaced with −w without violating the amplitude constraints
or changing the secrecy rate results.
First, we recall our assumption in Section II-B that both
x and j are uniformly distributed over the interval [−A, A].
Thus,
(h
AB
X) = log 2h
AB
A, (15a)
(h
AE
X) = log 2h
AE
A, (15b)
h
T
JE
wJ
= log 2h
T
JE
wA. (15c)
Then, using (14), (15b), and (15c), the differential entropy of
V (see Fig. 2) is given by
(V ) = min
log 2h
AE
A +
h
T
JE
w
h
AE
log
√
e ,
log 2h
T
JE
wA +
h
AE
h
T
JE
w
log
√
e
. (16)
Therefore, the secrecy capacity of the wiretap channel in (6)
can be lower-bounded as
C
s
= max
p
X
(I(X; Y ) − I(X; Z))
(a)
≥ I(X; Y ) − I(X; Z)
(b)
≥ I(X; Y ) − I(X; V )
= (Y ) − (Y |X) − (V ) + (V |X)
(c)
≥
1
2
log
2
2 (h
AB
X)
+ 2
2 (W
B
)
− (W
B
) − (V )
+
h
T
JE
wJ
(d)
=
1
2
log
4h
2
AB
A
2
+ 2πeσ
2
−
1
2
log 2πeσ
2
− (V )
+ log 2h
T
JE
wA
(e)
=
1
2
log
1 +
2h
2
AB
A
2
πeσ
2
− min
log
h
AE
h
T
JE
w
+
h
T
JE
w
h
AE
log
√
e ,
h
AE
h
T
JE
w
log
√
e
(17)
where (a) follows from dropping the maximization, (b) from
the data-processing inequality [14, Theorem 2.8.1], (c) from
lower-bounding (Y ) using the entropy-power inequality [14,
Theorem 17.7.3], (d) by substituting from (15a) and (15c), and
(e) by substituting from (16).
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