Optimal Relay Selection with a Full-Duplex Active Eavesdropper in Cooperative Wireless Networks

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Optimal Relay Selection with a Full-duplex Active
Eavesdropper in Cooperative Wireless Networks
He Zhou, Dongxuan He, Hua Wang, and Dewei Yang
Beijing Institute of Technology, China
Abstract—In this paper, we investigate the physical layer
security of a dual-hop cooperative network in the presence
of a full-duplex active eavesdropper, which can overhear the
confidential signals and transmit jamming signals simultaneously.
We utilize the optimal relay selection scheme to improve the
secrecy performance, where the relay maximizing the secrecy
capacity will be selected to forward the information. To evaluate
the secrecy performance of our system, we derive a compact
closed-form expression of the secrecy outage probability. Besides,
we also analyze the asymptotic performance related to the
position of the nodes. Finally, we verify our analysis through the
numerical results, and demonstrate that there exists a secrecy
protection region where the secrecy outage probability is below
a target probability.
I. INTRODUCTION
Due to the broadcast nature of wireless medium, communi-
cations in wireless networks are vulnerable to be attacked by
eavesdroppers, leading to severe threat to information security.
Thus, as an alternative to the conventional cryptography-based
methods at the upper layers, physical layer security (PLS),
which is based on information-theoretic security, is becoming
an effective way to combat the illegal interception [1].
To improve the security of the wireless networks, coopera-
tive transmission has been regarded as an effective method
to improve the security of the system. With the help of
the external node, the difference of the channel capacity
between legitimate link and wiretap link can be increased, thus
improving the secrecy performance of the wireless networks.
Among all the external node aided schemes, relay selection
[3]-[9] has been regarded as a promising method due to its
lower complexity and high diversity gain [2,3]. In [4], optimal
relay selection schemes without power constraints and direct
links had been proposed in terms of decreasing the intercept
probability. In [5] and [6], the authors considered the situation
that the decode-and-forward (DF) relays were not always able
to decode the messages, and the secrecy outage probability
(SOP) had been explored, where multiple eavesdroppers can
overhear the messages from both source and relays with
maximal ratio combining (MRC) [5] or selection combining
(SC) [6] technique.
In addition, the relay selection scheme has also been
combined with other techniques [7]-[9]. In [7], the joint
relay selection and cooperative jamming scheme was studied
to maximize the secrecy rate, where the source cooperated
with the destination in the first phase and the selected relay
cooperated with the source in the second phase to confound
the eavesdropper. The authors of [8] used the idea of nodes
cooperation by hybrid cooperative beamforming and jamming
scheme in order to protect the data transmissions during
the two phases. Moreover, [9] proposed to select a pair of
opportunistic relays, where one was selected as a relay to
forward the information and the other worked as a friendly
jammer.
On the other hand, full-duplex has been regarded as a novel
technique [10,11], which can double the spectral efficiency by
transmitting and receiving simultaneously. [10] studied a full-
duplex receiver scheme where an optimal jamming covariance
matrix that maximized the secrecy rate was designed. The
impact of full-duplex relay on the secrecy performance was
analyzed in [11], and a full-duplex relay jamming scheme was
proposed to improve the secrecy.
The aforementioned works all considered the passive eaves-
droppers, where the eavesdroppers only try to overhear the
confidential information. However, there exist some active
eavesdroppers in practice who can eavesdrop and jam simul-
taneously, which are called full-duplex eavesdroppers in some
works [12]-[14]. And the secure transmission without external
nodes against the active eavesdropper has been explored. For
example, [12] studied how to utilize the artificial noise to en-
hance the security in the presence of an active eavesdropper. In
[13], the authors proposed to use a Game-Theoretic framework
and derived the optimal transmission and jamming strategy to
obtain equilibrium for the game. Besides, [14] considered the
secrecy issue under an active eavesdropper in the multiple-
input-multiple-output (MIMO) system.
Motivated by these studies, in this paper, we explore the
secrecy transmission in the cooperative networks in the pres-
ence of a full-duplex active eavesdropper. Here, the optimal
relay that leads to the maximum secrecy capacity is selected to
forward the information in our system. To evaluate the secrecy
performance, the expression of secrecy outage probability is
derived, as well as the asymptotic performance. We also utilize
the numerical simulation to validate our analysis.
The remainder of this paper is organized as follows. In
Section II, we describe the system model of the multi-relay
cooperative network. In Section III, we provide the secrecy
outage probability and the asymptotic analysis. In Section IV,
we give the numerical results. Finally, Section V concludes
the paper.
II. SY ST EM MO DE L
As shown in Fig. 1, we consider a cooperative wireless
network consisting of one source S, one destination D, and
978-1-7281-1217-6/19/$31.00 ©2019 IEEE

Fig. 1: System model of a multi-relay cooperative network
under a full-duplex active eavesdropper.
Ntrusted DF relays Rn(n= 1,2, . . . ,N) in the presence of
an active eavesdropper E. We assume that the eavesdropper
is a full-duplex node, which can perform eavesdropping and
jamming simultaneously, and the source, destination, and
relays all work in half-duplex mode with single antenna. We
also assume that there is no direct link between Sand Ddue
to the deep fading and path-loss. Besides, we assume that all
the relays are close together and have the same distance to
other nodes.
It is assumed that all the channels in the system expe-
rience identical and independent distributed (i.i.d) Rayleigh
fading together with a large-scale fading and the CSIs of
all links are available. We denote hijd−α/2
ij as the channel
between the arbitrarily two nodes iand j, where hij ,dij
(i∈(S, Rn, E), j ∈(Rn, D, E )) and αare the small-scale
fading corresponding to a complex Gaussian random variable
with zero mean and unit variance, the distance from node
ito node jand the path-loss exponent, respectively. Due
to the full-duplex characteristics, the eavesdropper remains
a residual self-interference to itself, which can be modeled
as link between the two antennas of E, denoted by hEE ∼
CN (0,1). Moreover, ρ∈(0,1) is a linear coefficient after
self-interference cancellation.
The transmission can be divided into two phases. The relays
receive signal from the source in the first phase and then
forward the received signal to the destination in the second
phase. We assume the eavesdropper keeps eavesdropping and
jamming during the two phases. Here, we define the transmit
power of the source, the relays, and the eavesdropper as PS,
PR, and PE, respectively. As such, we express the received
signal at node Rnand Ein the first phase as
yRn=PShSRnd−α/2
SRnxs+PEhERnd−α/2
ERnxe+nRn,(1)
yE,1=PShS E d−α/2
SE xs+ρPEhE E d−α/2
EE xe+nE,1,(2)
where xsand xeare the information signal from the source and
jamming signal from the eavesdropper, respectively. Here, we
assume that E|xs|2= 1 and E|xe|2= 1.nRnand nE,1
are AWGN at Rnand Ein the first phase, respectively.
In the second phase, one relay is selected from the N
relays to forward the information signal. To this end, we can
formulate the received signal at node Dand Ein the second
phase as
yD=PRhRnDd−α/2
RnDxs+PEhED d−α/2
ED xe+nD,(3)
yE,2=PRhRnEd−α/2
RnExs+ρPEhEE d−α/2
EE xe+nE,2,
(4)
where nDand nE,2are AWGN at Dand Ein the second
phase, respectively.
It is assumed that the noise received at any node has
the same variance N0. Consequently, from (1) and (2), the
corresponding signal-to-interference-plus-noise ratio (SINR) at
node Rand Ecan respectively be expressed as
γSRn=γS|hSRn|2d−α
SRn
γE|hERn|2d−α
ERn+ 1 ,(5)
γSE =γS|hS E |2d−α
SE
ργE|hEE |2d−α
EE + 1 ,(6)
where γS=PS/N0and γE=PE/N0.
Similarly, according to (3) and (4), the obtained SINR
at node Dand Ein the second phase can respectively be
expressed as
γRnD=γR|hRnD|2d−α
RnD
γE|hED |2d−α
ED + 1 ,(7)
γRnE=γR|hRnE|2d−α
RnE
ργE|hEE |2d−α
EE + 1 ,(8)
where γR=PR/N0.
In this work, we assume that the eavesdropper adopts the
MRC [5] to decode the received signal. To this end, when the
n-th relay is selected, the secrecy capacity can be formulated
as
CS(n) = [CD(n)−CE(n)]+,(9)
where CD(n) = 1
2log (1 + min (γSRn, γRnD)),CE(n) =
1
2log (1 + γSE +γRnE)are capacities of the legitimate link
and the wiretap link when the n-th relay is selected to forward
the information, respectively. And [·]+=max(·,0).
III. SEC RE CY PERFORMANCE ANA LYSIS
In this section, we quantify the system secrecy performance
in terms of the secrecy outage probability (SOP). First, we
derive the analytical expression of SOP in closed-form. Then,
the asymptotic approximations will be derived, revealing the
influence of node relative position intuitively. Specifically, we
assume the relay that maximizes the secrecy capacity will be
selected in the considered system. As such, we can express
the relay selection scheme as
n∗= arg max
n=1,...,N CS(n).(10)

A. Secrecy Outage Probability Analysis
Once CS(n∗)< Rs, where Rsis a target secrecy transmis-
sion rate, the information can not be confidentially transmitted
to D. As such, we define the secrecy outage probability as
the probability that CS(n∗)is below Rs[15], which can be
formulated as
Pout = Pr [CS(n∗)< RS]
= Pr arg max
n=1,...,N 1
2log 1+min(γSRn,γRnD)
1+γSE +γRnE< RS.
(11)
For Nindependent relays are deployed in this system, we
can further express the SOP as
Pout =
N

n=1
Pr 1 + Z1
1 + Z2
< γth
=
N

n=1
(1−Pr [Z2< T (1 + Z1)−1])
=
N

n=1



1−∞
0
FZ2[T(1+x)−1]fZ1(x)dx
  
P



,(12)
where rth = 22RS,T= 1/γth,Z1=min(γS Rn, γRnD)and
Z2=γSE +γRnE.
Since the channel is Rayleigh channel, we know that |hij|2
(as well as |hij |2d−α
ij ) follows exponential distribution. For the
convenience of expression, we define σ2
ij =d−α
ij . As such, the
cumulative distribution function (CDF) and probability density
function (PDF) of |hij |2d−α
ij can be respectively expressed as
F|hij |2d−α
ij (x) = 1 −exp(−x
σ2
ij
),(13)
f|hij |2d−α
ij (x) = 1
σ2
ij
exp(−x
σ2
ij
).(14)
According to the above analysis, we can further obtain the
PDF of variable Z1as
fZ1(x)= a1a2
a2−a1a3x+a1a3+ 1
(x+a1)2−a3x+a2a3+ 1
(x+a2)2e
−a3x,
(15)
where a1=γSσ2
SRn
γEσ2
ERn
,a2=γRσ2
RnD
γEσ2
ED
and a3=1
γSσ2
SRn
+1
γRσ2
RnD
.
Proof: see Appendix.
Meanwhile, from (6) and (8), the CDF of variable Z2can
be expressed as
FZ2(x) = Pr ω3+ω4
ω5
< x
= Pr (ω3< xω5−ω4)(16)
=∞
1xω5
0
Fω3(xω5−ω4)fω4(ω4)fω5(ω5)dω4dω5,
where ω3=γS|hSE |2d−α
SE ,ω4=γR|hRnE|2d−α
RnEand ω5=
ργE|hEE |2d−α
EE + 1.
Substituting (13) and (14) into (16), we can re-express the
CDF of Z2as
FZ2(x) = 1 −(b1−b2)exp(−b3x)
x+b1
−b4
exp(−b6x)
x+b5
,(17)
with









b1=γRσ2
RE
ργEσ2
EE
, b2=γSγRσ2
SE σ2
RE
ργEσ2
EE (γSσ2
SE −γRσ2
RE )
b3=1
γRσ2
RE
, b4=(γSσ2
SE )2
ργEσ2
EE (γSσ2
SE −γRσ2
RE )
b5=γSσ2
SE
ργEσ2
EE
, b6=1
γSσ2
SE
(18)
With the aid of the exponential integral function Ei(·)[16,
eq.(3.352.4),(3.353.3)], we can obtain the expression of Pin
(12) by substituting (15) and (17) into the P, which is now
shown on the top of the next page. And the parameters in (19)
can be expressed as follows









c1= (T−1 + b1)/T , c2= (T−1 + b5)/T
c3=a3+b3T , c4=a3+b6T
d1=(b1−b2) exp(−b3(T−1))
T(c1−a1), d2=(b1−b2) exp(−b3(T−1))
T(c1−a2)
d3=b4exp(−b6(T−1))
T(c2−a1), d4=b4exp(−b6(T−1))
T(c2−a2)
(20)
Utilizing the above analysis, we can obtain a compact
expression of the SOP as
Pout =
N

n=1
(1 −P).(21)
B. Asymptotic Analysis
In this subsection, we give out the asymptotic analysis
of SOP, allowing us to observe the behavior of this metric
intuitively.
Here, two asymptotic expressions, both of which are ac-
cording to the distance relation, are discussing. Specifically,
we first consider that either the destination or the source is
closely located to the relays, and then we consider that the
eavesdropper is close to the transmitters (either the source or
the relays).
In the first case, we first assume the SNR of Rn−Dlink
is so high that σ2
RnD→ ∞, due to the short distance between
the relays and the destination. To this end, we can rewrite
(21) as (22), which is shown on the top of the next page. To
be noticed, this assumption implies the lower bound for the
secrecy performance with fixed average SNR σ2
SRn. Next, we
assume that σ2
SRn→ ∞ for the situation that the distance
between Sand Rnis sufficiently short. As such, we set
γSσ2
SRn=γRσ2
RnDand a1=a2in (22), which is observed
tends to a constant value. The above analysis demonstrates
that the quality of the S−Rnlink and that of the Rn−D
link have the similar impact on the secrecy performance.
In the second case, we first assume γEσ2
SE → ∞ for the
situation that the eavesdropper locates close to the source,
causing the power of the received signal at the eavesdropper to

P= 1 + a1a2
a2−a1{−d1a3−1
c1−a1exp(c1c3)Ei(−c1c3)+(b3T+1
c1−a1) exp(a1c3)Ei(−a1c3) + 1
a1
+d2a3−1
c1−a2exp(c1c3)Ei(−c1c3)+(b3T+1
c1−a2) exp(a2c3)Ei(−a2c3) + 1
a2
−d3a3−1
c2−a1exp(c2c4)Ei(−c2c4)+(b6T+1
c2−a1) exp(a1c4)Ei(−a1c4) + 1
a1
+d4a3−1
c2−a2exp(c2c4)Ei(−c2c4)+(b6T+1
c2−a2) exp(a2c4)Ei(−a2c4) + 1
a2}.
(19)
Pout ≃
N

n=1
{a1{d1 1
γSσ2
SRn
−1
c1−a1exp(c1c3)Ei(−c1c3)+(b3T+1
c1−a1) exp(a1c3)Ei(−a1c3) + 1
a1
+d3 1
γSσ2
SRn
−1
c2−a1exp(c2c4)Ei(−c2c4)+(b6T+1
c2−a1) exp(a1c4)Ei(−a1c4) + 1
a1}.
(22)
0.6 1 1.5 2 2.5 3 3.5
dRnD
10-4
10-3
10-2
10-1
100
Pout
S = 5dB
10dB
15dB
Simulation
Analytical curve
Asymptotic curve
Fig. 2: Pout versus dRnDfor different values of γSwith γR=
20 dB, γE= 5 dB, S(0, 0), Rn(0.5, 0), and E(-0.8, -1).
be sufficiently high. Under this assumption, the SOP will be 1
without doubt. The similar conclusion can be drawn once the
eavesdropper locates close to the relays. The above analysis
indicates that it is crucial to establish a secrecy protection
region to ensure the secure transmission.
IV. NUM ER IC AL RE SU LTS
In this section, we provide numerical results to validate our
analysis. Specifically, we first detail the impact of the distance
from the relays to the destination and the distance form the
eavesdropper to the legitimate nodes, respectively. And then,
we show the impact of the distribution of the eavesdropper
on the secrecy outage probability. Throughout this section, we
consider that Rs= 0.1bits/s/Hz, N= 3, and ρ= 0.01.
And we utilize the relative distance to represent the position
relation, where the distance from the source to the relays is 1.
First, we show how the distance between Rnand Dinflu-
ences the secrecy outage probability in Fig. 2. We can see that
the simulation points match with the analytical curve, which
validates our derivation of the expression of the secrecy outage
probability. We also observe that Pout increases as the increase
of dRnD. When the distance tends to zero, Pout tends to a con-
stant value which is determined by the quality of the channel
between the source and the relays. And Pout tends to 1 when
the distance is sufficiently large, this is because the relays can
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x
E
10-3
10-2
10-1
100
Pout
N = 3
5
7
Analytical curve
Optimal eavesdropping position
Fig. 3: Pout versus xEfor different values of Nwith γS= 10
dB, γR= 20 dB, γE= 5 dB, S(0, 0), Rn(0.7, 0), and D
(1.2, 0.5).
not forward the information to the destination successfully.
Specifically, the short distance region corresponds to the high
SNR regime, where σ2
RnD→ ∞. Here, we can see that
the analytical curve and the asymptotic curve overlap, which
demonstrates the effectiveness of our asymptotic analysis. In
addition, we can find that Pout decreases drastically in the
high SNR regime with the increase of γS, while it degrades
slightly in the low SNR regime. This can be explained by the
fact that the increasing transmit power enhances the wiretap
link more than the legitimate link due to the large distance
between Rnand D.
Then, we plot Pout versus xEfor different values of Nin
Fig. 3. To be noticed, the position of the eavesdropper here
refers to the x-coordinate of the eavesdropper, where the y-
coordinate of the eavesdropper always keeps to be −1. It is
observed that we can improve the secrecy performance by
increasing the number of the relays in this system. We can
also see that Pout first increases and then decreases, and there
exists an optimal eavesdropping position for the eavesdropper.
To this end, we can evaluate the worst secrecy performance
of our system by finding the optimal eavesdropping position.
Finally, we draw Fig. 4 to show how the Pout will be
influenced by the position of the eavesdropper. We can see
from this figure that once the eavesdropper is close to the

Fig. 4: Pout versus the position of Ewith γS= 10 dB, γR=
20 dB, γE= 5 dB, S(0, 0), Rn(0.7, 0), and D(1.2, 0.5).
legitimate nodes (including the source, the relays, and the
destination), the secrecy outage probability will be sufficiently
high. This demonstrates that there exists a secrecy protection
region where Pout is below a target outage probability, and
we can ensure the security by keeping the eavesdropper out
of this region in our system. Besides, the positions of the
legitimate nodes also influence the shape and scale of the
secrecy protection region. As such, it will be significant to
further design the deployment of the legitimate nodes, which
will be investigated in the later works.
V. CONCLUSION
In this paper, we analyzed the secure transmission in the
presence of a full-duplex active eavesdropper. To enhance the
security, we considered the optimal relay selection scheme,
where the relay corresponding to the maximum secrecy capac-
ity is selected to forward the signal, is adopted in this system.
To evaluate the secrecy performance, we derived the closed-
form expression of the secrecy outage probability, as well as
the asymptotic performance. Through our analysis, we found
that there exists a secrecy protection region, where the secrecy
performance will be ensured once we keep the eavesdropper
out of this region. The simulation results validated our analysis
and further demonstrated the existence of the secrecy protec-
tion region.
APP EN DI X
Since Z1=min (γSRn, γRnD), the CDF of Z1can be given
by FZ1(x) = 1 −Pr(γS Rn≥x)·Pr(γRnD≥x)
= 1 −(1 −FγS Rn(x))(1 −FγRnD(x)).(23)
To derive the CDF of Z1, we first rewrite the CDF of γSRn
by substituting (5) as
FγSRn(x) = Pr( γS|hSRn|2d−α
SRn
γE|hERn|2d−α
ERn+ 1 < x)
= Pr(ω1< x(ω2+ 1))
=∞
0
Fω1[x(ω2+ 1)] fω2(ω2)dω2,(24)
where ω1=γS|hSRn|2d−α
SRnand ω2=γE|hERn|2d−α
ERn.
By substituting the CDF and PDF into (24) with (13) and
(14), (24) can be calculated as
FγSRn(x) = 1 −γSσ2
SRn
γSσ2
SRn+γEσ2
ERnxexp(−x
γSσ2
SRn
).
(25)
Similarly, the CDF of γRnDcan be obtained in the same
way. Next, we can derive the CDF of Z1as
FZ1(x) = 1 −a1a2
(x+a1)(x+a2)exp(−a3x).(26)
Then, employing the derivative of FZ1(x)in (26) with
respect to x, we obtain the PDF of Z1as shown in (15).
ACKNOWLEDGEMENT
This work was supported by the National Natural Sci-
ence Foundation of China under Grant No. 61471037, No.
61771048, and No. 61201181.
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