Cooperative-Jamming-Aided Secrecy Enhancement in Wireless Networks With Passive Eavesdroppers

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2108 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 3, MARCH 2018
Cooperative-Jamming-Aided Secrecy Enhancement
in Wireless Networks With Passive Eavesdroppers
Lin Hu , Hong Wen , Bin Wu, Jie Tang, Fei Pan, and Run-Fa Liao
Abstract—This paper investigates cooperative security in wire-
less networks, where a source (Alice) intends to transmit a con-
fidential message to a legitimate destination (Bob), with the help
of a cooperative jammer (Charlie), coexisting with multiple pas-
sive eavesdroppers (Eves). In particular, by assuming knowledge
of Bob’s perfect channel state information (CSI) but only Eves’
statistical CSIs, secrecy beamforming with artificial noise (AN) is
utilized for secure transmission, and cooperative jamming (CJ) is
explored to further enhance secrecy. We first derive an accurate
closed-form expression for the secrecy outage probability (SOP),
and establish the condition under which positive secrecy rate is
achievable. Then, we provide a secure transmit design for max-
imizing the SOP constrained secrecy rate. Moreover, based on a
strict mathematical analysis, we characterize the impact of the
main channel quality and the number of Eves on transmit design
and secrecy performance. Specifically, optimal power allocation
ratio between the information-bearing signal and the AN signal
increases as the main channel quality improves, and decreases
with the number of Eves. Numerical results confirm that our de-
sign achieves performance improvement in terms of both secrecy
rate and secure energy efficiency, as compared to the approach
without CJ.
Index Terms—Artificial noise (AN), cooperative jamming (CJ),
physical layer security, secrecy rate, secrecy outage probability
(SOP), secure energy efficiency (EE).
I. INTRODUCTION
WIRELESS communication is particularly vulnerable
to eavesdropping due to the open nature of wireless
medium, and thus guarantying secrecy is one of the top issues in
wireless networks. By exploring physical properties of wireless
channel, physical layer security [1]–[3] can maintain confiden-
Manuscript received May 14, 2017; revised July 15, 2017; accepted Au-
gust 17, 2017. Date of publication August 25, 2017; date of current version
March 15, 2018. This work was supported in parts by the 863 High Technol-
ogy Plan under Grant 2015AA01A707 and the NSFC under Grants 61572114
and 61372085. This paper was presented at the IEEE Vehicular Technology
Conference, Toronto, Canada, September 2017. The review of this paper was
coordinated by Prof. X. Fang. (Corresponding author: Hong Wen.)
L. Hu, H. Wen, J. Tang, F. Pan, and R.-F. Liao are with the National Key
Laboratory of Science and Technology on Communications, University of
Electronic Science and Technology of China, Chengdu 611731, China (e-mail:
lin.hu.uestc@gmail.com; sunlike@uestc.edu.cn; cs.tan@163.com; panfeivivi@
hotmail.com; runfa.liao@std.uestc.edu.cn). (e-mail:,lin.hu.uestc@gmail.com;
sunlike@uestc.edu.cn; cs.tan@163.com; panfeivivi@hotmail.com;
runfa.liao@std.uestc.edu.cn).
B. Wu is with the Tianjin Key Laboratory of Advanced Networking (TANK),
School of Computer Science and Technology, Tianjin University, Tianjin
300350, China (e-mail: binw@tju.edu.cn). (e-mail:,binw@tju.edu.cn).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVT.2017.2744660
tiality against eavesdroppers, and is identified as a significant
complement to cryptographic techniques.
The pioneering work on physical layer security can be traced
back to the wiretap channel model [4], which is then extended
to the Gaussian degraded wiretap channel [5] and the gen-
eral non-degraded wiretap channel [6]. Based on these works,
secrecy-enhancing techniques are extensively studied, e.g., se-
cure multi-antenna transmission in [7]–[9], and cooperative se-
curity in [10]–[19]. The key issue is to improve the secrecy
rate, defined as the nonnegative rate difference between the
main channel (from source to destination) and the eavesdropper
channel (from source to eavesdropper) [20]. When the eaves-
dropper’s channel state information (CSI) is unknown, secrecy
beamforming with artificial noise (AN) is proposed in [7]. In
particular, power allocation between the information-bearing
signal and the AN signal is examined in [9] for networks with
randomly distributed eavesdroppers.
For cooperative networks, several protocols are designed,
such as amplify-and-forward (AF), decode-and-forward (DF)
[14], and cooperative jamming (CJ) [15]–[17]. Specifically, the
secrecy rate maximization (SRM) problem under the secrecy
outage probability (SOP) constraint is investigated for single-
input single-output (SISO) [15], single-input multiple-output
(SIMO) [16], and multiple-input single-output (MISO) wiretap
channels [17]. In addition, combinations of relaying (AF or DF)
and CJ scheme are studied in [18], [19].
However, the works on cooperative security [14]–[16], [18]
and [19] assume that both source and relay nodes are equipped
with a single antenna, leading to a limited application scope.
Our work is different from these studies in that we focus on
multi-antenna secure transmission. Although the work in [17]
provides a CJ scheme for secrecy enhancement in multi-antenna
wireless systems, the number of transmit antennas is predeter-
mined, and only a single eavesdropper is considered. Rather
than fixing the number of transmit antennas to specific values,
we extend the investigation in [17] to more general antenna
configurations, and extend CJ scheme to the case with multiple
passive eavesdroppers.
Furthermore, recent studies on cooperative secure transmis-
sion are mainly concerning about secrecy rate performance.
However, a greedy pursuit for secrecy performance improve-
ment may lead to excessively high power consumption, and
increase system complexity by involving more nodes for co-
operation. Therefore, secure energy efficiency (EE) should be
considered for cooperative transmit design. To the best of our
knowledge, there are few works on secure EE for cooperative
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HU et al.: COOPERATIVE-JAMMING-AIDED SECRECY ENHANCEMENT IN WIRELESS NETWORKS WITH PASSIVE EAVESDROPPERS 2109
Fig. 1. Cooperative jamming for secrecy enhancement in wireless networks.
networks. In particular, the problem of secure EE maximiza-
tion is investigated in [21]. However, the eavesdropper’s perfect
CSI is assumed to be available at transmitters, which is usually
impractical for general applications. To alleviate this strong as-
sumption, we propose a CJ scheme with only the statistical CSIs
of eavesdroppers, and evaluate both the secrecy rate and secure
EE performance.
Our contributions can be summarized as follows:
1) We present an accurate closed form expression for SOP,
and further establish the condition under which positive
secrecy rate can be achieved.
2) The explicit transmit design for the SOP constrained SRM
problem is obtained by optimizing power allocation and
rate parameters of the wiretap code. Moreover, we demon-
strate the impact of specific parameters (e.g., the number
of Eves and the main channel quality) on transmit designs,
through strict mathematical proofs.
3) We analyze the secure transmit strategy in the high signal-
to-noise ratio (SNR) regime. Specifically, based on the
asymptotic analysis, we obtain an accurate approximation
for explicit design solutions.
Notations: null(X)is the null space of X, and Π⊥
Xis the
orthogonal complement projector of X.CN (µ,Q),Γ(k, μ),
and Exp(λ)denote the circularly symmetric complex Gaussian
distribution with mean µand covariance Q, gamma distribution
with shape kand scale μ, and exponential distribution with rate
parameter λ, respectively. ·is the 2norm. log(·)and ln(·)de-
note the base-2 and natural logarithms, respectively. Γ(α)is the
gamma function [22, 8.310.1], 1F1(α;γ;z)denotes the degener-
ate hypergeometric function [22, 9.210.1], and 2F1(α, β;γ;z)
denotes the Gauss hypergeometric function [22, 9.100]. The
symbols ,⇒, and ⇐⇒ denote “defined as”, “implies”, and
the equivalence relation, respectively. The right limit of f(x)at
xis defined as f(x+)limρ→0f(x+ρ).
II. SYSTEM MODEL AND PROBLEM FORMULATION
System model is shown in Fig. 1, where a source (Alice)
transmits a confidential message to a destination (Bob) with
AN-assisted secrecy beamforming, in the presence of passive
eavesdroppers (Eves). Besides, there is a cooperative jammer
(Charlie) which emits jamming signals to confuse Eves. Alice
and Charlie are equipped with Naand Ncantennas, respectively,
while Bob and each Eve are equipped with only a single antenna.
The set of Eves is defined as K{1,2, ..., K}. Channels from
Alice to Bob and the kth Eve, k∈K, are denoted by hba ∈CNa
and he,k ∈CNa, respectively, and those from Charlie to Bob
and the kth Eve are denoted by hbc ∈CNcand ge,k ∈CNc,
respectively. All channels are assumed to be independent under
slow flat fading conditions.1Bob’s perfect CSI is assumed to
be available, and he,k and ge,k,k∈K, are modeled as random
vectors with distributions CN(0,INa)and CN (0,INc), respec-
tively. These statistical distributions are assumed to be available
at both Alice and Charlie.
Given that Alice transmits signal sa∈CNaand Charlie trans-
mits signal sc∈CNc, then received signals at Bob and the kth
Eve can be expressed as
yb=hH
basa+hH
bcsc+nb,(1)
ye,k =hH
e,ksa+gH
e,ksc+ne,k ,∀k∈K,(2)
where nb,ne,k ∼CN(0,1)represent additive complex white
Gaussian noise (AWGN) at Bob and the kth Eve, respectively.
Note that the jamming signal scinterferes both Eves and Bob.
Therefore, secrecy is compromised if the transmission of scis
inappropriately designed. To utilize this interference in a posi-
tive way, a zero-forcing (ZF) constraint is imposed at Charlie,
and sccan be constructed as
sc=Pc/(Nc−1)Wzc,(3)
where Pcis the transmit power of Charlie; Wis an orthonor-
mal basis for null hH
bc;zc∈CNc−1denotes a Gaussian noise
vector with distribution CN(0,INc−1).
For secure transmission from Alice to Bob, AN-assisted se-
crecy beamforming is developed. Let Pabe the transmit power
of Alice, and let φ∈(0,1]be the fraction of Paallocated to the
information-bearing signal. Then sais given by
sa=Paφvx+Pa(1−φ)/(Na−1)Vza.(4)
The first term on the right hand side of (4) represents the
information-bearing signal, where x∼CN(0,1)corresponds
to the data symbol for Bob; v=hba/hba is the secrecy
beamforming vector. The second term represents the AN sig-
nal, where za∈CNa−1is a Gaussian noise vector with dis-
tribution CN(0,INa−1);Vdenotes an orthonormal basis for
null hH
ba. From (1)–(4), the signal-to-interference-plus-noise
ratios (SINRs) at Bob and the kth Eve, k∈K,aregivenby
γb=Paφhba2,(5)
γe,k =Paφ|hH
e,kv|2
1+Pa(1−φ)
Na−1hH
e,kV2+Pc
Nc−1gH
e,kW2.(6)
Utilizing Wyner’s wiretap code [4], the confidential informa-
tion is encoded before transmission. Alice determines two rates,
namely, the overall codeword rate Rblog(1+γb), and the se-
crecy data rate Rs, with the rate redundancy Re=Rb−Rsfor
1For slow fading channels, the SOP and SOP constrained secrecy rate are two
important indicators for secrecy performance [2], [8].

2110 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 3, MARCH 2018
anti-eavesdropping. According to the compound wiretap chan-
nel model with multiple Eves [23], as long as one Eve is able
to decode the secret message, i.e., log (1+ maxk∈K γe,k)>R
e,
perfect secrecy cannot be guaranteed and secrecy outage occurs.
According to [8], the SOP can be defined as
εPr maxk∈K γe,k >2Rb−Rs−1.(7)
Let εth be a maximum allowable SOP. Then the SOP constrained
SRM problem can be formally formulated as
max Rss.t. ε≤εth;0<φ≤1.(8)
III. SECRECY RATE MAXIMIZATION
The SOP constrained SRM Problem in (8) is difficult to solve
directly. The main obstacle lies in the probabilistic SOP con-
straint (i.e., ε≤εth). In this section, we divide the solution
process into three steps. First, we derive a closed form expres-
sion for SOP constraint. Then, we transfer problem (8) into
a tractable power allocation problem. Finally, we develop an
efficient numerical method to obtain the optimal solution.
A. Formulation of the Explicit SOP Constraint
To simplify the analysis, we define new variables as follows
XkPaφ|hH
e,kv|2,Y
kPa(1−φ)
Na−1hH
e,kV2,(9)
ZkPc
Nc−1gH
e,kW2,ΥkYk+Zk,μ2Rb−Rs−1,
(10)
where k∈K. Consequently, the SOP defined in (7) can be
represented as
ε=Prmax
k∈K
Xk
1+Υ
k
>μ
.(11)
Note that hek ∼CN(0,INa)and gek ∼CN(0,INc). Then, it
can be verified that Xk∼Exp(λ),Yk∼Γ(α1,λ1), and Zk∼
Γ(α2,λ2), where α1Na−1, α2Nc−1, λ1/(Paφ),
λ1α1/(Pa(1−φ)), and λ2α2/Pc.
According to [24, Th. 2 in ch. 2.7], the SINRs at Eves (i.e.,
γe,1,γ
e,2, ..., γe,K ) are independent and identically distributed
(i.i.d.) random variables. Therefore, the SOP in (11) is given by
ε=1−Pr max
k∈K
Xk
1+Υ
k
≤μ
=1−Pr Xk
1+Υ
k
≤μK
.(12)
Proposition 1: A closed form expression for the SOP in (12)
can be expressed as
ε=1−1−λ1
λ1+λμα1λ2
λ2+λμα21
eλμK
.(13)
Proof: Please refer to Appendix A. 
From (7), it can be shown that εincreases with Rsfor any fixed
Rb, and we can conclude that the SOP constraint in problem (8)
holds with equality at the maximum secrecy rate, i.e., ε=εth.
Rearranging and simplifying (13) leads to
ln 1
1−(1−εth)1
K=w
Pa
+α1ln 1+(1−φ)w
α1
+α2ln 1+w
d,(14)
where wμ/φ and dPaλ2.
B. Reformulation of the SRM Problem (8)
Let γBPahba2be the maximum SNR at Bob. Then the
objective function of (8) can be expressed as
Rs=Rb−log(1+μ) = log 1+φγB
1+φw .(15)
Note that Rsdepends on both φand w. As given in (14), wcan
be taken as an implicit function of φ, denoted by w(φ). It can
be verified that w(φ)>0. Then, Rsin (15) can be taken as a
function of φ, denoted by Rs(φ). Hence, problem (8) can be
equivalently reformulated as
max Rs(φ) = log 1+φγB
1+φw(φ)s.t. (14); 0<φ≤1.
(16)
C. Power Allocation for SRM Problem (16)
By analyzing (14), we can obtain the following results.
Lemma 1: The function w(φ)determined by (14) is mono-
tonically increasing with φ.
Proof: Taking the derivative on both sides of (14) with re-
spect to φ, we can derive w(φ)as in (17) shown at the bottom
of this page. It is positive, and hence establishing the desired
result. 
Corollary 1: The function (1−φ)w(φ)determined by (14)
is monotonically decreasing with φ.
Proof: According to Lemma 1, both the first and third terms
on the right-hand side of (14) increase with φ. Hence, in order to
satisfy the SOP constraint in (14), (1−φ)w(φ)must decrease
with φ, leading to the desired result. 
w(φ)= Paα1w(φ)+dw(φ)
α1+(1−φ)w(φ)w(φ)+d+Paα1(1−φ)w(φ)+d+Paα2α1+(1−φ)w(φ).(17)
w(φ)
w(φ)=Paα1
α1+(1−φ)w(φ)+Paα1(1−φ)+Paα2
α1+(1−φ)w(φ)
w(φ)+d
.(18)

HU et al.: COOPERATIVE-JAMMING-AIDED SECRECY ENHANCEMENT IN WIRELESS NETWORKS WITH PASSIVE EAVESDROPPERS 2111
Corollary 2: The function w(φ)/w(φ)is monotonically in-
creasing with φ.
Proof: According to (17), w(φ)/w(φ)can be expressed as
in (18), shown at the bottom of the previous page. It can be shown
that the denominator strictly decreases with φ. This establishes
the desired result. 
Proposition 2: The objective function of problem (16) is
strictly concave on φ.
Proof: According to (15), R
s(φ)can be calculated as
R
s(φ)= 1
ln(2)γB
1+φγB
−w(φ)+φw(φ)
1+φw(φ)
=γB−w(φ)−φ(1+γBφ)w(φ)
(1+γBφ)1+φw(φ)ln(2)
=1
ln(2)⎛
⎜
⎜
⎝
γB−w(φ)
(1+φγB)1+φw(φ)−
w(φ)
w(φ)
1+1
φw(φ)
⎞
⎟
⎟
⎠
.
(19)
Note that the first and second terms on the right-hand side of
(19) are monotonically decreasing and increasing functions of
φ, respectively. As a result, R
s(φ)<0, and hence by the second
order condition [25] we obtain Proposition 2. 
Corollary 3: Rs(φ)in (16) is monotonically increasing with
φwhen R
s(1)≥0, and decreasing with φwhen R
s(0+)≤0.
Proof: By Proposition 2, it follows that when R
s(1)≥
0, R
s(φ)>R

s(1)≥0 for all φ∈(0,1). In contrast, when
R
s(0+)≤0, it follows that R
s(φ)<R

s(0+)≤0 for all φ∈
(0,1]. This completes the proof of Corollary 3. 
With the above results, an efficient numerical method for
solving problem (16) can be developed, where three different
cases are characterized as follows.
1) Case 1: R
s(0+)≤0.
According to Corollary 3, it is optimal for Alice to
suspend secure transmission (i.e., Rs=0). Therefore,
a positive secrecy rate cannot be guaranteed in this
case.
2) Case 2: R
s(0+)>0 and R
s(1)<0.
From Proposition 2, it entails that there must ex-
ists an unique optimal solution φ∗∈(0,1)such that
R
s(φ∗)=0. Furthermore, it can be verified that R
s(φ)
>0 when φ∈(0,φ
∗), and thus Rs(φ∗)>R
s(0+)
=0, i.e., a positive secrecy rate is achievable in this
case.
3) Case 3: R
s(1)≥0.
By Corollary 3, φ∗=1 is the optimal solution. Moreover,
it can be shown that Rs(1)>R
s(0+)=0, and hence a
positive secrecy rate can be guaranteed in this case.
Remark 1: For Case 1, the same solution can also be ob-
tained as follows. According to (19), we can conclude that
R
s(0+)=γB−w(0+)
ln(2)≤0⇐⇒ γB≤w(0+).(20)
Then by Lemma 1, γB≤w(φ)for all φ∈(0,1]. Therefore, by
(15), it is optimal for Alice to suspend secure transmission.
Remark 2: From Cases 1–3, a positive secrecy rate is achiev-
able only when R
s(0+)>0. According to Remark 1, it is equiv-
alent to γB>w(0+), where w(0+)can be obtained by solving
the following equation:
ln 1
1−(1−εth)1
K=w(0+)
Pa
+α1ln 1+w(0+)
α1
+α2ln 1+Pcw(0+)
Paα2.(21)
In other words, the maximum SNR at Bob is larger than a thresh-
old associated with the SOP constraint. When other parameters
are fixed in (21), w(0+)decreases with Pc. Therefore, the jam-
ming signal transmitted by Charlie is helpful in achieving a
positive secrecy rate.
Remark 3: For Case 2, it is difficult to obtain an analytical
expression for φ∗, and numerical methods can be developed
by solving simultaneous equations composed of R
s(φ∗)=0
and (14). In the high SNR region (i.e., Pais large), φ∗can be
approximated as in (22) (see Appendix B for details).
φ∗≈1
1+α1
1−(1−εth)1
K1
α1
−α1
.(22)
Remark 4: When Pcis much larger than Pa,φ∗=1isthe
optimal solution. In other words, all transmit power of Alice is
allocated to the information-bearing signal. A brief analysis is
given as follows. From (19), R
s(1)≥0 is equivalent to
γB≥w(1)+(1+γB)w(1).(23)
Then, substituting φ=1 into (14) and (17), we obtain that
ln 1
1−(1−εth)1
K=w(1)
Pa
+α2ln 1+w(1)
d,(24)
w(1)= Paα1w(1)(w(1)+d)
α1(w(1)+d)+Paα1α2
.(25)
Since d=α2Pa/Pc→0asPc/Pa→∞, (24) entails w(1)→
0, and (25) entails w(1)→0. Therefore, the condition in (23)
can be satisfied, leading to φ∗=1.
Based on the above discussion, the following proposition
characterizes relationships between φ∗and specific parameters,
including the number of Eves (denoted by K) and the main
channel quality (denoted by hba2).
Proposition 3: For problem (16), when the number of Eves
increases, more power should be allocated to the AN signal,
and the secrecy rate decreases. However, when the main chan-
nel quality improves, more power should be allocated to the
information-bearing signal, and the secrecy rate increases.
Proof: Please refer to Appendix C. 
IV. NUMERICAL RESULTS
In this section, we verify the performance of the proposed
CJ scheme. System parameters are set as Na=Nb=4, εth =
0.1. Unless otherwise specified, we set hba=1. Besides, the
secrecy rate is measured by bits per channel use (bpcu), and the

2112 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 3, MARCH 2018
Fig. 2. Secrecy rate versus transmit power of Alice.
Fig. 3. Secure EE versus the total transmit power.
Fig. 4. Optimal power allocation ratio versus transmit power of Alice.
secure EE (defined as the ratio of secrecy rate to total power
consumption.) is measured by bits per joule.
Fig. 2 provides the secrecy rate performance. Obviously, se-
crecy rate performance is enhanced with the help of Charlie.
Specifically, a positive secrecy rate is guaranteed in low SNR
region. This is mainly due to the interference generated by
Fig. 5. The impact of Kon Rsand φ∗, with Pa=20 dB.
Fig. 6. The impact of the hba2on Rsand φ∗, with Pa=20 dB, K=16.
Charlie, which causes significant performance degradation at
Eves. Another reason is that our proposed CJ scheme consumes
more power than the approach without CJ. However, as shown
in Fig. 3, a higher secure EE performance can be achieved in
a wide range of total transmit power. Fig. 2 also shows that in-
creasing power consumption may cause secure EE performance
degradation. This reveals that there exists a trade-off between
secrecy rate and secure EE performance.
Fig. 4 presents the optimal power allocation ratio. It shows
that with the help of Charlie, more power should be allocated
to transmit the confidential message. In particular, full power
is used for information-bearing signal when Pcis much larger
than Pa, as shown in Remark 4. In this case, the interference
created by Charlie is strong enough to mask the transmission of
the confidential message. As a result, the secrecy beamforming
without AN is performed at Alice. However, as Pakeeps in-
creasing, the probability of information leakage increases, and
hence it is preferable to perform AN-assisted secrecy beam-
forming at Alice. In addition, when Pabecomes very large, the
optimal power allocation ratio φ∗converges to a constant value,

HU et al.: COOPERATIVE-JAMMING-AIDED SECRECY ENHANCEMENT IN WIRELESS NETWORKS WITH PASSIVE EAVESDROPPERS 2113
as discussed in Remark 3. It can also be shown that our proposed
approximation is quite accurate.
Figs. 5 and 6 describe the impact of the number of Eves and
the main channel quality on power allocation and secrecy rate.
It can be shown that more power is allocated to the AN sig-
nal as Kincreases. In contrast, more power is allocated to the
information-bearing signal as hba2increases. This is consis-
tent with the result presented in Proposition 3.
V. C ONCLUSION
In this paper, we investigated a cooperative jamming (CJ)
scheme for secrecy enhancement. Specifically, we provided an
explicit transmit design for secrecy rate maximization (SRM)
problem, subject to the secrecy outage probability (SOP) con-
straint. Moreover, for the high signal-to-noise ratio (SNR)
regime, an accurate approximation for design solution was ob-
tained based on asymptotic analysis. By deriving a closed form
expression for SOP, we established a condition under which
positive secrecy rate can be guaranteed. Through strict math-
ematical proofs, we clearly revealed the impacts of specific
parameters on the optimal power allocation, including the num-
ber of eavesdroppers and the main channel quality. Numerical
results confirmed that both the security performance and the
secure energy efficiency (EE) can be enhanced, compared with
the approach without CJ.
APPENDIX A
PROOF OF PROPOSITION 1
Proof: The proof consists of two steps. First, the probability
density function (PDF) of Υkis calculated. Then, we provide
the complementary cumulative distribution function (CCDF) of
γe,k, which can be utilized to derive a closed form expression
for the SOP.
1) First step: According to [22, 3.383.1] and [24], the PDF of
Υkcan be calculated as
fΥk(υ)=∞
−∞
fYk(υ−z)fZk(z)dz
=λα1
1λα2
2e−λ1υ
Γ(α1)Γ(α2)υ
0
zα2−1(υ−z)α1−1e(λ1−λ2)zdz
=λα1
1λα2
2υα1+α2−1
eλ1υΓ(α1+α2)1F1(α2;α1+α2;(λ1−λ2)υ),
(26)
where υ>0. The last equation follows from α1,α
2>0.
2) Second step: From (26), the CCDF of γe,k is given by
¯
Fγe,k (τ)= Pr(γe,k >τ)
=Pr(Xk>τ+τΥk)
=∞
0∞
τ+τυ
λe−λxdxfΥk(υ)dυ
=λα1
1λα2
2e−λτ
Γ(α1+α2)∞
0
υα1+α2−1
e(λ1+λτ)υ
×1F1α2;α1+α2;(λ1−λ2)υdυ, (27)
where τ>0. By [22, 7.621.4], we can obtain a closed form for
(27), and three different cases are characterized as follows.
1) Case 1: λ1=λ2.
Since α1+α2>0, it follows that
¯
Fγe,k (τ)=λα1+α2
1e−λτ
Γ(α1+α2)∞
0
υα1+α2−1
e(λ1+λτ)υdυ
=λα1+α2
1e−λτ
(λ1+λτ)α1+α2.(28)
2) Case 2: λ1>λ2.
In this case, we can infer that
λ1+λτ>λ1−λ2>0⇒|λ1+λτ|>|λ1−λ2|.
(29)
Then, we can obtain that
¯
Fγe,k (τ)
=λα1
1λα2
2e−λτ
(λ1+λτ)α1+α22F1α2,α
1+α2;α1+α2;λ1−λ2
λ1+λτ
=λ1
λ1+λτα1λ2
λ2+λτα21
eλτ.(30)
3) Case 3: λ1<λ2.
In this case, we can infer that
λ2+λτ>λ2−λ1>0⇒|λ2+λτ|>|λ2−λ1|.
(31)
Thus, we can obtain that
¯
Fγe,k (τ)=λα1
1λα2
2e−λτ
Γ(α1+α2)∞
0
e(λ1−λ2)υ
e(λ1+λτ)υυα1+α2−1
×1F1α1;α1+α2;(λ2−λ1)υdυ
=λα1
1λα2
2e−λτ
(λ2+λτ)α1+α22F1
×α1,α
1+α2;α1+α2;λ2−λ1
λ2+λτ
=λ1
λ1+λτα1λ2
λ2+λτα21
eλτ.(32)
By (28), (30), and (32), the CCDF of γe,k is written as
¯
Fγe,k (τ)=λ1
λ1+λτα1λ2
λ2+λτα21
eλτ.(33)
Then, the SOP in (12) can be rewritten as
ε=1−1−¯
Fγe,k (μ)K.(34)
Substituting (33) into (34), we have Proposition 1. 
APPENDIX B
APPROXIMATION FOR THE OPTIMAL SOLUTION IN (22)
Recall that γBPahba2, it follows that γBincreases with
Pa, and hence the condition γB>w(0+)is satisfied when Pa
is large. Then by Remark 2, a positive secrecy rate is achievable
under this condition, meaning that φ∗>0.

2114 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 3, MARCH 2018
In addition, since dPaλ2→∞as Pa→∞,itfollowsthat
the SOP constraint in (14) can be approximated as
ln 1
1−(1−εth)1
K=α1ln 1+(1−φ)w(φ)
α1.(35)
After some rearrangement of terms, we can obtain that
(1−φ)w(φ)=α1
1−(1−εth)1
K1
α1
−α1
⇒w(φ)= 1
(1−φ)2α1
1−(1−εth)1
K1
α1
−α1.
(36)
Besides, as Pabecomes large, R
s(φ)in (19) can be approxi-
mated as
R
s(φ)= 1−φ2w(φ)
φ1+φw(φ)ln(2).(37)
It can be shown from (36) that w(φ)→∞ as φ→1, and
consequently by (37) that R
s(1)<0 holds true. Then according
to Case 2 discussed in Section III-C, we can obtain that φ∗<1.
Finally, by substituting (36) into (37) we obtain that
R
s(φ)=
1−φ
1−φ2α1
1−(1−εth)1
K1
α1
−α1
φ1+φw(φ)ln(2).
(38)
Then by solving R
s(φ)=0, we obtain the expression in (22).
APPENDIX C
PROOF OF PROPOSITION 3
Proof: The proof is divided into two parts. First, we prove
that optimal power allocation ratio and secrecy rate decrease
with K. Then, we prove that they are increasing with hba2.
1) First step: For simplicity, we define a function of Kas
ρ(K)ln 1
1−(1−εth)1
K.(39)
It can be shown that ρ(K)is a monotonically increasing function
of K. In addition, the SOP constraint in (14) can be rewritten as
ρ(K)= w
Pa
+α1ln 1+(1−φ)w
α1+α2ln 1+w
d.
(40)
w(K, φ)=
α1
1
Pa(α1
w(K,φ)+1−φ)+α1(1−φ)
w(K,φ)+α2
w(K,φ)+d(α1
w(K,φ)+1−φ).
(41)
As given in (40), the parameter wcan be taken as an implicit
function of (K, φ), which can be represented as w(K, φ). Then,
according to (17), the derivative of w(K, φ)at φ, denoted by
w(K, φ), can be expressed as in (41).
Lemma 2: The function w(K, φ)in (41) is monotonically
increasing with w(K, φ)and φ.
Proof: It can be shown that the denominator of (41) mono-
tonically decreases with w(ρ(K),φ)and φ, and hence the mono-
tonicity of w(K, φ)is established. 
Besides, the objective function of (16) can be rewritten as
Rs(K, φ) = log 1+φγB
1+φw(K, φ).(42)
R
s(K, φ)=γB−w(K, φ)−φ(1+γBφ)w(K, φ)
(1+γBφ)1+φw(K, φ)ln(2).(43)
ρ(K1)=w(K1,φ
∗
1)
Pa
+α1ln 1+(1−φ∗
1)w(K1,φ
∗
1)
α1
+α2ln 1+w(K1,φ
∗
1)
d,(44)
ρ(K2)=w(K2,φ
∗
2)
Pa
+α1ln 1+(1−φ∗
2)w(K2,φ
∗
2)
α1
+α2ln 1+w(K2,φ
∗
2)
d.(45)
Then the derivative of Rs(K, φ)at φ, denoted by R
s(K, φ),
can be expressed as in (43).
Lemma 3: Rs(K, φ)in (42) is strictly concave over φ.
Proof: For any fixed K, this result directly follows from
Proposition 2. 
Suppose that φ∗
1and φ∗
2are optimal power allocation ratios
for K=K1and K=K2, respectively. We also assume without
loss of generality that K1<K
2.
Lemma 4: If φ∗
1=φ∗
2, then we have w(K2,φ
∗
2)>w(K1,φ
∗
1)
and w(K2,φ
∗
2)>w
(K1,φ
∗
1).
Proof: According to (40), SOP constraints for K=K1and
K=K2can be rewritten as in (44) and (45), respectively.
By the monotonicity of ρ(K), we obtain that ρ(K2)>ρ(K1).
Then comparing (44) with (45), it follows that w(K2,φ
∗
2)>
w(K1,φ
∗
1). Then, it can be shown that the denominator of (41)
decreases with K, and hence w(K2,φ
∗
2)>w
(K1,φ
∗
1),imply-
ing the desired result. 
In the next, a comparison between φ∗
1and φ∗
2, and that between
Rs(K1,φ
∗
1)and Rs(K2,φ
∗
2)can be made. Specifically, four
different cases are characterized as follows.
1) Case 1: γB≥w(K2,1)+(1+γB)w(K2,1).
By (43), it can be shown that this condition is equivalent
to R
s(K2,1)≥0, and hence, by Lemma 3, φ∗
2=1. Then
by Lemma 4 , it follows that w(K2,1)>w(K1,1)and
w(K2,1)>w
(K1,1), and hence we obtain that
γB>w(K1,1)+(1+γB)w(K1,1),(46)
which results in φ∗
1=1, i.e., φ∗
1=φ∗
2=1. Moreover, by
(42), we can obtain that

HU et al.: COOPERATIVE-JAMMING-AIDED SECRECY ENHANCEMENT IN WIRELESS NETWORKS WITH PASSIVE EAVESDROPPERS 2115
Rs(K2,φ
∗
2) = log 1+γB
1+w(K2,1)
<log 1+γB
1+w(K1,1)
=Rs(K1,φ
∗
1).(47)
2) Case 2: γB≤w(K1,0+).
By (43) this condition is equivalent to R
s(K1,0+)≤0,
and consequently, by Lemma 3, φ∗
1=0. Besides, by
Lemma 4, we can obtain that γB<w(K2,0+), and
φ∗
2=0 follows by a similar argument. Then, by (42), we
can obtain that positive secrecy rate cannot be guaranteed
in this case, i.e., Rs(K1,φ
∗
1)=Rs(K2,φ
∗
2)=0
3) Case 3: w(K1,0+)<γ
B≤w(K2,0+).
Combining Case 1,Case 2 stated above and Remark 2 in
Section III-C, it follows that φ∗
1>φ
∗
2=0, and hence we
conclude that Rs(K1,φ
∗
1)>R
s(K2,φ
∗
2)=0.
4) Case 4: w(K2,0+)<γ
B<w(K2,1)+(1+γB)w
(K2,1).
Note that by Lemma 4, we have that γB>w(K1,0+),
and thus, by Remark 2, φ∗
1,φ
∗
2∈(0,1]. In addition, it
can be shown that γB<w(K2,1)+(1+γB)w(K2,1)
is equivalent to R
s(K2,1)<0, and thus, by Lemma 3, we
have that φ∗
2<1. Combining above information, we can
conclude that 0 <φ
∗
1≤1, Rs(K1,φ
∗
1)>0, 0 <φ
∗
2<1
(or equivalently R
s(K2,φ
∗
2)=0), and Rs(K2,φ
∗
2)>0.
Next, two different situations are considered below.
–φ∗
1=1.
Obviously, we have φ∗
1>φ
∗
2, and in addition, φ∗
1=1
is equivalent to R
s(K1,φ
∗
1)≥0, and then we have
γB≥w(K1,1)+(1+γB)w(K1,1),(48)
⇒w(K2,φ
∗
2)+(1+γB)w(K2,φ
∗
2)
>γ
B≥w(K1,1)+(1+γB)w(K1,1).(49)
To prove w(K2,φ
∗
2)>w(K1,1), assume in contra-
diction that w(K2,φ
∗
2)≤w(K1,1). Then by (49) it
follows that w(K2,φ
∗
2)>w
(K1,1), contradicting
the monotonicity of w(K, φ)(see Lemma 2). We
thus conclude that w(K2,φ
∗
2)>w(K1,1), and then
by (42), we obtain that
Rs(K2,φ
∗
2) = log 1+γB−w(K2,φ
∗
2)
1
φ∗
2+w(K2,φ
∗
2)
<log 1+γB−w(K1,φ
∗
1)
1
φ∗
1+w(K1,φ
∗
1)
=Rs(K1,φ
∗
1).(50)
–0<φ
∗
1<1.
According to Lemma 3, this condition is equivalent
to R
s(K1,φ
∗
1)=0, then by (43) we obtain that
γB=w(K1,φ
∗
1)+φ∗
1(1+γBφ∗
1)w(K1,φ
∗
1)
=w(K2,φ
∗
2)+φ∗
2(1+γBφ∗
2)w(K2,φ
∗
2).
(51)
To prove φ∗
2<φ
∗
1, we assume in contradiction that
φ∗
2≥φ∗
1. Since ρ(K2)>ρ(K1), then by (44)–(45),
w(K2,φ
∗
2)>w(K1,φ
∗
1), and hence by (51) we have
φ∗
1(1+γBφ∗
1
)w(K1,φ
∗
1)>φ
∗
2(1+γBφ∗
2)w(K2,φ
∗
2)
⇒w(K1,φ
∗
1)>w
(K2,φ
∗
2),(52)
which is contradicting the monotonicity of w(K, φ)
(see Lemma 2). Thus we have φ∗
2<φ
∗
1.
To show w(K2,φ
∗
2)>w(K1,φ
∗
1), suppose in contra-
diction that w(K2,φ
∗
2)≤w(K1,φ
∗
1). Then by (51),
it follows that
φ∗
1(1+γBφ∗
1)w(K1,φ
∗
1)≤φ∗
2(1+γBφ∗
2)w(K2,φ
∗
2)
⇒w(K1,φ
∗
1)<w
(K2,φ
∗
2),(53)
contradicting the monotonicity of w(K, φ)(see
Lemma 2). Thus we have w(K2,φ
∗
2)>w(K1,φ
∗
1).
Then by (42) we obtain Rs(K2,φ
∗
2)<R
s(K1,φ
∗
1).
Hence, we conclude that if K2>K
1, then φ∗
2≤φ∗
1and
Rs(K2,φ
∗
2)≤Rs(K1,φ
∗
1). A brief summary is given in (54)
shown at the bottom of this page.
2) Second step: Recall that γBPahba2, and it can be
taken as a function of hba, denoted by γB(hba ). By (14),
it follows that the SOP constraint is irrelevant to hba , and in
addition, the objective function of (16) can be rewritten as
Rs(hba,φ) = log 1+φγB(hba)
1+φw(φ),(55)
R
s(hba,φ)
=γB(hba)−w(φ)−φ(1+γB(hba )φ)w(φ)
(1+γB(hba)φ)1+φw(φ)ln(2).(56)
and the derivative of Rs(hba,φ)at φ, which is denoted by
R
s(hba,φ), can be expressed as in (56).
Lemma 5: Rs(hba,φ)in (55) is strictly concave over φ.
Proof: This result directly follows from Proposition 2. 
Suppose that φ∗
1and φ∗
2are optimal power allocation ratios for
different qualities of the main channel (denoted by h(1)
ba 2and
h(2)
ba 2), and assume h(1)
ba <h(2)
ba , which is equivalent to
h(1)
ba 2<h(2)
ba 2. Then obviously, we have that γB(h(1)
ba )<
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
φ∗
2=φ∗
1=1,0<R
s(K2,φ
∗
2)<R
s(K1,φ
∗
1),when γB≥w(K2,1)+(1+γB)w(K2,1);
0<φ
∗
2<φ
∗
1≤1,0<R
s(K2,φ
∗
2)<R
s(K1,φ
∗
1),when w(K2,0+)<γ
B<w(K2,1)+(1+γB)w(K2,1);
0=φ∗
2<φ
∗
1≤1,0=Rs(K2,φ
∗
2)<R
s(K1,φ
∗
1),when w(K1,0+)<γ
B≤w(K2,0+);
φ∗
2=φ∗
1=0,R
s(K2,φ
∗
2)=Rs(K1,φ
∗
1)=0,when γB≤w(K1,0+).
(54)

2116 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 67, NO. 3, MARCH 2018
γB(h(2)
ba ). In the following, we will prove that φ∗
1≤φ∗
2and
Rs(h(1)
ba ,φ
∗
1)≤Rs(h(2)
ba ,φ
∗
2). Specifically, three different
cases are characterized as follows.
1) Case 1: γB(h(1)
ba )≤w(0+).
By (56) this condition is equivalent to R
s(h(1)
ba ,0+)≤
0, and consequently, by Lemma 5, we can obtain that
φ∗
1=0 and Rs(h(1)
ba ,φ
∗
1)=0. Then obviously, we have
that φ∗
2≥φ∗
1and Rs(h(2)
ba ,φ
∗
2)≥Rs(h(1)
ba ,φ
∗
1).
2) Case 2: w(0+)<γ
B(h(1)
ba )<w(1)+(γB(h(1)
ba )+
1)w(1).
By (56) this condition is equivalent to R
s(h(1)
ba ,0+)>0
and R
s(h(1)
ba ,1)<0, and thus, by Lemma 5, 0 <φ
∗
1<
1 and R
s(h(1)
ba ,φ
∗
1)=0. Then by (56) we have
γB(h(1)
ba )−w(φ∗
1)−φ∗
1(1+γB(h(1)
ba )φ∗
1)w(φ∗
1)=0
⇐⇒ γB(h(1)
ba )=w(φ∗
1)+φ∗
1w(φ∗
1)
1−(φ∗
1)2w(φ∗
1).(57)
Substituting (57) into (55), we can obtain that
Rs(h(1)
ba ,φ
∗
1) = log 1
1−(φ∗
1)2w(φ∗
1).(58)
Since γB(h(2)
ba )>γ
B(h(1)
ba ),wehaveγB(h(2)
ba )>
w(0+), which is equivalent to R
s(h(2)
ba ,0+)>0, and
hence by Lemma 5, we obtain that 0 <φ
∗
2≤1. Next, two
different situations are considered below.
–φ∗
2=1
Obviously, we can obtain that φ∗
2=1>φ
∗
1, and in
addition, it can be shown that φ∗
2=1 is equivalent
to R
s(h(2)
ba ,1)≥0, and then by (56) we have that
γB(h(2)
ba )≥w(1)+(1+γB(h(2)
ba ))w(1),
(59)
⇒(1−w(1))γB(h(2)
ba )≥w(1)+w(1).
(60)
Note that since γB(h(2)
ba ),w(1), and w(1)are
positive, it follows that 1 >w
(1). Then combining
(55) and (60), we can obtain that
Rs(h(2)
ba ,φ
∗
2)≥log 1
1−w(1).(61)
Note that by (17), φw(φ)can be rewritten as in (62)
shown at the bottom of this page. Then by Lemma 1
and Corollary 1, it follows that the denominator of
(62) strictly decreases with φ, and the numerator of
(62) strictly increase with φ. Hence, φw(φ)strictly
increases with φ. Then, we can conclude that
1>w
(1)>φ
∗
1w(φ∗
1)>(φ∗
1)2w(φ∗
1)>0
⇒1
1−w(1)>1
1−(φ∗
1)2w(φ∗
1).(63)
Combining (58), (61), and (63), we can obtain that
Rs(h(2)
ba ,φ
∗
2)>log 1
1−(φ∗
1)2w(φ∗
1)
=Rs(h(1)
ba ,φ
∗
1).(64)
–0<φ
∗
2<1
According to Lemma 5, this condition is equivalent
to R
s(h(2)
ba ,φ
∗
2)=0, then by (56) we have
γB(h(2)
ba )=w(φ∗
2)+φ∗
2(1+γB(h(2)
ba )φ∗
2)w
(φ∗
2)
⇐⇒ γB(h(2)
ba )=w(φ∗
2)+φ∗
2w(φ∗
2)
1−(φ∗
2)2w(φ∗
2).(65)
Substituting (65) into (55), we obtain that
Rs(h(2)
ba ,φ
∗
2) = log 1
1−(φ∗
2)2w(φ∗
2).(66)
By (62), it can be shown that the denominator of
w(φ)strictly decreases with φ, and the numerator of
w(φ)strictly increases with φ. Hence w(φ)strictly
increases with φ. To prove φ∗
2>φ
∗
1, we assume in
contradiction that φ∗
2≤φ∗
1. Then by the monotonic-
ity of w(φ),itfollowsthatw(φ∗
2)≤w(φ∗
1), and
hence we obtain that (φ∗
2)2w(φ∗
2)≤(φ∗
1)2w(φ∗
1).
φw(φ)= Paα1φw(φ)
α1+(1−φ)w(φ)+Paα1(1−φ)+ Paα2α1+(1−φ)w(φ)
w(φ)+d
.(62)
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
φ∗
2=φ∗
1=1,0<R
s(h(1)
ba ,φ
∗
1)<R
s(h(2)
ba ,φ
∗
2),when γB(h(1)
ba )≥w(1)+(1+γB(h(1)
ba ))w(1);
0<φ
∗
1<φ
∗
2≤1,0<R
s(h(1)
ba ,φ
∗
1)<R
s(h(2)
ba ,φ
∗
2),when w(0)<γ
B(h(1)
ba )<w(1)+(1+γB(h(1)
ba ))w(1);
0=φ∗
1≤φ∗
2≤1,0=Rs(h(1)
ba ,φ
∗
1)≤Rs(h(2)
ba ,φ
∗
2),when γB(h(1)
ba )≤w(0).
(70)

HU et al.: COOPERATIVE-JAMMING-AIDED SECRECY ENHANCEMENT IN WIRELESS NETWORKS WITH PASSIVE EAVESDROPPERS 2117
Then by (57) and (65), it follows that
γB(h(2)
ba )=w(φ∗
2)+φ∗
2w(φ∗
2)
1−(φ∗
2)2w(φ∗
2)
≤w(φ∗
1)+φ∗
1w(φ∗
1)
1−(φ∗
1)2w(φ∗
1).
=γB(h(1)
ba ),(67)
leading to a contradiction to the assumption that
γB(h(2)
ba )>γ
B(h(1)
ba ). Thus we have φ∗
2>
φ∗
1. Combining (58) and (66) and using the
monotonicity of w(φ),wehaveRs(h(2)
ba ,φ
∗
2)>
Rs(h(1)
ba ,φ
∗
1).
3) Case 3: γB(h(1)
ba )≥w(1)+(1+γB(h(1)
ba ))
w(1).
By (56), this condition is equivalent to
R
s(h(1)
ba ,1)≥0, and consequently, by Lemma 5,
we can obtain that φ∗
1=1.
Since γB(h(2)
ba )>γ
B(h(1)
ba ), it follows that
(1−w(1))γB(h(2)
ba )>(1−w(1))γB(h(1)
ba )
ew(1)+w(1),(68)
which implies that
γB(h(2)
ba )w(1)+w(1)(1+γB(h(2)
ba )).(69)
By (56) and using Lemma 5, we have φ∗
2=1.
Then by (55), we can obtain Rs(h(2)
ba ,φ
∗
2)>
Rs(h(1)
ba ,φ
∗
1).
We thus conclude that if h(2)
ba >h(1)
ba , then φ∗
2≥φ∗
1and
Rs(h(2)
ba ,φ
∗
2)≥Rs(h(1)
ba ,φ
∗
1). A brief summary is given in
(70) shown at the bottom of previous page.
This completes the proof of Proposition 3. 
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Authors’ photographs and biographies not available at the time of publication.