Secure Downlink Massive MIMO NOMA Network in the Presence of a Multiple-Antenna Eavesdropper

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Secure Downlink Massive MIMO NOMA Network
in the Presence of a Multiple-Antenna Eavesdropper
Nam-Phong Nguyen∗‡, Octavia A. Dobre∗, Long D. Nguyen†, Chuyen T. Nguyen‡, and H. Vincent Poor§
∗Memorial University, NL, Canada (e-mail: {nnguyen, odobre}@mun.ca)
†Department of Science and Technology Dong Nai, Vietnam (e-mail: dinhlonghcmut@gmail.com)
‡Hanoi University of Science and Technology, Hanoi, Vietnam (e-mail: {phong.nguyennam, chuyen.nguyenthanh}@hust.edu.vn)
§Princeton University, Princeton, NJ, USA (e-mail: poor@princeton.edu)
Abstract—In this paper, the secrecy performance of a massive
multiple-input multiple-output (MIMO) non-orthogonal multiple
access (NOMA) network is studied in the presence of a multiple-
antenna eavesdropper. The ergodic secrecy rates for the downlink
transmission in the considered system are derived to provide
important insights. Then, by using these results, a joint power
allocation scheme is proposed for both uplink training and down-
link data transmission phases to maximize the sum ergodic secrecy
rates. Because the utility function of interest is non-concave and
the involved constraints are non-convex, a new iterative algorithm
is proposed, which can find at least a local optimum. The obtained
results reveal that the secrecy performance of NOMA networks
benefits from deploying massive MIMO techniques. They also
indicate that the proposed optimization algorithm enhances the
secrecy performance of the considered system.
Index Terms—massive MIMO, NOMA, physical layer security,
multiple-antenna eavesdropper.
I. INTRODUCTION
The advent of the Internet-of-Things is driving the need to
support a massive number of connected devices over limited
spectral resources. This is also one of the most important
requirements of the fifth-generation (5G) wireless networks [1],
[2]. In this context, non-orthogonal multiple access (NOMA)
has attracted significant attention, as it allows a large number
of users to be served in the same time-frequency resources
[3], [4]. A key idea of NOMA is to use the power domain for
multiple access by taking advantage of the successive interfer-
ence cancelation (SIC) technique [5]. Meanwhile, the massive
multiple-input multiple-output (MIMO) technology has been
considered for 5G, to enable increased spectral efficiency and
deal with interference-limited scenarios [6]. Therefore, the
combination of massive MIMO and NOMA is promising to
enhance spectral efficiency and the number of users that can
be served.
However, sharing the same time-frequency resources among
a large number of users imposes urgent security problems.
Recently, some studies have considered the physical layer
security of NOMA networks [7]–[9]. In [7], the authors studied
a secure beamforming optimization problem for downlink
NOMA networks with untrusted cell-edge users. In [8], the
authors investigated the optimal decoding order and user
power allocation in the presence of a passive single-antenna
eavesdropper. In [9], inter-user interference has been utilized
to confuse single-antenna eavesdroppers in a massive MIMO
NOMA network. However, the secrecy performance of massive
MIMO NOMA networks in the presence of a multiple antenna
eavesdropper has not been addressed in the literature.
Fig. 1: Massive MIMO NOMA system model with one
M-antenna BS, Ksingle-antenna UEs, and one N-antenna
eavesdropper.
In this paper, the secrecy performance of a massive MIMO
NOMA network in the presence of a multiple-antenna eaves-
dropper is investigated. In the considered scenario, it is
assumed that the eavesdropper passively obtains important
training information between the base station (BS) and user-
equipments (UEs) and employs this knowledge with multiple
antennas to overhear confidential information intended for
legitimate users. In particular, the ergodic secrecy rates of the
network are derived, and results show that NOMA networks
benefit from the massive MIMO technology in terms of se-
crecy performance. These results also reveal that one of the
traditional physical layer protection methods, i.e., increasing
the BS transmit power at the BS, does not guarantee a security
improvement. Further, a joint optimization algorithm of power
allocation for the uplink training and downlink transmission
phases is proposed to enhance the secrecy performance. Com-
puter simulations validate the analysis and the efficiency of the
power allocation algorithm.
The rest of this paper is organized as follows. The system
and channel models are described in Section II. Analytical
expressions for the ergodic secrecy rates are obtained in Section
III. In Section IV, the optimization algorithm is introduced.
Numerical results and discussions are presented in Section V.
Finally, the paper is concluded in Section VI.
II. SY ST EM A ND CHANNEL MO DE LS
A massive MIMO NOMA system as shown in Fig. 1 is
considered. The system includes one M-antenna BS, Ksingle-
antenna UEs, and one N-antenna eavesdropper. Due to the lack
of orthogonal pilot sequences, the UEs are grouped into K/2
clusters [10]. The k-th cluster includes one cell center user,
UEc
k, and one cell edge user, UEe
k. These two UEs share the

same pilot sequence in the uplink training. In a cluster, the
corresponding large-scale and small-scale fading parameters of
two UEs are denoted by βc
k,βe
kand hc
k,he
k(k= 1, ..., K/2),
respectively. It is considered that hc
k,he
k∼ CN(0,IM), and
βc
k|hc
k|2> βe
k|he
k|2. An eavesdropper with Nantennas tries to
overhear the confidential information from the BS through the
channel √βEG∈CM×N, where the elements of the matrix G
are independent and identically distributed CN(0,1), and βE
models the large-scale fading.
In the uplink training, the BS can obtain a linear combination
of the channels between the BS and the two UEs in the k-th
cluster, i.e., hk=pαc
kβc
khc
k+pαe
kβe
khe
k, where αc
kand αe
k
are the power control parameters at UEc
kand UEe
k, respectively.
In the downlink transmission phase, the BS uses hkfor
precoding. In particular, the transmitted signal vector from the
BS is given by
SBS =rρd2
KW x,(1)
where ρdis the downlink transmission power, x=
[x1...xK/2]T, with E|xk|2= 1, is the vector of K/2sym-
bols intended for the K/2groups. Each superposition symbol
xkof the k-th group is pγc
kxc
k+pγe
kxe
k, with γc
kand γe
kas
the power control coefficients, and xc
kand xe
kas the desired
signals for the cell center and cell edge UEs, respectively.
E|xc
k|2=E|xe
k|2= 1. Therefore, the power constraint
is PK/2
k(γc
k+γe
k)≤1.W= [w1...wK/2]∈CM×K/2
is the precoding matrix. In this paper, the maximum ratio
transmission (MRT) beamforming technique is considered as
follows: wk=h∗
k
qE{khkk2}.
1) At End Users: The received signals at UEc
kand UEe
k,
respectively, are
yc
k=pβc
khc
k
TSBS +nc
k
=rρdβc
k
2
K
K/2
X
l=1
hc
k
Twlpγc
lxc
l+pγe
lxe
l+nc
k,(2)
and
ye
k=pβe
khe
k
TSBS +ne
k
=rρdβe
k
2
K
K/2
X
l=1
he
k
Twlpγe
lxe
l+pγc
lxc
l+ne
k,(3)
where nc
k, ne
k,∼ CN (0, σ2
0)are additive white Gaussian noises
(AWGNs) at UEc
kand UEe
k, respectively. The superscript T
indicates the transpose.
2) At the Eavesdroppers: The received signal at the eaves-
dropper is denoted by yE∈CN×1, and is given as
yE=pβEGTSBS +nE
=rρdβE
2
K
K/2
X
l=1
GTwlpγc
lxc
l+pγe
lxe
l+nE,(4)
where nE∼ CN (0, σ2
EIN)is AWGN at the eavesdropper.
The worst case is considered, when the eavesdropper obtains
knowledge of its own channel Gand the precoder wkduring
downlink training, when the BS beams the pilot to clusters.
Using this knowledge, the eavesdropper performs maximal
ratio combining (MRC) on the received signal by multiplying
its received signal with
¯gH
k
k¯gkkkwkk. The received signal after
performing MRC at the eavesdropper is
˜yE=rρdβE
2
Kpγc
kk¯gkkxc
k+pγe
kk¯gkkxe
k
+
K/2
X
l6=krρdβE
2
K˜
gH
kwlpγc
lxc
l+pγe
lxe
l+ ˜nE,(5)
where ˜yE=
¯gH
k
k¯gkkkwkkyE,¯gk=GTwk
kwkk,˜
gH
k=
¯gH
kGT
k¯gkkkwkk, and
˜nE=
¯gH
k
k¯gkkkwkknE.
III. ERGODIC SEC RE CY RATE
A. Legitimate Rates
In the downlink training, the BS beams the pilots to the
clusters using the precoder wk. It is assumed that UEc
kand UEe
k
have perfect knowledge of hc
k
Twkand he
k
Twk, respectively.
The ergodic rates at UEc
kand UEe
k, respectively, are shown in
(6) and (7), and the ergodic rate of the desired data for UEe
k
at UEc
kis expressed in (9) below.
¯
Rc
k=E{Rc
k}=E(log2 1 + ρdβc
kγc
k2
K|hc
k
Twk|2
Θc
k+ 1 !)
≈log2 1 + γc
kφ2
k(αc
kAc
1,k +αe
kAc
2,k)
Ac
3,k PK/2
l6=k(γe
l+γc
l)+1!
= log2 1 + Υc
1,k
Υc
2,k != log2(1 + Υc
k),(6)
where Θc
k=PK/2
l6=kρdβc
k2
K(γe
l+γc
l)En|hc
k
Twl|2o,φk=
1/p(αc
kβc
k+αe
kβe
k),Ac
1,k =ρd(βc
k)22
K(M+ 1),Ac
2,k =
ρdβc
kβe
k2
K, and Ac
3,k =ρdβc
k2
K.
¯
Re
k=E{Re
k}=E(log2 1 + ρdβe
kγe
k2
K|he
k
Twk|2
Θe
k+ 1 !)
= log2(1 + Υe
k),(7)
where Θe
k=EnPK/2
l6=kρdβe
k2
Kγe
l|he
k
Twl|2o+
EnPK/2
l=1 ρdβe
k2
Kγc
l|he
k
Twl|2o, and where
Υe
k=Υe
1,k
Υe
2,k
=
γe
kφ2
k(αe
kAe
1,k +αc
kAe
2,k)
Ae
3,k PK/2
l6=k(γe
l+γc
l) + γc
kφ2
k(αe
kAe
1,k +αc
kAe
2,k)+1,(8)
where Ae
1,k =ρd(βe
k)22
K(M+ 1),Ae
2,k =ρdβe
kβc
k2
K, and
Ae
3,k =ρdβe
k2
K.
˜
Rc
k=E(log2 1 + ρdβc
kγe
k2
K|hc
k
Twk|2
˜
Θc
k+ 1 !)
≈log2(1 + Ψc
k),(9)

where
˜
Θc
k=E


K/2
X
l6=k
ρdβc
k
2
Kγe
l|hc
k
Twl|2


+E


K/2
X
l=1
ρdβc
k
2
Kγc
l|hc
k
Twl|2


,(10)
Ψc
k=γe
kφ2
k(αc
kZc
1,k +αe
kZc
2,k)
hZc
3,k PK/2
l6=k(γe
l+γc
l) + φ2
kγc
k(αc
kZc
1,k +αe
kZc
2,k)+1i,
(11)
Zc
1,k =ρd(βc
k)22
K(M+ 1),Zc
2,k =ρdβc
kβe
k2
K, and Zc
3,k =
ρdβc
k2
K. In order to perform SIC, the k-th center user needs to
decode the desired data for the k-th edge user. The condition
for that is ˜
Rc
k≥¯
Re
k[10]. Equivalently, the condition can be
rewritten as
Ψc
k≥Υe
k.(12)
B. Eavesdropping Rates
Similar to the previous subsection, the ergodic eavesdrop-
ping rates corresponding to UEe
kand UEc
kare shown below in
(13) and (14), respectively.
¯
Re
E,k =E(log2 1 + ρdβE2
Kγe
kk¯gkk2
Θe
E,k !)
≈log2 1 + γe
kB1
γc
kB1+B2PK/2
l6=k(γc
l+γe
l) + M!
= log2 1 + Υe
E,1,k
Υe
E,2,k != log2(1 + Υe
E,k),(13)
where Θe
E,k =ρdβE2
Kγc
kk¯gkk2+PK/2
l6=kρdβE2
K(γc
l+
γe
l)En˜
gH
kwl
2o+E˜
nE
2,B1=γEβE2
KNM,B2=
γEβE2
K(M+N+1), and γEis the normalized received signal-
to-noise ratio (SNR) at the eavesdropper.
¯
Rc
E,k ≈log2 1 + γc
kB1
γe
kB1+B2PK/2
l6=k(γc
l+γe
l) + M!
= log2 Υc
E,1,k
Υc
E,2,k != log2(1 + Υc
E,k).(14)
C. Ergodic Secrecy Rates
The ergodic secrecy rates of the UEc
kand UEe
k, respectively,
can be approximated as
Rc
S,k =E[Rc
k−Rc
E,k]+≈[¯
Rc
k−¯
Rc
E,k]+,(15)
Re
S,k =E[Re
k−Re
E,k]+≈[¯
Re
k−¯
Re
E,k]+,(16)
where [x]+= max(0, x). These approximations are reasonable
in the case of a massive number of antennas at the BS [9]. The
secrecy performance of the considered system is analyzed in
two scenarios: (i) the number of antennas is large, and (ii) the
transmit power at the BS is large, respectively.
(i) When the number of antennas at the BS is large, it is
observed that
Rc
S,k
M→∞
=∞,(17)
Re
S,k
M→∞
=log21 + γe
k
γc
k
−log2 1 + γe
kN
γc
kN+PK/2
l6=k(γc
l+γe
l)+1!#+
>0.
(18)
Remark 1: The downlink transmissions to UEc
kand UEe
k
are secured when the number of antennas at the BS is suffi-
ciently large. The secrecy performance for UEe
kcan be further
enhanced by controlling the transmit power of the downlink
transmission.
(ii) When increasing the transmit power at the BS, the secrecy
rates corresponding to UEc
kand UEe
kare, respectively,
Rc
S,k
ρd→∞
=
"log2 1 + γc
kφ2
k[αc
k¯
Ac
1,k +αe
k¯
Ac
2,k]
PK/2
l6=k(γe
l+γc
l)!−
log2 1 + γc
k¯
B1
γe
k¯
B1+¯
B2PK/2
l6=k(γc
l+γe
l)!#+
,(19)
where ¯
Ac
1,k =βc
kM(M+ 1),¯
Ac
2,k =Mβe
k,¯
B1=NM, and
¯
B2= (M+N+ 1), and
Re
S,k
ρd→∞
=
"log2 1 + γe
kφ2
k(αe
k¯
Ae
1,k +αc
k¯
Ae
2,k)
PK/2
l6=k(γe
l+γc
l) + γc
kφ2
k(αe
k¯
Ae
1,k +αc
k¯
Ae
2,k)!−
log2 1 + γe
k¯
B1
γc
k¯
B1+¯
B2PK/2
l6=k(γc
l+γe
l)!#+
,(20)
where ¯
Ae
1,k =βe
kM(M+ 1) and ¯
Ae
2,k =Mβc
k.
Remark 2: The secrecy rates converge to constant values
when the transmit power at the BS is sufficiently high. In other
words, a higher transmit power does not necessarily bring a
higher secrecy performance. The secrecy performance can be
enhanced by utilizing the power control parameters of uplink
training and downlink transmission phases.
IV. SEC RE CY RATE MAXIMIZATION PROB LE MS
The max-min secrecy rate problem among the K/2cell-
center UEs and K/2cell-edge UEs is formulated as
max
αc
k,αe
k,γe
k,γc
kmin
k=1,...,K/2(Re
S,k +Rc
S,k)(21a)
s.t.(12) (21b)
0≤αc
k, αe
k≤1,∀k= 1, ..., K/2,(21c)
K/2
X
k=1
(γc
k+γe
k)≤1, γc
k≥0, γe
k≥0.(21d)
It is easily seen that the problem (21) is non-convex since
the objective function (21a) is non-concave and the constraints
(12) are non-convex. We rewrite Re
k,Rc
k,Re
E,k, and Rc
E,k as
follows:

¯
Re
k= log2Υe
1,k + Υe
2,k−log2Υe
2,k
=Re,1
k−Re,2
k,(22)
¯
Rc
k= log2Υc
1,k + Υc
2,k−log2Υc
2,k
=Rc,1
k−Rc,2
k,(23)
¯
Re
E,k = log2Υe
E,1,k + Υe
E,2,k−log2Υe
E,2,k
=Re,1
E,k −Re,2
E,k,(24)
¯
Rc
E,k = log2Υc
E,1,k + Υc
E,2,k−log2Υc
E,2,k
=Rc,1
E,k −Rc,2
E,k.(25)
Then, the secrecy rates of UEc
kand UEe
k(15) and (16) can
be rewritten as
Rc
S,k = [Rc,1
k+Rc,2
E,k −(Rc,2
k+Rc,1
E,k)]+,(26)
Re
S,k = [Re,1
k+Re,2
E,k −(Re,2
k+Re,1
E,k)]+.(27)
By introducing new variables teand tc, the problem (21)
can be rewritten as follows:
max
αc
k,αe
k,γe
k,γc
k,te,tctc+ηte(28a)
s.t.(12),(21c),(21d)(28b)
te≤Re
S,k,∀k= 1, ..., K/2(28c)
tc≤Rc
S,k,∀k= 1, ..., K/2,(28d)
where η > 0is a parameter to control the optimal solution
such that the secrecy rates are not dominated by the secrecy
rates of UEc
k.1
Following the concavity of ln(1 + x)and the convexity of
ln(1 + 1/x)[11], the following inequality is used as in [12]
for all x≥0and ¯x≥0:
−ln(1 + x)≥ −ln(1 + ¯x) + ¯x
1 + ¯x−x
1 + ¯x,(29)
ln(1 + x)≥ln(1 + ¯x) + ¯x
1 + ¯x−¯x2
1 + ¯x
1
x.(30)
For 0< x = [xi]4
i=1 and 0<¯x= [¯xi]4
i=1, applying (29)
and (30) yields
f(x) = ln(1 + x1) + ln(1 + x2)−ln(1 + x3)−ln(1 + x4)
≥f(κ)(x) := a(κ)−
2
X
i=1
b(κ)
i
xi−
4
X
i=3
b(κ)
ixi(31)
for
0< a(κ)=f(¯x) +
4
X
i=1
¯xi
¯xi+ 1 ,
0< b(κ)
1=¯x2
1
1 + ¯x1
,0< b(κ)
2=¯x2
2
1 + ¯x2
,
0< b(κ)
3=1
1 + ¯x3
,0< b(κ)
4=1
1 + ¯x4
.(32)
For γe= [γe
k]K/2
k=1,γc= [γc
k]K/2
k=1,αe= [αe
k]K/2
k=1 and αc=
[αc
k]K/2
k=1, suppose that (γe(κ), γc(κ), αe(κ), αc(κ))is a feasible
1ηis a parameter used to adjust the secrecy rate of cell center and cell edge
UEs in the max-min algorithm. When η > 1, the algorithm focuses more on
increasing the secrecy rate of the cell edge UEs. In practice, the value of η
can be chosen by statistical analysis.
point for satisfying the constraint of problem (21). We compute
the function of Re
S,k by iterating (αe(κ+1), αc(κ+1) )in (28c)
for
x1= (γe
k
(κ)+γc
k
(κ))φ2
k(αe
kAe
1,k +αc
kAe
2,k)
+Ae
3,k
K/2
X
l6=k
(γe
l
(κ)+γc
l
(κ)),
x2=γc
k
(κ)B1+B2
K/2
X
l6=k
(γc
l
(κ)+γe
l
(κ)) + M−1,
x3=Ae
3,k
K/2
X
l6=k
(γe
l
(κ)+γc
l
(κ)) + γc
k
(κ)φ2
k(αe
kAe
1,k +αc
kAe
2,k),
x4= (γe
k
(κ)+γc
k
(κ))B1+B2
K/2
X
l6=k
(γc
l
(κ)+γe
l
(κ)) + M−1,
(33)
and (28d) for
x1=γc
k
(κ)φ2
k(αc
kAc
1,k +αe
kAc
2,k) + Ac
3,k
K/2
X
l6=k
(γe
l
(κ)+γc
l
(κ)),
x2=γe
k
(κ)B1+B2
K/2
X
l6=k
(γc
l
(κ)+γe
l
(κ)) + M−1,
x3=Ac
3,k
K/2
X
l6=k
(γe
l
(κ)+γc
l
(κ)),
x4= (γe
k
(κ)+γc
k
(κ))B1+B2
K/2
X
l6=k
(γc
l
(κ)+γe
l
(κ)) + M−1,
(34)
and ¯xi=xi(γe(κ), γc(κ), αe(κ), αc(κ)), i = 1,2,3,4.
To address the presence of φ2
k(αe
kAe
1,k +αc
kAe
2,k)in x1and
x3of (33), the following inequalities are applied [13]:
x
t≥2√x(κ)√x
t(κ)−x(κ)
(t(κ))2t,
x
t≤1
21
x(κ)t(κ)x2+x(κ)t(κ)1
t2(35)
to obtain upper and lower bounds as follows:
αe
kAe
1,k +αc
kAe
2,k
αc
kβc
k+αe
kβe
k≥



2qαe
k
(κ)Ae
1,k +αc
k
(κ)Ae
2,kqαe
kAe
1,k +αc
kAe
2,k
αc
k
(κ)βc
k+αe
k
(κ)βe
k
−αe
k
(κ)Ae
1,k +αc
k
(κ)Ae
2,k
(αc
k
(κ)βc
k+αe
k
(κ)βe
k)2(αc
kβc
k+αe
kβe
k)!,(36)

αe
kAe
1,k +αc
kAe
2,k
αc
kβc
k+αe
kβe
k≤
1
2 (αe
kAe
1,k +αc
kAe
2,k)2
(αe
k
(κ)Ae
1,k +αc
k
(κ)Ae
2,k)(αc
k
(κ)βc
k+αe
k
(κ)βe
k)
+(αe
k
(κ)Ae
1,k +αc
k
(κ)Ae
2,k)(αe
k
(κ)βc
k+αc
k
(κ)βe
k)
(αc
kβc
k+αe
kβe
k)2!.(37)
Next, in iterating (γe(κ+1), γc(κ+1)), (28c) is considered for
x1= (γe
k+γc
k)φ2
k
(κ+1)(αe
k
(κ+1)Ae
1,k +αc
k
(κ+1)Ae
2,k)
+Ae
3,k
K/2
X
l6=k
(γe
l+γc
l),
x2=γc
kB1+B2
K/2
X
l6=k
(γc
l+γe
l) + M−1,
x3=Ae
3,k
K/2
X
l6=k
(γe
l+γc
l)
+γc
kφ2
k
(κ+1)(αe
k
(κ+1)Ae
1,k +αc
k
(κ+1)Ae
2,k),
x4= (γe
k+γc
k)B1+B2
K/2
X
l6=k
(γc
l+γe
l) + M−1,(38)
and (28d) is considered for
x1=γc
kφ2
k
(κ+1)(αc
k
(κ+1)Ac
1,k +αe
k
(κ+1)Ac
2,k)
+Ac
3,k
K/2
X
l6=k
(γe
l+γc
l),
x2=γe
kB1+B2
K/2
X
l6=k
(γc
l+γe
l) + M,
x3=Ac
3,k
K/2
X
l6=k
(γe
l+γc
l),
x4=(γe
k+γc
k)B1+B2
K/2
X
l6=k
(γc
l+γe
l) + M, (39)
and
¯xi=xi(γe(κ), γc(κ), αe(κ+1) , αc(κ+1)), i = 1,2,3,4.
Recalling (31), it is apparent that
(αe(κ+1), αc(κ+1) , γe(κ+1) , γc(κ+1) )is a better feasible
point than (αe(κ), αc(κ), γe(κ), γ c(κ))[14]:
Re
S,k(γe(κ+1) , γc(κ+1) , αe(κ+1), αc(κ+1) )
> Re
S,k(γe(κ), γ c(κ), αe(κ+1), αc(κ+1)),(40)
Rc
S,k(γe(κ+1) , γc(κ+1) , αe(κ+1), αc(κ+1) )
> Rc
S,k(γe(κ), γ c(κ), αe(κ+1), αc(κ+1)).(41)
Then, the constraints (28c) and (28d) can be rewritten as
te≤Re
S,k
(κ),∀k= 1, ..., K/2,(42)
tc≤Rc
S,k
(κ),∀k= 1, ..., K/2,(43)
40 80 120 160 200 240 280 320 360
0
1
2
3
4
5
6
7
S e c r e c y r a t e ( b i t s / s / H z )

O p t i m i z e d S u m S e c r e c y R a t e
S u m S e c r e c y R a t e
C e n te r U s e r A p p r o x i m a t i o n
E d g e U s e r A p p r o x i m a t i o n
S i m u l a t i o n
A s y m p to ti c
Fig. 2: Secrecy rates at the cell-center and cell-edge UEs and
the sum secrecy rate of the 3rd cluster vs. M.
where Re
S,k
(κ)=a(κ)−P2
i=1
b(κ)
i
xi−P4
i=3 b(κ)
ixicorresponds
to (γe, γc, αe, αc)for (40), and Rc
S,k
(κ)=a(κ)−P2
i=1
b(κ)
i
xi−
P4
i=3 b(κ)
ixicorresponds to (γe, γc, αe, αc)for (41).
At the κ-th iteration, the following convex
program is solved to generate the next iterative point
(γe(κ+1), γ c(κ+1), αe(κ+1), αc(κ+1) )for (28):
max
αc
k,αe
k,γe
k,γc
k,te,tctc+ηte(44a)
s.t. (12),(21c),(21d),(42),(43).(44b)
Thus, in Algorithm 1, a path-following computational pro-
cedure for solving (28) is proposed.
Algorithm 1 Path-following algorithm for solving problem
(28)
1: Initialization: Set a feasible point
(γe(0), γ c(0), αe(0), αc(0) )for (28). Set κ:= 0.
2: Repeat
3: Iterate (αe(κ+1), αc(κ+1) )in solving (44) according to (33)
and (34).
4: Iterate (γe(κ+1), γ c(κ+1))in solving (44) according to (38)
and (39).
5: Set κ:= κ+ 1.
6: Until convergence of the objectives in (28).
V. NUMERICAL RESULTS
In this section, numerical results confirm the correctness of
our analysis and the improvement provided by the optimization
algorithm in terms of secrecy rates. Without loss of generality,
the parameters are set as follows:
•Secrecy rate vs. M(Scenario 1):
ρd= 10 dB, γE= 1 dB, N= 20,K= 8,βE= 1, β c
k= 2,
βe
k= 0.4,η= 2
•Secrecy rate vs. ρd(Scenario 2):
γE= 1 dB, M= 100,N= 20,K= 8,βE= 1, β c
k= 2,
βe
k= 0.4,η= 1.
In Figs. 2 and 3, we examine the correctness of our analysis.
As can be observed, our approximations for the ergodic secrecy
rates at the cell-center and cell-edge UEs are very tight. In
addition, the asymptotic analysis is also confirmed.
In Fig. 2, the secrecy rates are shown when the number of
antennas at the BS increases. In this case, it is observed that
the secrecy rate at the cell-center user increases. Meanwhile,
although suffering from intra-cluster and inter-cluster interfer-
ence, the secrecy rate at the cell-edge UE remains positive.

0 5 1 0 1 5 2 0
0
1
2
3
4
5
S e c r e c y r a t e ( b i t s / s / H z )
d ( d B )
O p t i m i z i e d S u m S e c r e c y R a t e
S u m S e c r e c y R a t e
C e n t e r U s e r A p p r o x i m a t i o n
E d g e U s e r A p p r o x i m a t i o n
S i m u l a t i o n
A s y m p t o t i c
Fig. 3: Secrecy rates at the cell-center and cell-edge UEs and
the sum secrecy rate of the 3rd cluster vs. ρd.
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2
4 . 5
4 . 6
4 . 7
4 . 8
S e c r e c y r a t e ( b i t s / s / H z )
N u m b e r o f i t e r a t i o n s
Fig. 4: Convergence of the proposed algorithm.
The improvement results from the channel hardening effect of
the massive MIMO technology that significantly reduces the
inter-cluster interference. Besides, the cell-center UE benefits
from SIC that can cancel out the intra-cluster interference.
In Fig. 3, the secrecy performance is shown versus the level
of transmit power at the BS. The secrecy rates of both UEs
first increase, then saturate. This is because the more power
is used, the more interference is imposed. Therefore, it can be
concluded that in the high SNR regime, increasing the transmit
power at the BS does not guarantee an improvement of the
secrecy rates. Thus, the transmit power should be carefully
designed to attain the highest efficiency.
In Figs. 2 and 3, the optimization results for the sum secrecy
rate maximization are also shown. The algorithm improves
the sum secrecy rate, especially in the higher transmit power
scenario. This is because the intra-cluster and inter-cluster
interference is restrained by allocating the transmit power at
the BS and the uplink transmit power of the UEs.
The convergence of the proposed algorithm is shown in
Fig. 4. Parameters are set as in Scenario 1 with M= 120.
It is observed that the sum secrecy rate of the NOMA pair that
has the smallest sum secrecy rate, converges to its maximum
after a few iterations.
VI. CONCLUSION
In this paper, the secrecy performance of a massive MIMO
NOMA network in the presence of a multiple-antenna eaves-
dropper has been investigated. Results have revealed that by
employing a large number of transmit antennas at the BS,
the secrecy rate of cell-center users is significantly enhanced,
while cell-edge users can achieve secure communications. In
contrast, increasing the transmit power at the BS does not
guarantee an improvement of the secrecy rate in the high SNR
regime. Furthermore, the proposed joint power allocation in the
uplink training and downlink transmission phases has shown
an improvement of the sum secrecy rate, especially in the high
SNR regime.
APPENDIX A
PROO F OF EQ UATION (6)
Equation (6) is obtained as follows:
En|hc
k
Twk|2o=E(|hc
k
T(pαc
kβc
khc
k
∗+pαe
kβe
khe
k
∗)|2
M(αc
kβc
k+αe
kβe
k))
=E(|(pαc
kβc
kkhc
kk2+pαe
kβe
khc
k
The
k
∗)|2
M(αc
kβc
k+αe
kβe
k))
=E(αc
kβc
kkhc
kk4+αe
kβe
khc
k
The
k
∗he
k
Thc
k
∗)
M(αc
kβc
k+αe
kβe
k))
=αc
kβc
k(M+ 1) + αe
kβe
k
(αc
kβc
k+αe
kβe
k).(B.1)
A similar approach is used to obtain (7) and (9).
ACKNOWLEDGMENT
This work has been supported in part by the Natural Sci-
ences and Engineering Research Council of Canada (NSERC),
through its Discovery program, and in part by the U.S. Na-
tional Science Foundation under Grants CCF-093970 and CCF-
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