Data-Aided Secure Massive MIMO Transmission
with Active Eavesdropping

arXiv:1801.07076v1 [] 22 Jan 2018

Yongpeng Wu, Chao-Kai Wen, Wen Chen, Shi Jin, Robert Schober, and Giuseppe Caire

Abstract—In this paper, we study the design of secure communication for time division duplexing multi-cell multi-user massive multiple-input multiple-output (MIMO) systems with active
eavesdropping. We assume that the eavesdropper actively attacks
the uplink pilot transmission and the uplink data transmission
before eavesdropping the downlink data transmission phase of
the desired users. We exploit both the received pilots and data
signals for uplink channel estimation. We show analytically that
when the number of transmit antennas and the length of the data
vector both tend to infinity, the signals of the desired user and the
eavesdropper lie in different eigenspaces of the received signal
matrix at the base station if their signal powers are different. This
finding reveals that decreasing (instead of increasing) the desire
user’s signal power might be an effective approach to combat a
strong active attack from an eavesdropper. Inspired by this result,
we propose a data-aided secure downlink transmission scheme
and derive an asymptotic achievable secrecy sum-rate expression
for the proposed design. Numerical results indicate that under
strong active attacks, the proposed design achieves significant
secrecy rate gains compared to the conventional design employing
matched filter precoding and artificial noise generation.

I. I NTRODUCTION
Wireless networks are widely used in civilian and military
applications and have become an indispensable part of our
daily lifes. Therefore, security is a critical issue for future
wireless networks. Conventional security approaches based on
cryptographic techniques have many well-known weaknesses.
Therefore, new approaches to security based on information
theoretical concepts, such as the secrecy capacity of the
propagation channel, have been developed and are collectively
referred to as physical layer security [1]–[4].
Massive MIMO is a promising approach for efficient transmission of massive amounts of information and is regarded
as one of the “big three” 5G technologies [5]. Most studies
on physical layer security in massive MIMO systems assume
that the eavesdropper is passive and does not attack the
The work of Y. Wu was supported in part by NSFC No. 61701301.
The work of C.-K. Wen was supported by the Ministry of Science and
Technology of Taiwan under Grant MOST 106-2221-E-110-019. The work of
W. Chen is supported by Shanghai STCSM 16JC1402900 and 17510740700,
by National Science and Technology Major Project 2017ZX03001002-005
and 2018ZX03001009-002, by NSF China 61671294, and by Guangxi NSF
2015GXNSFDA139037. The work of S. Jin was supported in part by the
NSFC under Grant 61531011.
Y. Wu and W. Chen are with the Department of Electronic Engineering,
Shanghai Jiao Tong University, Minhang 200240, China (e-mail: yongpeng.wu@sjtu.edu.cn; wenchen@sjtu.edu.cn).
C. K. Wen is with the Institute of Communications Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan (Email:
chaokai.wen@mail.nsysu.edu.tw).
S. Jin is with the National Mobile Communications Research Laboratory, Southeast University, Nanjing, 210096, P. R. China. (Emails: jinshi@seu.edu.cn).
R. Schober is with Institute for Digital Communications, Universität
Erlangen-Nürnberg, Cauerstrasse 7, D-91058 Erlangen, Germany (Email:
schober@lnt.de).
G. Caire is with Institute for Telecommunication Systems, Technical
University Berlin, Einsteinufer 25, 10587 Berlin, Germany (Email: caire@tuberlin.de).

communication process of the systems [6]–[9]. However, a
smart eavesdropper can perform the pilot contamination attack
to jeopardize the channel estimation process at the base station
[10]. Due to the channel hardening effect caused by large
antenna arrays, the pilot contamination attack results in a
serious secrecy threat to time division duplexing (TDD)-based
massive MIMO systems [10].
The authors of [11] propose a secret key agreement protocol
for single-cell multi-user massive MIMO systems under the
pilot contamination attack. An estimator for the base station
(BS) is designed to evaluate the information leakage. Then,
the BS and the desired users perform secure communication
by adjusting the length of the secrecy key based on the
estimated information leakage. Other works have studied how
to combat the pilot contamination attack. The authors of
[12] investigate the pilot contamination attack problem for
single-cell multi-user massive MIMO systems over independent and identically distributed (i.i.d.) fading channels. The
eavesdropper is assumed to only know the pilot signal set
whose size scales polynomially with the number of transmit
antenna. For each transmission, the desired users randomly
select certain pilot signals from this set, which are unknown
to the eavesdropper. In this case, it is proved that the impact of
the pilot contamination attack can be eliminated as the number
of transmit antenna goes to infinity. For the more pessimistic
assumption that the eavesdropper knows the exact pilot signals
of the desired users for each transmission, the secrecy threat
caused by the pilot contamination attack in multi-cell multiuser massive MIMO systems over correlated fading channels
is analyzed in [10]. Based on this, three transmission strategies
for combating the pilot contamination attack are proposed.
However, the designs in [10] are not able to guarantee a high
(or even a non-zero) secrecy rate for weakly correlated or i.i.d.
fading channels under a strong pilot contamination attack.
In this paper, we investigate secure transmission for i.i.d.
fading1 TDD multi-cell multi-user massive MIMO systems under a strong active attack. We assume the system performs first
uplink training followed by an uplink data transmission phase
and a downlink data transmission phase. The eavesdropper
jams the uplink training phase and the uplink data transmission
phase and then eavesdrops the downlink data transmission2 .
We utilize the uplink transmission data to aid the channel
estimation at the BS. Then, based on the estimated channels,
the BS designs precoders for the downlink transmission.
This paper makes the following key contributions:
1 For simplicity of presentation, we assume i.i.d. fading to present the basic
idea of data-aided secure massive MIMO transmission. The results can be
extended to the general case of correlated fading channels by combining the
techniques in [10] with those in this paper. This will be considered in extended
journal version of this paper.
2

1) We prove that when the number of transmit antennas
and the amount transmitted data both approach infinity,
the desired users’ and the eavesdropper’s signals lie in
different eigenspaces of the uplink received signal matrix
due to their power differences. Our results reveal that
increasing the power gap between the desired users’ and
the eavesdropper’s signals is beneficial for separating the
desired users and the eavesdropper. This implies that
when facing a strong active attack, decreasing (instead
of increasing) the desired users’ signal power could be
an effective approach to enable secrete communication.
2) Inspired by this observation, we propose a joint uplink and downlink data-aided transmission scheme to
combat strong active attacks from an eavesdropper.
Then, we derive an asymptotic achievable secrecy sumrate expression for this scheme. The derived expression
indicates that the impact of an active attack on the
uplink transmission can be completely eliminated by the
proposed design.
3) Our numerical results reveal that the proposed design
achieves a good secrecy performance under strong active attacks, while the conventional design employing
matched filter precoding and artificial noise generation
(MF-AN) [10] is not able to guarantee secure communication in this case.
Notation: Vectors are denoted by lower-case bold-face letters; matrices are denoted by upper-case bold-face letters.
Superscripts (·)T , (·)∗ , and (·)H stand for the matrix transpose,
conjugate, and conjugate-transpose operations, respectively.
We use tr(A) and A−1 to denote the trace and the inverse
of matrix A, respectively. diag {b} denotes a diagonal matrix
with the elements of vector b on its main diagonal. Diag {B}
denotes a diagonal matrix containing the diagonal elements of
matrix B on the main diagonal. The M × M identity matrix
is denoted by IM , and the M × N all-zero matrix and the
N × 1 all-zero vector are denoted by 0. The fields of complex
and real numbers are denoted by C and R, respectively. E [·]
denotes statistical expectation. [A]mn denotes the element in
the mth row and nth column of matrix A. [a]m denotes the
mth entry of vector a. ⊗ denotes the Kronecker product.
x ∼ CN (0, RN ) denotes a circularly symmetric complex
vector x ∈ CN ×1 with zero mean and covariance matrix RN .
+
var(a) denotes the variance of random variable a. [x] stands
for max {0, x}. a ≫ b means that a is much larger than b.
II. U PLINK T RANSMISSION
Throughout the paper, we adopt the following transmission
protocol. We assume the uplink transmission phase, composing
the uplink training and the uplink data transmission, which is
followed by a downlink data transmission phase.
We assume the main objective of the eavesdropper is to
eavesdrop the downlink data. The eavesdropper chooses to
attack the uplink transmission phase to impair the channel
estimation phase at the BS. The resulting mismatched channel
estimation will increase the information leakage in the subsequent downlink transmission. In the downlink transmission
phase, the eavesdropper does not attack but focuses on eavesdropping the data.
We study a multi-cell multi-user system with L + 1 cells.
We assume an Nt -antenna BS and K single-antenna users are
present in each cell. The cells are index by l = (0, . . . , L),

where cell l = 0 is the cell of interest. We assume an Ne antenna active eavesdropper3 is located in the cell of interest
and attempts to eavesdrop the data intended for all users in the
cell. The eavesdropper sends pilot signals and artificial noise
to interfere channel estimation and uplink data transmission4 ,
respectively. Let T and τ denote the coherence time of the
channel and the length of the pilot signal, respectively. Then,
for uplink transmission, the received pilot signal matrix Ypm ∈
CNt ×τ and the received data signal matrix Ydm ∈ CNt ×(T −τ )
at the BS in cell m are given by5
K
L X
K p
X
p X
T
T
Pl hm
hm
ω
+
Ypm = P0
lk ωk
0k k
k=1

l=1 k=1

K
X

Pe
Hm
W k + Nm
p
KNe e
k=1
K
L X
K p
X
p X
T
T
= P0
Pl hm
hm
d
+
lk dlk
0k 0k
+

Ydm

r

r

k=1

(1)

l=1 k=1

Pe m
H A + Nm
(2)
d
Ne e
where P0 , ωk ∈ Cτ ×1 , and d0k ∼ CN (0, IT −τ ) denote
the average transmit power, the pilot sequence, and the uplink transmission data of the kth user in cell of interest,
respectively. It is assumed that the same K orthogonal pilot
sequences are used in each cell where ωkH ωk = τ and
ωkH ωl = 0. Pl and dlk denote the average transmit power
and the uplink transmission data of the kth user in the lth
p
cell, respectively. hplk ∼ CN (0, βlk
INt ) denotes the channel
between the kth user in the lth cell and the BS in the pth cell,
p
where βlk
is the corresponding large-scale path loss. Hle and
Pe denote the channel between the eavesdropper and the base
station in the lth cell and the average transmit power of the
eavesdropper, respectively. We assume the columns
of Hle are

l
i.i.d. with Gaussian distribution CN 0, βe INt , where βel is
the large-scale path loss for the eavesdropper. For the training
phase, the eavesdropper attacks all the users in cell P
of interest.
K
Therefore, it uses the attacking pilot sequences
k=1 Wk
T
Nt ×τ
[12], where Wk = [ωk · · · ωk ] ∈ C
. For the uplink data
transmission phase, the eavesdropper generates artificial noise
A ∈ CNt ×T −τ , whose elements conform i.i.d. standard GausNt ×τ
Nt ×(T −τ )
sian distribution. Nm
and Nm
are
p ∈ C
d ∈ C
noise matrices whose columns are i.i.d. Gaussian distributed
with CN (0, N0 INt ).

We define Y0 = Yp0 Yd0 and the eigenvalue decomposition T 1Nt Y0 Y0H = [v1 , · · · , vNt ] Σ[v1 , · · · , vNt ]H , where
the eigenvalues on the main diagonal of matrix Σ are originated in ascending order. For the following, we make the
important assumption that due to the strong active attack
and the large-scale path loss difference between the cell of
0
0
interest and other cells, Pe βe0 , P0 β0k
, and Pl βlk
have the
+

3 An N -antenna eavesdropper is equivalent to N cooperative singlee
e
antenna eavesdroppers.
4 We note that if the eavesdropper only attacks the channel estimation phase
and remains silent during the uplink data transmission, then the impact of
this attack can be easily eliminated with the joint channel estimation and
data detection scheme in [13]. Therefore, a smart eavesdropper will attack
the entire uplink transmission.
5 For notation simplicity, we assume the users in each cell use the same
transmit power [6]. Following the similar techniques in this paper, the results
can be easily extended to the case of different transmit powers of the users
in each cell.

0
0
relationship Pe βe0 ≫ P0 β0k
≫ Pl βlk
. Let M = (L + 1)K +
Ne and vector (θ1 , · · · , θM ) has the same element as vec-
0
0
0
0
, · · · , PL βLK
, P0 β01
, · · · , P0 β0K
, Pe βe , · · · Pe βe
tor P1 β11
but with the elements originated in ascending order whose
index 1 ≤ i1 ≤ i2 · · · ≤ iK ≤ M satis0
0
fies θik = P0 β0k
, k = 1, 2, · · · , K. Define Veq
=
Define H0 =
K ].
[vN0 t −M+i10, vNt −M+i2 , · · · , 0vNt −M+i

h01 , · · · h0K and HI = h11 , · · · h01K , · · · , h0L1 , · · · , h0LK .
Then, we have the following theorem.

0 H
√1
Theorem 1. Let Z0p
=
Veq
Yp0
=
T Nt

H
1
0
0
√
Veq H0 =
[z0p,1 , · · · , z0p,K ] and Heq =
T Nt
[heq,01 , · · · , heq,0K ]. Then, when T → ∞ and Nt → ∞,
b eq,0k of
the minimum mean square error (MMSE) estimate h
heq,0k based on Z0p is
by
√ given 

p
P
0
b eq,0k =
(3)
P0 τ heq,0k + neq
h
P0 τ + N0
0
where neq = Veq
ñeq and ñeq ∼ CN (0, τ N0 INt ).

Proof. Please refer to Appendix A.

Remark 1: The basic intuition behind Theorem 1 is that
when T → ∞ and Nt → ∞, each channel tends to be an
eigenvector of the received signal matrix. As a result, we
project the received signal matrix along the eigenspace which
corresponds to the desired users’ channel. In this case, the
impact of the strong active attack can be effectively eliminated.
Remark 2: In Theorem 1, we assume that the coherence time
of the channel is significantly larger than the symbol duration
[14]. This assumption can be justified based on the expression
for the coherence time in [14, Eq. (1)]. For typical speeds of
mobile users and typical symbol duration, the coherence time
can be more than hundreds symbol durations or even more.
Remark 3: The simulation results in Section IV indicate
that a sufficient power gap between P0 and Pe can guarantee
a good secrecy performance when the number of transmit
antennas and the coherence time of the channel are large
but not infinite. We note that allocating more power to the
desired users to combat a strong active attack is not needed.
0
In contrast, the larger gap between P0 β0k
and Pe βe0 will
be beneficial to approach the channel estimation result in
Theorem 1. This implies that decreasing the power of the
desire users can be an effective secure transmission strategy
under a strong active attack.
Remark 4: We can use large dimension random matrix
theory [15] to obtain a more accurate approximation for the
eigenvalue distribution of T 1Nt YYH for the case when Nt and
T are large but not infinite. Then, power design policies for
P0 , Pl , and Pe can be obtained. This will be discussed in the
extended journal version of this work.
Based on Theorem 1, we can design the precoders for
downlink transmission.
III. D OWNLINK T RANSMISSION
In this section, we consider the downlink transmission. We
assume the BSs in all L + 1 cells perform channel estimation
b eq,0k , heq,0k , P0 , and
according to Theorem 1 by replacing h
0
l
b
Veq by heq,lk , heq,lk , Pl , and Veq , respectively. Then, the lth
BS designs the transmit signal as follows
K
√ X
tlk slk , l = 0, · · · , L,
(4)
xl = P
k=1

where P is the downlink transmission power, tlk =

l H ĥeq,lk
, and slk is the downlink transmitted signal
Veq
kĥeq,lk k
for the kth user in the lth cell.
For the proposed precoder design, the base station only
needs to know the statistical channel state information of the
eavesdropper Pe βe0 in order to construct V0 . This assumption
is justified in [10].
Because each user in the cell of interest has the risk of being
eavesdropped, an achievable ergodic secrecy sum-rate can be
expressed as [16]
K
X
+
Rsec =
[Rk − Ckeve ]
(5)
k=1

where Rk and Ckeve denote an achievable ergodic rate between
the BS and the kth user and the ergodic capacity between the
BS and the eavesdropper seeking to decode the information
of the kth user, respectively.
The received signal y0k at the kth user in the cell of interest
is given by
L
X
H
y0k =
h0lk xl + nd
l=0

=

+

√

P h00k

H

0
Veq

H ĥeq,0k

√
H

0 H
Veq
P h00k

s0k

ĥeq,0k

K
X

t=1,t6=k

ĥeq,0t

s0t

ĥeq,0t

L
K
√ X
H
 X
ĥeq,lt
l H
P
h0lk
Veq
slt + nd . (6)
t=1 ĥeq,lt
l=1
where nd ∼ CN (0, N0d ) is the noise in the downlink
transmission.
We use a lower bound for the achievable ergodic rate Rk
as follows [17]
R̄k = log (1 + γk )
(7)
where
γk =
i2
h
0
E g0k,k






K
L P
K
2
2
P
P
0
0
0
E glt,k
E g0t,k
+
N0d + var g0k,k +

+

l=1 t=1

t=1,t6=k

and

0
glt,k

√
H

l H
= P h0lk
Veq

(8)

ĥeq,lt
.
kĥeq,lt k

For Ckeve , we adopt the same pessimistic assumption as in
[10], i.e., we assume that the eavesdropper can eliminate all
interference from intra and inter-cell users to obtain an upper
bound of Ckeve as follows



P


eve
Ck,upper
= E log2 1 +
N0

geve

ĥeq,0k



2 

(9)

where 
H


H

0 H
0
ĥeq,0k . (10)
H0e Veq
H0e
geve = ĥeq,0k
Veq
Based on (5), (7), and (9), we have the following theorem.
Theorem 2. For the considered multi-cell multi-user massive
MIMO system, an asymptotic achievable secrecy sum-rate for

the transmit signal design in (4) is given by
K
Nt →∞ X
log (1 + γ̄k )
Rsec, ach →

18
16
Secrecy Rate (b/s/Hz)

k=1

where
γ̄k =

a1

N0d + P (a2 − a1 ) + P (K −
a1 =
a2 =

20

(11)

0
1)β0k

+ PK

0
P0 τ P0 τ β0k
(Nt + K − 1) + KN0

(P0 τ + N0 )

2

L
P

l=1



0
βlk

(12)

14
12
10
8
6
Exact Secrecy Rate for K = 5
Asymptotic Secrecy Rate for K = 5
Exact Secrecy Rate for K = 3
Asymptotic Secrecy Rate for K =3

4

(13)


2
0
0
0
0
+ 3 (K − 1) β0k
P0 τ β0k
Nt + β0k
(K − 1) + N0 Nt β0k
0 (N + K − 1) + N
P0 τ β0k
t
0
(14)

2
0
−20

Fig. 1:

−15

−10

−5
SNR (dB)

0

5

10

Secrecy rate vs. SNR for T = 1024, P0 /N0 = 5
dB, ρ = 30, and different numbers of users

Proof. Please refer to Appendix B.
20

Theorem 2 is a general expression which is valid for
arbitrary K and L. Also, Theorem 2 indicates that when
Nt tends to infinity, the impact of the active attack from
the eavesdropper disappears if the proposed joint uplink and
downlink transmission design is adopted.

V. C ONCLUSIONS
In this paper, we have proposed a data-aided secure transmission scheme for multi-cell multi-user massive MIMO systems which are under a strong active attack. We exploit the
received uplink data signal for joint uplink channel estimation
and secure downlink transmission. We show analytically that
when the number of transmit antennas and the length of
the data vector both approach infinity, the proposed design
can effectively eliminate the impact of an active attack by
an eavesdropper. Numerical results validate our theoretical

16
Secrecy Rate (b/s/Hz)

IV. N UMERICAL R ESULTS
In this section, we present numerical results to examine the
proposed design and the obtained analytical results. We set
0
0
L = 3, Nt = 128, β0k
= 1, k = 1, · · · , K, βlk
= 0.2,
k = 1, · · · , K, l = 1, · · · , L, and P0 = P1 = ... = PL .
We define the signal-to-noise ratio (SNR) as SNR = P/N0d .
Also, we define ρ = PE /P0 K.
Figure 1 plots the asymptotic and exact secrecy rate performance vs. the SNR for T = 1024, P0 /N0 = 5 dB, ρ = 30,
and different numbers of users, respectively. The exact secrecy
rate is obtained based on Monte Carlo simulation of (8) and
(9). We note from Figure 1 that the asymptotic secrecy rate
in Theorem 2 provides a good estimate for the exact secrecy
rate.
Figure 2 compares the secrecy performance of the proposed
design and the MF-AN design in [10] for large but finite Nt
and T as a function of ρ for K = 5, P0 /N0 = 5 dB, SNR
= 5dB, and different values of T . We keep P0 constant and
increase Pe to increase ρ. We observe from Figure 2 that when
the power of the active attack is strong, the MF-AN design
cannot provide a non-zero secrecy rate. However, our proposed
design performs well in the entire considered range of ρ.
0
As ρ increases, the gap between Pe βe and P0 β0k
increases
as well. Therefore, the secrecy rate increases with ρ for the
proposed design. Moreover, Figure 2 reveals that increasing
T is beneficial for the secrecy performance of the proposed
design.

18

14
12
10
8
6
4

Proposed Design for T = 4096
Proposed Design for T = 1024
MF−AN Design

2
0

Fig. 2:

10

20

30
ρ

40

50

60

Secrecy rate vs. ρ for K = 5, P0 /N0 = 5 dB, SNR
= 5dB, and different values of T

analysis and demonstrate the effectiveness of the proposed
design under strong active attacks.
A PPENDIX A
P ROOF OF T HEOREM 1
We define Ω0
=
[ω , · · · , ωK ]T , D0
=

√ 1 T
√
√
T
T T
P0 [d01 , · · · , d0K ] , ΩL =
P1 Ω0 , · · · , PL Ω0 , DL
T
√
√
√
√
=
P1 d11 , · ·√
· , P1 d1K , · · · , PL dL1 , · · · , PL dLK ,
X0  =
=
[ΩL DL ],
P0 Ω0 D0 , XI
q
q
K
P
PE
PE
Wk
Xe =
KNe
Ne A .
k=1

Based on (1) and (2), the received signal Y0 can be reexpressed as
0
Y
(15)
 0 0= H00X
 0 + HI XI + He Xe + N
where N = Np Nd .
When T → ∞, based on [18, Corollary 1], we obtain (16)
given at the
i
h top of the next page, where
(17)
UY = UW HI BI−1/2 He βe−1/2 H0 B0−1/2

0
0
B0 = diag β01
, · · · , β0K
(18)

0
0
0
0
(19)
BI = diag β11 , · · · , β1K , · · · , βL1 , · · · , βLK
PI = diag (P1 , · · · , P1 , · · · , PL , · · · , PL )
(20)
and UW ∈ CNt ×(Nt −M) has orthogonal columns.
When Nt → ∞, we have
1 H
Nt →∞
(21)
U UY → INt .
Nt Y


1
N0
1
1
T →∞ 1
0 H
H
H
+
IN
Y0 Y0H →
H0 X0 XH
HI XI XH
H0 Xe XH
0 H0 +
I HI +
e He
Nt T
Nt T
Nt T
Nt T e
Nt t
i
h
1
−1/2
−1/2
−1/2
=
UW HI BI
H e βe
H 0 B0
Nt
 N I

0 Nt −M
UH
W
1/2
H 1/2
BI XI XI BI
−1/2

 B
+ N0 I(L−1)K
HH


T
I
I


−1/2 H
βe Xe XH
e



β
H
+
N
I
e
0
N
e
e
T
1/2
1/2
−1/2 H
B0 X0 XH
0 B0
H
B
0
+
N
I
0
0 K
T


N0 INt −M
PI BI + N0 I(L−1)K
T →∞ 1

 H

→
UY 
 UY
βe0 Pe + N0 INe
Nt
P0 B0 + N0 IK
From (16)–(21), we know that for T → ∞, Nt → ∞, UY
is the right singular matrix of Y0 . Therefore, we obtain

1
→∞
0 H 0 Nt→
Veq
Yp
Z= √
T Nt
 p

1
1
0 H
0 H 0
√
Np .
(22)
Veq
Veq
P0 Ω0 X0 + √
T Nt
T Nt
Define z = vec (Z0p ), where Z0p is defined in Theorem 1.
From (22), we can re-express the equivalent received signal
during the pilot transmission phase as follows
K
p X
z = P0
(ωt ⊗ IK ) heq,0t + n
(23)

where

t=1



 .
n=
 ..

0
Veq

H

H

n0p1






(24)

0
Veq
n0pτ
and n0pt in (24) is the tth column of N0p .
Based onp(23), the MMSE estimate of heq,0k is given by
b eq,0k = P0 (P0 τ IK + N0 IK )−1 (ωk ⊗ IK )H z
h
√

P0 p
P0 τ heq,0k + (ωk ⊗ IK )H n . (25)
=
P0 τ + N0
For the noise term in (25), we have
τ
 X
H
∗
0 H
ωkt
wt
(ωk ⊗ IK ) n = Veq
t=1

0
= Veq

τ
H X

∗
0
ωkt
wt = Veq

t=1

H

ñeq

(26)

where ωkt is the tth element of ωk . Combining (25) and (26)
completes the proof.
A PPENDIX B
P ROOF OF T HEOREM 2
First,h basedi on the property
of iMMSE estimates, we know
h
√
0
= P E ĥeq,0k .
that E g0k,k
Based on (3) and (16), we have
2
P0
ĥeq,0k =
2
(P0 τ + N0 )
H p

p
0
0
P0 τ heq,0k + Veq
ñeq
P0 τ heq,0k + Veq
ñeq
×







(16)


H  0
1
h01 , · · · h00K B−1
h00k
0
Nt
(P0 τ + N0 )
H


1 H 0
× h001 , · · · h00K h00k +
ñeq h01 , · · · h00K B−1
0
i Nt

 0
H
× h01 , · · · h00K ñeq
"
K
X
H 0 −1 0 0 H 0
P0
2 1
h0t h0t h0k
β0t
=
h00k
2 P0 τ N
t t=1
(P0 τ + N0 )
#
K
1 X H 0 0 −1 0 H
(27)
h0t ñeq .
+
ñ h β
Nt t=1 eq 0t 0t
=

P0

2

P0 τ 2

When Nt → ∞, based on [18, Corollary 1], we have
H 0 0 −1 0 H 0
1
h0t h0k
h0t β0t
h0
Nt 0k


0
0 −1
H  Nt →∞ 0
Nt →∞ β0k β0t
→ β0k
→
(28)
tr h00t h00t
Nt



1
Nt →∞ 0
H
H 0
0 −1
Nt
(29)
h00k h00k → β0k
h0k β0k
h0
Nt 0k
1 H 0 0 −1 0 H
Nt →∞
(30)
h0t ñeq → τ N0 .
ñeq h0t β0t
Nt

Substituting (28)–(30) into (27), we have
2 Nt →∞
P0
2 0
→
ĥeq,0k
2 P0 τ β0k Nt +
(P0 τ + N0 )

0
(31)
P0 τ 2 β0k
(K − 1) + Kτ N0 .


0
. First, we obtain
Next, we evaluate var g0k,k



P0
H
H
0
0
0
0 H
ĥeq,0k ĥH
Veq
h00k
eq,0k Veq h0k =
2 h0k
(P0 τ + N0 )
p


1  0
−1/2
0
P0 τ heq,0k + Veq
h01 , · · · h00K B0
×√
ñeq
Nt
H 1
p
H

0
√ B0−1/2 h001 , · · · h00K h00k
P0 τ heq,0k + Veq
ñeq
×
Nt

 −1/2 1
 0

P0
1
Nt →∞
H
0
0
→
√
h , · · · h0K B0
2 h0k
Nt
Nt 01
(P0 τ + N0 )

H 0  0 H  0
 0
0
2 −1/2
0
h01 , · · · h0K
h0k h0k
× P0 τ B0
h01 , · · · h0K

 0
H
1 −1/2  0
−1/2
0
× B0
+
h01 , · · · h00K ñeq ñH
B0
eq h01 , · · · h0K
Nt
(32)

i


H
1
√ B0−1/2 h001 , · · · h00K h00k
Nt


P0
P0 τ 2 0 H  0
=
h0k
h01 , · · · h00K B−1
0
2
2
Nt
(P0 τ + N0 )





 0
H
H
h001 , · · · h00K B0−1
h00k h00k
× h01 , · · · h00K
H


H  0
1
× h001 , · · · h00K h00k + 2 h00k
h01 , · · · h00K B−1
0
Nt



 0
H
−1
0
0
× h01 , · · · h00K ñeq ñH
eq h01 , · · · h0K B0
i
H

(33)
× h001 , · · · h00K h00k .
From (28) and (29), we have
H 0 

 0
H  0
1
h0k
h01 , · · · h00K
h01 , · · · h00K B−1
h00k
0
Nt
−1/2

×B0

Nt →∞

0
0
→ β0k
Nt + β0k
(K − 1)
H

 0


1
H
h01 , · · · h00K ñeq
h001 , · · · h00K B−1
h0
0
Nt2 0k
 0
 −1  0
H
0
× ñH
h01 , · · · h00K h00k
eq h01 , · · · h0K B0

E

h

0
glt,k

2

i

0
= P β0k


eve
For Ck,upper
in


0 H
0
→
Veq He

 (40)

H


 ĥeq,lt ĥeq,lt

H
= PE 
tr
ĥ
ĥ

eq,lk
eq,lk
2
ĥeq,lt

0
= P βlk
.
(41)
(9), we know from (16) that when Nt → ∞,

0. Therefore, we have
N →∞

t
eve
→ 0.
(42)
Ck,upper
Substituting (31), (39), (40), (41), and (42) into (5) completes the proof.

(34)

Also, we have
H
H
 −1  0
 0
1
0
h01 , · · · h00K h00k h00k
= 2 ñH
eq h01 , · · · h0K B0
Nt
H

 0
 0
h01 , · · · h00K ñeq
× h01 , · · · h00K B−1
0

H
 0

Nt →∞ 1
→
h01 , · · · h00K h00k
τ
N
tr
h001 , · · · h00K B−1
0
0
2
Nt
  0
 0

H 
0 H
0
× h0k
h01 , · · · h00K B−1
h
,
·
·
·
h
01
0K
0
H

 0
τ N0 0 H  0
h01 , · · · h00K
h01 , · · · h00K B−1
=
h
0
Nt2 0k

 0

H
× h001 , · · · h00K B−1
h01 , · · · h00K h00k
0
K
τ N0 0 H X 0 −1 0 0 H
h0t h0t
β0t
h0k
2
Nt
t=1
K
X
H

0 −1 0
h0p h00p h00k
β0p
×

=

p=1

H 0 −1 0 0 H 0 −1 0 H 0 0
1
h00k
β0t
h0t h0t
β0p
h0p h0p h0k (35)
2
Nt
When k 6= t 6= p, we have


0




Nt →∞ β0k
0 H 0
0 −1
0 H
0 −1 0
→
h
h
β
h
h
β
tr
0p
0p
0p
0t
0t
0t
Nt2
0
Nt →∞ β0k
→
(36)
Nt
When k = t = p, we have
H
H 0 −1 0
H 0 −1 0
1
h0k h00k h00k
β0k
h0k h00k
β0k
h00k
2
Nt
Nt →∞

0
→ Nt β0k
(37)
When k = t 6= p, k = p 6= t, k 6= t = p, k 6= p = t, we
have
H 0 −1 0 0 H 0 −1 0 H 0 0
1
h0t h0t h0k
β0t
h0t h0t
β0t
h00k
2
Nt
Nt →∞

where a2and a1 
are defined
 and (14).
 in (13)
2
2
0
0
and E glt,k , we have
For E g0t,k


H

h

i
ĥ
ĥ
2

 eq,0t eq,0t
0
H
E g0t,k
= PE 
2 tr ĥeq,0k ĥeq,0k 
ĥeq,0t

0
→ β0k
(38)
0
Combining (31), (33)–(38), and the definition of g0k,k
, we
have

0
= a2 − a1 .
(39)
var g0k,k

R EFERENCES
[1] A. D. Wyner, “The wiretap channel,” Bell Syst. Tech. J., vol. 54, pp.
1355–1387, Oct. 1975.
[2] F. Oggier and B. Hassibi, “The secrecy capacity of the MIMO wiretap
channel,” IEEE Trans. Inf. Theory, vol. 57, pp. 4961–4972, Aug. 2011.
[3] Y. Wu, C. Xiao, Z. Ding, X. Gao, and S. Jin, “Linear precoding for finitealphabet signaling over MIMOME wiretap channels,” IEEE Trans. Veh.
Technol., vol. 61, pp. 2599–2612, Jul. 2012.
[4] Y. Wu, J.-B. Wang, J. Wang, R. Schober, and C. Xiao, “Secure
transmission with large numbers of antennas and finite alphabet inputs,”
IEEE Trans. Commun., vol. 65, pp. 3614–3628, Aug. 2017.
[5] J. G. Andrews, S. Buzzi, W. Choi, S. Hanly, A. Lozano, A. C. K. Soong,
and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Commun.,
vol. 32, pp. 1065–1082, Jun. 2014.
[6] J. Zhu, R. Schober, and V. K. Bhargava, “Secure transmission in
multicell massive MIMO systems,” IEEE Trans. Wireless Commun.,
vol. 13, pp. 4766–4781, Sep. 2014.
[7] X. Chen, L. Lei, H. Zhang, and C. Yuen, “Large-scale MIMO relaying
techniques for physical layer security: AF or DF?” IEEE Trans. Wireless
Commun., vol. 14, pp. 5135–5146, Sep. 2015.
[8] J. Chen, X. Chen, W. H. Gerstacker, and D. W. K. Ng, “Resource
allocation for a massive MIMO relay aided secure communication,”
IEEE Trans. Inf. Foresics Security, vol. 11, pp. 1700–1711, Aug. 2016.
[9] J. Zhu, D. W. K. Ng, N. Wang, R. Schober, and V. K. Bhargava,
“Analysis and design of secure massive MIMO systems in the presence
of hardware impairments,” IEEE Trans. Wireless Commun., vol. 16, pp.
2001–2016, Mar. 2017.
[10] Y. Wu, R. Schober, D. W. K. Ng, C. Xiao, and G. Caire, “Secure
massive MIMO transmission with an active eavesdropper,” IEEE Trans.
Inf. Theory, vol. 62, pp. 3880–3900, Jul. 2016.
[11] S. Im, H. Jeon, J. Choi, and J. Ha, “Secret key agreement with
large antenna arrays under the pilot contamination attack,” IEEE Trans.
Wireless Commun., vol. 14, pp. 6579–6594, Dec. 2015.
[12] Y. O. Basciftci, C. E. Koksal, and A. Ashikhmin, “Securing massive MIMO at the physical layer,” [Online]. Available:
http://arxiv.org/abs/1505.00396.
[13] C.-K. Wen, Y. Wu, K.-K. Wong, R. Schober, and P. Ting, “Performance
limits of massive MIMO systems based on Bayes-optimal inference,”
in Proc. Int. Conf. Commun. (ICC’2015), London, UK, Jun. 2015, pp.
1783–1788.
[14] R. R. Müller, L. Cottatellucci, and M. Vehkaperä, “Blind pilot decontamination,” IEEE J. Sel. Topic Signal Process., vol. 8, pp. 773–786,
Oct. 2014.
[15] R. Couillet and M. Debbah, Random Matrix Methods for Wireless
Communications. Cambridge University Press, 2011.
[16] G. Geraci, M. Egan, J. Yuan, A. Razi, and I. Collings, “Secrecy sumrates for multi-user MIMO regularized channel inversion precoding,”
IEEE Trans. Commun., vol. 60, pp. 3472–3482, Nov. 2012.
[17] J. Jose, A. Ashikhmin, T. L. Marzetta, and S. Vishwanath, “Pilot
contamination and precoding in multi-cell TDD systems,” IEEE Trans.
Wireless Commun., vol. 10, pp. 2640–2651, Aug. 2011.
[18] J. Evans and D. N. C. Tse, “Large system performance of linear
multiuser receivers in multipath fading channels,” IEEE Trans. Inf.
Theory, vol. 46, pp. 2059–2078, Sep. 2000.

