In this paper, a secure multilevel quadrature amplitude modulation (MQAM) scheme is proposed for the physical layer security (PLS) of the wireless communications. In the proposed scheme, each transmitted symbol’s signal constellation (SC) is hopping with the control of two unique factors:
amplitude distortion (AD) factor and phase hopping (PH) factor
. With unknown the two factors, the eavesdropper cannot extract effective information from the received signal. We first introduce a security metric, referred to as
secrecy gain
, and drive a lower bound on the gain that the secrecy capacity can be improved. Then, we investigate the relationship among the secrecy gain, the signal to noise power ratios (SNRs) of the main and wiretap channels, and the secrecy capacity. Next, we analyze the security of the proposed scheme, and the results indicate that the secrecy capacity is improved by our scheme. Specifically, a positive secrecy capacity is always obtained, whether the quality of the main channel is better than that of the wiretap channel or not. Finally, the numerical results are provided to prove the analytical work, which further suggests the security of the proposed scheme..
1. Introduction
I
n 1975, Wyner introduced a general model for the physical layer security (PLS) of communications
[1]
, where a transmitter (Alice) sends information to a legitimate user (Bob) through the main channel. Meanwhile, the eavesdropper (Eve) can extract the information from the output signal of the wiretap channel. He proposed a security metric for the system, i.e.,
secrecy capacity
, which is defined as the difference capacity of main and wiretap channel, and its general expression was further proposed in
[2]
. Since then, PLS has received considerable attentions
[3]

[27]
.
In
[3]

[7]
, authors proved that the system with multipleinput multipleoutput (MIMO) channel can obtain the secrecy capacity by optimizing beamforming
[3]

[5]
, proper signal power allocation
[6]

[7]
and spacetime coding
[7]
, when the channel state information (CSI) is known. While in
[8]

[12]
, secure cooperation communications were investigated. With known the perfect CSI, authors proposed optimal relay selection and weight schemes for different numbers of eavesdroppers and different relay strategies. In
[13]

[14]
, authors further proved that the orthogonal frequency division multiplexing (OFDM) modulation system can also achieve the secrecy capacity by optimizing the carrier power. And in
[15]

[16]
, the radio fingerprinting (RF) tachniques were applied to the communications authorization, where the RF feature of the received signal is utilized to detect the malicious intruders. In
[17]

[19]
, artificial noise (AN) were explored to the PLS of the wireless communications. Authors in
[17]
proved that, when Alice and her helpping relays have more antennas than Eve, PLS can be guaranteed by injecting the AN at the transmitter. And authors in
[18]

[19]
proposed the schemes that the AN is sent by Bob or relays. The methods are both robust for their security do not depend on the feedback of CSI and the condition that Eve’s antennas are less than Bob’s. In addition, some researchers propose the usage of coding to achieve the maximum secrecy rate. As shown in
[20]

[24]
, authors proved that we can utilize the existing coding schemes, e.g., low density parity check (LDPC) coding
[20]
, network coding
[21]
, polar coding
[22]

[23]
and lattice coding
[24]
, to design constructive codes which satisfy both reliability and security conditions
[1]
, while the secrecy rate approaches the secrecy capacity. Some researchers also investigate cryptography security enhancement schemes based on the physical layer techniques
[25]

[26]
. As shown in
[25]

[26]
, the secret key can be extracted from the fading channel coefficients. Based on the difference of the main and wiretap channels, Eve cannot intercept the key. And in
[27]
, a joint encryption, error correction and modulation scheme was proposed to improve both the security and reliability of the communications.
In deed, all the techniques in
[3]

[27]
have provided effective solutions for different secure communication scenarios. However, there exist some challenges for these techniques. As shown in
[3]

[14]
, CSI is needed for transmission strategy optimization, while it is usually difficult in practice. Due to the dynamic environment, the signal RF feature is always changing, which makes it hard to extract the signal RF
[15]

[16]
. When AN is utilized for the security, it needs additional power to transmit AN. However, it is not desirable for some power constraint communication scenario, e.g., satellite communication. While coding schemes will lead to the decrease of transmission efficiency, as shown in
[20]

[24]
. When it exploits the crosslayer techniques proposed in
[25]

[26]
, the system complexity will increase significantly.
Motivated by above observations, we propose a secure multilevel quadrature amplitude modulation (MQAM) scheme based on signal constellation hopping (SCH) in this paper. Different from traditional modulation scheme, each transmitted symbols’ signal constellation (SC) is hopping with the control of two unique factors:
amplitude distortion (AD) factor
and
phase hopping (PH) factor
, which are both generated by an ADPH generator. At the receiver, with known the two factors, the demodulator can rebuild the correct SC. Otherwise, the receiver will get uncorrect SC and decision regions, which will lead to the demodulation failure, i.e., the communication security is guaranteed. The main contributions of this paper are summarized as follows.

1) We introduce a security metric, referred to assecrecy gain, and drive a lower bound on the gain that the secrecy capacity can be improved. And the relationship among the secrecy gain, the signal to noise power ratios (SNRs) of the main and wiretap channels, and the secrecy capacity is also investigated.

2) We analyze the security of our scheme and prove that, when the secrecy gain is larger enough, a positive secrecy capacity can be always obtained, whether the quality of the main channel is better than that of the wiretap channel or not. Additionally, we also discuss the antiattack ability of the proposed scheme with a conclusion that, it is difficult to estimate AD and PH factors when it exists noise in the channel.

3) Three sets of simulations with 16QAM and 64QAM are provided to prove the analytical results, where both the main channel and the wiretap channel are assumed as additive white Gaussian noise (AWGN) channels.
The rest of the paper is organized as follows. In Section 2, we introduce the proposed scheme, and analyze the theory error performance. And in Section 3, we first propose the secrecy gain and drive a lower bound on the gain that the secrecy capacity can be improved. Then, relationship among the secrecy gain and secrecy capacity is investigated. And the antiattack ability of the proposed scheme is also discussed. Next, in Section 4, three sets of simulations are set up to evaluate the security performance of our scheme. Finally, we make some concluding in Section 5.
2. Proposed SCH MQAM Scheme
 2.1 System Model
The general ideal of our proposed scheme is to change each transmitted symbol’s SC with the AD and PH factors and to map all the symbols onto hopping SCs. With unknown the AD and PH factors, the receiver cannot make correct decision on each received symbols, i.e., it fails to extract the information from the received signal. The block diagram SCH MQAM system is illustrated in
Fig.1
.
Block diagram of the SCH MQAM system.
We can observe from the figure that, at the transmitter, the information bits (
D_{m}
) are first mapped onto MQAM symbols, i.e.,
X_{m}
. Then, through the processing of SCH with the AD and PH factors, i.e., (
α_{m}
, Δ
θ_{m}
), we canget the transmitted SCH MQAM symbol as
X'_{m}
. Next, through modulating, the final transmitted signal
s
(
t
) is given by
where
A_{m}
,
θ_{m}
are amplitude and phase of
X_{m}
,
g
(
t
) is the shape filter,
T_{s}
is the symbol period, and
f_{c}
is the carrier frequency. For each MQAM symbol, there has
where
A_{mi}
,
A_{mq}
are the informationbearing signal amplitudes of quadrature carriers.
At the receiver, the received signal
r
(
t
) will be
where ‘ * ’ is the convolution,
h
(
t
) is the channel impulse response,
n
(
t
) is the additive Gaussian noise with power spectral density of
N
_{0}
/ 2 . For the AWGN channel, there has
From (1), (3) and (4),
r
(
t
) can be written as
Suppose that there is no frequency and phase offsets, the received SCH MQAM symbol, denoted as
Y'_{m}
, will be
where
n'_{m}
is a complex i.i.d Gaussian variable with mean 0 and variance
N
_{0}
.
Finally, through the processing of the deSCH with the synchronization AD and PH factors, i.e., (
α'_{m}
, Δ
θ'_{m}
), the received MQAM symbol, denoted as
Y_{m}
, is given by
Then, we can get the estimated information bits (
D'_{m}
) based on
Y_{m}
and the decision rules of the MQAM.
It is noticed from the above presentation that, the security of our scheme is closely related to (
α_{m}
, Δ
θ_{m}
) and (
α'_{m}
, Δ
θ'_{m}
), which are all obtained from the ADPH generator. In this paper, we assume that the ADPH generator is driven by a random sequence, and its
m
th element is denoted as
K_{m}
. And it is also assumed that
K_{m}
is known to both Alice and Bob but unknown to Eve.
Remark 1:
The role of
K_{m}
in our scheme is the same as that of the secret key in the cryptography scheme. While in this paper, we do not introduce how to keep
K_{m}
secret in detail.
 2.2 Signal Constellation Hopping
In this subsection, we introduce how to make the MQAM symbols’ SC hopping with the control of the AD and PH factors. As shown in
Fig.2
, with the modification of (
α_{m}
, Δ
θ_{m}
), i.e.,
X'_{m}
=
α_{m}A_{m}e
^{j}
^{(θm + Δθm)}
, the constellation of a 16QAM symbol is changed. Since the AD and PH factors are both unique to each 16QAM symbol, its constellation will be different. Hence, when the symbol number increases, the transmitted SC will be fuzzy, correspondingly, just like the SC is hopping, as shown in
Fig.3
.
(a) Original 16QAM symbol constellation. (b) SCH 16QAM symbol constellation.
(a) Original 16QAM SC. (b) Fuzzy 16QAM SC.
 2.3 Error Performance
In this subsection, we analyze the error performance of the legitimate receiver (Bob) and the eavesdropper (Eve), where both the main channel and the wiretap channel are assumed as AWGN channels, and their noise samples are denoted as
, respectively. In addition, we refer the synchronization AD and PH factors of Bob and Eve to be different as
and
.
Since the AD and PH factors are known to Bob, we have
According to (7) and (8), Bob’s received symbol, denoted as
Y_{m}^{B}
, is given by
where
is the equivalent noise sample of the Bob’s channel.
Suppose that Eve know Alice’s signal procesing method, and she will try to recover the SC with
. Therefore, from (7), Eve’s received symbol, denoted as
Y_{m}^{E}
, will be given by
where
is the equivalent noise sample of the Eve’s channel, and
β
,
φ
are the estimation offsets of the AD and PH factors, i.e.,
In order to analyze the error performance of Bob and Eve, we first introduce the statistic characteristics of
and
with the following theorem.
Theorem 1:
Suppose that the random variables
X'
,
Y
and
θ
are mutual independent, and
X'
~
N
(0,
N
_{0}
),
Y
~
U
(0,
A
),
θ
~
U
(0, 2π). The complex variable
V'
=
X'Ye^{jθ}
will be a Gaussian variable with mean 0 and variance
A
^{2}
N
_{0}
.
Proof:
The proof of this theorem is implemented in AppendixA.
Theorem 1
shows that both
and
follow the Gaussian distribution. Hence, we can exploit the analysis method presented in
0
to evaluate the error performance of Bob and Eve.
It is shown in
0
that, the symbol error rate (SER) of a digital modulation scheme is equal to the average pairwise error probability, i.e., the probability of event that the transmit symbol is
X_{m}
but detected symbol is
X_{n}
.
According to (9), the pairwise error probability of Bob, denoted as
, is given by
To simplify (12), we define three vectors as
n
= (
n
_{1}
,
n
_{2}
),
s
_{m}
= (
s
_{m1}
,
s
_{m2}
) and
s
_{n}
= (
s
_{n1}
,
s
_{n2}
) , where
n
_{1}
= Re[
],
n
_{2}
= Im[
],
s
_{m1}
= Re[
A_{m}e
^{j(θm)}
],
s
_{m2}
= Im[
A_{m}e
^{j(θm)}
],
s
_{n1}
= Re[
A_{n}e
^{j(θn)}
],
s
_{n2}
= Im[
A_{n}e
^{j(θn)}
. Then, (12) can be written as
where
d_{mn}
is the Euclidean distance of
X_{m}
to
X_{m}
, and
n
(
s
_{n}

s
_{m}
) follows Gaussian distribution with mean 0 and variance
. Therefore, the SER of Bob will be
And from (10), the pairwise error probability of Eve, denoted as
, is given by
Based on the similar simplification as (12), we can get the SER of Eve, i.e.,
where
d_{em}
,
d_{en}
are the Euclidean distances of
βA_{m}e
^{j(θm + φ)}
to
X_{m}
and
X_{n}
, respectively, and
are obtained by
It is noticed from (16)(18) that the SER of Eve depends on
β
,
φ
. We assume that
β
and
φ
both follow the uniform distribution, i.e.,
β
~
U
(0.
T
_{0}
) and
φ
~
U
(0,2π). Then, the average SER of Eve will be
where
T
_{0}
is the maximum estimation offset of the AD factor.
3. Security Analysis
In this section, we analyze the security of the proposed scheme, in which the secrecy gain is introduced and the secrecy capacity is investigated. Additionally, we also discuss the antiattack ability of our scheme.
 3.1 Secrecy Gain
It is proven in
[2]
that the secrecy capacity of the wiretap system depends the SNR difference between the main channel and wiretap channel, which are both considered as AWGN channels. We can observe from (14) and (16) that the minimum Euclidean distance of transmitted symbol is changed by the proposed scheme, just as the SNR of channel has been modified. To quantify this “modified” SNR, here, we introduce a metric, referred to as
secrecy gain
, and its definition is given as follows.
Definition 1:
Let
SNR_{B}
,
SNR_{E}
be the actual SNRs of the main and wiretap channel, and
SNR'_{B}
,
SNR'_{B}
be their
equivalent
SNRs through the processing of the PLS technique. Suppose that
SNR'_{B}
/
SNR_{B}
=
γ_{B}
,
SNR'_{E}
/
SNR_{E}
=
γ_{E}
, the secrecy gain is defined as
γ_{s}
=
γ_{B}
/
γ_{E}
.
According to above definition and (14), (16), we can obtain the secrecy gain of our scheme as
Substituting (17) and (18) into (20), we have
For the 4QAM, the secrecy gain will be
Furthermore, when
M
→ ∞ , there has
Additionally, from (19), (20), we can get the average secrecy gain as
We notice from
Definition 1
that the security of the PLS technique can be measured via its secrecy gain. To elaborate this point, we introduce the following theorem to give a lower bound on the secrecy gain that the secrecy capacity can be improved.
Theorem 2:
If the secrecy gain satisfy
γ_{s}
> 1, the secrecy capacity of the wiretap system can be improved.
Proof:
The proof of this theorem is implemented in AppendixB.
Based on (22), (23) and
Theorem 2
, we can conclude that the secrecy capacity of the wiretap system can be improved by the proposed secure MQAM scheme.
 3.2 Secrecy Capacity
It is shown in
[2]
that, the secrecy capacity of the AWGN wiretap system is given by
Therefore, from (14), (16) and (25), we can obtain the secrecy capacity of our scheme as
Then, the average secrecy capacity will be
We notice from (26), (27) that the average secrecy capacity of the proposed scheme depends on its secrecy gain and
SNR_{B}
,
SNR_{E}
. To elaborate the relationship among them, here, we introduce another theorem as follows.
Theorem 3:
When the secrecy gain and
SNR_{B}
,
SNR_{E}
satisfy the condition of
γ_{s}
SNR_{B}
>
SNR_{E}
, a positive secrecy capacity can be achieved.
Proof:
The proof of this theorem is implemented in AppendixC.
Theorem 3
shows that, when the secrecy gain is large enough, a positive secrecy capacity can be always obtained, even
SNR_{E}
is larger than
SNR_{B}
. Hence, we can conclude from (23) that our SCH MQAM scheme can achieve a positive secrecy capacity.
 3.3 AntiAttack Ability
It is noticed from the previous subsections that the security of our scheme depends on the assumption that Eve is unknown about the AD and PH factors. To achieve this goal, we exploit an ADPH generator driven by a random sequence to generate these factors. In general, the driven sequence can be obtained from the coefficients of the main channel
[25]

[26]
, or from a homogenous, stationary and acyclic sequence. For the former case, Eve needs to know the CSI of the main channel, which is difficult in practice. While for the latter case, Eve needs to estimate the whole acyclic sequence, and it is more difficult. In fact, even the driven sequence is obtained from a periodical sequence, our scheme is still security. The reason is that, due to the existence of the channel noise, it cannot accurately estimate the AD and PH factors. Here, we take the estimation of AD sequence for example, as shown in
Fig.4
.
Estimation of AD factor sequence, and T_{α} is the sequence period.
We can observe from
Fig.4
that, with accurate AD factors, we can estimate the AD sequence correctly by
Though it needs long time to observe the AD factor sample. However, with the noised AD factors, there may exists
which means that it is impossible to estimate the AD sequence accurately. Hence, we can conclude that our proposed scheme is robust for Eve’s passive attacking.
4. Numerical Results
In this section, three sets of simulations are set up to evaluate the performance of our scheme. In these simulations, 16QAM and 64QAM are exploited, both the main and wiretap channels are assumed as AWGN channels, the maximum value of
β
is set to 1,i.e.,
T
_{0}
= 1.
 4.1 Simulations for Secrecy Gain
In the first set, we investigate relationship between the secrecy gain and
β
,
φ
, where the results are shown in
Fig.5
. It is noticed from the figure that, the secrecy gain of SCH 16QAM scheme is larger than 10dB. Specifically, when
β
< 0.1, the secrecy gain is more than 30dB. Even
β
is close to 1, the positive secrecy gain still exists, i.e., 12.5dB. In addition,
Table 1
provides the average secrecy gains of the SCH 16QAM and SCH 64QAM schemes. We can observe that the average secrecy gains of the two schemes are both larger than 450dB.
Secrecy gain of the SCH 16QAM scheme with different β and φ.
Secrecy Gains of SCH 16QAM and SCH 64QAM Schemes
Secrecy Gains of SCH 16QAM and SCH 64QAM Schemes
 4.2 Simulations for Secrecy Capacity
In the second set, we first investigate the relationship between the secrecy capacity and
β
,
φ
, where the results are shown in
Fig.6
. It is assumed that
SNR_{B}
=
SNR_{E}
= 6
dB
, we know from (25) that, with the original MQAM scheme, the secrecy capacity will be zero in this case. However, it is noticed from
Fig.6
that, there exists a positive secrecy capacity with the SCH 16QAM scheme. Specifically, even
β
is close to 1 and
φ
is close to 0, the positive secrecy capacity still exists, e.g., when
φ
= 0.04π,
β
= 1, the secrecy capacity is 0.048, and when
β
= 0.02,
φ
= 2π, it will be 0.435.
Secrecy capacity of the SCH 16QAM scheme with different β and φ.
Then, we explore the average secrecy capacities of the SCH 16QAM and SCH 64QAM schemes with
SNR_{E}
=
SNR_{B}
and
SNR_{E}
>
SNR_{B}
, where the results are illustrated in
Fig.7
. We observe from the figure that, the average secrecy capacities of two schemes are both larger than 0.27. When
SNR_{E}
=
SNR_{B}
=
SNR
> 0 dB , the average secrecy capacities are increasing with
SNR
. It is also noticed that, even
SNR_{E}
>
SNR_{B}
, there still exists a positive secrecy capacities for the two schemes, e.g., when
SNR_{E}

SNR_{B}
is equal to 50dB, the corresponding secrecy capacities are 0.0022 and 0.0018, respectively. And when
SNR_{E}

SNR_{B}
increases, the secrecy capacity will decrease.
Average secrecy capacities of SCH 16QAM and SCH 64QAM schemes.
 4.3 Simulations for Error Performance
In the third set, we first investigate the error performance of our scheme via bit error rate (BER), where the results are shown in
Fig.8
. It is noticed from the figure that, when SC is not hopping or the AD and PH factors are exactly synchronized, the BER is almost equal to the theory value. However, when the factors are not synchronized, the BER is around 0.13, which means that the receiver cannot extract the information. In addition, to evaluate the robust of our scheme, we also consider the case that Eve can get partial AD and PH factors. We notice that, even the ratio of the factors obtained by Eve is high to 60%, its BER is still close to the case that the factors are not synchronized.
BER comparison among the SCH 16QAM scheme with different cases.
Then, we explore the theory bounds on the SERs of the SCH 16QAM and SCH 64QAM schemes without the AD and PH factors, where the results are shown in
Fig.9
. It is noticed from the figure that, when SNR is less than 65 dB, the SER is increasing with SNR. Otherwise, it will keep at around 0.01, which indicates the security of the SCH MQAM scheme.
Bounds on the SERs of the SCH MQAM scheme without the AD and PH factors.
5. Conclusion
In this paper, a secure MQAM scheme was proposed for the PLS of the wireless communications. In our scheme, each transmitted symbol’s SC was hopping with the control of two unique factors: amplitude distortion factor and phase hopping factor. With unknown those two factors, the eavesdropper could not extract the information from the received signal. We first introduced a security metric, i.e., secrecy gain, and drove a lower bound on the gain that the secrecy capacity can be improved. Then, we investigated the relationship among the secrecy gain, the SNRs of the main and wiretap channels, and the secrecy capacity. The analytical and simulation results showed that the secrecy capacity can be improved by our scheme. Specifically, a positive secrecy capacity can be always obtained, whether the quality of the main channel is better than that of the wiretap channel or not, which indicates the security of our scheme.
Finally, One important point should be noted that, the security of our scheme is closely related to the security of the driven sequence of the ADPH generator. While in this paper, we did not introduce the detailed methods to generate the driven sequence and to keep it secret. Hence, the research on the driven sequence should be further conducted.
BIO
Yingxian Zhang, received his B.S. degree in information engineering, M.S. degree in communications and information system from College of Communications Engineering, PLA University of Science and Technology (PLAUST), Nanjing, China, in 2009 and 2011, respectively. He is currently pursuing the Ph.D. degree in communications and information system in College of Communications Engineering, PLAUST. His research interests focus on satellite communication, physical layer security, channel coding, and information theory. He was an Exemplary Reviewer for the IEEE Communications Letters and IET Communications in 2013 and 2014, respectively. Email: zhangyingxian@126.com.
Aijun Liu, received the B.S. degree in microwave communications, M.S. degree and Ph.D. degree in communications engineering and information systems from College of Communications Engineering, Nanjing, China, in 1990, 1994 and 1997, respectively. Since 1986, Dr. Liu has been with the College of Communications Engineering, PLA University of Science and Technology, where he is currently a Full Professor and the Head of the Department of Teaching and Research, College of Communications Engineering. He has published over 50 papers in refereed mainstream journals and reputed international conferences and has been granted over 10 patents in his research areas. His current research interests are satellite communication system theory, satellite communication antijamming, signal processing, space heterogeneous networks, channel coding, and information theory. Email: liuaj.cn@163.com.
Xiaofei Pan, received the B.S. degree in communications engineering, M.S. degree and Ph.D. degree in communications engineering and information systems from College of Communications Engineering, Nanjing, China, in 2001, 2004 and 2007, respectively. His major research interests include satellite communication antijamming and space heterogeneous networks. Email: motonula@163.com.
Zhan Ye, received the B.S. degree in communications engineering, M.S. degree and Ph.D. degree in communications engineering and information systems from College of Communications Engineering, Nanjing, China, in 1999, 2004 and 2011, respectively. His major research interests include satellite communication antijamming and signal process. Email: yezhi5223@163.com.
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