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Wireless covert communication is an emerging communication technique that prevents eavesdropping. This paper considers the bit error ratio (BER) problem of covert communication based on constellation shaping modulation (CSM). The impact of carrier-secret ratio (CSR) on BER is studied and the approximate solution of optimal CSR is obtained. Then, we extended the conclusion to typical communication scenarios with one and more relays where the undetectability and reliability were analyzed and inspected. It is proved that there also exists the optimal CSR in scenarios with relays. Additionally, it is found that the undetectability under the constraints of constant total power depends on the eavesdropper’s position, and we found an undetectability deterioration area (UDA) in the scenario of relays. Simulation results show the existence of optimal CSR and its impact on transmission performance.

Due to the openness of wireless channels, wireless communication systems are extremely vulnerable to attacks, counterfeiting, and eavesdropping. With the advent of the Internet of Things (IoT) era, a large number of smart devices are connected and controlled to meet various requirements. Hence, it is very important to safeguard the information against security breaches and to ensure the privacy of communication.

To ensure the security of personal information, some efficient anonymous authentication schemes have been proposed to adapt to different scenarios [

However, the challenges of information security and privacy are not limited to the above. Count on the rapid growth of telecommunication field new challenges arises [

Based on the ubiquitous channel noise phenomenon, modulation-type wireless covert communication modulates the secret information into an artificial noise signal, which is superimposed on the normal communication signal. It is the most widely used physical-layer wireless covert communication at present. The basic theory and performance limit of the covert communication in AWGN channels are discussed in Reference [

In modulation-type wireless covert communication, the bit error ratio (BER) of covert information is usually much greater than that of the carrier signal. The problems of BER are always solved by means of coding or increasing the power of covert signals. Yet, the difficulty of decryption and the transmission rate of the covert messages will deteriorate with encoding. By means of increasing the transmission power, undetectability will deteriorate [

The contributions of our work are as follows:

We investigated the relationship between BER, CSR, and SNR in wireless covert channels with constellation shaping modulation. We obtained the approximate solution of optimal CSR and extended it to several scenarios with relays. With the approximate solution of optimal CSR, the process of searching for an actual optimal CSR can be accelerated when some optimization algorithms are adopted such as gradient descent and conjugate gradient.

We found an undetectability deterioration area (UDA) in the scenario of one relay and two relays, and the undetectability deteriorates when an eavesdropper is in it. The UDA can be used to avoid the deterioration of undetectability with an improper set of relays. Otherwise, eavesdroppers can detect in the UDA to improve detection efficiency.

The remainder of this paper is organized as follows: in the next section, some background including wireless covert channel with dirty constellation and wireless covert channel with constellation shaping modulation is introduced; in Section

Wireless covert communication mainly involves three factors of inspection: undetectability, reliability, and communication rate. At present, there is no special detection work to measure undetectability for noisy wireless covert communication. References [

Reliability refers to the ability of wireless covert communication to resist channel interference. Channel interference may come from the natural fading of the channel, or from the jammer. To resist channel interference, multihop relaying is a frequently used method [

The researchers further analyzed the covert communication capacity of multiple scenarios with multiple unfavorable factors to the eavesdropper, including three aspects of the transmitter [

In the wireless covert channel with dirty constellation (WCC-DC), the secret message bits can be transmitted as the constellation error of the normal signal in order to reduce the suspicion by all uninformed detectors.

The framework of a wireless covert channel with dirty constellation is shown in Figure

The schematic diagram of WCC-DC: (a) the framework of WCC-DC and (b) rotation of covert constellation.

However, the wireless covert channel with dirty constellation has a high BER when the power of covert signal is low. When we increase the power of covert signal, the undetectability of covert communication deteriorates. Therefore, Cao et al. [

The general framework for the wireless communication system with constellation shaping modulation is demonstrated in Figure _{c} of the OFDM wireless communication is modulated by QPSK. In the proposed scheme, we can use all subcarriers to establish the wireless covert communication. With constellation shaping modulation, the secret information _{s} is modulated into an artificial noise signal _{s}. Then, the artificial noise signal _{s} is superimposed on the carrier signal _{c} to generate the secret subcarrier _{ct}.

The framework of WCC-CSM wireless covert channel.

To generate the secret artificial noise signal _{s}, the cumulative distribution function (CDF) _{CDF} of noise is estimated with the reference channel noise data _{0}.

The secret information is denoted by

For shaping modulation, the transmitter firstly transforms the secret information _{s} into continuous variables _{i}, and then _{i} are mapping to artificial noise signal _{s} with CDF of the reference channel noise _{normal}.

The transform function of _{i} is defined as follows:

We denote _{s} into _{s} is defined as

The mapping function _{normal}. Then, the artificial noise signal _{s} is superimposed on carrier signal _{c} to generate secret subcarrier _{ct}.

The received secret subcarrier is denoted by

We denote

We denote the

The receiver can demodulate the secret information

Covert demodulation constellation of WCC-CSM.

The four black points are the ideal constellation points; the red regions are the distribution areas of secret subcarrier with artificial noise. The function of covert demodulation constellation is denoted by

Similar to the famous Alice–Bob model [

Willie observes the channel to detect whether Alice transmits or not. Willie’s probability of detection error consists of two components: the probability of missed detection and the probability of false alarm.

The literature as seen in the aforementioned works only mentioned the impact of finite samples (i.e., finite

According to the system model shown in Figure

The framework of wireless covert communication.

In communication, Alice totally transmits

The main purpose of Willie is to confirm whether Alice transmits or not. We define two hypotheses,

Suppose Willie performs the optimal detect. Following Pinsker’s inequality [_{1}(_{0}(_{c}[

The KL divergence is always used to calculate the correlation between distributions. Except KL divergence, we can also use KS distance (also called Kolmogorov–Smirnov statistic) to calculate the distance between distributions. The KS distance is defined as follows:

_{1}[_{0}[_{1}[_{0}[_{1}[_{0}[

Willie always set threshold

To measure the reliability of the communication, we denote BER as follows:

After the secret subcarrier

The detector (i.e., Willie) can receive the output as

The undetectability of wireless covert communication deteriorates with the increase of CSR.

As is mentioned above, the probability of detection error must satisfy a lower boundary of

Considering equations (_{1}(_{0}(_{0}. The artificial noise _{s} is the mapping of _{0} and

As is illustrated in equation (

With the increase of _{F} will be great. If the _{M} will be great. We need to choose a suitable value of

Analysis: Considering expression equations (

The KL divergence increases with the increase of _{Willie}. When the KL divergence is greater than _{0} can be expressed as

If SNR < SNR_{0}, the covert communication will not be detected. When the transmission power is constant, the threshold detection distance _{0} can be illustrated in Figure

The probability of undetected _{ud} can be expressed as

Location diagram of Alice, Bob, and Willie.

There exists an optimal ratio between the carrier signal and secret signal no matter what the value of SNR is. The BER minimizes at the optimal ratio.

The reliability of the system is inspected by the BER. The BER of QPSK is

_{ecov}, which can be expressed as

SNR denotes the signal-noise ratio of _{0}, CSR denotes the carrier-secret ratio of _{c} to _{s}. Considering the range of SNR and CSR, the expression (

Taking the partial derivative with respect to CSR, the equation (

Analysis: Let the

The equality can be established when

As can be seen from Figure _{ecov} minimizes at the optimal CSR. And the approximate solution we obtained is consistent with the theoretical _{ecov} in Figure

As wireless communication is affected by channel fading, it is often necessary to set one or more relays to extend the communication distance. Therefore, the relay communication scenarios are described in details in the following subsections.

BER curve in AWGN channel.

Covert information is always transmitted with low power; we can set relays to extend the transmission distance. Each relay employs the amplify-and-forward (AF) protocol and has two phases. Alice transmits signal in one phase; the relay amplifies the signal and forwards to Bob in another phase. We can set the positions of Alice, Bob, Willie, and relay as illustrated in Figure _{tr} denotes the distance between Alice and Bob; _{trr} denotes the distance between Alice and relay; and _{rb} denotes the distance between relay and Bob. If we keep the total transmission energy constant, the transmission energy of Alice and relay are both 0.5_{t}, and we can obtain the output of the covert channel:

Diagram of relay location: (a) relay at random position and (b) relay in middle.

As is illustrated in Figure

We can obtain the signal which is received by Bob:

In free space, the signal which is received by Bob is only about the energy and distance. So the optimization position of the relay is in the middle of Alice and Bob (i.e., _{rb} = _{trr}), as can be seen in Figure

The distance between Willie, Alice, and relay are denoted by _{W} and _{WR}. The probability of undetectability _{ud} can be expressed as

Construct a coordinate system with Alice as the origin of the coordinate axis. We can obtain the coordinates of Alice (0, 0), Bob (2_{trr}, 0), relay (_{trr}, 0), and Willie (_{Willie}, _{Willie}). In the AWGN channel, the power of signal received by Willie is just related to distance and transmit power. We can get “equipower lines” in the scenario of no relay and one relay.

Then, the power detected by Willie can be expressed as

Equation (

We denoted the circle expressed in equation (

It is illustrated in Figure _{ud} will decrease in the scenario of one relay and the consequent deterioration of undetectability. Correspondingly, the undetectability will ameliorate if Willie is outside the green dotted circle. When William is on the green dotted line, the _{ud} will be constant.

Power comparison of classic scenario and one relay scenario.

The BER of covert communication _{ecov.r1} can be expressed as

Referring expression (

Based on the above, we discuss the two-relay scenario. If we keep the total transmission energy constant, the transmission energy of Alice, relay1, and relay2 are all

As can be seen in Figure _{WR}, _{WR1}, and _{WR2}, respectively.

Diagram of two relays location.

The probability of undetected _{ud} can be expressed as

Further extension, the probability of undetectability for

The UDA of relay1 can be expressed as

Equation (

The UDA of relay2 can be expressed as

Equation (

As can be seen in Figure

Power comparison of the classic scenario and two-relay scenario.

The BER of covert communication _{ecov.r2} can be expressed as

It has been proved that there exists an optimal CSR, and we can obtain the minimum of _{ecov.r2} at

In this section, we inspect the undetectability and reliability to benchmark the proposed scheme. We set the wireless communication on an 802.11a/g PHY layer. The wireless covert channel is performed on all 100000 symbols. In transmissions, there are 48 subcarriers in a symbol. Simulation experiments are carried out in wireless channel models of AWGN channel models [

Willie (i.e., detector) observes the channel to judge whether Alice (i.e., transmitter) is transmitting in the covert channel or not. There must be a threshold _{F} will be too great. If the _{M} will be too great. In this paper, we set the

In this section, we set the number of bins

As can be seen in Figure

KL divergence of constellation errors with different CSRs: (a) KL divergence of I vectors, (b) KL divergence of Q vectors, (c) KL divergence of magnitudes, and (d) KL divergence of phase.

In Figure

As can be seen in Figure

KS distances of constellation errors with different CSRs: (a) KS distances of I vectors, (b) KS distances of Q vectors, (c) KS distances of magnitudes, and (d) KS distances of phase.

We can come to a stage conclusion, WCC-CSM meets the threshold of “KL divergence” and “KS distance” in the range of SNR = 10, …, 40 dB. Hence, we can regulate the CSR to reduce the BER without exceeding the threshold of “KL divergence” or “KS distance.”

The reliability of the system is measured by the BER. As is illustrated in expression (28), the BER decreases with the increase of SNR. In Section

BER of wireless covert channel in typical scenarios. (a) BER of classic scenario with different SNR. (b) BER of one relay with different SNR. (c) BER of two relays with different SNR. (d) Comparison of BER of several scenarios.

In Figure

It is proved that an optimal CSR exists and the BER minimizes at the optimal CSR. With the increase of SNR, the optimal CSR gradually increases. But the theoretical approximate value of optimal CSR is slightly lower than the simulation results. As can be seen in Figures

Comparing the BER under the constraints of constant total power, suppose that the SNR in the classic scenario is 20 dB. The SNR in the scenario of one relay is 23 dB, and the SNR in the scenario of two relays is 25 dB. As can be seen in Figure

Simulation experiments are carried out in wireless channel models of AWGN channel. As can be seen in Figure _{ecov}, _{ecov.r1}, and _{ecov.r2} at the optimal CSR in the AWGN channel. And the optimal CSR achieved in simulation is slightly higher than the theoretical approximation.

Reliability and undetectability are the main aspects of wireless covert communication. We considered the BER problem of covert communication based on WCC-CSM. We studied the impact of carrier-secret ratio (CSR) on the BER and investigated the relationship between SNR, CSR, and BER. We obtained the approximate solution of optimal CSR and extended it to the scenario of relays. With the approximate solution of optimal CSR, the process of searching for an actual optimal CSR can be accelerated. Furthermore, we found that the undetectability under the constraints of constant total power depends on the eavesdropper’s position. And we found an undetectability deterioration area (UDA) in the scenario of relays, and undetectability deteriorates with setting relays when an eavesdropper is in the UDA.

The simulation proved that there exists an optimal CSR in the AWGN channel. The transmitter can obtain greater reliability with great undetectability at the optimal CSR. Additionally, the reliability deteriorates with setting relays under the constraints of constant total power. Some error correction coding or other methods must be adopted, avoiding the deterioration of BER.

To improve the detection capability of Willie, it is necessary to find a better way to detect the covert communication except “KL divergence” or “KS distance” in our future work.

The data used to support the findings of this study are included within the supplementary information files.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grants nos. U1836104, 61772281, 61702235, 61801073, 61931004, and 62072250).

These files contain the KL divergence of four vectors, KS distance of four vectors, and BERs of wireless covert communication in typical scenarios. (i) KL divergence of constellation errors with different