Issue 
Security and Safety
Volume 2, 2023
Security and Safety in Physical Layer Systems



Article Number  2023015  
Number of page(s)  18  
Section  Information Network  
DOI  https://doi.org/10.1051/sands/2023015  
Published online  03 July 2023 
Research Article
Covert communication in relay and RIS networks
^{1}
College of Physics and Information Engineering, Fuzhou University, Fuzhou, Fujian, 350116, China
^{2}
Fujian Key Laboratory for Intelligent Processing and Wireless Transmission of Media Information, Fuzhou University, Fuzhou, Fujian, 350116, China
^{3}
School of Information and Communication Engineering, Hainan University, Haikou, Hainan, 570100, China
^{*} Corresponding authors (email: youjia.chen@fzu.edu.cn (Youjia Chen); shufeng@hainanu.edu.cn (Feng Shu))
Received:
21
December
2022
Revised:
10
March
2023
Accepted:
14
June
2023
Covert communication aims to prevent the warden from detecting the presence of communications, i.e. with a negligible detection probability. When the distance between the transmitter and the legitimate receiver is large, large transmission power is needed, which in turn increases the detection probability. Relay is an effective technique to tackle this problem, and various relaying strategies have been proposed for longdistance covert communication in these years. In this article, we first offer a tutorial on the relaying strategies utilized in covert transmission. With the emergence of reflecting intelligent surface and its application in covert communications, we propose a hybrid relayreflecting intelligent surface (HRRIS)assisted strategy to further enhance the performance of covert communications, which simultaneously improves the signal strength received at the legitimate receiver and degrades that at the warden relying on optimizing both the phase and the amplitude of the HRRIS elements. The numerical results show that the proposed HRRISassisted strategy significantly outperforms the conventional RISaided strategy in terms of covert rate.
Key words: Covert communication / relay / reflecting intelligent surface
Citation: Hu J, Shi X and Chen Y et al. Covert communication in relay and RIS networks. Security and Safety 2023; 2: 2023015. https://doi.org/10.1051/sands/2023015
© The Author(s) 2023. Published by EDP Sciences and China Science Publishing & Media Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Security and privacy are two critical issues in current communication systems. The development of classical cryptography, informationtheoretic security, and quantum cryptography have greatly improved the security levels of data transmission. Cryptography presents the adversary with a problem that it is assumed not to be able to solve because of computational constraints, while informationtheoretic security presents the adversary with a signal from which no information can be extracted. They addressed many security issues by protecting the content of the message, they cannot mitigate the threat of communication detection. Adversary/warden detects a potential data transmission, it may disturb it. To tackle this issue, covert communication arose, which focuses on transmitting the confidential information from a transmitter to legitimate receivers, while avoiding being detected by the potential warden [1, 2].
For instance, the transmitter Alice tends to transmit a confidential message to the receiver Bob with in the presence of warden Willie. That is, Alice has to provide a reliable transmission to Bob while the transmission remains hidden from Willie. Willie, on the other hand, is not interested in the content of the message and only wants to determine whether Alice transmits any message to Bob or not. We note that this is in strong contrast to the role of an eavesdropper in traditional physical layer security schemes where the eavesdropper is aware of the presence of a message transmission and looks to decode the information transmitted from Alice to Bob.
Consequently, while Bob has to decode the information sent by Alice, Willie has to confront a detection problem, attempting to distinguish between the two potential states of Alice’s transmission. Recent research efforts in the domain of covert communications have explored different problems in this field, from establishing the achievable fundamental limits to exploiting any uncertainties at the potential adversary Willie, including noise, channel, network interference, and jamming signals/artificial noise introducing uncertainties in Willie’s observations.
Noise uncertainty: The lack of knowledge of the exact noise power is called noise uncertainty. In practice, the sources of background noise include thermal noise, quantization noise, imperfect filters, etc. Noise uncertainty is almost unavoidable due to, e.g., temperature change, environmental noise change, and calibration error. Therefore, the consideration of noise uncertainty is practical and necessary for the study related to power detection. The authors in [3] considered two models of noise uncertainty at the Willie, i.e., the bounded uncertainty model and the unbounded uncertainty model. In [4], the noise uncertainty is used to analyze the minimum error detection probability of the warden to realize covert wireless communication in spaceairground integrated vehicular networks. Considering the noise uncertainty at warden, the authors in [5] studied the IoT covert communication network with the channel state information (CSI) unawareness at Bob.
Channel uncertainty: All the node experience uncertainty about their channel knowledge. One of the main assumptions in most covert communications literature is that the CSI of both the covert link is perfectly known by both the legitimate receiver and the transmitter. Usually, the CSI is obtained at the receiver by channel estimation during pilot transmission. Then, a feedback link (if available) is used to send the CSI to the transmitter. Hence, the accuracy of the channel estimation at the receiver affects the quality of CSI at the transmitter. However, in covert communication scenarios, transmitting pilots and acquiring feedback is often infeasible, especially as the transmission of pilots will also enable the adversary to acquire channel information from the covert transmitter. The authors in [6] considered the scenario where a public link is used to hide a cover link while users including the legitimate receiver and warden Willie suffer from uncertainty in their channel knowledge from the transmitter. Under the CSI uncertainty scenario, Willie’s optimal detection performance is derived and then the optimal transmission rates are determined for both the legitimate and covert links under certain transmission outage probabilities. In [7], the authors analyzed the ability of the suspicious receiver to detect the artificial noise under the channel uncertainty and optimized the performance of the covert surveillance performance.
Interference: In practical wireless networks, a major source of the uncertain interference at the receiver is the ambient signals from other transmitters, and the uncertainty of the aggregate received interference at the warden will help to achieve the positive covert throughput in covert communications. The uncertainty of the aggregated interference experienced by the warden is beneficial to the potential transmitters for covert communication [8]. The authors in [9] studied covert communication in wireless networks with the aid of stochastic geometry. Instead of assuming that all communication nodes in the network are friendly helpers, it is assumed that all nodes in the network, which are distributed according to a homogeneous Poisson point process (PPP), randomly transmit without the intention to help the covert communication. When the interference is sufficiently small and comparable with the receiver noise, the covert throughput increases as the density or the transmit power of the concurrent interferers increases. The authors of [10] derived the exact covert capacity region in the covert communication network and proved that the scheme using interference as noise is optimal.
Jamming signals/artificial noise: The performance of covert communication can be improved by letting a friendly jammer deliberately broadcast a jamming signal to degrade the detection performance of the adversary. If this jammer randomly varies its transmit power appropriately or if timevarying multipath fading causes sufficient variation, channel estimation during periods outside the period time when Willie is attempting to detect Alice’s transmission cannot be used to estimate the statistics of the noise impacting Willie’s receiver during the period of interest [11]. The authors consider the use of an FD receiver to achieve covert communication [12]. Specifically, the fullduplex (FD) receiver generates AN with a randomized transmit power, causing deliberate confusion and affecting the decisions at Willie regarding the presence of any covert transmissions. The use of an FD receiver generating AN provides a cover for covert transmission and offers a multitude of benefits as compared to the use of a separate, independent jammer. Being equipped with an FD receiver, we can exercise better control over the power used for transmitting AN, hence better management of system resources to achieve a higher covert rate. Furthermore, while Willie will face strong interference, the selfinterference at the FD receiver can be greatly suppressed by the welldeveloped selfinterference cancellation techniques. Furthermore, a finite blocklength covert communication scheme with artificial noise was proposed in [13]. In [14], the authors investigated the influence of the jammer equipped with multiple antennas on Alice’s transmit power, and consequently on covert communication performance. The authors in [15] adopted the channel inversion power control to maintain the received power with a constant value for covert communication, while the receiver sends the fullduplex artificial noise with random power, which allows the transmitter to ensure security with higher transmit power.
The aforementioned contributions are mainly focused on improving wireless covertness without paying much attention to communication reliability. When Alice and Bob are located far from each other, in order to make the probability of error at Bob sufficiently small, Alice should use a high transmit power. However, this increases the probability of being detected by Willie, especially if Willie is close to Alice and thus receives a strong signal. The pointtopoint covert communication in the presence of a single warden that is discussed in the literature should be extended to scenarios with multiple receivers (hops) for relaying the covert message.
2. Typical applications of relay for covert communications
The following discussions will show that the covertness of relayassisted networks can be enhanced by exploiting the distributed diversity and shortening the access distance with the relay. Figure 1 shows several typical scenarios of relayassisted covert communications. The existing research works on the relay in covert communications are demonstrated to show how to achieve the demands of longdistance wireless communication with low detection probability and reliable transmission.
Figure 1. Typical applications of relay network in covert communication systems: a) Greedy relay with blockage; b) Relay with different duplex mode; c) Relay networks with relay selection ; d) MultiHop against UAV surveillance; e) MultiHop against multiple collaborating Willies 
2.1. Greedy relay with blockage
As shown in Fig. 1a, a oneway relay network over a Rayleigh fading channel is considered, in which the source transmits information to the destination with the aid of the relay, since a direct link from source to destination is not available [16]. In the considered scenario, the source allocates some resources (e.g., power [17] and spectrum) to relay in order to seek its help to relay the message to the destination. However, in some scenarios, the relay may intend to use this resource to transmit its message to a destination as well, which is forbidden by the source and thus should be kept covert from the source. As such, in the considered system model the source is also the warden which is detecting whether the relay transmits its information to the destination when it is aiding the transmission from the source to the destination. The covert transmission from the relay to the destination is similar to steganography, in which covert information is transmitted by hiding in innocuous objects. These innocuous objects are utilized as “cover medium" to carry the covert information. In [16] and [17], the innocuous objects are the forwarding transmissions from relay to the destination. The main difference between this work and steganography is that in this work the covert information is shielded by the forwarding transmissions from the relay to the destination at the physical layer, while in steganography the covert information is hidden and transmitted by encoding or modifying some contents (e.g., shared videos or images) at the application layer. The covert communication from a relay to a destination only occurs when the successful transmission from source to destination is guaranteed. As such, when the covert message is transmitted via the relay, successive interference cancellation (SIC) that allows a receiver to decode different signals arriving simultaneously is implemented at the destination. Following SIC, the destination decodes the stronger signal (i.e., original message) first, subtracts it from the combined signal, and finally decodes the weaker one (i.e., covert message) from the residue. More recently, the authors in [18] proposed two covert transmission schemes, named random beamforming and maximumratio transmission (MRT) beamforming to guarantee reception reliability at the destination, when the greedy relay is equipped with multiple antennas.
2.2. Relay with different duplex mode
Figure 1b illustrates that the relay can work in either the fullduplex (FD) mode or the halfduplex (HD) mode [19]. Under the FD mode, the relay can simultaneously receive and forward information on the same channel, but the communication is negatively affected by the selfinterference of the Relay. Under the HD mode, Relay receives and forwards information to users at two orthogonal time slots, which experience two phases. Therefore, the relay to flexibly switch between the FD and HD modes for improving the covert rate performance, and optimize the relay transmit power to achieve the covert rate maximization under such a joint mode. It is worth noting that Relay always sends the jamming signal to Willie when Alice does not transmit. Under the FD mode, although suffering from selfinterference, can also enhance the covert rate by careful power control. As for the HD mode, it can avoid the negative effect of selfinterference. However, it may reduce the covert rate due to the different receiving and forwarding time slots at the relay.
2.3. Relay networks with relay selection
In multiplerelay networks, relay selection has been regarded as an effective technique to achieve spacial diversity gain. In [20], a covert transmission scheme in a relay selection system was proposed, where the relay with the best relaytodestination link is selected to forward the information, it can also opportunistically transmit its message covertly to the destination, which is shown in Fig. 1c. This relay network consists of one source, one destination, and multiple decodeandforward (DF) relays. When forwarding the source’s message, the selected relay decodes the data received in the first phase, encodes them with another codebook, and then transmits them to the destination. Hence, the received SNR at the relay decides the success of decoding. The authors in [20] investigated the tradeoff between covertness and reliability in multiple relay systems, where the probability of detection error (i.e., covertness) is quantified in terms of the probability that the warden fails in detecting the power relay’s covert signal, while reliability represents the probability that an outage event is encountered at the relay for decoding the original message. This work showed that the diversity gain provided by relay selection will lead to a decrease in the probability of detection error.
2.4. MultiHop against UAV’s surveillance
As shown in Fig. 1d, the authors in [21] considered an unmanned aerial vehicle (UAV) in the air to act as a warden to monitor any covert communication and an eavesdropper to wiretap the transmitted signal. Compared with the terrestrial channels, the characteristics of the airtoground channels make legitimate information particularly vulnerable to being detected and wiretapped. Compared with the terrestrial channels, the advantages of the airtoground channels lead to the proneness of legitimate information leakage and detection. In order to reduce the information leakage for a pair of terrestrial nodes against the UAV surveillance while maximizing the throughput, the optimal designs of coding rate, transmission power, and the number of hops are exploited in this work. This strategy is especially suitable for networks with limited energy and longdistance between the source and the destination.
2.5. MultiHop against multiple collaborating willies
In covert communication, when the distance between Alice and Bob becomes large compared with the distance between Alice and Willie(s) shown in Fig. 1e, the performance of covert communication degrades. In this case, multihop transmission via intermediate relays can help to improve performance. The multihop covert communication over a moderatesize network and in the presence of multiple collaborating Willies is studied in [22]. For covert communication, the source and the intermediate relays use a key to encode the message. The relays can transmit covertly using either a single key for all relays or different independent keys at the relays. The routing algorithms for maximizing the covert throughput and minimizing the endtoend delay are developed for two relaying approaches.
Typical Applications of relay for Covert Communications
3. Intelligent reflecting surface assisted covert communication
Since the relay forwards signals to assist sourcedestination transmission in an active mode, leading to a high power consumption problem when compared with the RIS. In addition, the amplifying noise at the amplifyandforward (AF) relay will increase the probability of being detected at Willie and the reduction of the signaltonoise ratio at the DF relay will lead to a higher decoding error probability.
As a special kind of relay, the authors in [23] presented the potential of using the intelligent reflecting surface (IRS), also known as reflecting intelligent surface (RIS), to improve covert communication performance. In this section, we propose a case study of RIS to assist the signal transmission from source to destination while defending against Willie’s detection.
3.1. Related work
IRS/RIS is a flat surface composed of a large number of reconfigurable and lowcost passive reflective elements, each of which is capable of controlling the phase and amplitude of the incident signal for optimal reflection, making the wireless channel between the transmitter and receiver more favorable for communication [24]. Following [23], several IRS/RISaided covert communication approaches have been developed. The authors in [25] considered the design of a latencyconstrained covert RISassisted communication system when with the global CSI and without Willie’s instantaneous CSI, respectively. The authors examined RISassisted covert communications by considering only Willie’s statistical CSI is available [26]. The authors of [27] investigated the multipleinputmultipleoutput (MIMO) covert communication assisted by RIS, where the covert rate was maximized by jointly designing the transmit covariance matrix and phase shift matrix. In [28], the authors proved that the reliability of communication transmission can be improved by adding RIS elements. Under the assumption that the number of channels used is infinite, the influence of AN on IRSassisted covert communication is studied [29]. The authors in [30] adopted a twoway relaying protocol in the fullduplex relaying network assisted by RIS to realize covert communication. In [31], the authors studied the covert beamforming design of the IoT network assisted by the RIS. The information freshness maximization problem in IRSaided fullduplex covert communications was studied in [32], where the nonretransmission protocol and the automatic repeatrequest (ARQ) protocol are considered. In [33], UAV and IRS are combined to improve covert communication performance.
It should be emphasized that the existing studies on RISassisted covert communication all adopted fullypassive beamforming. In traditional RISassisted covert communication, due to the “double fading" effect of RIS, it’s gain for covert performance is small [34], and the passive reflection of RIS limits the freedom of beamforming. The traditional RIS can be easily outperformed by a halfduplex relay when the number of elements in the RIS is not sufficiently large. Motivated by this, if a few passive elements of the IRS/RIS are replaced by active ones, the gain from active relaying can be achieved. The idea is to activate some elements of the RIS by connecting them to radio frequency (RF) chains and power amplifiers. This implies that if a few passive elements of the RIS are replaced by active ones, such that the traditional IRS becomes a hybrid relayreflecting intelligent surface (HRRIS) [35]. The active elements can not only modify the phase but also amplify the incident signal, improving the degrees of freedom in the beamforming. However, the amplitude of the passive element is generally set to 1, which means that it can only adjust the phase of the incident signal. HRRIS requires extra cost in hardware implementation of active elements and signal processing. However, these only require a single or several active elements. Hardware and computing costs increase only slightly, given that the total number of elements in a traditional RIS is very large. It is not difficult to find that the active beamforming in HRRIS is similar to the relay.
In HRRISassisted covert communications, the reconfigurability of HRRIS can be used to establish a favorable environment to enhance the quality of legitimate communications. Besides, the passive elements do not use a transmitter module, which incurs no additional power consumption and hardware cost compared with existing technologies based on active elements [36]. The active elements exploit the extra power consumption to overcome the “doublefading" effect. Compared to the AF relay, active elements utilize the principle of electromagnetic scattering to amplify signals directly, without the RF chain. Furthermore, HRRIS works in FD mode without selfinterference elimination.
In this paper, we consider covert communication from a transmitter (Alice) to a receiver (Bob) with the help of HRRIS. In order to help Bob realize the low probability of being detected by Willie and improve the covert rate, an alternate optimization method was proposed to obtain the optimal reflection coefficient and transmit power. Our novelty is summarized as follows:

(1)
We propose covert communication assisted by HRRIS. The HRIRS only needs to activate one or more elements of the IRS to act as an active relay. In this system, we consider the maximization of Bob’s (covert user) covert rate under the covert constraint and total power constraint. For obtaining an efficient solution, matrix decomposition is used to deal with the nonconvexity of the problem. On this basis, an alternating optimization (AO) method is proposed to obtain the optimal reflection phase shift of HRIRS.

(2)
Considering covert communication with finite block length, we used the KullbackLeibler (KL) divergence on Willie as the covert constraint. In this case, the closedform expression of the covertness constraint is derived, and it is proved that the KL divergence used in the covertness constraint is a monotonically increasing function of Willie’s received power, based on which the optimal transmit power of Alice is obtained.

(3)
We evaluate the performance of the proposed HRIRSassisted covert communication through simulations. Particularly, it shows that having a small number of active elements in the HRIRS can significantly improve the achievable covert rate. In addition, compared with traditional IRS/RISassisted covert communication systems, our proposed system can obtain a large amount of covert rate gain.
3.2. Proposed HRRIS assisted covert communication
As shown in Fig. 2, we propose a covert communication transmission scheme assisted by HRRIS, where a transmitter (Alice) intends to send confidential information to a legitimate receiver (Bob) with the aid of the HRRIS, while a warden (Willie) attempts to detect the existence of this transmission. There are two paths from Alice to Bob, i.e., one is the direct link, and the other is the reflection/relaying link via HRRIS. Similarly, there are also two paths from Alice to Willie. In addition, it is assumed that the signals reflected by the HRRIS twice or more are ignored due to the significant path loss.
Figure 2. HRRIS aided covert communication 
Alice, Bob, and Willie are assumed to be equipped with N _{ a }, N _{ b }, and N _{ w } antennas, respectively. The HRRIS is assumed to be equipped with N elements, including M passive reflecting elements and K active relaying elements (i.e., M + K = N). The passive reflecting elements are implemented by a phase shifter, while the active relaying elements can tune the phase and amplitude of the incident signal. We assume that the active elements work in the AF mode. Therefore, for K = 0, HRRIS returns to the traditional RIS. For K = N, by contrast, it becomes a relay station equipped with N antennas. Hence, in this work, we have 1 ≤ K ≤ N. Furthermore, similar to the conventional RIS, we assume that each (active/passive) element of HRRIS can independently reflect the received signals.
For the HRRIS, ℚ represents the set of active relay elements. We define Θ = Φ + Ψ, where $\mathrm{\Theta}=\mathrm{diag}\{{\theta}_{1},\dots ,{\theta}_{N}\}\in {\mathbb{C}}^{N\times N}$, $\mathrm{\Phi}=\mathrm{diag}\{{\varphi}_{1},\dots ,{\varphi}_{N}\}\in {\mathbb{C}}^{N\times N}$, and $\mathrm{\Psi}=\mathrm{diag}\{{\psi}_{1},\dots ,{\psi}_{N}\}\in {\mathbb{C}}^{N\times N}$, where Φ and Ψ denote the reflection coefficients of passive elements and active elements, respectively. Therefore, we have
$$\begin{array}{c}\hfill {\theta}_{n}=\{\begin{array}{cc}{\beta}_{n}{e}^{j{\mu}_{n}},\hfill & if\phantom{\rule{1em}{0ex}}n\in \mathbb{Q},\hfill \\ {e}^{j{\mu}_{n}},\hfill & otherwise,\hfill \end{array}\end{array}$$(1)
where μ _{ n } ∈ [0, 2π) represents the phase shift. We notice that β _{ n }=1 for n ∉ ℚ, and β _{ n } for n ∈ ℚ is determined by the total power of the active elements, which will be discussed later.
3.2.1. Transmission from Alice to Bob
When Alice transmits a confidential message, the received signal at Bob is given by
$$\begin{array}{cc}\hfill {\mathbf{y}}_{b}& =\sqrt{{P}_{a}}{\mathbf{H}}_{\mathit{rb}}\mathrm{\Psi}{\mathbf{H}}_{\mathit{ar}}\mathbf{x}+\sqrt{{P}_{a}}{\mathbf{H}}_{\mathit{rb}}\mathrm{\Phi}{\mathbf{H}}_{\mathit{ar}}\mathbf{x}+\sqrt{{P}_{a}}{\mathbf{H}}_{\mathit{ab}}\mathbf{x}+{\mathbf{H}}_{\mathit{rb}}\mathrm{\Psi}{\mathbf{n}}_{r}+{\mathbf{n}}_{b}\hfill \\ \hfill & =\sqrt{{P}_{a}}({\mathbf{H}}_{\mathit{rb}}\mathrm{\Theta}{\mathbf{H}}_{\mathit{ar}}+{\mathbf{H}}_{\mathit{ab}})\mathbf{x}+{\mathbf{n}}_{\mathit{bt}},\hfill \end{array}$$(2)
where n _{ b t } = H _{ r b } Ψ n _{ r } + n _{ b } represents the total effective noise at the Bob. H _{ a r } and H _{ a b } are the steering vectors from Alice to HRRIS and the steering vector from Alice to Bob, respectively, and H _{ r b } denotes the steering vector from HRRIS to Bob. In addition, n _{ r } ∼ $\mathcal{CN}(\mathbf{0},{\sigma}_{r}^{2}{\mathbf{I}}_{K})$ and n _{ b } ∼ $\mathcal{CN}(\mathbf{0},{\sigma}_{b}^{2}{\mathbf{I}}_{{N}_{b}})$ are the complex additive white Gaussian noise (AWGN) space vectors at the K active elements of the HRRIS and the Bob, respectively. For simplicity, we assume that ${\sigma}_{r}^{2}={\sigma}_{b}^{2}$ and n _{ b t } ∼ $\mathcal{CN}(\mathbf{0},{\sigma}_{b}^{2}({\mathbf{I}}_{{N}_{b}}+{\mathbf{H}}_{\mathit{rb}}\mathrm{\Psi}{\mathrm{\Psi}}^{H}{\mathbf{H}}_{\mathit{rb}}^{H}))$. x is the signal vector transmitted by Alice, with each element following 𝒞𝒩(0, 1), and P _{ a } is Alice’s transmit power, which should meet the constraint condition ${P}_{a}\le {P}_{a}^{\mathrm{max}}$, where ${P}_{a}^{\mathrm{max}}$ is the maximum transmit power of Alice.
We assume that Bob’s CSI is perfectly known by himself through channel estimation [37]. Following (2), Bob’s covert rate in the HRRISaided covert communication system can be expressed as [35]
$$\begin{array}{c}\hfill f(\mathrm{\Theta},{P}_{a})={log}_{2}{\mathbf{I}}_{\mathit{Nb}}+{\mathrm{\Omega}}_{b},\end{array}$$(3)
where ${\mathrm{\Omega}}_{b}={P}_{a}{\mathbf{U}}_{b}{\mathbf{R}}^{1}/{\sigma}_{b}^{2}$, the signal covariance matrix is given by U _{ b } = (H _{ r b } Θ H _{ a r } + H _{ a b })(H _{ r b } Θ H _{ a r } + H _{ a b })^{ H }, and the aggregate noise covariance matrix is given by $\mathbf{R}=({\mathbf{I}}_{{N}_{b}}+{\mathbf{H}}_{\mathit{rb}}\mathrm{\Psi}{\mathrm{\Psi}}^{H}{\mathbf{H}}_{\mathit{rb}}^{H})$ ∈ ℂ^{ N b × N b }.
The transmit power of the active elements at the HRRIS can be expressed as
$$\begin{array}{c}\hfill {P}_{r}=Tr(\mathrm{\Psi}({\mathbf{H}}_{\mathit{rb}}{\mathbf{H}}_{\mathit{rb}}^{H}{P}_{a}+{\sigma}_{b}^{2}){\mathrm{\Psi}}^{H}),\end{array}$$(4)
which should meet the constraint condition ${P}_{r}\le {P}_{r}^{\mathrm{max}}$, where ${P}_{r}^{\mathrm{max}}$ is the maximum transmit power of all the K active elements.
3.2.2. Binary hypothesis testing at Willie
In this work, we focus on delayconstrained covert communication, that is, the number of channel uses L is finite. In order to detect the existence of a transmission, Willie attempts to distinguish the following two hypotheses:
$$\begin{array}{c}\hfill {\mathbf{y}}_{w}=\{\begin{array}{cc}{\mathbf{n}}_{w},\hfill & {\mathcal{H}}_{0},\hfill \\ \sqrt{{P}_{a}}({\mathbf{H}}_{\mathit{rw}}\mathrm{\Theta}{\mathbf{H}}_{\mathit{ar}}+{\mathbf{H}}_{\mathit{aw}})\mathbf{x}+{\mathbf{H}}_{\mathit{rw}}\mathrm{\Psi}{\mathbf{n}}_{r}+{\mathbf{n}}_{w},\hfill & {\mathcal{H}}_{1},\hfill \end{array}\end{array}$$(5)
where H _{ a w } and H _{ r w } are the steering vector from Alice to Willie and the steering vector from HRRIS to Willie, respectively, and n _{ w } ∼ $\mathcal{CN}(\mathbf{0},{\sigma}_{w}^{2}{\mathbf{I}}_{{N}_{w}})$ is the AWGN space vector at Willie. ℋ_{0} denotes the null hypothesis in which Alice does not transmit, and ℋ_{1} denotes the alternative hypothesis where Alice transmits signals. Similarly, we assume that ${\sigma}_{w}^{2}={\sigma}_{r}^{2}$. Therefore, the total noise power at Willie is n _{ w t } = (H _{ r w } Ψ n _{ r } + n _{ w }) ∼ $\mathcal{CN}(\mathbf{0},{\sigma}_{w}^{2}({\mathbf{I}}_{{N}_{w}}+{\mathbf{H}}_{\mathit{rw}}\mathrm{\Psi}{\mathrm{\Psi}}^{H}{\mathbf{H}}_{\mathit{rw}}^{H}))$ under ℋ_{1}.
Considering the worstcase scenario for covert communication, we assume that Willie knows the channels perfectly. In this case, Willie has the maximum detection probability, and the corresponding covert performance can be used as a lower bound. In practice, the covert performance will be better than or equal to the worst case. The probability density function of y _{ w } under ℋ_{0} and ℋ_{1} can be respectively given by
$$\begin{array}{cc}& f({\mathbf{y}}_{w}{\mathcal{H}}_{0})=\mathcal{CN}(\mathbf{0},{\sigma}_{w}^{2}{\mathbf{I}}_{{N}_{w}}),\hfill \\ \hfill & f({\mathbf{y}}_{w}{\mathcal{H}}_{1})=\mathcal{CN}(\mathbf{0},{P}_{a}{\mathbf{U}}_{w}+{\sigma}_{w}^{2}({\mathbf{I}}_{{N}_{w}}+{\mathbf{H}}_{\mathit{rw}}\mathrm{\Psi}{\mathrm{\Psi}}^{H}{\mathbf{H}}_{\mathit{rw}}^{H})),\hfill \end{array}$$(6)
where U _{ w } = (H _{ r w } Θ H _{ a r } + H _{ a w })(H _{ r w } Θ H _{ a r } + H _{ a w })^{ H }.
Based on (6), the optimal detection threshold and the corresponding minimum detection error rate ξ ^{*} at Willie can be derived [38]. However, due to the expression for ξ ^{*} involves an incomplete gamma function, which cannot be handled by subsequent analysis and design. In order to deal with this difficulty, we present a lower bound of ξ ^{*}. which is given by [39]
$$\begin{array}{c}\hfill {\xi}^{\ast}\ge 1\sqrt{\frac{1}{2}{\mathcal{D}}_{01}},\end{array}$$(7)
where 𝒟_{01} is the KullbackLeibler (KL) divergence from ℙ_{0} to ℙ_{1}, which is given by [38]
$$\begin{array}{cc}& {\mathcal{D}}_{01}=L[ln(1+{\gamma}_{w})\frac{{\gamma}_{w}}{1+{\gamma}_{w}}],\hfill \end{array}$$(8)
where γ _{ w } is the signaltointerferenceplusnoise ratio (SINR) at Willie under ℋ_{1} is given by
$$\begin{array}{c}\hfill {\gamma}_{w}=\frac{{\mathbf{U}}_{w}{P}_{a}}{(\mathbf{M}{\sigma}_{w}^{2})},\end{array}$$(9)
where $\mathbf{M}=({\mathbf{I}}_{{N}_{w}}+{\mathbf{H}}_{\mathit{rw}}\mathrm{\Psi}{\mathrm{\Psi}}^{H}{\mathbf{H}}_{\mathit{rw}}^{H})$. We note that U _{ w } and M are Hermitian matrices. Then, we perform the eigenvalue decomposition (EVD) on the above two matrices, which can be written as ${\mathbf{U}}_{w}=\mathbf{G}\mathrm{\Xi}{\mathbf{G}}^{1}$ and M = J Λ J ^{−1}, where G and J are matrices of eigenvectors of U _{ w } and M, respectively, and G and J ∈ℂ^{ N w × N w }, Ξ = diag{ω _{1},ω _{2},…,ω _{ N w }}, ω _{ n } is the nth eigenvalue of U _{ w }, Λ = diag{κ _{1},κ _{2},…,κ _{ N w }}, κ _{ n } is the nth eigenvalue of M. As such, we have ${\mathbf{U}}_{w}=\prod _{i=1}^{{N}_{w}}{\omega}_{i}$ and $\mathbf{M}=\prod _{i=1}^{{N}_{w}}{\kappa}_{i}$.
In covert communications, ξ ^{*} > 1 − ϵ is generally adopted as the covertness constraint, where ϵ is a small value to determine the required covertness level. Therefore, according to (7) and (8), we obtained the covertness constraint of Willie, which can be rewritten as
$$\begin{array}{c}\hfill {\mathcal{D}}_{01}\le 2{\u03f5}^{2}.\end{array}$$(10)
3.2.3. Problem formulation and solution
In this part, we jointly design the transmit power at Alice and relay/reflection coefficients of the HRRIS to maximize the covert rate at Bob subject to the covertness and other constraints, of which the optimization problem can be formulated as
$$\begin{array}{cc}\hfill (\mathrm{P}1):& \underset{\mathrm{\Theta},{\mathbf{P}}_{\mathbf{a}}}{\mathrm{max}}f(\mathrm{\Theta},{P}_{a}),\hfill \end{array}$$(11a) $$\begin{array}{cc}\hfill & \mathbf{s}.t.\phantom{\rule{3.33333pt}{0ex}}{\mathcal{D}}_{01}\le 2{\u03f5}^{2},\hfill \end{array}$$(11b) $$\begin{array}{cc}\hfill & \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\beta}_{n}=1,\phantom{\rule{3.33333pt}{0ex}}for\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}n\notin \mathbb{Q},\hfill \end{array}$$(11c) $$\begin{array}{cc}\hfill & \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{P}_{a}\le {P}_{a}^{\mathrm{max}}.\hfill \end{array}$$(11d)
Our goal is to maximize the covert rate at Bob by jointly designing P _{ a } and Θ. We propose an alternating optimization algorithm to optimize P _{ a } and Θ. Specifically, we first optimize Θ for a given P _{ a }, and the objective function is transformed into a form that is easy to handle. Then, we optimize P _{ a } for a given Θ.
First, we randomly generate the coefficient of HRRIS and use 𝒟_{01} = 2ϵ ^{2} to get feasible P _{ a }. The objective function f(Θ, P _{ a }) is nonconvex with respect to Θ. In addition, the feasible set of (P1) is nonconvex due to the unitmodulus constraint (11c). Therefore, (P1) is difficult to be tackled. Thus, we approximate the objective function f(Θ, P _{ a }) by using its upper bound f _{0}(Θ, P _{ a }), which can be written as
$$\begin{array}{cc}& f(\mathrm{\Theta},{P}_{a})={log}_{2}{\mathbf{I}}_{{N}_{b}}+\frac{{P}_{a}{\mathbf{U}}_{b}{\mathbf{R}}^{1}}{{\sigma}_{b}^{2}}\hfill \\ \hfill & \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}={log}_{2}\mathbf{R}+\rho {\mathbf{U}}_{b}{log}_{2}\mathbf{R}\hfill \\ \hfill & \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\stackrel{a}{\le}{log}_{2}\mathbf{R}+\rho {\mathbf{U}}_{b}\hfill \\ \hfill & \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}={log}_{2}\mathbf{R}+\rho ({\mathbf{H}}_{\mathit{rb}}\mathrm{\Theta}{\mathbf{H}}_{\mathit{ar}}+{\mathbf{H}}_{\mathit{ab}}){({\mathbf{H}}_{\mathit{rb}}\mathrm{\Theta}{\mathbf{H}}_{\mathit{ar}}+{\mathbf{H}}_{\mathit{ab}})}^{H}\hfill \\ \hfill & \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}={f}_{0}(\mathrm{\Theta},{P}_{a}),\hfill \end{array}$$(12)
where $\rho ={P}_{a}/{\sigma}_{b}^{2}$ and a is achieved by set ℚ = ∅. We note that this upper bound becomes tighter as log_{2}R decreases.
Generally, the proposed solution is a sequential procedure where in each iteration, a specific coefficient of HRRIS is updated when the others are fixed. Specifically, we let ${\mathbf{a}}_{n}^{H}\in {\mathbb{C}}^{{N}_{b}\times 1}$ denote the nth row of H _{ a b }, and b _{ n } ∈ ℂ^{ N b × 1} denote the nth column of H _{ r b }, i.e., H _{ a r } = [a _{1}, a _{2}, …, a _{ N }]^{ H }, and H _{ r b } = [b _{1}, b _{2}, …, b _{ N }]. Since Ψ and Θ are diagonal matrices, we have ${\mathbf{H}}_{\mathit{rb}}\mathrm{\Theta}{\mathbf{H}}_{\mathit{ar}}=\sum _{n=1}^{N}{\theta}_{n}{\mathbf{b}}_{n}{\mathbf{a}}_{n}^{H}$ and H _{ r b } Ψ = ∑_{ n ∈ ℚ} θ _{ n } b _{ n }. Hence, we can rewrite f _{0}(Θ, P _{ a }) as
$$\begin{array}{cc}\hfill {f}_{0}(\mathrm{\Theta},{P}_{a})& ={log}_{2}\mathbf{R}+\rho ({\mathbf{H}}_{\mathit{rb}}\mathrm{\Theta}{\mathbf{H}}_{\mathit{ar}}+{\mathbf{H}}_{\mathit{ab}}){({\mathbf{H}}_{\mathit{rb}}\mathrm{\Theta}{\mathbf{H}}_{\mathit{ar}}+{\mathbf{H}}_{\mathit{ab}})}^{H}\hfill \\ \hfill & ={log}_{2}{\mathbf{I}}_{{N}_{b}}+\sum _{i\in Q}{\theta}_{i}{\mathbf{b}}_{i}{\theta}_{i}^{\ast}{\mathbf{b}}_{i}^{H}+\rho \sum _{i=1}^{N}{{\theta}_{i}}^{2}{\mathbf{b}}_{i}{\mathbf{a}}_{i}^{H}{\mathbf{a}}_{i}{\mathbf{b}}_{i}^{H}\hfill \\ \hfill & +\rho {\mathbf{H}}_{\mathit{ab}}{\mathbf{H}}_{\mathit{ab}}^{H}+\rho \sum _{i=1}^{N}\sum _{j=1,j\ne i}^{N}{\theta}_{i}{\theta}_{j}^{\ast}{\mathbf{b}}_{i}{\mathbf{a}}_{i}^{H}{\mathbf{a}}_{j}{\mathbf{b}}_{j}^{H}\hfill \\ \hfill & +\rho \sum _{i=1}^{N}({\mathbf{H}}_{\mathit{ab}}{\theta}_{i}^{\ast}{\mathbf{a}}_{i}{\mathbf{b}}_{i}^{H}+{\theta}_{i}{\mathbf{b}}_{i}{\mathbf{a}}_{i}^{H}{\mathbf{H}}_{\mathit{ab}}^{H}).\hfill \end{array}$$(13)
The objective function f _{0}(Θ, P _{ a }) can be rewritten as
$$\begin{array}{cc}\hfill {f}_{0}(\mathrm{\Theta},{P}_{a})& ={log}_{2}{\mathbf{A}}_{n}+{{\theta}_{n}}^{2}{\mathbf{B}}_{n}+{\theta}_{n}{\mathbf{C}}_{n}+{\theta}_{n}^{\ast}{\mathbf{C}}_{n}^{H}\hfill \\ \hfill & ={log}_{2}{\mathbf{A}}_{n}+{f}_{1}(\mathrm{\Theta},{P}_{a}),\hfill \end{array}$$(14)
where A _{ n }, B _{ n }, C _{ n } are obtained by some transformation.
Since A _{ n } is an invertible matrix satisfying rank(A _{ n })=N _{ b }. Moreover, log_{2}(A _{ n }) is a constant, and f _{1}(Θ, P _{ a }) is given by
$$\begin{array}{cc}\hfill {f}_{1}(\mathrm{\Theta},{P}_{a})=& {log}_{2}{\mathbf{I}}_{{N}_{b}}+{{\theta}_{n}}^{2}{\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n}+{\theta}_{n}{\mathbf{A}}_{n}^{1}{\mathbf{C}}_{n}+{\theta}_{n}^{\ast}{\mathbf{A}}_{n}^{1}{\mathbf{C}}_{n}^{H}.\hfill \end{array}$$(15)
Similarly, the relationship between the transmit power of the relay and Ψ can be determined as
$$\begin{array}{cc}\hfill {P}_{r}& =Tr(\mathrm{\Psi}({\mathbf{H}}_{\mathit{rb}}{\mathbf{H}}_{\mathit{rb}}^{H}{P}_{a}+{\sigma}_{b}^{2}){\mathrm{\Psi}}^{H})\hfill \\ \hfill & ={P}_{a}\sum _{n\in \mathbb{Q}}{\psi}_{n}{}^{2}{\mathbf{b}}_{n}{}^{2}+{\sigma}_{b}^{2}\sum _{n\in \mathbb{Q}}{{\psi}_{n}}^{2}\hfill \\ \hfill & =\sum _{n\in \mathbb{Q}}{\psi}_{n}{}^{2}[{P}_{a}{\mathbf{b}}_{n}{}^{2}+{\sigma}_{b}^{2}].\hfill \end{array}$$(16)
Denote ${\stackrel{~}{P}}_{r}$ = $\sum _{i\in \mathbb{Q},i\ne n}{\psi}_{i}{}^{2}[{\sigma}_{b}^{2}+{P}_{a}{\mathbf{b}}_{n}{}^{2}]$, which is a constant due to that the variables ∑_{ i ∈ ℚ, i ≠ n } ψ _{ i } are fixed. Therefore, (15) can be rewritten as
$$\begin{array}{cc}\hfill {P}_{r}& =\sum _{n\in \mathbb{Q}}{\psi}_{n}{}^{2}[{P}_{a}{\mathbf{b}}_{n}{}^{2}+{\sigma}_{b}^{2}]+{\stackrel{~}{P}}_{r}\hfill \\ \hfill & =\sum _{n\in \mathbb{Q}}{\beta}_{n}{}^{2}[{P}_{a}{\mathbf{b}}_{n}{}^{2}+{\sigma}_{b}^{2}]+{\stackrel{~}{P}}_{r}.\hfill \end{array}$$(17)
Here, we notice that ${\psi}_{n}{}^{2}={{\beta}_{n}}^{2}$ for n ∈ ℚ.
Following the above, the problem of updating Θ, denoted by (P2), is given by
$$\begin{array}{cc}\hfill (\mathrm{P}2):& \underset{\mathrm{\Theta}}{\mathit{max}}\phantom{\rule{3.33333pt}{0ex}}{f}_{1}(\mathrm{\Theta},{P}_{a})\hfill \\ \hfill & \mathbf{s}.t.\phantom{\rule{3.33333pt}{0ex}}{\beta}_{n}=1,\hfill & \hfill for\phantom{\rule{3.33333pt}{0ex}}n\notin \mathbb{Q},\\ \hfill & \phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\beta}_{n}{}^{2}\le \frac{{P}_{r}^{\mathrm{max}}{\stackrel{~}{P}}_{r}}{[{\sigma}_{b}^{2}+{P}_{a}{\mathbf{b}}_{n}{}^{2}]},\hfill & \hfill for\phantom{\rule{3.33333pt}{0ex}}n\in \mathbb{Q}.\end{array}$$(18)
In order to efficiently determine the optimal closedform solution to (P2), the objective function f _{1}(Θ, P _{ a }) can be rewritten as
$$\begin{array}{cc}\hfill {f}_{1}(\mathrm{\Theta},{P}_{a})& ={log}_{2}{\mathbf{D}}_{n}+{\theta}_{n}{\mathbf{A}}_{n}^{1}{\mathbf{C}}_{n}+{\theta}_{n}^{\ast}{\mathbf{A}}_{n}^{1}{\mathbf{C}}_{n}^{H}\hfill \\ \hfill & ={log}_{2}{\mathbf{D}}_{n}+lo{g}_{2}{\mathbf{I}}_{{N}_{b}}+{\theta}_{n}{\mathbf{E}}_{n}^{1}{\mathbf{C}}_{n}+{\theta}_{n}^{\ast}{\mathbf{E}}_{n}^{1}{\mathbf{C}}_{n}^{H},\hfill \end{array}$$(19)
where D _{ n } = ${\mathbf{I}}_{{N}_{b}}+{{\theta}_{n}}^{2}{\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n}$ and E _{ n } = A _{ n } D _{ n }.
We next analyse the objective function f _{1}(Θ) by considering the first term in (18), i.e., log_{2}D _{ n }. Specifically, for D _{ n }, we note that $\mathrm{rank}({\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n})$ < rank(B _{ n })=1. Moreover, the probability of $\mathrm{rank}({\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n})$ is close to zero (it only happens when ${\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n}=0$). Thus, we have $\mathrm{rank}({\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n})=1$. Similarly, we find that ${\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n}$ is not diagonalizable when $\mathrm{rank}({\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n})=0$, which usually rarely happens. Based on this, we have $({\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n})\ne 0$ with a high probability and ${\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n}$ is diagonalizable. Hence, we can rewrite ${\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n}={\mathbf{W}}_{n}{\mathrm{\Sigma}}_{n}{\mathbf{W}}_{n}^{1}$ based on EVD, where Σ _{ n } = diag{ι _{ n },0,…,0}, ι _{ n } is the only nonzero eigenvalue of $({\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n})$. Finally, since both A _{ n } and B _{ n } are positive semidefinite, ι _{ n } is nonnegative and real. Thus, we have
$$\begin{array}{c}\hfill {log}_{2}{\mathbf{D}}_{n}={log}_{2}\left1+{\theta}_{n}{}^{2}{\iota}_{n}\right,\end{array}$$(20)
where ι _{ n } is the only nonzero eigenvalue of ${\mathbf{A}}_{n}^{1}{\mathbf{B}}_{n}$.
We are now focusing on the second term of (18). Thus, we have ${\mathbf{E}}_{n}^{1}{\mathbf{C}}_{n}={\mathbf{T}}_{n}{\mathrm{\Gamma}}_{n}{\mathbf{T}}_{n}^{1}$ based on the EVD, where T _{ n } ∈ ℂ^{ N b × N b }, Γ _{ n } = diag{λ _{ n },0,…,0}, λ _{ n } is the sole nonzero eigenvalue of ${\mathbf{E}}_{n}^{1}{\mathbf{C}}_{n}$. Let ${\mathbf{V}}_{n}={\mathbf{T}}_{n}{\mathbf{A}}_{n}{\mathbf{T}}_{n}^{1}$, and v _{ n } denote first element of the first column of ${\mathbf{V}}_{n}^{1}$ and ${v}_{n}\prime $ denote first element of the first row of V _{ n }. Note that it follows that $v{\prime}_{n}{v}_{n}=1$. So, according to the [40], we can write
$$\begin{array}{cc}\hfill {log}_{2}& {\mathbf{I}}_{{N}_{b}}+{\theta}_{n}{\mathbf{E}}_{n}^{1}{\mathbf{C}}_{n}+{\theta}_{n}^{\ast}{\mathbf{E}}_{n}^{1}{\mathbf{C}}_{n}^{H}={log}_{2}(1+{\theta}_{n}{}^{2}{\lambda}_{n}{}^{2}+2\left\}\right\}R({\theta}_{n}{\lambda}_{n}){v}_{n}^{{}^{\prime}}{v}_{n}{\lambda}_{n}{}^{2}),\hfill \end{array}$$(21)
where ℝ denotes the real part of a complex number. We note that the additional coefficient θ _{ n }^{2} is related to the active relay elements in HRRIS, which does not exist in traditional RIS.
In summary, based on (19) and (20), we have
$$\begin{array}{cc}\hfill {f}_{1}(& \mathrm{\Theta},{P}_{a})={log}_{2}(1+{\theta}_{n}{}^{2}{\iota}_{n})+{log}_{2}(1+{\theta}_{n}{}^{2}{\lambda}_{n}{}^{2}+2\left\}\right\}R({\theta}_{n}{\lambda}_{n}){v}_{n}^{{}^{\prime}}{v}_{n}{\lambda}_{n}{}^{2}).\hfill \end{array}$$(22)
Hence, according to (21) we have ${\mu}_{n}^{\ast}=\mathrm{arg}({\lambda}_{n})$. So the optimal solution of the problem (P2) is given by
$$\begin{array}{c}\hfill {\theta}_{n}^{\ast}=\{\begin{array}{cc}{\beta}_{n}{e}^{j\mathrm{arg}({\lambda}_{n})},\hfill & n\in \mathbb{Q},\hfill \\ {e}^{j\mathrm{arg}({\lambda}_{n})},\hfill & n\notin \mathbb{Q}.\hfill \end{array}\end{array}$$(23)
In the HRRIS, ℚ is available to determine {β _{ n }}_{ n ∈ ℚ}. Therefore, from (P2), we obtain
$$\begin{array}{c}\hfill {\beta}_{n}=\sqrt{\frac{{P}_{r}^{\mathrm{max}}{\stackrel{~}{P}}_{r}}{[{\sigma}_{b}^{2}+{P}_{a}{\mathbf{b}}_{n}{}^{2}]}},\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}n\in \mathbb{Q}.\end{array}$$(24)
As a result, the optimal solution to (P2) is given as
$$\begin{array}{c}\hfill {\theta}_{n}^{\ast}=\{\begin{array}{cc}\sqrt{\frac{{P}_{r}^{\mathrm{max}}{\stackrel{~}{P}}_{r}}{[{\sigma}_{b}^{2}+{P}_{a}{\mathbf{b}}_{n}{}^{2}]}}{e}^{j\mathrm{arg}({\lambda}_{n})},\hfill & n\in \mathbb{Q},\hfill \\ {e}^{j\mathrm{arg}({\lambda}_{n})},\hfill & n\notin \mathbb{Q}.\hfill \end{array}\end{array}$$(25)
Substituting ${\stackrel{~}{P}}_{r}=\sum _{i\in \mathbb{Q},i\ne n}{\beta}_{i}{}^{2}[{\sigma}_{b}^{2}+{P}_{a}{\mathbf{b}}_{n}{}^{2}],n\in \mathbb{Q}$ to (23), it is observed that a larger K results in a smaller β _{ n }. Therefore, increasing the number of active elements (i.e., K) does not always guarantee the covert rate improvement of HRRIS over traditional IRS. In particular, with a limited power budget ${P}_{r}^{\mathrm{max}}$, the HRRIS can have β _{ n }< 1, which attenuates the signal and degrades the covert rate. In this case, HRRIS with a smaller K is more likely to attain a covert rate gain than those with more active elements. This conclusion will be further demonstrated numerically.
For a given Θ, we proved that the KL divergence adopted in the covertness constraint is a monotonically increasing function of the transmit power at Alice. Based on this, we can find the optimal transmit power of Alice ${P}_{a}^{\ast}$ by solving 𝒟_{01} = 2ϵ ^{2}. According to ((11)d), ${P}_{a}^{\ast}=min({P}_{a},{P}_{a}^{\mathrm{max}})$ is the global optimal solution. Take ${P}_{a}^{\ast}$ into (23) to get the amplitude coefficient of the optimized HRRIS.
The details of the process are shown in Algorithm 1.
According to (16), the amplitude of the active element will increase with its power, which increases the received power at Bob and Willie. Besides, from (8), when the covertness constraint satisfies 𝒟_{01} = 2ϵ^{2}, we can infer that the γ_{w} at Willie is a fixed value, therefore the transmit power should be decreased. In addition, the active element will amplify the noise at Willie and reduce γ_{w}, therefore, the transmit power will increase. Therefore, the increase in active power will cause a change in the transmit power.
The optimal number of active elements K can be solved by numerical search, which maximizes the objective function. From (16), for a given ${P}_{r}^{\mathrm{max}}$, the optimal number of active elements K can be solved by a onedimensional bisection search over the interval [1, N − 1], where 1 and N − 1 are the lower and upper bounds of the search interval. Given the accuracy 1, N − 1 represents the number of elements that need to be compared by the bisection method, which means that the maximum iteration number is ($lo{g}_{2}(N1)$).
1: Input: H_{ar}, H_{rb}, H_{ab}, H_{rw}, H_{aw}, ℚ.
2: Output: {θ^{*}_{1}, θ^{*}_{2}, …, θ^{*}_{N}}, P^{*}_{a}.
3: Randomly generate a matrix θ_{n} where β_{n} = 1, n ∉ ℚ, and ∑_{n∈ℚ}β_{n}^{2}[P_{a}b_{n}^{2} + σ^{2}_{b}].
4: while The objective function does not converge ^{do}
5: for n = 1 → N do
6: Compute A_{n}, B_{n}, and C_{n}.
7: D_{n} = I_{Nb} + θ_{n}^{2} A_{n}^{−1} B_{n}, E_{n} = A_{n}D_{n}.
8: Find λ_{n} as the sole nonzero eigenvalue of E_{n}^{−1}C_{n}.
9: Update θ^{*}_{N} as (24).
10: end for
11: end while
12: Find the optimal P^{*}_{a} based on 𝒟(ℙ_{0}ℙ_{1}) = 2∈^{2}.
13: Update β_{n} as (23).
3.2.4. Convergence and complexity of the proposed algorithm
From Algorithm 1, we aim to find the unique nonzero eigenvalue of the matrix ${\mathbf{E}}_{n}^{1}{\mathbf{C}}_{n}$. In each iteration, we find the unique nonzero eigenvalue of ${\mathbf{E}}_{n}^{1}{\mathbf{C}}_{n}$ as a reflection coefficient of each HRRIS element, so we can ensure that the objective function f(Θ, P_{a}) is nondecreasing, defined as
$$\begin{array}{c}\hfill f({\mathrm{\Theta}}^{(t)},{P}_{a}^{(t)})\le f({\mathrm{\Theta}}^{(t)},{P}_{a}^{(t+1)})\le f({\mathrm{\Theta}}^{(t+1)},{P}_{a}^{(t+1)}),\end{array}$$(26)
where Θ^{(t + 1)} and ${P}_{a}^{(t+1)}$ are the optimal solutions of HRRIS and the optimal solutions of transmit power at the Alice. Therefore, the proposed AO algorithm monotonically converges to the local optimum of (P1) [41].
3.2.5. The challenge to design a covert communication system by using the HRRIS
In relayassisted covert communication, it is generally only designed the relay beamforming vector aligns to Bob. In HRRISassisted covert communication, the phase and amplitude of each element should be optimized, and the constant modulus constraint of passive elements is nonconvex, improving the complexity of optimization. Compared with passive RIS, the active elements of HRRIS will amplify the noise, which increases the design challenge of realizing covert transmission.
3.2.6. Numerical results
In Fig. 3, we plot the covert rate of Bob versus the total reflection elements N of the HRRIS under different covert constraints ϵ. In this figure, we first observe that Bob’s covert rate increases as N increases. We also note that for a given N, Bob’s covert rate decreases as the ϵ decreases due to the fact that the covert constraint gets tighter. As expected, the figure illustrates that the use of active elements (i,e., K = 3) can achieve significant performance improvement with a higher covert rate at Bob compared with the traditional RIS, which demonstrates the benefits of introducing active elements to RIS in covert communications.
Figure 3. Covert rate versus the total number of reflecting elements at HRRIS for different values of the covertness level ϵ, where K = 3, ${P}_{r}^{\mathrm{max}}=30$ dBm 
In Fig. 4, we investigate the covert rate versus the number of active elements K with different covert constraints. In this figure, it can be seen that Bob’s covert transmission rate increases as K increases. For a small K, the HRRIS performs far better than the RIS, even with a limited power budget. However, increasing K does not guarantee the covert rate improvement, especially for low ${P}_{r}^{\mathrm{max}}$, which is explained in Remark 1. As a result, when the maximum covert rate is obtained, we will not continue to increase the number of active elements K. Therefore, we can get the optimal number of active elements from this figure. Based on this, we can conclude that a small amount of active elements (e.g., 5) is sufficient for HRRIS to achieve a significant improvement in terms of the covert rate when compared to conventional IRS/RISaided covert communication schemes.
Figure 4. Covert rate versus the number of the active elements at the HRRIS K for different values of the covertness level ϵ, where N = 64, ${P}_{r}^{\mathrm{max}}=30$ dBm 
In Fig. 5, we investigate the covert rate versus the different distance between Alice and the IRS/HRRIS X_{r}. For ease of comparison between different schemes, IRS and HRRIS are assumed to be used in the same location. It can be observed that the optimal HRRIS position to achieve the maximum covert rate is close to Bob (i.e., (50 m, 2 m)), which is because the path loss from HRRIS to Bob is small. Likewise, this figure shows that the HRRIS system with active components achieves a higher covert rate when compared to the traditional RIS scheme.
Figure 5. Covert rate versus different distance between Alice and the IRS/HRRIS, where N = 64, K = 2, and ${P}_{r}^{\mathrm{max}}=30$ dBm 
In Fig. 6, we plot the transmit power versus the active elements’ power with different covertness constrains. From the figure, when ${P}_{r}^{\mathrm{max}}$ increases, the transmit power P_{a} will increase and approaches a limit value, which is explained in Remark 2. Therefore, employing active elements can use a higher transmit power to improve covert communication.
Figure 6. Transmit power versus the power of the active relay elements of the covertness level ϵ, where K = 3 
In Fig. 7, we plot the covert rate versus the number of total elements with different schemes, where ϵ = 0.01, K = 3. From the figure, we can observe the active RIS is significantly better than the passive RIS, meanwhile, the performance of the proposed scheme is superior to that of the other two cases, which validates the effectiveness of our algorithm.
Figure 7. Covert rate versus different scheme with ϵ = 0.01, K = 3 ${P}_{r}^{\mathrm{max}}=30$ dBm 
4. Conclusions
In this article, we first presented typical applications of relays in covert communication. Then, we proposed a covert communication scheme with the help of HRRIS, where several elements are active elements, and the remaining ones as passive reflecting elements. We used KL divergence to represent Willie’s detection performance, based on which, we obtained the optimal value of Alice’s transmit power. Then, we used an alternating optimization algorithm to obtain the optimal reflection coefficients of HRRIS to improve covert communication performance. The numerical results demonstrated that the proposed scheme significantly outperforms the conventional IRS/RISaided covert communication schemes in terms of covert rate by using a small number of active elements.
Conflict of Interest
The author declare no conflict of interest.
Data Availability
No data are associated with this article.
Authors’ Contributions
Jinsong Hu mainly surveyed relayassisted covert communication. Xiaoqiang Shi mainly surveyed covert communication assisted by hybrid relayreflecting intelligent surfaces. Youjia Chen mainly surveyed the future challenges of HRRISassisted covert communication and improved the readability of the paper by grammatical modification and polishing. Tiesong Zhao and Feng Shu designed the whole framework of the paper and the structure of the paper.
Acknowledgments
We thank the anonymous reviewers for their helpful comments.
Funding
This work was supported in part by the National Natural Science Foundation of China (Nos. 62001116, 62271150, 62171134, U22A2002 and 62071234), the Natural Science Foundation of Fujian Province under Grant (Nos.2020J05106 and 2022J01081), the Major Science and Technology plan of Hainan Province under Grant ZDKJ2021022, and the Scientific Research Fund Project of Hainan University under Grant KYQD(ZR)21008.
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Jinsong Hu received his B.S. degree and Ph.D. degree from the School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, China in 2013 and 2018, respectively. From 2017 to 2018, he was a Visiting Ph.D. Student at the Research School of Engineering, Australian National University, Canberra, ACT, Australia. He is an associate professor at the College of Physics and Information Engineering, Fuzhou University, Fuzhou, China. He served as a TPC member for the IEEE ICC2020/2019. His research interests include array signal processing, covert communications, and physical layer security.
Xiaoqiang Shi received the B.S. degree from the School of Physics and Electronic Information Engineering, Minjiang University, Fuzhou, China, in 2020. He is pursuing his M.S. degree in College of Physics and Information Engineering, Fuzhou University, Fuzhou, China. His research interests include covert communication and intelligent reecting surface.
Youjia Chen received the B.S. and M.S. degrees in communication engineering from Nanjing University, Nanjing, China, and Ph.D. degree in wireless engineering from the University of Sydney, Australia, in 2005, 2008 and 2017, respectively. Currently, she is a professor at the College of Physics and Information Engineering, Fuzhou University, China. She has published over 30 research papers in leading international journals and conference, and contributed to a WileyIEEE Press book. Her current research interests include wireless caching/computing, intelligent reecting surfaces, internet of video things, and wireless AI.
Tiesong Zhao received the B.S. degree in electrical engineering from the University of Science and Technology of China, Hefei, China, in 2006, and the Ph.D. degree in computer science from the City University of Hong Kong, Hong Kong, in 2011. He served as a Research Associate with the Department of Computer Science, City University of Hong Kong, from 2011 to 2012; a Postdoctoral Fellow with the Department of Electrical and Computer Engineering, University of Waterloo, from 2012 to 2013; and a Research Scientist with the Ubiquitous Multimedia Laboratory, The State University of New York at Buffalo, from 2014 to 2015. He is currently a Minjiang Distinguished Professor with the College of Physics and Information Engineering, Fuzhou University, China. He also works with the coding, quality assessment, and transmission. Due to his contributions in video coding and transmission, he received the Fujian Science and Multimedia Signal Processing Studio, Peng Cheng Laboratory, Shenzhen, China. His research interests include multimedia signal processing. Technology Award for Young Scholars in 2017. He has also been serving as an Associate Editor of IET Electronics Letters since 2019.
Feng Shu received the B.S. degree from Fuyang Teaching University, Fuyang, China, in 1994, the M.S. degree from Xidian University, Xi'an, China, in 1997, and the Ph.D. degree from Southeast University, Nanjing, China, in 2002. Since November 2020, he has been with the School of Information and Communication Engineering, Hainan University, where he is currently a Professor of the third level, and also a Supervisor of the Ph.D. students. From 2009 to 2010, he was a PostDoctoral Researcher with The University of Texas at Dallas. He has published about 300 articles, of which over 200 are in archival journals, including more than 100 articles in the IEEE journals and more than 150 SCIindexed articles. His research interests include wireless networks and array signal processing. He is currently recognized as a Plan of Leading Talents in Hainan Province. He was also the Mingjiang Chair Professor and a HundredsofTalented Plan in Fujian Province. He is currently an Associate Editor of IEEE Systems Journal, IEEE Wireless Communications Letters, and IEEE Access.
All Tables
All Figures
Figure 1. Typical applications of relay network in covert communication systems: a) Greedy relay with blockage; b) Relay with different duplex mode; c) Relay networks with relay selection ; d) MultiHop against UAV surveillance; e) MultiHop against multiple collaborating Willies 

In the text 
Figure 2. HRRIS aided covert communication 

In the text 
Figure 3. Covert rate versus the total number of reflecting elements at HRRIS for different values of the covertness level ϵ, where K = 3, ${P}_{r}^{\mathrm{max}}=30$ dBm 

In the text 
Figure 4. Covert rate versus the number of the active elements at the HRRIS K for different values of the covertness level ϵ, where N = 64, ${P}_{r}^{\mathrm{max}}=30$ dBm 

In the text 
Figure 5. Covert rate versus different distance between Alice and the IRS/HRRIS, where N = 64, K = 2, and ${P}_{r}^{\mathrm{max}}=30$ dBm 

In the text 
Figure 6. Transmit power versus the power of the active relay elements of the covertness level ϵ, where K = 3 

In the text 
Figure 7. Covert rate versus different scheme with ϵ = 0.01, K = 3 ${P}_{r}^{\mathrm{max}}=30$ dBm 

In the text 
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