Covert Communication Achieved by A Greedy Relay in Wireless Networks

# Covert Communication Achieved by A Greedy Relay in Wireless Networks

Jinsong Hu,  Shihao Yan,  Xiangyun Zhou,
Feng Shu,  Jun Li,  and Jiangzhou Wang,
J. Hu, F. Shu, and J. Li are with the School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, China. F. Shu is also with National Key Laboratory of Electromagnetic Environment, China Research Institute of Radiowave Propagation, China, and with National Mobile Communications Research Laboratory, Southeast University, Nanjing, China (emails: {jinsong_hu, shufeng, jun.li}@njust.edu.cn). S. Yan and X. Zhou are with the Research School of Engineering, Australian National University, Canberra, ACT, Australia (emails: {shihao.yan, xiangyun.zhou}@anu.edu.au). J. Wang is with the School of Engineering and Digital Arts, University of Kent, Canterbury CT2 7NT, U.K. (email: j.z.wang@kent.ac.uk).This work was supported in part by the National Natural Science Foundation of China (Nos. 61472190 and 61501238), the Australian Research Council’s Discovery Projects (DP150103905), the open research fund of National Mobile Communications Research Laboratory, Southeast University, China (Nos. 2013D02 and 2017D04), the Jiangsu Provincial Science Foundation Project (BK20150786), the Specially Appointed Professor Program in Jiangsu Province, 2015, and the Fundamental Research Funds for the Central Universities (No. 30916011205). Part of this work will be presented at IEEE Global Communication Conference (GLOBECOM 2017), Singapore, Dec. 2017 [1].
###### Abstract

Covert communication aims to hide the very existence of wireless transmissions in order to guarantee a strong security in wireless networks. In this work, we examine the possibility and achievable performance of covert communication in one-way relay networks. Specifically, the relay is greedy and opportunistically transmits its own information to the destination covertly on top of forwarding the source’s message, while the source tries to detect this covert transmission to discover the illegitimate usage of the resource (e.g., power, spectrum) allocated only for the purpose of forwarding source’s information. We propose two strategies for the relay to transmit its covert information, namely fixed-rate and fixed-power transmission schemes, for which the source’s detection limits are analysed in terms of the false alarm and miss detection rates and the achievable effective covert rates from the relay to destination are derived. Our examination determines the conditions under which the fixed-rate transmission scheme outperforms the fixed-power transmission scheme, and vice versa, which enables the relay to achieve the maximum effective covert rate. Our analysis indicates that the relay has to forward the source’s message to shield its covert transmission and the effective covert rate increases with its forwarding ability (e.g., its maximum transmit power).

Physical layer security, covert communication, wireless relay networks, detection, transmission schemes.

## I Introduction

### I-a Background and Related Works

Security and privacy are critical in existing and future wireless networks since a large amount of confidential information (e.g., credit card information, physiological information for e-health) is transferred over the open wireless medium [2, 3, 4]. Against this background, conventional cryptography [5, 6] and information-theoretic secrecy technologies [7, 8, 9, 10, 11] have been developed to offer progressively higher levels of security by protecting the content of the message against eavesdropping. However, these technologies cannot mitigate the threat to a user’s security and privacy from discovering the presence of the user or communication. Meanwhile, this strong security (i.e., hiding the wireless transmission) is desired in many application scenarios of wireless communications, such as covert military operations and avoiding to be tracked in vehicular ad hoc networks. As such, the hiding of communication termed covert communication or low probability of detection communication, which aims to shield the very existence of wireless transmissions against a warden to achieve security, has recently drawn significant research interests and is emerging as a cutting-edge technique in the context of wireless communication security[12, 13, 14].

Although spread-spectrum techniques are widely used to achieve covertness in military applications of wireless communications [15], many fundamental problems have not been well addressed. This leads to the fact that when, or the probability that the spread-spectrum techniques fail to hide wireless transmissions is unknown, significantly limiting its application. The fundamental limit of covert communication has been studied under various channel conditions, such as additive white Gaussian noise (AWGN) channel [16], binary symmetric channel [17], discrete memoryless channel [18], and multiple input multiple output (MIMO) AWGN channel [19]. It is proved that bits of information can be transmitted to a legitimate receiver reliably and covertly in channel uses as . This means that the associated covert rate is zero due to . Following these pioneering works on covert communication, a positive rate has been proved to be achievable when the warden has uncertainty on his receiver noise power [20, 21, 22], the warden does not know when the covert communication happens [23], or an uniformed jammer comes in to help [24, 25]. Most recently, [26] has examined the impact of noise uncertainty on covert communication by considering two practical uncertainty models in order to debunk the myth of this impact. In addition, the effect of the imperfect channel state information and finite blocklength (i.e., finite ) on covert communication has been investigated in [27] and [28], respectively.

### I-B Motivation and Our Contributions

The ultimate goal of covert communication is to establish shadow wireless networks [13], in which each hop transmission should be kept covert to enable the end-to-end covert communication, in order to guarantee the “invisibility” of the transmitters. Following the previous works that only focused on covert transmissions in point-to-point communication scenarios, in this work, for the first time, we consider covert communications in the context of one-way relay networks. This is motivated by the scenario where the relay (R) tries to transmit its own information to the destination (D) on top of forwarding the information from the source (S) to D, while S forbids R’s transmission of its own message, since the resource (e.g., power, spectrum) allocated to R by S is dedicated to be solely used on aiding the transmission from S to D. As such, R’s transmission of its own message to D should be kept covert from S, where S acts as the warden trying to detect this covert communication. Our main contributions are summarized below.

• We examine the possibility and achievable performance of covert communications in one-way relay networks. Specifically, we propose two strategies for R to transmit the covert information to D, namely the fixed-rate and fixed-power transmission schemes, in which the transmission rate and transmit power of the covert message are fixed and to be optimized regardless of the channel quality from R to D, respectively. We also identify the necessary conditions that the covert transmission from R to D can possibly occur without being detected by S with probability one and clarify how R hides its covert transmission in forwarding S’s message to D in these two schemes.

• We derive the detection limits at S in terms of the false alarm rate and miss detection rate are in closed-from expressions for the proposed two transmission schemes. Then, we determine the optimal detection threshold at S, which minimizes the detection error and obtain the associated minimum detection error . Our analysis leads to many useful insights. For example, we analytically prove that increases with R’s maximum transmit power, which indicates that boosting the forwarding ability of R also increases its capacity to perform covert transmissions. This demonstrates a tradeoff between the achievable effective covert rate and R’s ability to aid the transmission from S to D.

• We analyze the effective covert rates achieved by these two schemes subject to the covert constraint , where is predetermined to specify the covert constraint. Our analysis indicates that the achievable effective covert rate approaches zero as the transmission rate from S to D approaches zero, which demonstrates that covert transmission from D to R is only feasible with the legitimate transmission from S to D as the shield. Our examination shows that the fixed-rate transmission scheme outperforms the fixed-power transmission scheme under some specific conditions, and vise versa. Our examination enables R to switch between these two schemes in order to achieve a higher effective covert rate.

The rest of this paper is organized as follows. Section II details our system model and adopted assumptions. Section III and IV present the fixed-rate and fixed-power transmission schemes, respectively. Thorough analysis on the performance of these two transmission are provided in these two sections as well. Section V provides numerical results to confirm our analysis and provide useful insights on the impact of some parameters. Section VI draws conclusions.

Notation: Scalar variables are denoted by italic symbols. Vectors is denoted by lower-case boldface symbols. Given a complex number, denotes the modulus. Given a complex vector, denotes the conjugate transpose. denotes expectation operation.

## Ii System Model

### Ii-a Considered Scenario and Adopted Assumptions

As shown in Fig. 1, in this work we consider a one-way relay network, in which S transmits information to D with the aid of R, since a direct link from S to D is not available. As mentioned in the introduction, S allocates some resource to R in order to seek its help to relay the message to D. However, in some scenarios R may intend to use this resource to transmit its own message to D as well, which is forbidden by S and thus should be kept covert from S. As such, in the considered system model S is also the warden to detect whether R transmits its own information to D when it is aiding the transmission from S to D.

We assume the wireless channels within our system model are subject to independent quasi-static Rayleigh fading with equal block length and the channel coefficients are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian random variables with zero-mean and unit-variance. We also assume that each node is equipped with a single antenna. The channel from S to R is denoted by and the channel from R to D is denoted by . We assume R knows both and perfectly, while S only knows and D only knows . Considering channel reciprocity, we assume the channel from R to S (denoted by ) is the same as and thus it is perfectly known by S. We further assume that R operates in the half-duplex mode and thus the transmission from S to D occurs in two phases: phase 1 (S transmits to R) and phase 2 (R transmits to D).

### Ii-B Transmission from Source to Relay (Phase 1)

In phase 1, the received signal at R is given by

 yr[i]=√Pshsrxb[i]+nr[i], (1)

where is the fixed transmit power of source, is the transmitted signal by S satisfying , is the index of each channel use ( is the total number of channel uses in each phase), and is the AWGN at relay with as its variance, i.e., .

In this work, we consider that R operates in the amplify-and-forward mode and thus R will forward a linearly amplified version of the received signal to D in phase 2. As such, the forwarded (transmitted) signal by R is given by

 xr[i]=Gyr[i]=G(√Pshsrxb[i]+nr[i]), (2)

which is a linear scaled version of the received signal by a scalar . In order to guarantee the power constraint at R, the value of G is chosen such that , which leads to .

In this work, we also consider that the transmission rate from S to D is predetermined, which is denoted by . We also consider a maximum power constraint at R, i.e., . As such, although R knows both and perfectly, transmission outage from S to D still incurs when , where is the channel capacity from S to D for . Then, the transmission outage probability is given by , which has been derived in a closed-form expression [29]. We assume that all the nodes in the network do not transmit signals when the outage occurs. In practice, the pair of and determines the specific aid (i.e., the value of ) required by S from R, which relates to the amount of resource allocated to R by S as a payback. In this work, we assume both and are predetermined, which leads to a predetermined .

### Ii-C Transmission Strategies at Relay (Phase 2)

In this subsection, we detail the transmission strategies of R when it does and does not transmit its own information to D. We also determine the condition that R can transmit its own message to D without being detected by S with probability one, in which the probability to guarantee this condition is also derived.

#### Ii-C1 Relay’s Transmission without Covert Message

In the case when relay does not transmit its own message (i.e., covert message) to Bob, it only transmit to D. Accordingly, the received signal at D is given by

 yd[i] =√P0rhrdxr[i]+nd[i] (3) =√P0rGhrd√Pshsrxb[i]+√P0rGhrdnr[i]+nd[i],

where is the transmit power of at R in this case and is the AWGN at D with as its variance, i.e., . Accordingly, the signal-to-noise ratio (SNR) at the destination for is given by

 γd =Ps|hsr|2P0r|hrd|2G2P0r|hrd|2G2σ2r+σ2d=γ1γ2γ1+γ2+1, (4)

where and .

For a predetermined , R does not have to adopt the maximum transmit power for each channel realization in order to guarantee a specific transmission outage probability. When the transmission outage occurs (i.e., occurs), R will not transmit (i.e., ). When , R only has to ensure , where . Then, following (4) the transmit power of R when is given by , where

 μ≜(Ps|hsr|2+σ2r)(22Rsd−1)[Ps|hsr|2−σ2r(22Rsd−1)]. (5)

Noting , we have when . As such, given in (5) is nonnegative. Following (4), we note that requires . As such, the transmit power of R without covert message is given by

 P0r=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩μσ2d|hrd|2,|hrd|2≥μσ2dPmaxr,0,|hrd|2<μσ2dPmaxr. (6)

#### Ii-C2 Relay’s Transmission with Covert Message

In the case when R transmits the covert message to D on top of forwarding , the received signal at D is given by

 yd[i] =√P1rGhrd√Pshsrxb[i]+√PΔhrdxc[i] +√P1rGhrdnr[i]+nd[i]. (7)

where is R’s transmit power of in this case and is R’s transmit power of the covert message satisfying . We note that the covert transmission from R to D should not affect the transmission from S to D. Otherwise, S can easily observe this covert transmission. As such, here we assume D always first decodes with as interference. Following (II-C2), the signal-to-interference-plus-noise ratio (SINR) for is given by

 γd =Ps|hsr|2P1r|hrd|2G2P1r|hrd|2G2σ2r+PΔ|hrd|2+σ2d =γ1γ3γ3+(γ1+1)(γ3PΔ/P1r+1), (8)

where . We will determine based on different transmission strategies of the covert message from R to D.

### Ii-D Decoding of the Covert Message

As discussed above, the covert transmission from R to D should not affect the transmission from S to D. We also note that this covert transmission cannot happen when the transmission outage from S to D occurs. This is, for example, due to the fact that when the transmission outage occurs R will request a retransmission from S, which enables S to detect R’s covert transmission with probability one if the covert transmission happened. Therefore, the covert transmission from R to D only occur when the successful transmission from S to D is guaranteed (i.e., when is successfully decoded at D). As such, when the covert message is transmitted by R, D first decodes and subtracts the corresponding component from its received signal given in (II-C2). Then, the effective received signal used to decode the covert message is given by

 ~yd[i]=√PΔhrdxc[i]+√P1rhrdGnr[i]+nd[i]. (9)

Then, following (9) the SINR for is

 γc=PΔ|hrd|2P1r|hrd|2G2σ2r+σ2d. (10)

### Ii-E Binary Detection at Source and the Covert Constraint

In this subsection, we present the optimal detection strategy adopted by S (i.e., Source).

In phase 2 when R transmits to D, S is to detect whether R transmits the covert message on top of forwarding S’s message to D. R does not transmit in the null hypothesis while it does in the alternative hypothesis . Then, the received signal at S in phase 2 is given by

 ys[i]=⎧⎪⎨⎪⎩√P0rhrsxr[i]+ns[i],                      H0,√P1rhrsxr[i]+√PΔhrsxc[i]+ns[i],   H1. (11)

where is the AWGN at S with as its variance. We note that neither nor is known at S since it does not know , while the statistics of and are known since the distribution of is publicly known. The ultimate goal of S is to detect whether comes from or in one fading block. As proved in [27], the optimal decision rule that minimizes the detection error at S is given by

 TD1≷D0τ, (12)

where , is a predetermined threshold, and are the binary decisions that infer whether R transmits covert message or not, respectively. In this work, we consider infinite blocklength, i.e., . As such, we have

 T={P0r|hrs|2+σ2s,                 H0,P1r|hrs|2+PΔ|hrs|2+σ2s,    H1. (13)

The detection performance of S is normally measured by its detection error, which is defined as

 ξ≜α+β, (14)

where is S’s false alarm rate and is S’s miss detection rate. In covert communications, it is required that , where is predetermined to specify the covert constraint. In practice, it is hard, if not impossible, to know at R since the threshold adopted by S is unknown. In this work, we consider the worst-case scenario where is optimized at S to minimize . As such, the covert constraint considered in this work is , where is the minimum detection error achieved at S.

## Iii Fixed-Rate Transmission Scheme

In this section, we consider the fixed-rate transmission scheme, in which R transmits covert message to D with a constant rate if possible. To this end, R varies its transmit power as per such that is fixed as . Specifically, we first determine R’s transmit power in and then analyze the detection performance at S, based on which we also derive S’s optimal detection threshold. Furthermore, we derive the effective covert rate achieved by the fixed-rate transmission scheme.

### Iii-a Transmit Power at Relay under H1

Following (II-C2) and defining , in order to guarantee under , is given as

 P1r=μ(Q+σ2d)|hrd|2, (15)

which requires that leads to . Considering the maximum power constraint at R (i.e., under this case), R has to give up the transmission of the covert message (i.e., ) when and sets the same as given in (6). This is due to the fact that S knows and it can detect with probability one when the total transmit power of R is greater than . Then, the transmit power of under for the fixed-rate transmission scheme is given by

 P1r= (16)

As per (16), we note that R also not transmit covert message as well when it cannot support the transmission from S to D (i.e., when ). This is due to the fact that a transmission outage occurs when and D will request a retransmission from S, which enables S to detect R’s covert transmission with probability one if the covert transmission happened. In summary, S cannot detect R’s covert transmission with probability one (R could possibly transmit covert message without being detected) only when the condition is guaranteed. We denote this necessary condition for covert communication as . Considering Rayleigh fading, the cumulative distribution function (cdf) of is given by and thus the probability that is guaranteed is given by

 Pc=exp{−μσ2d+μQ+QPmaxr}. (17)

We note that is a monotonically decreasing function of (and thus the covert transmission rate), which indicates that the probability that R can transmit covert message (without being detected with probability one) decreases as increases.

### Iii-B Detection Performance at Source

In this subsection, we derive S’s false alarm rate, i.e., , and miss detection rate, i.e., .

###### Theorem 1

When the condition is guaranteed, for a given , the false alarm and miss detection rates at S are derived as

 α =⎧⎪⎨⎪⎩1,τ<σ2s,1−P−1cκ1(τ),σ2s≤τ≤ρ1,0,τ>ρ1. (18) β =⎧⎪⎨⎪⎩0,τ<σ2s,P−1cκ2(τ),σ2s≤τ≤ρ2,1,τ>ρ2. (19)

where

 ρ1≜Pmaxr|hrs|2μσ2dμσ2d+μQ+Q+σ2s, (20) ρ2≜Pmaxr|hrs|2+σ2s, κ1(τ)≜exp{−μσ2d|hrs|2τ−σ2s}, κ2(τ)≜exp{−(μσ2d+μQ+Q)|hrs|2τ−σ2s}.
###### Proof:

Considering the maximum power constraint at R under (i.e., ) and following (6), (12), and (13), the false alarm rate under the condition is given by

 α=P[μσ2d|hrd|2|hrs|2+σ2s≥τ∣∣C] (21) =⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩1,τ<σ2s,P[μσ2d+μQ+QPmaxr≤|hrd|2≤μσ2d|hrs|2τ−σ2s]Pc,σ2s≤τ≤ρ1,0,τ>ρ1.

Then, substituting into the above equation ( is perfectly known by S and thus it is not a random variable here) we achieve the desired result in (18).

Considering the maximum power constraint at R under (i.e., ) and following (12), (13), and (16), the miss detection rate under the condition is given by

 β=P[μ(Q+σ2d)|hrs|2|hrd|2+Q|hrs|2|hrd|2+σ2s<τ∣∣C] =⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩0,τ<σ2s,P[|hrd|2≥(μσ2d+μQ+Q)|hrs|2τ−σ2s]Pc,σ2s≤τ≤ρ2,1,τ>ρ2. (22)

Then, substituting into (22) we achieve the desired result in (19). \qed

We note that the false alarm and miss detection rates given in Theorem 1 are functions of the threshold and we next examine how S sets the value of it in order to minimize its detection error in the following subsection.

### Iii-C Optimization of the Detection Threshold at Source

In this subsection, we derive the optimal value of the detection threshold that minimizes the detection error for the fixed-rate transmission scheme.

###### Theorem 2

The optimal threshold that minimizes for the fixed-rate transmission scheme is given by

 τ∗=min⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩(μ+1)Q|hrs|2ln(1+(μ+1)Qμσ2d)+σ2s,ρ1⎫⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪⎭, (23)
###### Proof:

Since as given in Theorem 1, following (18) and (19), we have the detection error at S as

 ξ=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩1,τ≤σ2s,1−P−1c[κ1(τ)−κ2(τ)],σ2s<τ≤ρ1,P−1cκ2(τ),ρ1≤τ<ρ2,1,τ≥ρ2. (24)

We first note that is the worst case for S and thus S does not set or . We also note that given in (24) is a continuous function of following Theorem 1. Following (24), we derive the first derivative of with respect to when as

 ∂(ξ)∂τ=P−1c(μσ2d+μQ+Q)|hrs|2(τ−σ2s)2κ2(τ)>0. (25)

This demonstrates that is an increasing function of when . Thus, S will set as the threshold to minimize if . We next derive the first derivative of with respect to for as

 ∂(ξ)∂τ=P−1c|hrs|2(τ−σ2s)2[(μσ2d+μQ+Q)κ2(τ)−μσ2dκ1(τ)] =P−1c(μσ2d+μQ+Q)|hrs|2κ2(τ)(τ−σ2s)2× {1−μσ2dμσ2d+μQ+Qexp{(μ+1)Q|hrs|2τ−σ2s}}. (26)

We note that due to and as given in Theorem 1. As such, without the constraint , the value of that ensures in (III-C) is given by

 τ‡=(μ+1)Q|hrs|2ln(1+(μ+1)Qμσ2d)+σ2s. (27)

From (27), we can see that , which satisfies the constraint . We note that , for , and , for . This is due to the term in (III-C) is monotonically decreasing with respect to . This indicates that minimizes without the constraint . As such, if , the optimal threshold is

 τ∗=τ‡. (28)

If , following (25) and noting is a continuous function of , we can conclude that the optimal threshold is

 τ∗=ρ1. (29)

This completes the proof of Theorem 2. \qed

Following Theorem 2, we obtain the minimum detection error at S in the following corollary.

###### Corollary 1

The minimum value of at S is

 ξ∗= (30)
###### Proof:

Substituting into (24), we obtain the minimum value of as , which completes the proof of Corollary 1. \qed

Base on Theorem 1, Theorem 2, and Corollary 1, we draw the following useful insights.

###### Remark 1

We can conclude that monotonically decreases with . Based on (30), this conclusion is true for . We now prove this conclusion for . To this end, we next prove that in (30) monotonically increases with , since both and in (30) are monotonically increasing functions of . Setting , we have , where

 f3(x)=(1+x)−1/x. (31)

In order to determine the monotonicity of with respect to , we derive its first derivative as

 ∂f3(x)∂x=exp{−ln(1+x)x}(1+x)ln(1+x)−xx2(1+x). (32)

We note that whether or depends on . As such, we derive the first derivative of with respect to as

 ∂g(x)∂x=ln(1+x). (33)

Noting that and , we conclude that monotonically decreases with . Then, we have and thus . This leads to that monotonically increases with and thus monotonically decreases with for .

###### Remark 2

We also conclude that when . This follows from (30) for , since when we have as per (20) and thus (then ). These conclusions indicate that the covert constraint determines an upper bound on , which is denoted by and achieved by solving .

###### Remark 3

The minimum detection error increases with as shown in (30), but when . This later conclusion follows from (30) for , since when we have as per (20) and thus (then ). This result demonstrates that Willie can still possibly detect the covert communication even when R does not have the maximum power constraint, where Willie’s detection performance still depends on other system parameters (i.e., , , and ).

###### Remark 4

We have when or . As , as per (5) we have and thus (then ) following (20). Then, from (30) for we can see that as . As , following (5) again we note that will be negative and thus the transmission from S to D fails, which leads to as discussed in Section III-A. This result means that there exists an optimal value of that maximizes and thus maximizes the effective covert rate for given other system parameters. We will numerically examine the impact of on covert communications in Section V.

### Iii-D Optimization of Effective Covert Rate

In this section, we examine the effective covert rate achieved in the considered system subject to a covert constraint.

#### Iii-D1 Effective Covert Rate

From (10), the SINR of at D in the fixed-rate transmission scheme is given as

 γc =PΔ|hrd|2P1r|hrd|2G2σ2r+σ2d =Qμ(Q+σ2d)|hsr|2|hsr|2G2σ2r+σ2d, =Qμ(Q+σ2d)η|hsr|2+1+σ2d, (34)

where . Then, the covert rate achieved by R is . As such, we can see that the covert rate is fixed when Q is fixed as per (III-D1). We next derive the effective covert rate, i.e., the covert rate averaged over all realizations of , in the following theorem.

###### Theorem 3

The achievable effective covert rate by R in the fixed-rate transmission scheme is derived as a function of given by

 ¯¯¯¯Rc =RcPc =log2⎛⎜ ⎜ ⎜⎝1+Qμ(Q+σ2d)η|hsr|2+1+σ2d⎞⎟ ⎟ ⎟⎠× exp{−μσ2d+μQ+QPmaxr} (35)

Based on Theorem 3, we note that is not an increasing function of and thus , since as increases increases as per (3) while (i.e., the probability that the condition is guaranteed) decreases following (17). This indicates that there may exists an optimal value of (equivalently ) that maximizes the effective covert rate, which motivates our following optimization of in the considered system model.

#### Iii-D2 Maximization of ¯¯¯¯Rc with the Covert Constraint

Following Theorem 2, the optimal value of that maximizes subject to the covert constraint can be obtained through

 Q∗= argmax0≤Q≤Qϵ¯¯¯¯Rc, (36)

where is the upper bound on determined by the covert constraint as discussed in Remark 1. We note that the optimization problem (36) is of one dimension, which can be solved by efficient numerical search. The maximum value of is then achieved by substituting into (3), which is denoted by .

## Iv Fixed-Power Transmission Scheme

In this section, we consider the fixed-power transmission scheme, in which R transmits covert message to D with a constant transmit power if possible. Specifically, we first determine R’s transmit power in and then analyze the detection performance at S, based on which we also derive S’s optimal detection threshold. Furthermore, we derive the effective covert rate achieved by the fixed-power transmission scheme.

### Iv-a Transmit Power at Relay

Following (II-C2), when we have

 P1r=μPΔ+μσ2d|hrd|2. (37)

We note that requires and thus . Considering the maximum power constraint at R (i.e., under this case), R has to give up the transmission of the covert message (i.e., ) when and sets the same as given in (6). This is due to the fact that S knows and it can detect the covert transmission with probability one when the total transmit power of R is greater than . Then, the transmit power of under for the fixed-power transmission scheme is given by

 P1r= (38) ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩μPΔ+μσ2d|hrd|2,|hrd|2≥μσ2dPmaxr−(μ+1)PΔ,μσ2d|hrd|2,μσ2dPmaxr≤|hrd|2<μσ2dPmaxr−(μ+1)PΔ,0,|hrd|2<μσ2dPmaxr.

As per (38), we note that R also does not transmit covert message when it cannot support the transmission from S to D (i.e., when ). This is due to the fact that a transmission outage occurs when and D would request a retransmission from S, which enables S to detect R’s covert transmission with probability one if this covert transmission happened. In summary, S cannot detect R’s covert transmission with probability one (R could possibly transmit covert message without being detected) only when the condition is guaranteed. We again denote this necessary condition for covert communication as . Noting , the probability that is guaranteed is given by

 Pc=exp{−μσ2dPmaxr−(μ+1)PΔ}. (39)

We note that is a monotonically decreasing function of , which indicates that the probability that R can transmit covert message (without being detected with probability one) decreases as increases. Following (37) and noting , we have and thus .

### Iv-B Detection Performance at Source

In this subsection, we derive S’s false alarm rate, i.e., , and miss detection rate, i.e., .

###### Theorem 4

When the condition is guaranteed, for a given , the false alarm and miss detection rates at S are derived as

 α =⎧⎪⎨⎪⎩1,τ<σ2s,