# Securing Large-Scale D2D Networks Using Covert Communication and Friendly
Jamming

Shaohan Feng, , Xiao Lu, ,
Sumei Sun, , Dusit Niyato, ,
and Ekram Hossain S. Feng and S. Sun are with the Institute for Infocomm
Research, Singapore. Email: {fengs<EMAIL_ADDRESS>Lu is with
York University, Toronto, ON M3J 1P3, Canada. Email<EMAIL_ADDRESS>Niyato
is with the School of Computer Science and Engineering, Nanyang Technological
University, Singapore. Email<EMAIL_ADDRESS>Hossain is with the
Department of Electrical and Computer Engineering, University of Manitoba,
Canada. Email<EMAIL_ADDRESS>
###### Abstract

We exploit both covert communication and friendly jamming to propose a
friendly jamming-assisted covert communication and use it to doubly secure a
large-scale device-to-device (D2D) network against eavesdroppers (i.e.,
wardens). The D2D transmitters defend against the wardens by: 1) hiding their
transmissions with enhanced covert communication, and 2) leveraging friendly
jamming to ensure information secrecy even if the D2D transmissions are
detected. We model the combat between the wardens and the D2D network (the
transmitters and the friendly jammers) as a two-stage Stackelberg game.
Therein, the wardens are the followers at the lower stage aiming to minimize
their detection errors, and the D2D network is the leader at the upper stage
aiming to maximize its utility (in terms of link reliability and communication
security) subject to the constraint on communication covertness. We apply
stochastic geometry to model the network spatial configuration so as to
conduct a system-level study. We develop a bi-level optimization algorithm to
search for the equilibrium of the proposed Stackelberg game based on the
successive convex approximation (SCA) method and Rosenbrock method. Numerical
results reveal interesting insights. We observe that without the assistance
from the jammers, it is difficult to achieve covert communication on D2D
transmission. Moreover, we illustrate the advantages of the proposed friendly
jamming-assisted covert communication by comparing it with the information-
theoretical secrecy approach in terms of the secure communication probability
and network utility.

###### Index Terms:

Large-scale D2D network, covert communication, physical-layer security,
friendly jamming, Stackelberg game, successive convex approximation.

##  I Introduction

###  I-A Background and Motivation

Recently, D2D communication has attracted great attention due to its
capability of enabling direct communication among proximal mobile devices
without the involvement of network infrastructure such as a base station (BS)
and the core network [1]. By such, D2D communication has emerged as a
promising solution to the provision of energy-efficient communication with low
end-to-end latency in supporting miscellaneous location-based peer-to-peer
applications [2]. However, the D2D communications are vulnerable to malicious
attacks such as eavesdropping due to the broadcast nature of the wireless
medium. In general, it is challenging to secure D2D communication due to the
following reasons:

  * •

In various application scenarios, the D2D devices are resource-constrained and
hence cannot afford high overhead in computation and communication. In this
case, heavy resource-dependent methods such as the traditional cryptographic
methods may not be feasible [1].

  * •

The D2D networks may not have any centralized support to execute
functionalities such as authentication to achieve confidentiality of the
transmissions.

  * •

Merely achieving a positive rate difference between legitimate and wiretapping
channels is not sufficient from the perspective of privacy protection. For
example, in a military application, it may be required to prevent the enemy
from detecting transmissions so as to hide the presence and activities of the
legitimate army [3].

The above context raises the need for covert communication [4] (also referred
to as undetectable communication), which is an enabler for reliable
transmission from a legitimate transmitter to its intended receiver while
undetectable by a vigilant adversary as shown in Fig. 1(a). Covert
communication has a number of advantages. First, the implementation of covert
communication does not incur high resource consumption, which makes it
suitable for resource-constrained devices and networks. Second, the
implementation of covert communication does not rely on the execution of a
complicated authentication functionality. Third, the covert communication
promises a stronger protection on transmission compared to the information-
theoretical secrecy approach as it can hide the transmission itself and thus
the information carried by this transmission becomes immune to interception.
Also, the performance of covert communication is independent of the
adversary’s competence. Nevertheless, the covert communication cannot hide the
presence of transmission completely and there will be a non-trivial
probability of detection by the adversary. Therefore, the following two
questions arise: 1) how to reduce the detection probability and improve the
communication covertness so as to achieve enhanced covert communication? 2) is
there any countermeasure when the adversary attempts to wiretap the
transmission once it has detected the transmission?

Figure 1: (a) Covert communication, (b) friendly jamming, and (c) friendly
jamming-assisted covert communication including (d) enhanced covert
communication and (e) friendly jamming-enabled countermeasure.

To answer the above questions, we resort to the friendly jamming approach, in
which friendly jammers emit jamming signals to interfere with the reception at
the adversary (Fig. 1(b)). In this way, the jamming signals will increase the
uncertainty in the received signal at the adversary, and thereby, the
detection error of the adversary will increase and the communication
covertness will improve (Fig. 1(d)) [5]. In addition, since the jamming
signals emitted by the jammers will increase the interference power received
at the adversary, even in the case that the transmission has been detected,
the adversary may not be able to decode the received signal successfully. This
will ensure information secrecy. In this way, friendly jamming-assisted covert
communication (Fig. 1(c)) will provide an additional level of protection for
transmissions (Fig. 1(e)).

###  I-B Contributions

We propose a friendly jamming-assisted covert communication method to secure a
large-scale D2D network. As shown in Fig. 2(a), the network under our
consideration involves: 1) a large number of power-controlled D2D transmitters
cooperating with friendly jammers (which emit jamming signals), 2) their
dedicated D2D receivers, and 3) spatially distributed adversaries, namely,
wardens. The D2D transmitters randomly transmit information to their dedicated
D2D receivers. The wardens aim to detect the D2D transmissions by following a
threshold-based rule on their received signal powers, and once detect, they
will wiretap the D2D transmissions. In this case, the decision-making of the
wardens is after that of the D2D transmitters, which can be captured by using
the framework of a two-stage Stackelberg game with the wardens as the
followers at the lower stage and the D2D transmitters together with the
friendly jammers as the leaders at the upper stage. The main contributions of
this paper can be summarized as follows.

  * •

We propose a friendly jamming-assisted covert communication approach to secure
a large-scale D2D network. In this network, the friendly jammers are deployed
to cooperate with the D2D network so as to jointly enhance the communication
covertness and information secrecy for the D2D transmissions.

  * •

We model the combat between the wardens and the D2D network (i.e., the D2D
transmitters and the jammers) by a two-stage Stackelberg game. At the lower
stage, the wardens aim to minimize their own detection errors. At the upper
stage, the D2D network jointly determines the D2D transmissions power and
jamming power in order to maximize network utility subject to the constraint
on communication covertness.

  * •

We use stochastic geometry to model the spatial configuration of the large-
scale D2D network, namely, the locations of the D2D transmitters, the jammers,
and the wardens, and characterize their performance metrics in order to
quantitatively analyze the system-level network performance.

  * •

To obtain the equilibrium of our two-stage Stackelberg game, we first analyze
the characterizes of the lower-stage problem for the wardens and apply the
Rosenbrock method to solve it. The solution is regarded as the best response
from the lower stage. Given this best response, we develop a bi-level
optimization algorithm based on SCA method to obtain the optimal strategy for
the D2D network.

  * •

We present simulation results to verify the optimality of the obtained
strategy. We also evaluate and discuss the performance of our proposed
approach and demonstrate its advantages by comparing it with the conventional
information-theoretical secrecy approach.

###  I-C Organization

The rest of the paper is organized as follows. In Section II, we present the
related work. In Section III, we present the system model and the derivation
of the performance metrics. The problem statement and the game formulation are
presented in Section IV. In Section V, we develop a bi-level algorithm to
solve the problem. The numerical results and discussion are presented in
Section VI, which is followed by the conclusions in Section VII. The symbols
used in this paper together with their descriptions and default values are
given in Table I.

Table I: Symbols and descriptions Symbol | Description | Default value
---|---|---
$d$, $\left.w\right|_{d}$, $j$ |  Indexes of the typical D2D transmitter, the nearest warden of D2D transmitter $d$, and jammer, respectively. | —
$\cal{D}$, $\cal{W}$, $\cal{J}$ | Sets of the D2D transmitters, wardens, and jammers, respectively. | —
$\Phi_{\cal{D}}$, $\Phi_{\cal{W}}$, $\Phi_{\cal{J}}$, $\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|d\right\\}}$, $\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|\left.w\right|_{d}\right\\}}$, $\Phi_{\left\\{\left.{\cal{J}}\right|d\right\\}}$, $\Phi_{\left\\{\left.{\cal{J}}\right|\left.w\right|_{d}\right\\}}$ |  PP of D2D transmitters, PP of wardens, PP of jammers, PP of the D2D transmitters that excludes D2D transmitter $d$ with D2D transmitter $d$ as the observation point, PP of the D2D transmitters that excludes D2D transmitter $d$ with warden $\left.w\right|_{d}$ as the observation point, PP of the jammers with D2D transmitter $d$ as the observation point, and PP of the jammers with warden $\left.w\right|_{d}$ as the observation point, respectively. | —
$\lambda_{{\cal{D}}}$, $\lambda_{\cal{W}}$, $\lambda_{\cal{J}}$ | Densities of $\Phi_{{\cal{D}}}$, $\Phi_{\cal{W}}$, and $\Phi_{\cal{J}}$, respectively. |  $0.1/{\rm{m}}^{2}$, $0.01/{\rm{m}}^{2}$, $0.1/{\rm{m}}^{2}$.
${\cal{D}}_{0}$, ${\cal{D}}_{1}$ | Events that D2D transmitter is inactive and active, respectively. | —
${\mathbb{P}}^{{\cal{D}}_{0}}$, ${\mathbb{P}}^{{\cal{D}}_{1}}$ |  Probabilities of events ${\cal{D}}_{0}$ and ${\cal{D}}_{1}$, respectively. |  $0.5$, $0.5$.
$p^{\rm{D}}$, $p^{\rm{J}}$, $\tau$ |  D2D transmission power, jamming power, and power detection threshold, respectively. |  $\left[0,30\right]{\rm{dBm}}$, $\left[0,30\right]{\rm{dBm}}$ [6], —.
$g_{d^{\prime}d}$, $r_{d^{\prime}d}$, $R$ |  Channel and distance between D2D transmitter $d^{\prime}$ and the typical D2D receiver, and distance between D2D transmitter and its dedicated D2D receiver, respectively. |  $\exp\left(1\right)$ [7], —, $1{\rm{m}}$ [8].
$\alpha$, $\varepsilon$, $\xi^{\rm{D}}$, $\xi^{\rm{W}}$ |  Path-loss exponent, threshold of detection error, SINR threshold at D2D receiver, and SINR threshold at warden, respectively. |  $4$, $0.01$, $-10{\rm{dB}}$, $-10{\rm{dB}}$ [9].
$N_{d}$, $N_{\left.w\right|_{d}}$ | Noises at the typical D2D receiver and warden $\left.w\right|_{d}$, respectively. |  $-90{\rm{dBm}}$, $-90{\rm{dBm}}$ [6]
$S^{{\cal{D}}_{1}}_{d}$, $S^{{\cal{J}}}_{d}$, $S^{{\cal{D}}_{1}}_{\left.w\right|_{d}}$, $S^{{\cal{J}}}_{\left.w\right|_{d}}$ |  Sum of signal powers from D2D transmitter $d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|d\right\\}}$ received at the typical D2D receiver, sum of signal powers from jammer $j\in\Phi_{\left\\{\left.{\cal{J}}\right|d\right\\}}$ received at the typical D2D receiver, sum of signal powers from D2D transmitter $d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|\left.w\right|_{d}\right\\}}$ received at warden $\left.w\right|_{d}$, and sum of signal powers from jammer $j\in\Phi_{\left\\{\left.{\cal{J}}\right|\left.w\right|_{d}\right\\}}$ received at warden $\left.w\right|_{d}$, respectively. | —
$w^{\rm{D}}$, $w^{\rm{J}}$ |  Reward of stabilizing D2D link conditioned on secure communication and cost of unit jamming power, respectively. |  $1$, $1$
$f_{a}$, $F_{a}$ | PDF and CDF of a random variable $a$, respectively. | —
${\mathbb{P}}$, ${\mathbb{F}}$ | Probabilistic and expectation operators, respectively. | —
${\mathcal{L}}$, ${\mathcal{L}}^{-1}$ | Laplace and inverse Laplace transforms, respectively. | —

##  II Related Work

###  II-A Covert Communication

Covert communication promises reliable while undetectable transmission from
the legitimate transmitter to its intended receiver, which thereby completely
secures the legitimate transmission [10]. Covert communication was adopted to
secure a large-scale wireless network in [11], where the uncertainty caused by
the co-channel interference has been leveraged to enhance the communication
covertness. Similar idea was adopted in [12], [13], and [4]. In [12], a novel
backscatter communication was proposed, where an original message is divided
into two parts (i.e., active message and backscatter message). The backscatter
message is transmitted by backscattering the active signal (i.e., the signal
for transmitting the active message) via ambient backscatter tag. Therein, the
active signal can be used to shelter the presence of the backscatter signal
(i.e., the signal for transmitting the backscatter message), which thereby can
be used to carry secret information. The authors in [13] studied an Internet
of Things (IoT) network and used the overt channel as the “spectrum shelter”
of the covert channel. The interference from the overt channel was leveraged
to hide the presence of the IoT transmission so as to enhance the network
security. A large-scale IoT system involving massive IoT gateways (GWs) and
IoT devices was secured in [4] against the adversaries from transmission
detection. In the system, the in-band full-duplex (IBFD) IoT GWs emit
artificial noise, which together with the co-channel interference from the IoT
devices can mislead the adversaries in making their decisions. Different from
the aforementioned works that leverage the uncertainty caused by the
interference/noise, reconfigurable intelligent surface (RIS) was introduced in
[14] to improve the communication covertness. Therein, by using RIS to
strengthen the directionality of the intended signal, constructive and
destructive effects can be achieved on the signals received by the intended
receiver and the unintended receiver (i.e., an adversary), respectively. A
scenario in which the adversary is unaware of the time of the communication
attempt from the legitimate transmitter was investigated in [15]. In this
scenario, the legitimate transmitter selects one single slot out of multiple
slots for transmission. An upper bound of the achievable covert rate was
analytically derived as a performance metric to evaluate the effectiveness of
covert communication.

###  II-B Friendly Jamming

The main idea of friendly jamming is to emit jamming signal that will
interfere with the receptions at the eavesdropper to achieve confidentiality
of legitimate transmissions. Thus the jamming signals increase the decoding
difficulty for the eavesdropper and improve the secrecy outage probability, or
equivalently, the secrecy rate [16, 17]. Friendly jamming was adopted together
with bandwidth allocation in [18] to secure the unmanned aerial vehicle
(UAV)-enabled wireless communication. A UAV-enabled friendly jamming scheme
was proposed in [19] to secure industrial IoT (IIoT) against eavesdropping,
and its effectiveness in reducing the secrecy outage probability was verified
both analytically and numerically. Except for jamming the wiretap channel so
as to reduce the secrecy outage probability, friendly jamming can also be
applied to improve the communication covertness. This is due to the fact that
the jamming signal emitted by the friendly jammer can also increase the
uncertainty in the received signal power of the adversary and thereby increase
the detection error. For example, the authors in [5] deployed a friendly
jammer to transmit jamming signal and prevent the adversary from detecting
transmission of the local model updates in order to achieve a covert federated
learning (FL) process. As the jamming signal can also reduce transmission
reliability, an optimization problem was formulated to determine the optimal
jamming power such that both the link reliability and communication covertness
for the local model updates in the FL process can be guaranteed. [20]
considered a scenario where a transmitter intends to communicate with multiple
receivers via orthogonal frequency bands and the adversary is able to monitor
all the bands aiming at detecting the transmissions. A friendly jammer was
deployed to broadcast jamming signals to distort the adversary’s observation
in order to increase the detection error for the adversary over all the bands.
Different from other works, the authors in [21] considered a scenario where
the adversary can optimize not only its detection threshold but also its
location aiming to minimize the covert outage probability. To defend against
such a powerful adversary, a full-duplex transceiver was deployed, and the
jamming signal transmitted by this transceiver was leveraged to maximize the
throughput of the legitimate user subject to the constraint on communication
covertness.

##  III System Model, Assumptions, and Performance Metrics

###  III-A Network Model

We consider a large-scale D2D network under the threat of wardens (Fig. 2(a)).
In the network, the spatial distribution of the D2D transmitters follows a
homogeneous Poisson point process (PPP) $\Phi_{\cal{D}}$ with density
$\lambda_{\cal{D}}$, and each of which has a dedicated D2D receiver located at
distance $R$ in a random orientation. The D2D transmitters and their dedicated
D2D receivers form a Poisson bipolar network [7]. The ALOHA channel access
scheme is adopted for the D2D network, where each D2D transmitter becomes
active independently. This event is denoted by ${\cal{D}}_{1}$ and it occurs
with probability ${\mathbb{P}}^{{\cal{D}}_{1}}$ [22]. Accordingly, each D2D
transmitter will independently become inactive with probability
${\mathbb{P}}^{{\cal{D}}_{0}}\triangleq 1-{\mathbb{P}}^{{\cal{D}}_{1}}$, where
${\cal{D}}_{0}$ denotes the corresponding inactive event. To analyze the
performance of the D2D network, we condition on that there is a D2D receiver
at the origin which is considered to be the typical D2D receiver, and its
associating D2D transmitter is the typical D2D transmitter and denoted by D2D
transmitter $d$. Following [23], we consider that the wardens are spatially
distributed by following an independent homogeneous PPP $\Phi_{\cal{W}}$ with
density $\lambda_{\cal{W}}$. Also, the spatial distribution of the jammers
follows another independent homogeneous PPP $\Phi_{\cal{J}}$ with density
$\lambda_{\cal{J}}$.

For D2D transmitter $d$, since the nearest warden, denoted by
$\left.w\right|_{d}$, is the most threatening warden, we focus on a scenario
where D2D transmitter $d$ defends against warden $\left.w\right|_{d}$.
Accordingly, we consider that warden $\left.w\right|_{d}$ aims to detect, and
once detect, wiretaps the transmission of D2D transmitter $d$. All the
entities in the network are equipped with a single antenna and all the
channels are assumed to experience Rayleigh fading with unit mean [7]. Note
that similar to [24], the discussion in the rest of this paper is based on the
performance of the representative network nodes (i.e., D2D transmitter $d$,
the typical D2D receiver, and warden $\left.w\right|_{d}$).

Figure 2: (a) D2D transmitters, D2D receivers, friendly jammers, and the
wardens; (b) Stackelberg game-based problem formulation with the wardens at
the lower stage and the D2D network (i.e., D2D transmitters and jammers) at
the upper stage.

###  III-B Performance Metrics

####  III-B1 Successful Transmission Probability for the D2D Transmitter

The received signal power of the typical D2D receiver regarding the activation
status of its associating D2D transmitter (i.e., D2D transmitter $d$) is

$y_{d}=\left\\{\begin{aligned}
&p^{\rm{D}}g_{d}R^{-\alpha}+S_{d}^{{\cal{D}}_{1}}+S_{d}^{{\cal{J}}}+N_{d},&{\text{if}}\,{\cal{D}}_{1},\\\
&S_{d}^{{\cal{D}}_{1}}+S_{d}^{{\cal{J}}}+N_{d},&{\text{if}}\,{\cal{D}}_{0},\end{aligned}\right.$
(1)

where
$S_{d}^{{\cal{D}}_{1}}=p^{\rm{D}}\sum_{d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|d\right\\}}}{\mathbbm{1}}_{d^{\prime}}g_{d^{\prime}d}r_{d^{\prime}d}^{-\alpha}$
and
$S_{d}^{{\cal{J}}}=p^{\rm{J}}\sum_{j\in\Phi_{\left\\{\left.{\cal{J}}\right|d\right\\}}}g_{jd}r_{jd}^{-\alpha}$
are the sum of the signal powers from D2D transmitter
$d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|d\right\\}}$
and that from jammer $j\in\Phi_{\left\\{\left.{\cal{J}}\right|d\right\\}}$
received at the typical D2D receiver, respectively, $g_{d}$ is the channel
between D2D transmitter $d$ and the typical D2D receiver, and $N_{d}$ is the
noise power at the typical D2D receiver. Therein,
$\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|d\right\\}}$
and $\Phi_{\left\\{\left.{\cal{J}}\right|d\right\\}}$ are the point process
(PP) of the D2D transmitters that excludes D2D transmitter $d$ with the
typical D2D receiver as the observation point and that of the jammers with the
typical D2D receiver as the observation point, respectively,
${\mathbbm{1}}_{d^{\prime}}$ is the activation indicator of D2D transmitter
$d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|d\right\\}}$
and equal to $1$ if D2D transmitter $d^{\prime}$ is active and $0$ otherwise,
$p^{\rm{D}}$ and $p^{\rm{J}}$ are the D2D transmission power and jamming
power, respectively, $g_{d^{\prime}d}$ and $r_{d^{\prime}d}$ are the channel
and distance between D2D transmitter
$d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|d\right\\}}$
and the typical D2D receiver, respectively, and $g_{jd}$ and $r_{jd}$ are the
channel and distance between jammer
$j\in\Phi_{\left\\{\left.{\cal{J}}\right|d\right\\}}$ and the typical D2D
receiver, respectively. Accordingly, the SINR at the typical D2D receiver
regarding the activation status of its associating D2D transmitter (i.e., D2D
transmitter $d$) is

${\rm{SINR}}_{d}\left(p^{\rm{D}},p^{\rm{J}}\right)=\left\\{\begin{aligned}
&\frac{p^{\rm{D}}g_{d}R^{-\alpha}}{S_{d}^{{\cal{D}}_{1}}+S_{d}^{{\cal{J}}}+N_{d}},&{\text{if}}\,{\cal{D}}_{1},\\\
&0,&{\text{if}}\,{\cal{D}}_{0}.\\\ \end{aligned}\right.$ (2)

Denoting the SINR threshold at the D2D receiver by $\xi^{\rm{D}}$ and
conditioned on the active status of D2D transmitter $d$ (i.e.,
${\cal{D}}_{1}$), we have the successful transmission probability for the D2D
transmitter $d$ as follows:

${\mathbb{P}}\left[\left.{\rm{SINR}}_{d}\left(p^{\rm{D}},p^{\rm{J}}\right)>\xi^{\rm{D}}\right|{\cal{D}}_{1}\right].$
(3)

The detailed derivation of (3) is given in Appendix A. We validate the
derivation of (3) by comparing the analytical results with the simulation
results in Fig. 3. Note that the simulation results are generated by using
Monte Carlo method using the parameter setting given in Table I. We assume the
same parameter setting in the rest of this paper. It can be observed from Fig.
3 that our analytical results match with the simulation results.

Figure 3: Comparison between simulation results and analytical results on
successful transmission probability.

####  III-B2 Detection Error at the Warden

The wardens receive the signals transmitted from the D2D transmitters and
jammers and aim to detect the D2D transmission independently. Accordingly, for
warden $\left.w\right|_{d}\in\Phi_{\cal{W}}$, its received signal power
regarding the activation status of its target D2D transmitter (i.e., D2D
transmitter $d$) can be expressed as follows:

$y_{\left.w\right|_{d}}=\left\\{\begin{aligned}
&p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}+S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}+N_{\left.w\right|_{d}},&{\text{if}}\,{\cal{D}}_{1},\\\
&S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}+N_{\left.w\right|_{d}},&{\text{if}}\,{\cal{D}}_{0},\end{aligned}\right.$
(4)

where
$S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}=p^{\rm{D}}\sum_{d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|\left.w\right|_{d}\right\\}}}{\mathbbm{1}}_{d^{\prime}}g_{d^{\prime}\left.w\right|_{d}}r_{d^{\prime}\left.w\right|_{d}}^{-\alpha}$
and
$S_{\left.w\right|_{d}}^{{\cal{J}}}=p^{\rm{J}}\sum_{j\in\Phi_{\left\\{\left.{\cal{J}}\right|\left.w\right|_{d}\right\\}}}g_{j\left.w\right|_{d}}r_{j\left.w\right|_{d}}^{-\alpha}$
are the sum of the signal powers from D2D transmitter
$d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|\left.w\right|_{d}\right\\}}$
and that from jammer
$j\in\Phi_{\left\\{\left.{\cal{J}}\right|\left.w\right|_{d}\right\\}}$
received at warden $\left.w\right|_{d}$, respectively,
$g_{d\left.w\right|_{d}}$ and $r_{d\left.w\right|_{d}}$ are the channel and
distance between D2D transmitter $d$ and warden $\left.w\right|_{d}$,
respectively, and $N_{\left.w\right|_{d}}$ is the additive noise at warden
$\left.w\right|_{d}$. Therein,
$\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|\left.w\right|_{d}\right\\}}$
and $\Phi_{\left\\{\left.{\cal{J}}\right|\left.w\right|_{d}\right\\}}$ are the
PP of the D2D transmitters that excludes D2D transmitter $d$ with warden
$\left.w\right|_{d}$ as the observation point and that of the jammers with
warden $\left.w\right|_{d}$ as the observation point, respectively,
$g_{d^{\prime}\left.w\right|_{d}}$ and $r_{d^{\prime}\left.w\right|_{d}}$ are
the channel and distance between D2D transmitter
$d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|\left.w\right|_{d}\right\\}}$
and warden $\left.w\right|_{d}$, respectively, and $g_{j\left.w\right|_{d}}$
and $r_{j\left.w\right|_{d}}$ are the channel and distance between jammer
$j\in\Phi_{\left\\{\left.{\cal{J}}\right|\left.w\right|_{d}\right\\}}$ and
warden $\left.w\right|_{d}$, respectively.

The warden uses a threshold-based detection rule to detect the transmission of
its target D2D transmitter [14]. Specifically, the warden will assume that its
target D2D transmitter is active if its received signal power is larger than a
predetermined detection threshold, denoted by $\tau$. Otherwise, it will
assume that its target D2D transmitter is inactive. It will be a false alarm
(FA) for warden $\left.w\right|_{d}$ if its received signal power is larger
than the predetermined detection threshold $\tau$ (i.e.,
$y_{\left.w\right|_{d}}>\tau$) while D2D transmitter $d$ is inactive (i.e.,
${\cal{D}}_{0}$). On the other hand, it will be a miss detection (MD) for
warden $\left.w\right|_{d}$, if its received signal power is lower than the
predetermined detection threshold $\tau$ (i.e., $y_{\left.w\right|_{d}}<\tau$)
while D2D transmitter $d$ is active (i.e., ${\cal{D}}_{1}$).

The FA and MD probabilities for warden $\left.w\right|_{d}\in\Phi_{\cal{W}}$
are defined as

${\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)={\mathbb{P}}\left[\left.y_{\left.w\right|_{d}}>\tau\right|{\cal{D}}_{0}\right]={\mathbb{P}}\left[S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}+N_{\left.w\right|_{d}}>\tau\right]$
(5)

and

${\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)={\mathbb{P}}\left[\left.y_{\left.w\right|_{d}}<\tau\right|{\cal{D}}_{1}\right]={\mathbb{P}}\left[p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}+S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}+N_{\left.w\right|_{d}}<\tau\right],$
(6)

respectively, and the specific expressions of which are derived in Appendices
C and D, respectively. Again, we use simulation results to verify the
derivations of
${\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$
and
${\mathbb{P}}^{\text{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$.

As shown in Fig. 4(a), we first fix $p^{\rm{J}}$ to be $15$dBm and vary the
values of $p^{\rm{D}}$ and $\tau$ to compare the simulation results and
analytical results on
${\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$
and observe that these two results are consistent, which validates our
derivation of
${\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$.
The same conclusion can be drawn based on the results in Fig. 4(b), which
compares the simulation results and analytical results on
${\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$
by fixing $p^{\rm{D}}$ to be $15$dBm and varying the values of $p^{\rm{J}}$
and $\tau$. Similar outcomes can be observed in Fig. 5, which validates our
derivation of
${\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$.

Figure 4: Comparison between the simulation results and analytical results on
FA probability with (a) $p^{\rm{J}}=15{\rm{dBm}}$ and (b)
$p^{\rm{D}}=15{\rm{dBm}}$.

Figure 5: Comparison between the simulation results and analytical results on
MD probability with (a) $p^{\rm{J}}=15{\rm{dBm}}$ and (b)
$p^{\rm{D}}=15{\rm{dBm}}$.

####  III-B3 Secrecy Outage Probability for the D2D Transmitter

According to the received signal power of warden $\left.w\right|_{d}$ given in
(4) and conditioned on the active status of its target D2D transmitter (i.e.,
D2D transmitter $d$), the secrecy outage probability for D2D transmitter $d$
is

${\mathbb{P}}\left[\left.{\rm{SINR}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}}\right)>\xi^{\rm{W}}\right|{\cal{D}}_{1}\right],$
(7)

where $\xi^{\rm{W}}$ is the SINR threshold at warden for successfully decoding
the received signal and the expression of which is derived in Appendix B. The
consistency between the analytical results and simulation results on the
secrecy outage probability is shown in Fig. 6.

Figure 6: Comparison between simulation results and analytical results on
secrecy outage probability.

##  IV Problem Statement and Game Formulation

We model the combat between wardens and the D2D network including D2D
transmitters and the jammers in the framework of a two-stage Stackelberg game
as shown in Fig. 2(b). Therein, the D2D network jointly decides the D2D
transmission power and jamming power first, and the wardens make decision
based on their received signal powers later. Hence, the wardens are the
followers at the lower stage and the D2D network is the leader at the upper
stage. The formulation of the two-stage Stackelberg game regarding the
representative network nodes mentioned in Section III is a single-leader-
single-follower game and defined as follows:

  * •

At the lower stage, given the strategies of the D2D network (i.e., the D2D
transmission power $p^{\rm{D}}$ and jamming power $p^{\rm{J}}$) from the upper
stage, warden $\left.w\right|_{d}$ acts as a single follower and makes
decision on the detection threshold $\tau$ in order to minimize its detection
error [14] as follows:

$\tau^{*}=\arg\min_{\tau}{\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)+{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right),$
(8)

where
${\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$
and
${\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$
are defined in (5) and (6), respectively.

In the above, the reason that we omit ${\mathbb{P}}^{{\cal{D}}_{1}}$ and
${\mathbb{P}}^{{\cal{D}}_{0}}$ in the detection error can be explained as
follows. First, as warden $\left.w\right|_{d}$ is unaware of the activation
status of its target D2D transmitter (i.e., ${\cal{D}}_{1}$ and
${\cal{D}}_{0}$), we relax its detection error probability as follows:

$\displaystyle{\mathbb{P}}^{{\cal{D}}_{0}}{\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)+{\mathbb{P}}^{{\cal{D}}_{1}}{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$
(9) $\displaystyle\leq$
$\displaystyle\max\left\\{{\mathbb{P}}^{{\cal{D}}_{0}},{\mathbb{P}}^{{\cal{D}}_{1}}\right\\}\left[{\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)+{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)\right],$

where the second row of (9) is the upper bound of the detection error
probability (i.e., the first row of (9)) and can be further simplified as the
objective shown in (8). As such, the probabilistic terms related to the
activation status of the D2D transmitter are avoided in the warden’s
objective. Also, this enables us to treat FA and MD equivalently so as to
avoid any bias between them [25, 26].

  * •

At the upper stage, D2D transmitter $d$ together with the jammers work as a
single leader and aim to maximize the network utility (i.e., (10a)) subject to
the constraint on communication covertness with respect to (w.r.t.) warden
$\left.w\right|_{d}$ (i.e., (10b)). This can be expressed as a nonlinear
constrained optimization problem as follows:

$\displaystyle\begin{aligned}
\max_{p^{\rm{D}},p^{\rm{J}}}&\,w^{\rm{D}}{\mathbb{P}}\left[\left.{\rm{SINR}}_{d}\left(p^{\rm{D}},p^{\rm{J}}\right)>\xi^{\rm{D}}\right|{\cal{D}}_{1}\right]\left\\{{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau^{\star}\right)\right.\\\
&\,\left.+\left[1-{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau^{\star}\right)\right]\left[1-{\mathbb{P}}\left[\left.{\rm{SINR}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}}\right)>\xi^{\rm{W}}\right|{\cal{D}}_{1}\right]\right]\right\\}-\frac{\lambda_{\cal{J}}}{\lambda_{\cal{D}}}w^{\rm{J}}p^{\rm{J}}\end{aligned}$
(10a)
$\displaystyle\quad{\text{s.t.}}\,{\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau^{\star}\right)+{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau^{\star}\right)\geq
1-\varepsilon$ (10b)
$\displaystyle\quad\,\quad\,p^{\rm{D}}\in\left[{\underline{p}}^{\rm{D}},{\overline{p}}^{\rm{D}}\right],\,p^{\rm{J}}\in\left[{\underline{p}}^{\rm{J}},{\overline{p}}^{\rm{J}}\right],$
(10c)

where ${\underline{p}}^{\rm{D}}$ (${\underline{p}}^{\rm{J}}$) and
${\overline{p}}^{\rm{D}}$ (${\overline{p}}^{\rm{J}}$) are the lower bound and
upper bound of the D2D transmission (jamming) power, respectively. Therein,
$w^{\rm{D}}$ is the reward for guaranteeing D2D link reliability (i.e.,
${\rm{SINR}}_{d}\left(p^{\rm{D}},p^{\rm{J}}\right)>\xi^{\rm{D}}$) conditioned
on secure communication. The secure communication happens if and only if the
malicious attempt of warden $\left.w\right|_{d}$ is unsuccessful (i.e., miss
detect the D2D transmission or secrecy outage is avoided if the D2D
transmission is detected) when its target D2D transmitter (i.e., D2D
transmitter $d$) is active. In this case, the secure communication excludes
the FA, and the probability of which is defined as the sum of the probability
that MD occurs at warden $\left.w\right|_{d}$ (i.e.,
${\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau^{\star}\right)$)
and the probability that secrecy outage is avoided for D2D transmitter $d$
(i.e.,
$1-{\mathbb{P}}\left[\left.{\rm{SINR}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}}\right)>\xi^{\rm{W}}\right|{\cal{D}}_{1}\right]$)
in the case that its transmission is detected by warden $\left.w\right|_{d}$
(i.e.,
$1-{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau^{\star}\right)$).
Hence, the probability of secure communication is given by

${\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau^{\star}\right)+\left[1-{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau^{\star}\right)\right]\left[1-{\mathbb{P}}\left[\left.{\rm{SINR}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}}\right)>\xi^{\rm{W}}\right|{\cal{D}}_{1}\right]\right].$
(11)

In the objective of the optimization problem (i.e., (10a)), the term
$\frac{\lambda_{\cal{J}}}{\lambda_{\cal{D}}}w^{\rm{J}}p^{\rm{J}}$ is the cost
incurred by the assistance from jammers with $w^{\rm{J}}$ being the cost of
unit jamming power. Here, if the area of the region occupied by the large-
scale D2D network is $A$, the expected number of jammers in this region is
$\lambda_{\cal{W}}A$. In this case, the total jamming power cost is
$\lambda_{\cal{W}}Aw^{\rm{J}}p^{\rm{J}}$. Let D2D transmitters equally share
the total jamming power cost,
$\frac{\lambda_{\cal{W}}Aw^{\rm{J}}p^{\rm{J}}}{\lambda_{\cal{D}}A}=\frac{\lambda_{\cal{J}}}{\lambda_{\cal{D}}}w^{\rm{J}}p^{\rm{J}}$
is therefore the jamming power cost for each D2D transmitter, where
$\lambda_{\cal{D}}A$ is the expected number of D2D transmitters in this
region. The inequality constraint (i.e., (10b)) puts a lower bound on the
detection error of warden $\left.w\right|_{d}$ in order to maintain certain
communication covertness for the D2D transmission.

##  V Equilibrium Analysis and Algorithm Design of Two-Stage Stackelberg Game

In this section, we provide the equilibrium analysis for the two-stage
Stackelberg game formulated in Section IV and develop an algorithm to search
for the corresponding equilibrium. First, given the strategy from the upper
stage, we analyze the lower-stage detection error minimization problem for
warden $\left.w\right|_{d}$ in Section V-A and choose a suitable algorithm to
search for the optimal strategy of warden $\left.w\right|_{d}$ as the best
response of the lower stage. Taking into account the best response from the
lower stage, we design a bi-level algorithm based on the SCA method to search
for the optimal strategy of D2D transmitter $d$ and the jammers at the upper
stage, which together with the optimal strategy of warden $\left.w\right|_{d}$
constitute the equilibrium of the two-stage Stackelberg game.

###  V-A Solution to Lower-Stage Problem: Detection Error Minimization

Figure 7: Illustration of the uniqueness of the solution to the lower-stage
detection error minimization problem with $p^{\rm{D}}=20$dBm and
$p^{\rm{J}}=10$dBm.

For the lower-stage detection error minimization problem in (8), given any D2D
transmission power and jamming power (i.e., $p^{\rm{D}}$ and $p^{\rm{J}}$,
respectively), the objective of warden $\left.w\right|_{d}$ will be in a
trough shape w.r.t. $\tau$. In this case, we can apply the Rosenbrock method
[27] to obtain the optimal strategy for warden $\left.w\right|_{d}$. The
reason for the trough shape of the warden’s objective can be explained as
follows. First, we rewrite the objective function of warden
$\left.w\right|_{d}$ (i.e., detection error in (8)) as follows:

$\displaystyle{\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)+{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$
(12) $\displaystyle=$
$\displaystyle{\mathbb{P}}\left[S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}+N_{\left.w\right|_{d}}>\tau\right]+{\mathbb{P}}\left[p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}+S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}+N_{\left.w\right|_{d}}<\tau\right]$
$\displaystyle=$ $\displaystyle 1-\int_{0}^{\tau-
N_{\left.w\right|_{d}}}f_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}}\left(t\right){\rm{d}}t+\int_{0}^{\tau-
N_{\left.w\right|_{d}}}f_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}}\left(t\right)F_{p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}}\left(\tau-
N_{\left.w\right|_{d}}-t\right){\rm{d}}t.$

In (12) and given any D2D transmission power and jamming power (i.e.,
$p^{\rm{D}}$ and $p^{\rm{J}}$, respectively), as
$F_{p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}}\left(\cdot\right)\in\left[0,1\right]$,
$\int_{0}^{\tau-
N_{\left.w\right|_{d}}}f_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}}\left(t\right){\rm{d}}t$
will have a steeper increase w.r.t. $\tau$ than $\int_{0}^{\tau-
N_{\left.w\right|_{d}}}f_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}}\left(t\right)F_{p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}}\left(\tau-
N_{\left.w\right|_{d}}-t\right){\rm{d}}t$ when $\tau$ is small, which is
demonstrated with the numerical results in Fig. 7, e.g., $\tau=5$dBm. In this
case,
${\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)+{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$
will decrease w.r.t. $\tau$. Later, when $\tau$ further increases,
$\int_{0}^{\tau-
N_{\left.w\right|_{d}}}f_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}}\left(t\right){\rm{d}}t$
almost approaches its probabilistic upper bound, and its increasing speed
becomes slower than that of $\int_{0}^{\tau-
N_{\left.w\right|_{d}}}f_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}}\left(t\right)F_{p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}}\left(\tau-
N_{\left.w\right|_{d}}-t\right){\rm{d}}t$, which also can be observed from
Fig. 7, e.g., $\tau=25$dBm. In this case,
${\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)+{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$
will increase w.r.t. $\tau$. Consequently,
${\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)+{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)$
takes on a trough shape w.r.t. $\tau$, which induces a unique optimal strategy
(i.e., $\tau^{\star}$) that minimizes the detection error for warden
$\left.w\right|_{d}$. Such an optimal strategy is the best response of the
lower stage corresponding to the given D2D transmission power and jamming
power (i.e., $p^{\rm{D}}$ and $p^{\rm{J}}$, respectively).

###  V-B Solution to Upper-Stage Problem: Friendly Jamming-Assisted Covert
Communication

For the friendly jamming-assisted covert communication problem at the upper
stage, it is unwieldy to verify its convexity due to the multiple integrals in
both the objective and constraint. In this case, we design a bi-level
optimization algorithm based on the SCA method [28] and take into account the
best response from the lower stage (i.e., $\tau^{\star}$ in (8)). In
particular, let $U_{0}\left(p^{\rm{D}},p^{\rm{J}}\right)$ be the negative
value of (10a) and
$U_{1}\left(p^{\rm{D}},p^{\rm{J}}\right)=-{\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau^{\star}\right)-{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau^{\star}\right)+1-\varepsilon$,
we have the upper-stage friendly jamming-assisted covert communication problem
(i.e., (10)) rewritten as follows:

$\displaystyle\min_{p^{\rm{D}},p^{\rm{J}}}$
$\displaystyle\,U_{0}\left(p^{\rm{D}},p^{\rm{J}}\right)$ (13) s.t.
$\displaystyle\,U_{1}\left(p^{\rm{D}},p^{\rm{J}}\right)\leq 0$
$\displaystyle\,p^{\rm{D}}\in\left[{\underline{p}}^{\rm{D}},{\overline{p}}^{\rm{D}}\right],\,p^{\rm{J}}\in\left[{\underline{p}}^{\rm{J}},{\overline{p}}^{\rm{J}}\right].$

According to [28], we can follow Algorithm 1 and switch to solve an
approximation convex program of (13) (i.e., (15)) iteratively. The convergence
of Algorithm 1 has been proven in [29]. (15) is formulated by replacing
$U_{i}\left(p^{\rm{D}},p^{\rm{J}}\right)$ for all $i\in\left\\{0,\,1\right\\}$
of (13) with the surrogate function
${\hat{U}}_{i}\left(\left.{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right|{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right)$ for all
$i\in\left\\{0,\,1\right\\}$ that satisfies the following conditions [29]:

  1. 1.

Consistency in function value: ${\hat{U}}_{i}\left(\left.{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right|{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle
k\rangle}\right)=U_{i}\left({p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right)$;

  2. 2.

Consistency in gradient:
$\nabla_{p^{\rm{D}}}{\hat{U}}_{i}\left(\left.{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right|{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle
k\rangle}\right)=\nabla_{p^{\rm{D}}}U_{i}\left({p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right)$ and
$\nabla_{p^{\rm{J}}}{\hat{U}}_{i}\left(\left.{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right|{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle
k\rangle}\right)=\nabla_{p^{\rm{J}}}U_{i}\left({p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right)$;

  3. 3.

Convexity:
${\hat{U}}_{i}\left(\left.p^{\rm{D}},p^{\rm{J}}\right|{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right)$ is strongly convex w.r.t.
$\left[p^{\rm{D}},p^{\rm{J}}\right]$.

As stated in [29], the surrogate function
${\hat{U}}_{i}\left(\left.p^{\rm{D}},p^{\rm{J}}\right|{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right)$ for all
$i\in\left\\{0,1\right\\}$ in (15) can be designed by using the following
gradient descent update:

$\displaystyle{\hat{U}}_{i}\left(\left.p^{\rm{D}},p^{\rm{J}}\right|{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right)=$ $\displaystyle
U_{i}\left({p^{\rm{D}}}^{\langle k\rangle},{p^{\rm{J}}}^{\langle
k\rangle}\right)+\nabla U_{i}\left({p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle
k\rangle}\right)^{\top}\begin{pmatrix}p^{\rm{D}}-{p^{\rm{D}}}^{\langle
k\rangle}\\\ p^{\rm{J}}-{p^{\rm{J}}}^{\langle k\rangle}\\\ \end{pmatrix}$ (14)
$\displaystyle+\frac{\delta}{2}\left(p^{\rm{D}}-{p^{\rm{D}}}^{\langle
k\rangle}\right)^{2}+\frac{\delta}{2}\left(p^{\rm{J}}-{p^{\rm{J}}}^{\langle
k\rangle}\right)^{2}.$

Data: Input ${p^{\rm{D}}}^{\langle k\rangle}$ and ${p^{\rm{J}}}^{\langle
k\rangle}$.

1 Initialize $k=0$ and set error tolerance $\epsilon$;

2 while  _error $>\epsilon$_ do

3 Solve the following optimization problem
$\displaystyle\min_{p^{\rm{D}},p^{\rm{J}}}$
$\displaystyle\,{\hat{U}}_{0}\left(\left.p^{\rm{D}},p^{\rm{J}}\right|{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right)$ (15) s.t.
$\displaystyle\,{\hat{U}}_{1}\left(\left.p^{\rm{D}},p^{\rm{J}}\right|{p^{\rm{D}}}^{\langle
k\rangle},{p^{\rm{J}}}^{\langle k\rangle}\right)\leq 0$
$\displaystyle\,p^{\rm{D}}\in\left[{\underline{p}}^{\rm{D}},{\overline{p}}^{\rm{D}}\right],\,p^{\rm{J}}\in\left[{\underline{p}}^{\rm{J}},{\overline{p}}^{\rm{J}}\right]$
and obtain the solution ${\tilde{p}}^{\rm{D}}$ and ${\tilde{p}}^{\rm{J}}$;

4 ${p^{\rm{D}}}^{\langle k+1\rangle}={p^{\rm{D}}}^{\langle
k\rangle}+\gamma\left({\tilde{p}}^{\rm{D}}-{p^{\rm{D}}}^{\langle
k\rangle}\right)$ and ${p^{\rm{J}}}^{\langle k+1\rangle}={p^{\rm{J}}}^{\langle
k\rangle}+\gamma\left({\tilde{p}}^{\rm{J}}-{p^{\rm{J}}}^{\langle
k\rangle}\right)$ ;

5 $error=\left\|{p^{\rm{D}}}^{\langle k+1\rangle}-{p^{\rm{D}}}^{\langle
k\rangle}\right\|^{2}+\left\|{p^{\rm{J}}}^{\langle
k+1\rangle}-{p^{\rm{J}}}^{\langle k\rangle}\right\|^{2}$;

6 $k+1\mapsto k$;

7

8 end while

Algorithm 1 Bi-level Optimization Algorithm

##  VI Performance Evaluation

In this section, we provide the numerical results to evaluate the system
performance. The parameter setting is given in Table I. First, we verify the
optimality of the obtained strategy and the effectiveness of the friendly
jamming in Fig. 8. Second, we demonstrate the advantages of the proposed
friendly jamming-assisted covert communication by comparing it to the
information-theoretical secrecy approach in Fig. 9. Third, the impact of the
communication covertness requirement and the warden’s density on the network
utility are illustrated in Fig. 10. Fourth, we evaluate the impact of the
densities of the D2D transmitters and jammers on the network utility in Fig.
11. Finally, we present the impact of the SINR thresholds at the warden and
the D2D transmitter on the network utility in Fig. 12.

###  VI-A Effectiveness of Friendly Jamming

Figure 8: (a) Network utility and (b) detection error w.r.t. D2D transmission
power and jamming power (i.e., $p^{\rm{D}}$ and $p^{\rm{J}}$, respectively).

Figure 8 depicts the network utility and detection error as functions of the
D2D transmission power $p^{\rm{D}}$ and jamming power $p^{\rm{J}}$. In Fig.
8(a), we can observe that the optimal strategy (i.e., optimal D2D transmission
power and jamming power) can maximize the network utility within the feasible
domain, where the boundary of the feasible domain is determined by the
constraint on communication covertness (i.e., (10b)) as shown in Fig. 8(b).
The optimal strategy is obtained by the developed bi-level optimization
algorithm (i.e., Algorithm 1) and is consistent with that obtained by the
exhaustive search method. The exhaustive search method searches for the
optimal strategy via: 1) discretizing the domain of definition of the strategy
(i.e., the D2D transmission power and jamming power), 2) enumerating all
possible candidates for the strategy taking into account the corresponding
best response from the lower stage to evaluate the network utility and
detection error, and 3) finding the optimal strategy that jointly maximizes
the network utility and satisfies the constraint on communication covertness.
In this case, the optimality of the obtained equilibrium strategy is
validated. Moreover, as shown in Fig. 8(b), the detection error can satisfy
the constraint on communication covertness (i.e., (10b)) if and only if the
jamming power $p^{\rm{J}}\gtrsim 12$dBm111$p^{\rm{J}}$ is approximately larger
than $12$dBm.. The reason is that merely relying on the co-channel
interference in the D2D network is insufficient and hence difficult to confuse
and mislead the decision of the warden. In this case, without adequate
assistance from the friendly jammers, the covert communication cannot be
achieved.

###  VI-B Advantages of Friendly Jamming-Assisted Covert Communication

Figure 9: Friendly jamming-assisted covert communication versus information-
theoretical secrecy approach.

In Fig. 9, we compare our proposed friendly jamming-assisted covert
communication and the information-theoretic secrecy approach in terms of the
network utility and the secure communication probability (i.e., (11)). The
problem of the information-theoretic secrecy approach is the problem in (10)
with the removal of the constraint on communication covertness (i.e., (10b)).
By such, the problem of the information-theoretic secrecy approach aims at the
joint link reliability maximization and secrecy outage minimization. We can
observe that the friendly jamming-assisted covert communication starts to take
effect at $\frac{p^{\rm{J}}}{{\bar{p}}^{\rm{J}}}\approx-18$dB, where the
network utility becomes nonzero and also the optimal performance is achieved
(i.e., the network utility is maximized at around $0.5$ with a certain secure
communication probability at around $0.97$). In this case, the optimal
performance of the friendly jamming-assisted covert communication at
$\frac{p^{\rm{J}}}{{\bar{p}}^{\rm{J}}}\approx-18$dB will be used as a
“Baseline” for comparison in the following discussion. Regarding the
information-theoretical secrecy approach, initially it can achieve a higher
utility and a lower secure communication probability compared to the optimal
performance of the friendly jamming-assisted covert communication. Later, if
we further increase the jamming power, although the secure communication
probability of the information-theoretical secrecy approach significantly
increases, its network utility decreases dramatically. Until the jamming power
approaches the upper bound, the secure communication probability of the
information-theoretical secrecy approach is still lower than that of the
friendly jamming-assisted covert communication at
$\frac{p^{\rm{J}}}{{\bar{p}}^{\rm{J}}}\approx-18$dB (i.e., around $0.97$)
while achieving a negative utility, which is much lower than the utility of
the friendly jamming-assisted covert communication at
$\frac{p^{\rm{J}}}{{\bar{p}}^{\rm{J}}}\approx-18$dB (i.e., around $0.5$). In
this case, the friendly jamming-assisted covert communication significantly
outperforms the information-theoretical secrecy approach, and its advantages
are well demonstrated.

###  VI-C Impact of Warden’s Density and Communication Covertness Requirement

Figure 10: Network utility w.r.t. the warden’s density $\lambda_{{\cal{W}}}$
with varying communication covertness requirement $\varepsilon$.

We evaluate the impact of the warden’s density $\lambda_{{\cal{W}}}$ and the
communication covertness requirement $\varepsilon$, as given in the nonlinear
inequality constraint (10b), on the network utility in Fig. 10. The network
utility decreases w.r.t. the increase in the warden’s density
$\lambda_{{\cal{W}}}$. The reason is that when the warden’s density becomes
higher, the distance between the warden and its target D2D transmitter becomes
shorter. In this case, the large-scale fading caused by the path loss weakens,
the signal power received at the warden from its target D2D transmitter
becomes stronger, which simultaneously reduces the difficulty in the
transmission detection and increases the secrecy outage probability for D2D
transmission once being detected. As a result, the network utility
deteriorates accordingly. Additionally, the network utility decreases w.r.t.
the warden’s density $\lambda_{{\cal{W}}}$ with different rates corresponding
to different communication covertness requirements (i.e., $\varepsilon$). This
is due to the fact that when the value of $\varepsilon$ decreases, the
communication covertness requirement is tighter and it becomes more difficult
to achieve covert communication, which together with the more threatening
warden (i.e., higher warden densification) induce a faster decreasing rate in
the network utility.

###  VI-D Impact of Densities of Friendly Jammers and D2D Transmitters

Figure 11: Network utility w.r.t. the jammers’s density $\lambda_{{\cal{J}}}$
with varying D2D transmitters’ density $\lambda_{\cal{D}}$.

Figure 11 evaluates the impact of the densities of the friendly jammers and
D2D transmitters on the network utility. An interesting result can be observed
from Fig. 11 that the increase in the jammers’ density improves instead of
damaging the network utility. The reason is that although the increase in the
jammers’ density leads to stronger interference to the D2D communication and
weakens the SINR at the D2D receiver, this interference can also enhance the
communication covertness. Based on the results shown in Fig. 11, the gain
obtained from the enhancement of communication covertness overcompensates the
loss incurred by the weakened SINR. This again demonstrates the benefit of
using the friendly jammers in achieving covert communication. Another
interesting result can be observed in Fig. 11 that the improvement due to the
increase in jammers’ density becomes diminishing when the D2D transmitters’
density increases. The reason is that as the density of D2D transmitters
increases, the co-channel interference due to the concurrent transmission
becomes stronger, which can also distort the observation of the warden and
mislead its decision. In this case, it is not necessary to have massive
jammers as the assistants in achieving covert communication. Hence, the
improvement due to the increase in jammers’ density becomes less when the D2D
transmitters’ density increases.

###  VI-E Impact of SINR Thresholds at Warden and D2D Transmitter

Figure 12: Network utility w.r.t. the SINR threshold at warden
$\xi^{{\rm{W}}}$ with varying SINR threshold at D2D receiver $\xi^{\rm{D}}$.

We evaluate the impact of the SINR threshold at warden and D2D transmitter
(i.e., $\xi^{{\rm{W}}}$ and $\xi^{\rm{D}}$, respectively). As shown in Fig.
12, the network utility concavely increases w.r.t. $\xi^{{\rm{W}}}$. This is
due to the fact that to successfully decode the received signal is of more
difficulty for the warden along with the increase in the SINR threshold
$\xi^{{\rm{W}}}$. In this case, the secure communication probability (i.e.,
(11)) will improve, which induces a higher network utility. The concave
increasing trend in the network utility regarding the increase in
$\xi^{{\rm{W}}}$ can be explained as follows. With the increase in
$\xi^{{\rm{W}}}$, the secrecy outage probability gradually approaches its
probabilistic lower bound, and its decreasing speed becomes slower than
before. In this case, the increasing speed in the secure communication
probability becomes slower, which induces a concave increasing trend in the
network utility. Regarding the improvement in the network utility
corresponding to the decrease in the SINR threshold at D2D transmitter
$\xi^{{\rm{D}}}$. This is due to that the decrease in $\xi^{{\rm{D}}}$ can
improve the successful transmission probability for the D2D transmitter (i.e.,
(3)), which makes the objective of the D2D network easier in achieving higher
value.

##  VII Conclusion

We leverage both covert communication and friendly jamming to achieve
physical-layer security in a large-scale D2D network in order to defend
against the wardens. The objective is to jointly increase the detection error
at the warden and decrease secrecy outage for the D2D transmitters in case
transmissions are detected. The combat between the wardens and the D2D network
(i.e., the D2D transmitters and the jammers) is modeled by a Stackelberg game
with the wardens as the followers at the lower stage and the legitimate
entities as the leaders at the upper stage. For the problem of the warden at
the lower stage, we both analytically and numerically validate the existence
and uniqueness of its optimal strategy, which is regarded as the best response
from the lower stage. Given the best response from the lower stage, we develop
a bi-level optimization algorithm based on SCA method to search for the
optimal strategy of the D2D network, which together with that of the warden
constitute the Stackelberg equilibrium. We have presented numerical results to
evaluate the proposed approach and validate the optimality of the obtained
equilibrium strategy. Moreover, we have demonstrated the advantages of the
proposed friendly jamming-assisted covert communication by comparing it with
the information-theoretic secrecy approach in terms of network utility and
secure communication probability. Furthermore, we have evaluated the impact of
warden’s density and communication covertness requirement on system
performance. Considering multiple antennas at both the D2D transmitters and
receivers and also using the proposed approach to secure D2D communication in
a D2D-underlaid cellular network will be future extensions of this work.

##  Appendix A SINR Distribution for D2D Network

Based on (3) and conditioned on ${\cal{D}}_{1}$, the successful transmission
probability for D2D transmitter $d$ is

$\displaystyle{\mathbb{P}}\left[\left.{\rm{SINR}}_{d}\left(p^{\rm{D}},p^{\rm{J}}\right)\geq\xi^{\rm{D}}\right|{\cal{D}}_{1}\right]={\mathbb{P}}\left[\frac{p^{\rm{D}}g_{d}R^{-\alpha}}{S_{d}^{{\cal{D}}_{1}}+S_{d}^{{\cal{J}}}+N_{d}}\geq\xi^{\rm{D}}\right]$
(16) $\displaystyle=$
$\displaystyle{\mathbb{P}}\left[g_{d}\geq\frac{R^{\alpha}\xi^{\rm{D}}}{p^{\rm{D}}}\left(S_{d}^{{\cal{D}}_{1}}+S_{d}^{{\cal{J}}}+N_{d}\right)\right]$
$\displaystyle\mathop{=}\limits^{(a)}$
$\displaystyle{\mathbb{E}}\left[\exp\left(-\frac{R^{\alpha}\xi^{\rm{D}}}{p^{\rm{D}}}\left(S_{d}^{{\cal{D}}_{1}}+S_{d}^{{\cal{J}}}+N_{d}\right)\right)\right]$
$\displaystyle\mathop{=}\limits^{(b)}$
$\displaystyle\exp\left(-\frac{R^{\alpha}\xi^{\rm{D}}}{p^{\rm{D}}}N_{d}\right){\mathbb{E}}_{S_{d}^{{\cal{D}}_{1}}}\left[\exp\left(-\frac{R^{\alpha}\xi^{\rm{D}}}{p^{\rm{D}}}S_{d}^{{\cal{D}}_{1}}\right)\right]{\mathbb{E}}_{S_{d}^{{\cal{J}}}}\left[\exp\left(-\frac{R^{\alpha}\xi^{\rm{D}}}{p^{\rm{D}}}S_{d}^{{\cal{J}}}\right)\right],$

where $(a)$ follows from the Rayleigh fading of $g_{d}$ and $(b)$ follows the
independence between the random variables $S_{d}^{{\cal{D}}_{1}}$ and
$S_{d}^{{\cal{J}}}$. The second term of the final row of (16) (i.e.,
${\mathbb{E}}_{S_{d}^{{\cal{D}}_{1}}}\left[\exp\left(-\frac{R^{\alpha}\xi^{\rm{D}}}{p^{\rm{D}}}S_{d}^{{\cal{D}}_{1}}\right)\right]$)
is derived as follows:

$\displaystyle{\mathbb{E}}_{S_{d}^{{\cal{D}}_{1}}}\left[\exp\left(-\frac{R^{\alpha}\xi^{\rm{D}}}{p^{\rm{D}}}S_{d}^{{\cal{D}}_{1}}\right)\right]={\mathbb{E}}\left[\exp\left(-\frac{R^{\alpha}\xi^{\rm{D}}}{p^{\rm{D}}}p^{\rm{D}}\sum_{d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|d\right\\}}}{\mathbbm{1}}_{d^{\prime}}g_{d^{\prime}d}r_{d^{\prime}d}^{-\alpha}\right)\right]$
(17) $\displaystyle\mathop{=}\limits^{(a)}$
$\displaystyle{\mathbb{E}}_{\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|d\right\\}}}\left[\prod_{d^{\prime}\in\Phi_{\left\\{\left.{\cal{D}}\backslash\left\\{d\right\\}\right|d\right\\}}}\left(\frac{{\mathbb{P}}^{{\cal{D}}_{1}}}{R^{\alpha}\xi^{\rm{D}}r_{d^{\prime}d}^{-\alpha}+1}+1-{\mathbb{P}}^{{\cal{D}}_{1}}\right)\right]$
$\displaystyle\mathop{=}\limits^{(b)}$
$\displaystyle\exp\left(-2\pi\lambda_{\cal{D}}\int^{\infty}_{0}\left[1-\left(\frac{{\mathbb{P}}^{{\cal{D}}_{1}}}{R^{\alpha}\xi^{\rm{D}}r_{d^{\prime}d}^{-\alpha}+1}+1-{\mathbb{P}}^{{\cal{D}}_{1}}\right)\right]r_{d^{\prime}d}{\rm{d}}r_{d^{\prime}d}\right)$
$\displaystyle=$
$\displaystyle\exp\left(-2\pi\lambda_{\cal{D}}{\mathbb{P}}^{{\cal{D}}_{1}}R^{2}\left(\xi^{\rm{D}}\right)^{\frac{2}{\alpha}}\int^{\infty}_{0}\frac{r}{1+r^{\alpha}}{\rm{d}}r\right)$
$\displaystyle=$
$\displaystyle\exp\left(-2\pi\lambda_{\cal{D}}{\mathbb{P}}^{{\cal{D}}_{1}}R^{2}\left(\xi^{\rm{D}}\right)^{\frac{2}{\alpha}}\int^{\infty}_{0}r\int^{\infty}_{0}\exp\left(-t\left(1+r^{\alpha}\right)\right){\rm{d}}t{\rm{d}}r\right)$
$\displaystyle=$
$\displaystyle\exp\left(-2\pi\lambda_{\cal{D}}{\mathbb{P}}^{{\cal{D}}_{1}}R^{2}\left(\xi^{\rm{D}}\right)^{\frac{2}{\alpha}}\int^{\infty}_{0}\exp\left(-t\right)\int^{\infty}_{0}r\exp\left(-tr^{\alpha}\right){\rm{d}}r{\rm{d}}t\right)$
$\displaystyle=$
$\displaystyle\exp\left(-2\pi\lambda_{\cal{D}}{\mathbb{P}}^{{\cal{D}}_{1}}R^{2}\left(\xi^{\rm{D}}\right)^{\frac{2}{\alpha}}\int^{\infty}_{0}t^{-\frac{2}{\alpha}}\exp\left(-t\right){\rm{d}}t\int^{\infty}_{0}\frac{1}{\alpha}\theta^{\frac{2}{\alpha}-1}\exp\left(-\theta\right){\rm{d}}\theta\right)$
$\displaystyle=$
$\displaystyle\exp\left(-\pi\lambda_{\cal{D}}{\mathbb{P}}^{{\cal{D}}_{1}}R^{2}\left(\xi^{\rm{D}}\right)^{\frac{2}{\alpha}}\int^{\infty}_{0}t^{-\frac{2}{\alpha}}\exp\left(-t\right){\rm{d}}t\int^{\infty}_{0}\theta^{\frac{2}{\alpha}}\exp\left(-\theta\right){\rm{d}}\theta\right)$
$\displaystyle\mathop{=}\limits^{(c)}$
$\displaystyle\exp\left(-\frac{\pi\lambda_{\cal{D}}{\mathbb{P}}^{{\cal{D}}_{1}}R^{2}\left(\xi^{\rm{D}}\right)^{\frac{2}{\alpha}}}{{\rm{sinc}}\left(\frac{2}{\alpha}\right)}\right),$

where $(a)$ follows from the Rayleigh fading of $g_{d^{\prime}d}$, $(b)$
follows the probability generating functionals (PGFL) of PPP [30], and $(c)$
is obtained according to the reflection formula in Pi function (i.e.,
$\Pi\left(z\right)\Pi\left(-z\right)=\frac{\pi z}{\sin\left(\pi
z\right)}=\frac{1}{{\rm{sinc}}\left(z\right)}$ with
$\Pi\left(z\right)=z\Gamma\left(z\right)=\int_{0}^{\infty}t^{z}\exp\left(-t\right){\rm{d}}t$
being the Pi function and $\Gamma\left(z\right)$ being the gamma
function222Please refer to [31] and (43) in [24] for the definition and
details.). Similarly, the third term of the final row of (16) can be derived
as follows:

$\displaystyle{\mathbb{E}}_{S_{d}^{{\cal{J}}}}\left[\exp\left(-\frac{R^{\alpha}\xi^{\rm{D}}}{p^{\rm{D}}}S_{d}^{{\cal{J}}}\right)\right]={\mathbb{E}}\left[\exp\left(-\frac{R^{\alpha}\xi^{\rm{D}}}{p^{\rm{D}}}p^{\rm{J}}\sum_{j\in\Phi_{\left\\{\left.{\cal{J}}\right|d\right\\}}}g_{jd}r_{jd}^{-\alpha}\right)\right]$
(18)
$\displaystyle\mathop{=}\limits^{(\ref{eq:Prob_SINR_specific_second_term})}$
$\displaystyle\exp\left(-\frac{\pi\lambda_{\cal{J}}R^{2}\left(\frac{p^{\rm{J}}}{p^{\rm{D}}}\xi^{\rm{D}}\right)^{\frac{2}{\alpha}}}{{\rm{sinc}}\left(\frac{2}{\alpha}\right)}\right).$

##  Appendix B Secrecy Outage Probability for D2D Network

According to (7) and conditioned on ${\cal{D}}_{1}$, the secrecy outage
probability for D2D transmitter $d$ is

$\displaystyle{\mathbb{P}}\left[\left.{\rm{SINR}}_{d\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}}\right)\geq\xi^{\rm{W}}\right|{\cal{D}}_{1}\right]={\mathbb{P}}\left[\frac{p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}}{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}+N_{\left.w\right|_{d}}}\geq\xi^{\rm{W}}\right]$
(19) $\displaystyle=$
$\displaystyle{\mathbb{P}}\left[g_{d\left.w\right|_{d}}\geq\frac{\xi^{\rm{W}}r_{d\left.w\right|_{d}}^{\alpha}}{p^{\rm{D}}}\left(S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}+N_{\left.w\right|_{d}}\right)\right]$
$\displaystyle=$
$\displaystyle{\mathbb{E}}\left[\exp\left(-\frac{\xi^{\rm{W}}r_{d\left.w\right|_{d}}^{\alpha}}{p^{\rm{D}}}\left(S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}+N_{\left.w\right|_{d}}\right)\right)\right]$
$\displaystyle=$
$\displaystyle{\mathbb{E}}_{r_{d\left.w\right|_{d}}}\left[\exp\left(-\frac{\xi^{\rm{W}}r_{d\left.w\right|_{d}}^{\alpha}}{p^{\rm{D}}}N_{\left.w\right|_{d}}\right){\mathbb{E}}_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}}\left[\exp\left(-\frac{\xi^{\rm{W}}r_{d\left.w\right|_{d}}^{\alpha}}{p^{\rm{D}}}S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}\right)\right]\right.$
$\displaystyle\left.\times{\mathbb{E}}_{S_{\left.w\right|_{d}}^{{\cal{J}}}}\left[\exp\left(-\frac{\xi^{\rm{W}}r_{d\left.w\right|_{d}}^{\alpha}}{p^{\rm{D}}}S_{\left.w\right|_{d}}^{{\cal{J}}}\right)\right]\right],$

where the probability density function (PDF) of $r_{d\left.w\right|_{d}}$ is

$f_{r_{d\left.w\right|_{d}}}\left(r\right)=\exp\left(-\lambda_{\cal{W}}\pi
r^{2}\right)2\lambda_{\cal{W}}\pi r,$ (20)
${\mathbb{E}}_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}}\left[\exp\left(-\frac{\xi^{\rm{W}}r_{d\left.w\right|_{d}}^{\alpha}}{p^{\rm{D}}}S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}\right)\right]\mathop{=}\limits^{(\ref{eq:Prob_SINR_specific_second_term})}\exp\left(-\frac{\pi\lambda_{\cal{D}}{\mathbb{P}}^{{\cal{D}}_{1}}r_{d\left.w\right|_{d}}^{2}\left(\xi^{\rm{W}}\right)^{\frac{2}{\alpha}}}{{\rm{sinc}}\left(\frac{2}{\alpha}\right)}\right),$
(21)

and

${\mathbb{E}}_{S_{\left.w\right|_{d}}^{{\cal{J}}}}\left[\exp\left(-\frac{\xi^{\rm{W}}r_{d\left.w\right|_{d}}^{\alpha}}{p^{\rm{D}}}S_{\left.w\right|_{d}}^{{\cal{J}}}\right)\right]\mathop{=}\limits^{(\ref{eq:Prob_SINR_specific_second_term})}\exp\left(-\frac{\pi\lambda_{\cal{J}}r_{d\left.w\right|_{d}}^{2}\left(\frac{p^{\rm{J}}}{p^{\rm{D}}}\xi^{\rm{W}}\right)^{\frac{2}{\alpha}}}{{\rm{sinc}}\left(\frac{2}{\alpha}\right)}\right).$
(22)

##  Appendix C False Alarm Probability for Warden

According to (5) and conditioned on ${\cal{D}}_{0}$, the FA probability for
warden $\left.w\right|_{d}$ is

$\displaystyle{\mathbb{P}}^{\text{FA}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)={\mathbb{P}}\left[\left.y_{\left.w\right|_{d}}>\tau\right|{\cal{D}}_{0}\right]$
(23) $\displaystyle=$
$\displaystyle{\mathbb{P}}\left[S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}>\tau-
N_{\left.w\right|_{d}}\right]=1-F_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}}\left(\tau-
N_{\left.w\right|_{d}}\right).$

The cumulative distribution function (CDF) of
$S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}$
can be obtained as follows:

  1. 1.

We first calculate the Laplace transform of
$S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}$ as

$\displaystyle{\mathcal{L}}\left\\{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}\right\\}\left(s\right)={\mathbb{E}}\left[\exp\left(-s\left(S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}\right)\right)\right]$
(24) $\displaystyle=$
$\displaystyle{\mathbb{E}}\left[\exp\left(-sS_{\left.w\right|_{d}}^{{\cal{D}}_{1}}\right)\right]{\mathbb{E}}\left[\exp\left(-sS_{\left.w\right|_{d}}^{{\cal{J}}}\right)\right]$
$\displaystyle\mathop{=}\limits^{(\ref{eq:Prob_SINR_specific_second_term})}$
$\displaystyle\exp\left(-\frac{\pi\lambda_{\cal{D}}{\mathbb{P}}^{{\cal{D}}_{1}}\left(sp^{\rm{D}}\right)^{\frac{2}{\alpha}}}{{\rm{sinc}}\left(\frac{2}{\alpha}\right)}\right)\exp\left(-\frac{\pi\lambda_{\cal{J}}\left(sp^{\rm{J}}\right)^{\frac{2}{\alpha}}}{{\rm{sinc}}\left(\frac{2}{\alpha}\right)}\right).$

  2. 2.

The CDF of
$S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}$ is
the inverse Laplace transform of
${\mathcal{L}}\left\\{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}\right\\}\left(s\right)$
and can be obtained by using the Bromwich inversion theorem in Chapter 2 of
[32] as follows:

$\displaystyle
F_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}}\left(t\right)=$
$\displaystyle{\mathcal{L}}^{-1}\left\\{{\mathcal{L}}\left\\{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}\right\\}\left(s\right)\right\\}\left(t\right)$
(25) $\displaystyle=$ $\displaystyle
1-\frac{1}{\pi}\int_{0}^{\infty}\frac{1}{\theta}\sin\left(\nu\theta^{\frac{2}{\alpha}}\sin\left(\frac{2\pi}{\alpha}\right)\right)\exp\left(-\nu\theta^{\frac{2}{\alpha}}\cos\left(\frac{2\pi}{\alpha}\right)-t\theta\right){\rm{d}}\theta,$

where
$\nu=\frac{\pi}{{\rm{sinc}}\left(\frac{2}{\alpha}\right)}\left(\lambda_{\cal{D}}{\mathbb{P}}^{{\cal{D}}_{1}}\left(p^{\rm{D}}\right)^{\frac{2}{\alpha}}+\lambda_{\cal{J}}\left(p^{\rm{J}}\right)^{\frac{2}{\alpha}}\right)$.

##  Appendix D Miss Detection Probability for Warden

According to (6) and conditioned on ${\cal{D}}_{1}$, the MD probability for
warden $\left.w\right|_{d}$ is

$\displaystyle{\mathbb{P}}^{\rm{MD}}_{\left.w\right|_{d}}\left(p^{\rm{D}},p^{\rm{J}},\tau\right)=$
$\displaystyle{\mathbb{P}}\left[\left.y_{\left.w\right|_{d}}<\tau\right|{\cal{D}}_{1}\right]={\mathbb{P}}\left[p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}+S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}+N_{\left.w\right|_{d}}<\tau\right]$
(26) $\displaystyle=$ $\displaystyle\int_{0}^{\tau-
N_{\left.w\right|_{d}}}f_{p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}}\left(t\right)F_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}}\left(\tau-
N_{\left.w\right|_{d}}-t\right){\rm{d}}t,$

where
$F_{S_{\left.w\right|_{d}}^{{\cal{D}}_{1}}+S_{\left.w\right|_{d}}^{{\cal{J}}}}\left(\cdot\right)$
is given in (25) and

$\displaystyle
f_{p^{\rm{D}}g_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}}\left(t\right)=\frac{{\rm{d}}}{{\rm{d}}t}F_{p^{\rm{D}}h_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}}\left(t\right)=\frac{{\rm{d}}}{{\rm{d}}t}{\mathbb{P}}\left[p^{\rm{D}}h_{d\left.w\right|_{d}}r_{d\left.w\right|_{d}}^{-\alpha}\leq
t\right]$ (27) $\displaystyle=$
$\displaystyle\frac{{\rm{d}}}{{\rm{d}}t}{\mathbb{P}}\left[h_{d\left.w\right|_{d}}\leq\frac{tr_{d\left.w\right|_{d}}^{\alpha}}{p^{\rm{D}}}\right]=\frac{{\rm{d}}}{{\rm{d}}t}{\mathbb{E}}_{r_{d\left.w\right|_{d}}}\left[1-\exp\left(-\frac{tr_{d\left.w\right|_{d}}^{\alpha}}{p^{\rm{D}}}\right)\right]$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}f_{r_{d\left.w\right|_{d}}}\left(r\right)\exp\left(-\frac{tr^{\alpha}}{p^{\rm{D}}}\right)\frac{r^{\alpha}}{p^{\rm{D}}}{\rm{d}}r$

with $f_{r_{d\left.w\right|_{d}}}\left(r\right)$ given in (20).

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