# Opportunistic Relay in Multicast Channels with Generalized Shadowed Fading
Effects: A Physical Layer Security Perspective

S. M. S. Shahriyer  A. S. M. Badrudduza  S. Shabab  Department of Electrical &
Electronic Engineering, RUET  M. K. Kundu  Department of Electrical & Computer
Engineering, RUET  Heejung Yu  Department of Electronics and Information
Engineering, Korea University, Sejong 30019, South Korea.

###### Abstract

## Abstract

Through ordinary transmissions over wireless multicast networks are greatly
hampered due to the simultaneous presence of fading and shadowing of wireless
channels, secure transmissions can be enhanced by properly exploiting random
attributes of the propagation medium. This study focuses on the utilization of
those attributes to enhance the physical layer security (PLS) performance of a
dual-hop wireless multicast network over $\kappa-\mu$ shadow-fading channel
under the wiretapping attempts of multiple eavesdroppers. In order to improve
the secrecy level, the best relay selection strategy among multiple relays is
employed. Performance analysis is carried out based on the mathematical
modeling in terms of analytical expressions of non-zero secrecy capacity
probability, secure outage probability, and ergodic secrecy capacity over
multicast relay networks. Capitalizing on those expressions, the effects of
system parameters, i.e., fading, shadowing, the number of antennas,
destination receivers, eavesdroppers, and relays, on the secrecy performance
are investigated. Numerical results show that the detrimental impacts caused
by fading and shadowing can be remarkably mitigated using the well-known
opportunistic relaying technique. Moreover, the proposed model unifies secrecy
analysis of several classical models, thereby exhibiting enormous versatility
than the existing works.

## Keywords

$\kappa-\mu$ shadowed fading, opportunistic relaying, physical layer security,
secure outage probability, wireless multicasting.

##  1 Introduction

###  1.1 Background and Related Works

Data-carrying radio waves, which propagate through communication channels,
experience several constraints, such as diffraction, scattering of waves on
the object surface, shadowing, fading, limited bandwidth, and vulnerable
nature of the wireless medium, etc. Especially, random shadowing caused by
obstacles in the local scenarios or human body exhibits some variations in the
interaction pattern of radio wave propagation. Therefore, the aim to build a
fortified network not only involves the enhancement of communicating link
performance but also the use of optimized protocols in transmitter and
receiver circuitry to prevent shadowing and swift variations in multipath
propagation conditions. To understand the effect of dominant and scattered
components of the dual shadowing process, the authors in [1] derived
expressions of probability density function (PDF), cumulative distribution
function (CDF), and Moment Generating Function (MGF) of Rician-fading
envelopes. In [2], the authors analyzed outage performance over compound
$\eta-\mu$ fading-log-normal shadowing radio channels and derived a formula
for the PDF of $\eta-\mu$ fading distribution. With a view to unifying all
classic fading models, the authors of [3] investigated the ergodic capacity
(EC) and flexibility of channels. Similarly, a trade-off between mathematical
complexity and flexibility was represented in [4] by varying different fading
parameters under Rayleigh fading distribution. An optimal rate adaptation
(ORA) scheme under composite $\kappa-\mu$ / Inverse Gamma (I-Gamma) and
$\eta-\mu$ / I-Gamma [5] fading models was investigated in [6], where the
analysis incorporated expressions of channel capacity (CP) with a view to
developing highly attractive wireless communication systems. The authors in
[7] studied average symbol error probability (SEP), the CP under ORA, channel
inversion with fixed-rate (CIFR), and truncated CIFR under I-Gamma shadowed
fading channels. A qualified analysis among Log-Normal, Inverse Gaussian [8],
Gamma, and I-Gamma distribution was also depicted. Apart from these fading
channels, the $\kappa-\mu$ shadowed fading channel gained much popularity
because of its broad spectrum of flexibility and general characterization. The
researchers in [9] proposed a $\kappa-\mu$ shadowed fading model to improve
network quality, capacity, and spectral efficiency taking human body shadowing
under consideration. Maximum ratio combining (MRC) and square-law combining
schemes over $\kappa-\mu$ shadowed fading channel was performed in [10] to
observe energy detection in wireless communication scenarios.  The
$\kappa-\mu$ shadowed model was also employed in [11] to investigate the bit
error rate (BER) considering user mobility instead of static user position.
The outage probability (OP), average bit error probability and the effective
capacity was analyzed in [12] considering double shadowed $\kappa-\mu$ fading
model. With the interest of investigating the effective rate of the multiple-
input multiple-output (MIMO) systems, the authors in [13, 14] performed
higher-order statistics analysis and proved that the properties of the
considered fading distribution could be approximated by Gamma distribution.
Multiplicative shadowing was investigated in  many works such as [15, 16],
where the authors showed the superiority of $\kappa-\mu$ / Gamma composite
fading model over $\kappa-\mu$ / log-normal line-of-sight (LOS) shadowed
fading model in an indoor off-body communication system.

Presently, the need for security enhancement between communicating devices in
wireless medium has become a major concern [17]. To compensate for the
consequences due to several hindrances such as shadowing, fading, wiretapping,
various studies have been introduced to build reliable wireless networks to
make it impossible for any eavesdropper to decode any information from the
communicating network. Wyner’s classic wiretap structure is known as the
leading model which recapitulates the importance of security enhancement over
different fading channels. Among them, shadow-fading distribution is more
admissible for being amenable than any other modern fading model and vast span
of propagation conditions. In an extension of this fact, outage performance
over log-normal shadowed Rayleigh fading channel was analyzed [18, 19], where
the authors prosecuted the Gaussian-Hermite integration approach to show that
outage performance significantly enhances standard deviations of shadowing.
Authors in [20] analyzed the impact of shadowing on numbers of antennas and
propagation conditions over secrecy performance deriving several secrecy
measures. Free space optical (FSO) links undergoing shadowed Rician and
$\alpha-\mu$ fading in [21] were analyzed to achieve a perfect secrecy level
despite severe channel constraints. To get the more generalized picture,
security over $\kappa-\mu$ shadowed fading model at a physical layer was
examined in [22, 23, 24], where the authors demonstrated that MIMO system
diversity manifests superior performance over MRC and selecting diversity in
case of security. The authors in [25] considered $\kappa-\mu$ shadow-fading
channels to observe the effect of correlation and drew a conclusion on the
fact that a correlation coefficient works as a propitious parameter in case of
improving secrecy performance at a lower signal-to-noise ratio (SNR).

Cooperative relaying is another proficient technology to enhance security in
wireless systems [26]. The impact of shadowing is inevitably minimized using
the dual-hop network which in turn helps to improve wireless links. The
authors in [27] analyzed the OP over a non-identical log normal fading channel
employing the ”best relay” selection scheme. A multi-branch-multi-hop
cooperative relaying system in the presence of co-channel interferers was
considered in [28, 29] over shadowed Nakagami-$m$ channel, where the authors
evinced that a fading parameter affects the system performance greater than
the shadowing parameter. A multiple cooperative relay-based satellite-
terrestrial system over non-identical shadowed Rician and Nakagami-$m$ fading
channels was studied in [30, 31], whether the authors came to a conclusion
that EC of the system decreases with the number of relay nodes between
satellite and ground users. The best relay selection scheme was employed in
the performance analysis of $\kappa-\mu$ shadowed fading channel with multiple
relays in [32]. The authors examined the expressions of outage probability
(OP), EC, and average BER to manifest the superiority of multiple relay
systems over all other transmission techniques. The performance of a
$\kappa-\mu$ shadowed fading model with beamforming and amplify-and-forward
(AF) relaying technique was demonstrated in [33, 34]. The author validated the
fact that the increment in the number of antennas at the transmitting earth
station (ES) provides better performance than the increment in the number of
antennas at the receiving ES.

Recently, multicast channels have earned much popularity in wireless
communication due to its versatile nature to transmit data to multiple
destinations accommodating fewer network supplies [35]. Due to such wide
spread of wireless multicast networks, lots of researches have been are
undertaken to enhance the secrecy performance. Physical layer security (PLS)
in a multicasting scenario was analyzed in [36] over quasi-static Rayleigh
fading channel where the authors derived the expressions of the probability of
non-zero secrecy multicast capacity (PNSMC) and secure outage probability for
multicasting (SOPM). A virtual MIMO antenna array scheme was employed in a
multiple relay system in [37], where the authors provided a completed
description on cooperative spatial multiplexing and derived analytical
expressions of SOPM and ergodic secrecy multicast capacity (ESMC). The authors
proved that the secrecy performance of a cooperative network completely
outweighs that of a direct network.

###  1.2 Motivation and Contributions

According to the aforementioned works, most of the researches was performed to
investigate the system improvement applying numerous technologies over various
fading channels, i.e., both generalized and multipath fading. However, there
are few works that considered wireless multicasting scenarios and the impact
of shadowing on the secure wireless multicast schemes with opportunistic
relaying and multiple eavesdroppers over multipath/generalized shadowed fading
channels has not been investigated yet. Motivated from this perspective, this
paper investigates the mathematical modeling of a secure wireless multicasting
scheme over $\kappa-\mu$ shadow-fading channels with an opportunistic relaying
technique. Here, a single sender communicates with a set of multiple
destination receivers via a set of multiple cooperative relays in the
existence of multiple eavesdroppers. $\kappa-\mu$ shadow model assumes random
fluctuations of ling of sight (LOS) component and also matches well with
experimental data of land mobile satellite (LMS) communication channel.
Moreover, a $\kappa-\mu$ fading channel is a generalized fading model, and
hence a number of classical fading models, e.g., one-sided Gaussian,
Nakagami-$m$, Rayleigh, Rician-$K$, and shadowed Rician, can be obtained as
particular cases of the proposed model. The prime contributions of the authors
are as follows:

  * •

We derive the PDF and CDF of SNRs for multicast and eavesdropper channels by
first realizing the PDF of SNRs of each individual hop and then obtaining the
PDF of dual-hop SNR with the best relay selection algorithm. To the best of
the authors’ knowledge, the derived PDF and CDF are absolutely novel and have
not been reported yet in any existing literature.

  * •

We analyze the secrecy performance utilizing novel expressions of some well-
known secrecy metrics i.e. PNSMC, SOPM, and ESMC, and quantify the effects of
each system parameter, i.e., fading parameters, shadowing, number of receive
antennas, relays, destination receivers, and eavesdroppers, etc. In comparison
to the previous literature, only the proposed work demonstrates how the
detrimental impact of shadowing on secure multicasting can be mitigated by
employing an opportunistic relaying strategy.

  * •

Finally, we verify all the numerical results corresponding to the derived
analytical expressions of secrecy metrics via Monte-Carlo simulations.

###  1.3 Organization

The organization of this paper is summarized as follows: Section 2 discusses
the proposed system model and problem formulation. The derivations of the
expressions of secrecy metrics i.e. PNSMC, SOPM, and ESMC are demonstrated in
Section 3. Section 4 provides the numerical results analysis. Finally, the
conclusion of this work is illustrated in section 5.

##  2 System Model and Problem Formulation

Figure 1: Proposed system model.

A secure wireless multicast network is shown in Fig.1, where a source, $S$
with a single antenna sends secret information to a set of $Q$ destination
receivers via $P$ relays. A set of $W$ eavesdroppers are also present in that
network which is intended to decipher the secret messages. Each relay has a
single antenna while each destination receiver and each eavesdropper are
equipped with $G_{Q}$ and $G_{W}$ antennas, respectively. In this particular
communication scenario, we assume that the distances of $Q$ and $W$ from $S$
are too large and due to masking effect and severe shadowing there are no
direct communication paths between $S$ to $Q$ as well as $S$ to $W$. Hence the
only communication path that exists is through the relay. The overall process
is performed in two phases. In the first-hop, $S$ sends messages to the relay.
Then, in the second-hop, desired messages are received by the destination
receivers from the best relay only.  It is noteworthy that AF variable gain
relaying scheme has been adopted in this work. Meanwhile, at the same time
slot, the eavesdroppers also try to steal information from a best relay.

The best relay selection is performed using the method of distributed timers,
where all the relays use their timers to estimate own instantaneous channel
gains and compete to access the wireless medium according to their own channel
conditions. In an opportunistic relaying scheme, competition among cooperative
relays offers diversity benefits in the direction of destination that enhances
secrecy rate (i.e. minimises secure outage probability) and adhere to the
‘opportunistic’ cooperation rule giving priority to the ‘best’ available relay
even when they are not chosen to transmit but rather chosen to cooperatively
listen.

The direct channel coefficients for $S$ to $a$th $(a=1,2,3,\ldots,P)$ relay
link is $f_{s,a}\in\mathcal{C}^{1\times 1}$, for $a$th relay to $b$th
$(b=1,2,3,\ldots,Q)$ destination receiver link is
$\mathbf{g}_{a,b}\in\mathcal{C}^{G_{Q}\times 1}$ $(\text{i.e.
}\mathbf{g}_{ab}=$ $[g_{1ab}\quad$ $g_{2ab}\quad$ $g_{3ab}\quad$ $\ldots\quad
g_{G_{Q}ab}]^{T})$ and for $a$th relay to $c$th $(c=1,2,3,\ldots,W)$
eavesdropper link is $\mathbf{h}_{a,c}\in\mathcal{C}^{G_{W}\times 1}$
$(\text{i.e. }\mathbf{h}_{ac}=$ $[h_{1ac}\quad$ $h_{2ac}\quad$ $h_{3ac}\quad$
$\ldots\quad h_{G_{W}ac}]^{T})$.

In the first-hop, the received signal at $a$th relay is expressed as

$\displaystyle y_{s,a}=f_{s,a}x+z_{a},$ (1)

where $x\sim\mathcal{\widetilde{N}}(0,T_{s})$ is the transmitted message
signal from $S$, $T_{s}$ is the transmit power, $\mathcal{\widetilde{N}}$ is
circularly symmetric complex Gaussian distribution with mean 0 and variance
$T_{s}$, $z_{a}\sim\mathcal{\widetilde{N}}(0,N_{a})$ indicates the additive
white gaussian noise (AWGN) imposed on $a$th relay with noise power $N_{a}$.

In the second-hop, the received signal of the $a$th best relay will be
forwarded to the receivers. Hence, the received signals at the $b$th receiver
is denoted as

$\displaystyle\mathbf{y}_{a,b}$
$\displaystyle=\mathbf{g}_{a,b}y_{s,a}+\mathbf{j}_{b}=\mathbf{g}_{a,b}(f_{s,a}x+z_{a})+\mathbf{j}_{b}$
$\displaystyle=\mathbf{d}_{ab}x+\mathbf{u}_{b},$ (2)

whereas the received signal at $c$th eavesdropper can be written as

$\displaystyle\mathbf{y}_{a,c}$
$\displaystyle=\mathbf{h}_{a,c}y_{s,a}+\mathbf{k}_{c}=\mathbf{h}_{ac}(f_{s,a}x+z_{a})+\mathbf{k}_{c}$
$\displaystyle=\mathbf{d}_{ac}x+\mathbf{v}_{c}.$ (3)

Here, $\mathbf{u}_{b}\triangleq\mathbf{g}_{a,b}z_{a}+\mathbf{j}_{b}$,
$\mathbf{v}_{c}\triangleq\mathbf{h}_{a,c}z_{a}+\mathbf{k}_{c}$,
$\mathbf{d}_{a,b}=\mathbf{g}_{a,b}f_{s,a}$,
$\mathbf{d}_{a,c}=\mathbf{h}_{a,c}f_{s,a}$,
$\mathbf{j}_{b}\sim\mathcal{\widetilde{N}}(0,N_{b}\mathbf{I}_{G_{Q}})$ and
$\mathbf{k}_{c}\sim\mathcal{\widetilde{N}}(0,N_{c}\mathbf{I}_{G_{W}})$
symbolize the noises imposed on the $b$th receiver and $c$th eavesdropper,
$N_{b}$ and $N_{c}$ represent noise powers, and $\mathbf{I}_{G}(.)$ is the
identity matrix of order $G\times G$.

We assume all the channels between source to relays ($S\rightarrow P$ links),
relays to destination receivers ($P\rightarrow Q$ links), and relays to
eavesdroppers ($P\rightarrow W$ links) undergo independent and identically
distributed (i.i.d.) $\kappa-\mu$ shadowed fading i.e. all the LOS components
are subjected to shadowing. This channel is widely used to model land mobile
satellite communication systems. Moreover, this model exhibits extreme
versatility since a wide number of multipath/generalized fading models can be
obtained as special cases from this particular model as shown in Table 1.

Table 1: Special Cases of $\kappa-\mu$ shadowed fading channel [38]. Fading Channels |  $\kappa-\mu$ Shadowed Fading Parameters
---|---
$\kappa_{b}$=$\kappa_{c}$=$\kappa_{a}$ |  $\mu_{b}$=$\mu_{c}$=$\mu_{a}$ |  $m_{b}$=$m_{c}$=$m_{a}$
One Sided Gaussian | 0 | 0.5 | $\infty$
Rayleigh | 0 | 1 | $\infty$
Nakagami-$m$ | 0 | $m$ | $\infty$
Shadowed Rician | $K$ | 1 | $m$
Rician-$K$ | $K$ | 1 | $\infty$

###  2.1 Channel Model

The instantaneous SNRs of $S\rightarrow P$, $P\rightarrow Q$ and $P\rightarrow
W$ links are respectively given by
$\lambda_{s,a}=\frac{T_{s}}{N_{a}}\|f_{s,a}\|^{2}$,
$\lambda_{a,b}=\frac{P_{a}}{N_{b}}\|\mathbf{g}_{a,b}\|^{2}$, and
$\lambda_{a,c}=\frac{P_{a}}{N_{c}}\|\mathbf{h}_{a,c}\|^{2}$, where $P_{a}$ is
the transmit signal power from the relay. The PDF of respective SNRs are shown
below.

####  2.1.1 PDF of $\lambda_{s,a}$

The PDF of $\lambda_{s,a}$ is given by [39, eq. 1]

$\displaystyle
f_{s,a}(\lambda)=\alpha_{1}e^{-\mathcal{A}_{2}\lambda}\lambda^{\mu_{a}-1}{{}_{1}F_{1}}(m_{a},\mu_{a};\alpha_{3}\lambda),$
(4)

where
$\alpha_{1}=\frac{\mu_{a}^{\mu_{a}}m_{a}^{m_{a}}(1+\kappa_{a})^{\mu_{a}}}{\Gamma(\mu_{a})\bar{\lambda}_{sa}^{\mu_{a}}(\mu_{a}\kappa_{a}+m_{a})^{m_{a}}}$,
$\mathcal{A}_{2}=\frac{\mu_{a}(1+\kappa_{a})}{\bar{\lambda}_{sa}}$,
$\alpha_{3}=\frac{\mu_{a}^{2}\kappa_{a}(1+\kappa_{a})}{(\mu_{a}\kappa_{a}+m_{a})\bar{\lambda}_{sa}}$,
the average SNR of $S\rightarrow P$ link channel is $\bar{\lambda}_{sa}$,
$\kappa_{a}$ is the ratio of the powers between dominant and scattered
components, $\mu_{a}$ is the number of clusters, $m_{a}$ is the Nakagami-$m$
faded shadowing component and ${}_{1}F_{1}(.,.;.)$ is the confluent hyper-
geometric function which can be expressed as
${}_{1}F_{1}(x_{1},y_{1};z_{1})=\frac{\Gamma(y_{1})}{\Gamma(x_{1})}\sum_{d_{1}=0}^{\infty}\frac{\Gamma(x_{1}+d_{1})z_{1}^{d_{1}}}{\Gamma(y_{1}+d_{1})d_{1}!}$
[40, eq. 13]. Hence, finally $f_{s,a}(\lambda)$ can be written as

$\displaystyle
f_{s,a}(\lambda)=\sum_{e_{1}=0}^{\infty}\mathcal{A}_{1}e^{-\mathcal{A}_{2}\lambda}\lambda^{\mathcal{A}_{3}},$
(5)

where $\mathcal{A}_{1}=\alpha_{1}\alpha_{e_{1}}$,
$\alpha_{e_{1}}=\frac{\Gamma(\mu_{a})\Gamma(m_{a}+e_{1})\alpha_{3}^{e_{1}}}{\Gamma(m_{a})\Gamma(\mu_{a}+e_{1})e_{1}!}$,
and $\mathcal{A}_{3}=\mu_{a}-1+e_{1}$.

####  2.1.2 PDF of $\lambda_{a,b}$

Similar to (5), PDF of $\lambda_{a,b}$ can be written as [39, eq. 1]

$\displaystyle
f_{a,b}(\lambda)=\sum_{e_{2}=0}^{\infty}\mathcal{B}_{1}e^{-\mathcal{B}_{2}\lambda}\lambda^{\mathcal{B}_{3}},$
(6)

where
$\mathcal{B}_{1}=\beta_{1}\beta_{e_{2}}$,$\beta_{1}=\frac{(G_{Q}\mu_{b})^{G_{Q}\mu_{b}}(G_{Q}m_{b})^{G_{Q}m_{b}}(1+\kappa_{b})^{G_{Q}\mu_{b}}}{\Gamma(G_{Q}\mu_{b})(\bar{\lambda}_{ab})^{G_{Q}\mu_{b}}(G_{Q}\mu_{b}\kappa_{b}+G_{Q}m_{b})^{G_{Q}m_{b}}}$,
$\beta_{e_{2}}=\frac{\Gamma(G_{Q}\mu_{b})\Gamma(G_{Q}m_{b}+e_{2})\beta_{2}^{e_{2}}}{\Gamma(G_{Q}m_{b})\Gamma(G_{Q}\mu_{b}+e_{2})e_{2}!}$,$\beta_{2}=\frac{G_{Q}^{2}\mu_{b}^{2}\kappa_{b}(1+\kappa_{b})}{(G_{Q}\mu_{b}\kappa_{b}+G_{Q}m_{b})\bar{\lambda}_{ab}}$,
$\mathcal{B}_{2}=\frac{G_{Q}\mu_{b}(1+\kappa_{b})}{\bar{\lambda}_{ab}}$,
$\mathcal{B}_{3}=G_{Q}\mu_{b}-1+e_{2}$, the average SNR of $P\rightarrow Q$
link channel is $\bar{\lambda}_{ab}$ and shape parameters corresponding to
$P\rightarrow Q$ link are denoted by $\kappa_{b}$, $\mu_{b}$ and $m_{b}$.

####  2.1.3 PDF of $\lambda_{a,c}$

The PDF of $\lambda_{a,c}$ can be expressed as [39, eq. 1]

$\displaystyle
f_{a,c}(\lambda)=\sum_{e_{3}=0}^{\infty}\mathcal{C}_{1}e^{-\mathcal{C}_{2}\lambda}\lambda^{\mathcal{C}_{3}},$
(7)

where
$\mathcal{C}_{1}=\iota_{1}\iota_{e_{3}}$,$\iota_{1}=\frac{(G_{W}\mu_{c})^{G_{W}\mu_{c}}(G_{W}m_{c})^{G_{W}m_{c}}(1+\kappa_{c})^{G_{W}\mu_{c}}}{\Gamma(G_{W}\mu_{c})(\bar{\lambda}_{ac})^{G_{W}\mu_{c}}(G_{W}\mu_{c}\kappa_{c}+G_{W}m_{c})^{G_{W}m_{c}}}$,
$\iota_{e_{3}}=\frac{\Gamma(G_{W}\mu_{c})\Gamma(G_{W}m_{c}+e_{3})\iota_{2}^{e_{3}}}{\Gamma(G_{W}m_{c})\Gamma(G_{W}u_{c}+e_{3})e_{3}!}$,
$\iota_{2}=\frac{G_{W}^{2}\mu_{c}^{2}\kappa_{c}(1+\kappa_{c})}{(G_{W}\mu_{c}\kappa_{c}+G_{W}m_{c})\bar{\lambda}_{ac}}$,
$\mathcal{C}_{2}=\frac{G_{W}\mu_{c}(1+\kappa_{c})}{\bar{\lambda}_{ac}}$,
$\mathcal{C}_{3}=G_{W}\mu_{c}-1+e_{3}$, the average SNR of $P\rightarrow W$
link is $\bar{\lambda}_{ac}$ and $\kappa_{c}$, $\mu_{c}$ and $m_{c}$ symbolize
the shape parameters corresponding to $P\rightarrow W$ link.

###  2.2 PDFs of Dual-hop SNRs

Denoting SNRs of $S\rightarrow Q$ and $S\rightarrow W$ links by
$\lambda_{s,b}$ and $\lambda_{s,c}$, respectively, the PDFs of $\lambda_{s,b}$
and $\lambda_{s,c}$ are defined as

$\displaystyle f_{s,b}(\lambda)=\frac{dF_{s,b}(\lambda)}{d\lambda},$ (8)
$\displaystyle f_{s,c}(\lambda)=\frac{dF_{s,c}(\lambda)}{d\lambda},$ (9)

where $F_{s,b}(\lambda)$, and $F_{s,c}(\lambda)$ express the CDFs of
$\lambda_{s,b}$, and $\lambda_{s,c}$. The CDF of $\lambda_{s,b}$ is defined as

$\displaystyle
F_{s,b}(\lambda)=1-Pr(\lambda_{s,a}>\lambda_{s,b})Pr(\lambda_{a,b}>\lambda_{s,b}),$
(10)

where $Pr(\lambda_{s,a}>\lambda_{s,b})$ and $Pr(\lambda_{a,b}>\lambda_{s,b})$
are the complementary cumulative distribution functions (CCDFs) of
$\lambda_{s,a}$ and $\lambda_{a,b}$, and the CCDFs are respectively defined as

$\displaystyle
Pr(\lambda_{s,a}>\lambda_{s,b})=\int_{\lambda_{s,b}}^{\infty}f_{s,a}(\lambda)d\lambda,$
(11) $\displaystyle
Pr(\lambda_{a,b}>\lambda_{s,b})=\int_{\lambda_{s,b}}^{\infty}f_{a,b}(\lambda)d\lambda.$
(12)

Substituting (5) into (11) and executing integration using the following
identity of [41, eq. 3.351.2], we get

$\displaystyle
Pr(\lambda_{s,a}>\lambda_{s,b})=\sum_{e_{1}=0}^{\infty}\mathcal{A}_{1}\mathcal{A}_{2}^{-\mu_{a}-e_{1}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda_{s,b}),$
(13)

where $\Gamma(.,.)$ denotes the upper incomplete gamma function. Further,
substituting (6) into (12) and performing integration, we have

$\displaystyle
Pr(\lambda_{a,b}>\lambda_{s,b})=\sum_{e_{2}=0}^{\infty}\mathcal{B}_{1}\mathcal{B}_{2}^{-G_{Q}\mu_{b}-e_{2}}\Gamma(G_{Q}\mu_{b}+e_{2},\mathcal{B}_{2}\lambda_{s,b}).$
(14)

Deploying (13) and (14) into (10), the CDF of $\lambda_{s,b}$ is obtained as

$\displaystyle F_{s,b}(\lambda)$
$\displaystyle=1-\sum_{e_{2}=0}^{\infty}\sum_{e_{1}=0}^{\infty}\lambda_{e_{2}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda)$
$\displaystyle\times\Gamma(G_{Q}\mu_{b}+e_{2},\mathcal{B}_{2}\lambda),$ (15)

where
$\lambda_{e_{2}}=\mathcal{A}_{1}\mathcal{B}_{1}\mathcal{A}_{2}^{-\mu_{a}-e_{1}}\mathcal{B}_{2}^{-G_{Q}\mu_{b}-e_{2}}$.
Now, substituting (15) into (8) and performing differentiation with respect to
$\lambda_{s,b}$, the PDF of $\lambda_{s,b}$ is found as

$\displaystyle f_{s,b}(\lambda)$
$\displaystyle=\sum_{s_{2}=0}^{\infty}\sum_{e_{4}=0}^{\infty}\frac{\lambda_{e_{4}}\lambda^{\mu_{a}+s_{2}-1}}{e^{\mathcal{A}_{2}\lambda}}\Gamma(G_{Q}\mu_{b}+e_{4},\mathcal{B}_{2}\lambda)$
$\displaystyle+\sum_{s_{3}=0}^{\infty}\sum_{e_{5}=0}^{\infty}\frac{\lambda_{e_{5}}\lambda^{G_{Q}\mu_{b}+e_{5}-1}}{e^{\mathcal{B}_{2}\lambda}}\Gamma(\mu_{a}+s_{3},\mathcal{A}_{2}\lambda),$
(16)

where
$\lambda_{e_{4}}=\mathcal{A}_{1}\mathcal{B}_{1}\mathcal{A}_{2}^{-\mu_{a}-s_{2}}\mathcal{B}_{2}^{-G_{Q}\mu_{b}-e_{4}}$
and
$\lambda_{e_{5}}=\mathcal{A}_{1}\mathcal{B}_{1}\mathcal{A}_{2}^{-\mu_{a}-s_{3}}\mathcal{B}_{2}^{-G_{Q}\mu_{b}-e_{5}}$.
Similarly, the CDF of $\lambda_{s,c}$ can be obtained as

$\displaystyle F_{s,c}(\lambda)$
$\displaystyle=1-\sum_{e_{3}=0}^{\infty}\sum_{e_{1}=0}^{\infty}\lambda_{e_{3}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda)$
$\displaystyle\times\Gamma(G_{W}\mu_{c}+e_{3},\mathcal{C}_{2}\lambda),$ (17)

where
$\lambda_{e_{3}}=\mathcal{A}_{1}\mathcal{C}_{1}\mathcal{A}_{2}^{-\mu_{a}-e_{1}}\mathcal{C}_{2}^{-G_{W}\mu_{c}-e_{3}}$.
Furthermore, replacing (17) into (9), the PDF of $\lambda_{s,c}$ is obtained
as

$\displaystyle f_{s,c}(\lambda)$
$\displaystyle=\sum_{s_{4}=0}^{\infty}\sum_{e_{6}=0}^{\infty}\frac{\lambda_{e_{6}}\lambda^{\mu_{a}+s_{4}-1}}{e^{\mathcal{A}_{2}\lambda}}\Gamma(G_{W}\mu_{c}+e_{6},\mathcal{C}_{2}\lambda)$
$\displaystyle+\sum_{s_{5}=0}^{\infty}\sum_{e_{7}=0}^{\infty}\frac{\lambda_{e_{7}}\lambda^{G_{W}\mu_{c}+e_{7}-1}}{e^{\mathcal{C}_{2}\lambda}}\Gamma(\mu_{a}+s_{5},\mathcal{A}_{2}\lambda),$
(18)

where
$\lambda_{e_{6}}=\mathcal{A}_{1}\mathcal{C}_{1}\mathcal{A}_{2}^{-\mu_{a}-s_{4}}\mathcal{C}_{2}^{-G_{W}\mu_{c}-e_{6}}$
and
$\lambda_{e_{7}}=\mathcal{A}_{1}\mathcal{C}_{1}\mathcal{A}_{2}^{-\mu_{a}-s_{5}}\mathcal{C}_{2}^{-G_{W}\mu_{c}-e_{7}}$.

###  2.3 Best Relay Selection

Let $\lambda_{b}^{*}$ denote the SNR between best relay and $b$th receiver
which is expressed as

$\displaystyle\lambda_{b}^{*}=arg_{a\in\varpi}^{max}min(\lambda_{s,a},\lambda_{a,b}),$
(19)

where $\varpi={1,2,\ldots,P}$ is the relay set. The CDF of $\lambda_{b}^{*}$
is demonstrated as

$\displaystyle F_{*,b}(\lambda)=[F_{s,b}(\lambda)]^{P}.$ (20)

Hence, substituting (15) into (20), the CDF of $\lambda_{b}^{*}$ is derived as

$\displaystyle F_{*,b}(\lambda)$
$\displaystyle=\Big{[}1-\sum_{e_{2}=0}^{\infty}\sum_{e_{1}=0}^{\infty}\lambda_{e_{2}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda)$
$\displaystyle\times\Gamma(G_{Q}\mu_{b}+e_{2},\mathcal{B}_{2}\lambda)\Big{]}^{P}.$
(21)

Differentiating (20) with respect to $\lambda_{s,b}$, the PDF of
$\lambda_{b}^{*}$ is obtained as

$\displaystyle f_{*,b}(\lambda)=Pf_{s,b}(\lambda)[F_{s,b}(\lambda)]^{P-1}.$
(22)

Again, deploying (15) and (16) into (22), $f_{*,b}(\lambda)$ is given by

$\displaystyle f_{*,b}(\lambda)$
$\displaystyle=P\biggl{[}\sum_{s_{2}=0}^{\infty}\sum_{e_{4}=0}^{\infty}\frac{\lambda_{e_{4}}\lambda^{\mu_{a}+s_{2}-1}}{e^{\mathcal{A}_{2}\lambda}}\Gamma(G_{Q}\mu_{b}+e_{4},\mathcal{B}_{2}\lambda)$
$\displaystyle+\sum_{s_{3}=0}^{\infty}\sum_{e_{5}=0}^{\infty}\frac{\lambda_{e_{5}}\lambda^{G_{Q}\mu_{b}+e_{5}-1}}{e^{\mathcal{B}_{2}\lambda}}\Gamma(\mu_{a}+s_{3},\mathcal{A}_{2}\lambda)\biggl{]}\Big{[}1$
$\displaystyle-\sum_{e_{2}=0}^{\infty}\sum_{e_{1}=0}^{\infty}\lambda_{e_{2}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda)\Gamma(G_{Q}\mu_{b}+e_{2},\mathcal{B}_{2}\lambda)\Big{]}^{P-1}.$
(23)

Similar to (19), the SNR between best relay and $c$th eavesdropper denoted by
$\lambda_{c}^{*}$ is explained as

$\displaystyle\lambda_{c}^{*}=arg_{a\in\varpi}^{max}min(\lambda_{s,a},\lambda_{a,c}),$
(24)

the CDF of which is given by

$\displaystyle F_{*,c}(\lambda)=[F_{s,c}(\lambda)]^{P}.$ (25)

Substituting (17) into (25), the CDF of $\lambda_{c}^{*}$ is obtained as

$\displaystyle F_{*,c}(\lambda)$
$\displaystyle=\Big{[}1-\sum_{e_{3}=0}^{\infty}\sum_{e_{1}=0}^{\infty}\lambda_{e_{3}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda)$
$\displaystyle\times\Gamma(G_{W}\mu_{c}+e_{3},\mathcal{C}_{2}\lambda)\Big{]}^{P}.$
(26)

Further, differentiating (25) with respect to $\lambda_{s,c}$, and
substituting (17) and (18) into it, the PDF of $\lambda_{c}^{*}$ is derived as

$\displaystyle f_{*,c}(\lambda)$
$\displaystyle=P\biggl{[}\sum_{s_{4}=0}^{\infty}\sum_{e_{6}=0}^{\infty}\frac{\lambda_{e_{6}}\lambda^{\mu_{a}+s_{4}-1}}{e^{\mathcal{A}_{2}\lambda}}\Gamma(G_{W}\mu_{c}+e_{6},\mathcal{C}_{2}\lambda)$
$\displaystyle+\sum_{s_{5}=0}^{\infty}\sum_{e_{7}=0}^{\infty}\frac{\lambda_{e_{7}}\lambda^{G_{W}\mu_{c}+e_{7}-1}}{e^{\mathcal{C}_{2}\lambda}}\Gamma(\mu_{a}+s_{5},\mathcal{A}_{2}\lambda)\biggl{]}\Big{[}1$
$\displaystyle-\sum_{e_{3}=0}^{\infty}\sum_{e_{1}=0}^{\infty}\lambda_{e_{3}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda)\Gamma(G_{W}\mu_{c}+e_{3},\mathcal{C}_{2}\lambda)\Big{]}^{P-1}.$
(27)

###  2.4 Modeling of Multicast Channels

Note that we consider multiple destination receivers ($Q$) each of which can
receive the multicast messages at the same instant. To ascertain a secure
communication with each receiver, we demonstrate secrecy analysis considering
the worst possible scenario which includes taking into consideration the
minimum SNR among all receivers as denoted by
$\lambda_{min}=min_{1<b<Q}\lambda_{b}^{*}$. Hence, it is clear from this
consideration that if the proposed system is capable of protecting multicast
information from being eavesdropped for the worst case, then for all other
cases (i.e. better than worst case), the system will undoubtedly be secure.
Since, $\lambda_{1}^{*}$, $\lambda_{2}^{*}$, $\ldots$, $\lambda_{Q}^{*}$ are
all independent, using order statistics, the PDF of $\lambda_{min}$ can be
defined as [42]

$\displaystyle
f_{\lambda_{min}}(\lambda)=Qf_{*,b}(\lambda)[1-F_{*,b}(\lambda)]^{Q-1}.$ (28)

Now, substituting (21) and (23) into (28), $f_{\lambda_{min}}(\lambda)$ is
obtained as

$\displaystyle f_{\lambda_{min}}(\lambda)$
$\displaystyle=PQ\biggl{[}\sum_{s_{2}=0}^{\infty}\sum_{e_{4}=0}^{\infty}\frac{\lambda_{e_{4}}\lambda^{\mu_{a}+s_{2}-1}}{e^{\mathcal{A}_{2}\lambda}}\Gamma(G_{Q}\mu_{b}+e_{4},\mathcal{B}_{2}\lambda)$
$\displaystyle+\sum_{s_{3}=0}^{\infty}\sum_{e_{5}=0}^{\infty}\frac{\lambda_{e_{5}}\lambda^{G_{Q}\mu_{b}+e_{5}-1}}{e^{\mathcal{B}_{2}\lambda}}\Gamma(\mu_{a}+s_{3},\mathcal{A}_{2}\lambda)\biggl{]}\Big{[}1$
$\displaystyle-\sum_{e_{2}=0}^{\infty}\sum_{e_{1}=0}^{\infty}\lambda_{e_{2}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda)\Gamma(G_{Q}\mu_{b}+e_{2},\mathcal{B}_{2}\lambda)\Big{]}^{P-1}$
$\displaystyle\times\Big{[}1-\big{[}1-\sum_{e_{2}=0}^{\infty}\sum_{e_{1}=0}^{\infty}\lambda_{e_{2}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda)$
$\displaystyle\times\Gamma(G_{Q}\mu_{b}+e_{2},\mathcal{B}_{2}\lambda)\big{]}^{P}\Big{]}^{Q-1}.$
(29)

Utilizing the identity of [41, eq. 1.111], (29) can be simplified as

$\displaystyle f_{\lambda_{min}}(\lambda)$
$\displaystyle=PQ\sum_{e_{8}=0}^{Q-1}\sum_{e_{9}=0}^{P+Pe_{8}-1}\lambda_{e_{9}}\biggl{[}\sum_{s_{3}=0}^{\infty}\sum_{e_{5}=0}^{\infty}\frac{\lambda_{e_{5}}\lambda^{G_{Q}\mu_{b}+e_{5}-1}}{e^{\mathcal{B}_{2}\lambda}}$
$\displaystyle\times\Gamma(\mu_{a}+s_{3},\mathcal{A}_{2}\lambda)+\sum_{s_{2}=0}^{\infty}\sum_{e_{4}=0}^{\infty}\frac{\lambda_{e_{4}}\lambda^{\mu_{a}+s_{2}-1}}{e^{\mathcal{A}_{2}\lambda}}$
$\displaystyle\times\Gamma(G_{Q}\mu_{b}+e_{4},\mathcal{B}_{2}\lambda)\biggl{]}\Big{[}\sum_{e_{2}=0}^{\infty}\sum_{e_{1}=0}^{\infty}\lambda_{e_{2}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda)$
$\displaystyle\times\Gamma(G_{Q}\mu_{b}+e_{2},\mathcal{B}_{2}\lambda)\Big{]}^{e_{9}},$
(30)

where $\lambda_{e_{9}}=\frac{(_{\;e_{8}}^{Q-1})(_{\quad
e_{9}}^{P+Pe_{8}-1})}{(-1)^{-e_{8}-e_{9}}}$. Applying the identity of [41, eq.
8.352.7], (30) is further simplified as

$\displaystyle f_{\lambda_{min}}(\lambda)$
$\displaystyle=PQ\sum_{e_{8}=0}^{Q-1}\sum_{e_{9}=0}^{P+Pe_{8}-1}\lambda_{e_{9}}e^{-(\mathcal{A}_{2}+\mathcal{B}_{2})\lambda}\big{[}\gamma_{1}(\lambda)\big{]}^{e_{9}}$
$\displaystyle\times\Big{[}\sum_{s_{3}=0}^{\infty}\sum_{e_{5}=0}^{\infty}\sum_{e_{11}=0}^{\mu_{a}+s_{3}-1}\lambda_{e_{11}}\lambda^{G_{Q}\mu_{b}+e_{5}+e_{11}-1}$
$\displaystyle+\sum_{s_{2}=0}^{\infty}\sum_{e_{4}=0}^{\infty}\sum_{e_{10}=0}^{G_{Q}\mu_{b}+e_{4}-1}\lambda_{e_{10}}\lambda^{\mu_{a}+s_{2}+e_{10}-1}\Big{]},$
(31)

where
$\lambda_{e_{10}}=\frac{\lambda_{e_{4}}\Gamma(e_{4}+G_{Q}\mu_{b})}{e_{10}!\mathcal{B}_{2}^{-e_{10}}}$
and
$\lambda_{e_{11}}=\frac{\lambda_{e_{5}}\Gamma(e_{5}+\mu_{a})}{e_{11}!\mathcal{A}_{2}^{-e_{11}}}$.
Here $\gamma_{1}(\lambda)$ is denoted as

$\displaystyle\gamma_{1}(\lambda)$
$\displaystyle=\sum_{e_{1}=0}^{\infty}\sum_{e_{2}=0}^{\infty}\sum_{e_{12}=0}^{\mu_{a}+e_{1}-1}\sum_{e_{13}=0}^{G_{Q}\mu_{b}+e_{2}-1}\lambda_{e_{13}}\lambda^{(e_{12}+e_{13})}$
$\displaystyle\times e^{-(\mathcal{A}_{2}+\mathcal{B}_{2})\lambda},$ (32)

where
$\lambda_{e_{13}}=\frac{\mathcal{A}_{1}\mathcal{B}_{1}\Gamma(e_{1}+\mu_{a})\mathcal{B}_{2}^{-G_{Q}\mu_{b}-e_{2}+e_{13}}\Gamma(e_{2}+G_{Q}\mu_{b})}{e_{12}!e_{13}!\mathcal{A}_{2}^{\mu_{a}+e_{1}-e_{12}}}$.
Applying the multinomial theorem of [43, eq. 7], we obtain

$\displaystyle[\gamma_{1}(\lambda)]^{e_{9}}$
$\displaystyle=\sum_{\varpi_{e_{9}}}(_{g_{0,0,0,0},\cdots,g_{e_{1},e_{2},e_{12},e_{13}},\cdots,e_{\infty,\infty,\mu_{a}+e_{1}-1,G_{Q}\mu_{b}+e_{2}-1}}^{\quad\qquad\qquad\qquad
e_{9}})$
$\displaystyle\times\Omega_{\varpi_{e_{9}}}e^{-\Lambda_{\varpi_{e_{9}}}\lambda}\lambda^{\Psi_{\varpi_{e_{9}}}},$
(33)

where $(_{i_{1},i_{2},\cdots,i_{m}}^{\qquad i})=\frac{i!}{i_{1}!i_{2}!\cdots
i_{m}!}$ symbolizes the multinomial coefficients,
$\Omega_{\varpi_{e_{9}}}=\prod_{e_{1},e_{2},e_{12},e_{13}}\lambda_{e_{13}}^{g_{e_{1},e_{2},e_{12},e_{13}}}$,
$\Psi_{\varpi_{e_{9}}}=\sum_{e_{1}}$
$\sum_{e_{2}}\sum_{e_{12}}\sum_{e_{13}}(e_{12}+e_{13})g_{e_{1},e_{2},e_{12},e_{13}}$
and $\Lambda_{\varpi_{e_{9}}}=\sum_{e_{1}}\sum_{e_{2}}$
$\sum_{e_{12}}\sum_{e_{13}}(\mathcal{A}_{2}+\mathcal{B}_{2})g_{e_{1},e_{2},e_{12},e_{13}}$.
For each element of $\varpi_{e_{9}}$, the sum in (33) is to be performed,
which can be defined as

$\displaystyle\varpi_{e_{9}}$
$\displaystyle=[(g_{0,0,0,0},\cdots,g_{e_{1},e_{2},e_{12},e_{13}},\cdots,e_{\infty,\infty,\mu_{a}+e_{1}-1,G_{Q}\mu_{b}+e_{2}-1}):$
$\displaystyle g_{e_{1},e_{2},e_{12},e_{13}}\in\mathbb{N},0\leq
e_{1}\leq\infty,0\leq e_{2}\leq\infty,$ $\displaystyle 0\leq
e_{12}\leq\mu_{a}+e_{1}-1,0\leq e_{13}\leq G_{Q}\mu_{b}+e_{2}-1;$
$\displaystyle\sum_{e_{1},e_{2},e_{12},e_{13}}g_{e_{1},e_{2},e_{12},e_{13}}=e_{9}].$
(34)

Finally, substituting (33) into (31), we get

$\displaystyle f_{\lambda_{min}}(\lambda)$
$\displaystyle=\sum_{e_{8}=0}^{Q-1}\sum_{e_{9}=0}^{P+Pe_{8}-1}\sum_{\varpi_{e_{9}}}\Big{(}\sum_{s_{2}=0}^{\infty}\sum_{e_{4}=0}^{\infty}\sum_{e_{10}=0}^{G_{Q}\mu_{b}+e_{4}-1}\mathcal{J}_{1}\lambda^{\mathcal{J}_{3}}$
$\displaystyle+\sum_{s_{3}=0}^{\infty}\sum_{e_{5}=0}^{\infty}\sum_{e_{11}=0}^{\mu_{a}+s_{3}-1}\mathcal{J}_{2}\lambda^{\mathcal{J}_{4}}\Big{)}e^{-\mathcal{J}_{5}\lambda},$
(35)

where
$\lambda_{\varpi_{e_{9}}}=\big{(}_{g_{0,0,0,0},\cdots,g_{e_{1},e_{2},e_{12},e_{13}},\cdots,e_{\infty,\infty,\mu_{a}+e_{1}-1,G_{Q}\mu_{b}+e_{2}-1}}^{\quad\qquad\qquad\qquad
e_{9}}\big{)}$
$\times\Omega_{\varpi_{e_{9}}}$,
$\mathcal{J}_{1}=PQ\lambda_{e_{9}}\lambda_{e_{10}}$
$\lambda_{\varpi_{e_{9}}}$, $\mathcal{J}_{2}=PQ\lambda_{e_{9}}$
$\lambda_{e_{11}}\lambda_{\varpi_{e_{9}}}$,
$\mathcal{J}_{3}=\mu_{a}+\Psi_{\varpi_{e_{9}}}+s_{2}+e_{10}-1$,
$\mathcal{J}_{4}=G_{Q}\mu_{b}+\Psi_{\varpi_{e_{9}}}+e_{5}+e_{11}-1$, and
$\mathcal{J}_{5}=\mathcal{A}_{2}+\mathcal{B}_{2}+\Lambda_{\varpi_{e_{9}}}$.

###  2.5 Modeling of Eavesdropper Channels

In order to perform the secrecy analysis assuming the worst possible case
(i.e. maximum strength of the eavesdroppers), we herein, consider maximum SNR
among $W$ eavesdroppers as denoted by
$\lambda_{max}=max_{1<c<W}\lambda_{c}^{*}$. Similar to the multicast channels,
$\lambda_{1}^{*}$, $\lambda_{2}^{*}$, $\ldots$, $\lambda_{W}^{*}$ are
independent, and by means of order statistics, the PDF of $\lambda_{max}$ is
defined as [42]

$\displaystyle
f_{\lambda_{max}}(\lambda)=Wf_{*,c}(\lambda)[F_{*,c}(\lambda)]^{W-1}.$ (36)

Hence, substituting (26) and (27) into (36), and after some mathematical
manipulation, $f_{\lambda_{max}}(\lambda)$ is obtained as

$\displaystyle f_{\lambda_{max}}(\lambda)$
$\displaystyle=PW\biggl{[}\sum_{s_{4}=0}^{\infty}\sum_{e_{6}=0}^{\infty}\frac{\lambda_{e_{6}}\lambda^{\mu_{a}+s_{4}-1}}{e^{\mathcal{A}_{2}\lambda}}\Gamma(G_{W}\mu_{c}+e_{6},\mathcal{C}_{2}\lambda)$
$\displaystyle+\sum_{s_{5}=0}^{\infty}\sum_{e_{7}=0}^{\infty}\frac{\lambda_{e_{7}}\lambda^{G_{W}\mu_{c}+e_{7}-1}}{e^{\mathcal{C}_{2}\lambda}}\Gamma(\mu_{a}+s_{5},\mathcal{A}_{2}\lambda)\biggl{]}$
$\displaystyle\times\Big{[}1-\sum_{e_{3}=0}^{\infty}\sum_{e_{1}=0}^{\infty}\lambda_{e_{3}}\Gamma(\mu_{a}+e_{1},\mathcal{A}_{2}\lambda)$
$\displaystyle\times\Gamma(G_{W}\mu_{c}+e_{3},\mathcal{C}_{2}\lambda)\Big{]}^{PW-1}.$
(37)

Simplifying (37) similar to (31), we get

$\displaystyle f_{\lambda_{max}}(\lambda)$
$\displaystyle=PW\sum_{e_{14}=0}^{PW-1}\lambda_{e_{14}}e^{-(\mathcal{A}_{2}+\mathcal{C}_{2})\lambda}[\gamma_{3}(\lambda)]^{e_{14}}\biggl{[}\sum_{s_{4}=0}^{\infty}\sum_{e_{6}=0}^{\infty}$
$\displaystyle\times\sum_{e_{15}=0}^{G_{W}\mu_{c}+e_{6}-1}\lambda_{e_{15}}\lambda^{\mu_{a}+s_{4}+e_{15}-1}+\sum_{s_{5}=0}^{\infty}\sum_{e_{7}=0}^{\infty}$
$\displaystyle\times\sum_{e_{16}=0}^{\mu_{a}+s_{5}-1}\lambda_{e_{16}}\lambda^{G_{W}\mu_{c}+e_{7}+e_{16}-1}\biggl{]},$
(38)

where $\lambda_{e_{14}}=(-1)^{e_{14}}(_{\,\,\,\,e_{14}}^{PR-1})$,
$\lambda_{e_{15}}=\frac{\lambda_{e_{6}}\Gamma(e_{6}+G_{W}\mu_{c})}{e_{15}!\mathcal{C}_{2}^{-e_{15}}}$
and
$\lambda_{e_{16}}=\frac{\lambda_{e_{7}}\Gamma(s_{5}+\mu_{a})}{e_{16}!\mathcal{A}_{2}^{-e_{16}}}$.
Here,

$\displaystyle\gamma_{3}(\lambda)$
$\displaystyle=\sum_{e_{1}=0}^{\infty}\sum_{e_{3}=0}^{\infty}\sum_{e_{17}=0}^{\mu_{a}+e_{1}-1}\sum_{e_{18}=0}^{G_{W}\mu_{c}+e_{3}-1}\lambda_{e_{18}}\lambda^{e_{17}+e_{18}}$
$\displaystyle\times e^{-(\mathcal{A}_{2}+\mathcal{C}_{2})\lambda},$ (39)

where
$\lambda_{e_{18}}=\frac{\mathcal{A}_{1}\mathcal{C}_{1}\Gamma(e_{1}+\mu_{a})\mathcal{C}_{2}^{-G_{W}\mu_{c}-e_{3}+e_{18}}\Gamma(e_{3}+G_{W}\mu_{c})}{e_{17}!e_{18}!\mathcal{A}_{2}^{\mu_{a}+e_{1}-e_{17}}}$.
Implementing multinomial theorem, we get

$\displaystyle[\gamma_{3}(\lambda)]^{e_{14}}=\sum_{\varpi_{e_{14}}}\lambda_{\varpi_{e_{14}}}e^{-\Lambda_{\varpi_{e_{14}}}\lambda}\lambda^{\Psi_{\varpi_{e_{14}}}},$
(40)

where $\lambda_{\varpi_{e_{14}}}=\sum_{\varpi_{e_{14}}}$
$\times\big{(}_{h_{0,0,0,0},\cdots,h_{e_{1},e_{3},e_{17},e_{18}},\cdots,e_{\infty,\infty,\mu_{a}+s_{5}-1,G_{W}\mu_{c}+e_{3}-1}}^{\qquad\qquad\qquad\qquad\qquad
e_{14}}\big{)}\Omega_{\varpi_{e_{14}}}$,
$\Omega_{\varpi_{e_{14}}}=\prod_{e_{1},e_{3},e_{17},e_{18}}\lambda_{e_{18}}^{h_{e_{1},e_{3},e_{17},e_{18}}}$,
$\Psi_{\varpi_{e_{14}}}=\sum_{e_{1}}\sum_{e_{3}}$
$\sum_{e_{17}}\sum_{e_{18}}(e_{17}+e_{18})h_{e_{1},e_{3},e_{17},e_{18}}$,
$\Lambda_{\varpi_{e_{14}}}=\sum_{e_{1}}\sum_{e_{3}}\sum_{e_{17}}$
$\sum_{e_{18}}(\mathcal{A}_{2}+\mathcal{C}_{2})h_{e_{1},e_{3},e_{17},e_{18}}$.
Finally, substituting (40) into (38), we obtain

$\displaystyle f_{\lambda_{max}}(\lambda)$
$\displaystyle=\sum_{e_{14}=0}^{PW-1}\sum_{\varpi_{e_{14}}}\Big{(}\sum_{s_{4}=0}^{\infty}\sum_{e_{6}=0}^{\infty}\sum_{e_{15}=0}^{G_{W}\mu_{c}+e_{6}-1}\mathcal{J}_{6}\lambda^{\mathcal{J}_{8}}$
$\displaystyle+\sum_{s_{5}=0}^{\infty}\sum_{e_{7}=0}^{\infty}\sum_{e_{16}=0}^{\mu_{a}+s_{5}-1}\mathcal{J}_{7}\lambda^{\mathcal{J}_{9}}\Big{)}e^{-\mathcal{J}_{10}\lambda},$
(41)

where
$\mathcal{J}_{6}=PW\lambda_{e_{14}}\lambda_{e_{15}}\lambda_{\varpi_{e_{14}}}$,
$\mathcal{J}_{7}=PW\lambda_{e_{14}}\lambda_{e_{16}}\lambda_{\varpi_{e_{14}}}$,
$\mathcal{J}_{8}=\mu_{a}+\Psi_{\varpi_{e_{14}}}+s_{4}+e_{15}-1$,
$\mathcal{J}_{9}=G_{W}\mu_{c}+\Psi_{\varpi_{e_{14}}}+e_{7}+e_{16}-1$ and
$\mathcal{J}_{10}=\mathcal{A}_{2}+\mathcal{C}_{2}+\Lambda_{\varpi_{e_{14}}}$.

##  3 Performance Metrics

In the following parts, we utilize $f_{\lambda_{min}}(\lambda)$ and
$f_{\lambda_{max}}(\lambda)$ of (35) and (41) to derive analytical expressions
of three performance metrics i.e. SOPM, PNSMC, and ESMC.

$\displaystyle P_{out}(\xi_{s})$
$\displaystyle=1-\sum_{e_{8}=0}^{Q-1}\sum_{e_{9}=0}^{P+Pe_{8}-1}\sum_{\varpi_{e_{9}}}\sum_{e_{14}=0}^{PW-1}\sum_{\varpi_{e_{14}}}\Bigg{[}\sum_{s_{2}=0}^{\infty}\sum_{e_{4}=0}^{\infty}\sum_{e_{10}=0}^{G_{Q}\mu_{b}+e_{4}-1}\sum_{f_{2}=0}^{\mathcal{J}_{3}}\sum_{f_{4}=0}^{f_{2}}\bigg{[}\sum_{s_{4}=0}^{\infty}\sum_{e_{6}=0}^{\infty}\sum_{e_{15}=0}^{G_{W}\mu_{c}+e_{6}-1}\frac{\mathcal{J}_{6}\omega_{5}(\mathcal{J}_{8}+f_{4})!}{(\mathcal{J}_{10}+\mathcal{J}_{5}q_{\psi_{s}})^{\mathcal{J}_{8}+f_{4}+1}}$
$\displaystyle+\sum_{s_{5}=0}^{\infty}\sum_{e_{7}=0}^{\infty}\sum_{e_{16}=0}^{\mu_{a}+s_{5}-1}\frac{\mathcal{J}_{7}\omega_{5}(\mathcal{J}_{9}+f_{4})!}{(\mathcal{J}_{10}+\mathcal{J}_{5}q_{\psi_{s}})^{\mathcal{J}_{9}+f_{4}+1}}\bigg{]}-\sum_{s_{3}=0}^{\infty}\sum_{e_{5}=0}^{\infty}\sum_{e_{11}=0}^{\mu_{a}+s_{3}-1}\sum_{f_{3}=0}^{\mathcal{J}_{4}}\sum_{f_{5}=0}^{f_{3}}\bigg{[}\sum_{s_{4}=0}^{\infty}\sum_{e_{6}=0}^{\infty}\sum_{e_{15}=0}^{G_{W}\mu_{c}+e_{6}-1}$
$\displaystyle\times\frac{\mathcal{J}_{6}\omega_{6}(\mathcal{J}_{8}+f_{5})!}{(\mathcal{J}_{10}+\mathcal{J}_{5}q_{\psi_{s}})^{\mathcal{J}_{8}+f_{5}+1}}+\sum_{s_{5}=0}^{\infty}\sum_{e_{7}=0}^{\infty}\sum_{e_{16}=0}^{\mu_{a}+s_{5}-1}\frac{\mathcal{J}_{7}\omega_{6}(\mathcal{J}_{9}+f_{5})!}{(\mathcal{J}_{10}+\mathcal{J}_{5}q_{\psi_{s}})^{\mathcal{J}_{9}+f_{5}+1}}\bigg{]}\Bigg{]}.$
(44)

###  3.1 SOPM Analysis

Signifying the target secrecy rate by $\xi_{s}$, and the secrecy multicast
capacity as $\mathcal{C}_{s,m}$ [44], the SOPM is denoted as

$\displaystyle P_{out}(\xi_{s})$
$\displaystyle=\Pr(\mathcal{C}_{s,m}<\xi_{s})$
$\displaystyle=1-\int_{0}^{\infty}\int_{\psi_{s}}^{\infty}f_{\lambda_{min}}(\lambda_{b}^{*})f_{\lambda_{max}}(\lambda_{c}^{*})d\lambda_{b}^{*}d\lambda_{c}^{*},$
(42)

where $\psi_{s}=2^{\xi_{s}}(1+\lambda_{c}^{*})-1$ and $\xi_{s}>0$. This
definition specifies that, reliable transmission is achievable only if
$\mathcal{C}_{s,m}>\xi_{s}$, otherwise the security can not be guaranteed.
Substituting (35) and (41) into (42), we get

$\displaystyle P_{out}(\xi_{s})$
$\displaystyle=\int_{0}^{\infty}\int_{\psi_{s}}^{\infty}\sum_{e_{8}=0}^{Q-1}\sum_{e_{9}=0}^{P+Pe_{8}-1}\sum_{\varpi_{e_{9}}}\Big{(}\sum_{s_{2}=0}^{\infty}\sum_{e_{4}=0}^{\infty}\sum_{e_{10}=0}^{G_{Q}\mu_{b}+e_{4}-1}\mathcal{J}_{1}\lambda_{b}^{{*}^{\mathcal{J}_{3}}}$
$\displaystyle+\sum_{s_{3}=0}^{\infty}\sum_{e_{5}=0}^{\infty}\sum_{e_{11}=0}^{\mu_{a}+s_{3}-1}\mathcal{J}_{2}\lambda_{b}^{{*}^{\mathcal{J}_{4}}}\Big{)}e^{-\mathcal{J}_{5}\lambda_{b}^{*}}\sum_{e_{14}=0}^{PW-1}\sum_{\varpi_{e_{14}}}\Big{(}\sum_{s_{4}=0}^{\infty}\sum_{e_{6}=0}^{\infty}$
$\displaystyle\times\sum_{e_{15}=0}^{G_{W}\mu_{c}+e_{6}-1}\mathcal{J}_{6}\lambda_{c}^{{*}^{\mathcal{J}_{8}}}+\sum_{s_{5}=0}^{\infty}\sum_{e_{7}=0}^{\infty}\sum_{e_{16}=0}^{\mu_{a}+s_{5}-1}\mathcal{J}_{7}\lambda_{c}^{{*}^{\mathcal{J}_{9}}}\Big{)}$
$\displaystyle\times
e^{-\mathcal{J}_{10}\lambda_{c}^{*}}d\lambda_{c}^{*}d\lambda_{b}^{*}.$ (43)

Now, performing integration making use of [41, (eq. 3.351.2, 3.351.3)] in
(43), the closed form expression for the SOPM is given in (44), where
$p_{\psi_{s}}=2^{\xi_{s}}-1$, $q_{\psi_{s}}=2^{\xi_{s}}$,
$\omega_{5}=\frac{\mathcal{J}_{1}\mathcal{J}_{3}!(_{f_{4}}^{f_{2}})p_{\psi_{s}}^{f_{2}-f_{4}}}{f_{2}!\mathcal{J}_{5}^{\mathcal{J}_{3}-f_{2}+1}q_{\psi_{s}}^{-f_{4}}e^{\mathcal{J}_{5}p_{\psi_{s}}}}$
and
$\omega_{6}=\frac{\mathcal{J}_{2}\mathcal{J}_{4}!(_{f_{5}}^{f_{3}})p_{\psi_{s}}^{f_{3}-f_{5}}}{f_{3}!\mathcal{J}_{5}^{\mathcal{J}_{4}-f_{3}+1}q_{\psi_{s}}^{-f_{5}}e^{\mathcal{J}_{5}p_{\psi_{s}}}}$.

#### Significance of SOPM Expression

It can be seen that, (44) comprises of all the system parameters of the
proposed network which helps to evaluate the secrecy outage behaviour of the
proposed system. Hence (44) can explain and quantify the secrecy trade-off in
terms of the secrecy outage probability utilizing opportunistic relaying
mechanism. Besides, (44) also exhibits generic characteristics which helps the
design engineers to model more practical networks. Note that, as a special
case of the proposed network with $\kappa_{a}=K$, $\mu_{a}=1$ and $m_{a}=m$,
we obtain Shadowed Rician fading distribution for satellite links and for
$\kappa_{b}=\kappa_{c}=0$, $\mu_{b}=\mu_{c}=m$ and
$m_{b}=m_{c}\rightarrow\infty$, we obtain Nakagami-$m$ fading distribution for
terrestrial links. For this particular case our results with (44) totally
matches with [45, eq. 45]. Likewise, for a special case with
($\kappa_{b}=\kappa_{c}=\kappa_{a}=0$, $\mu_{b}=\mu_{c}=\mu_{a}=m$, and
$m_{b}=m_{c}=m_{a}\rightarrow\infty$), our results with (44) can be shown
similar to uncorrelated (correlation coefficient $\rightarrow$ 0 )
Nakagami-$m$ fading channel in [26].

###  3.2 PNSMC Analysis

The PNSMC can be expressed as

$\displaystyle\Pr(\mathcal{C}_{s,m}>0)=\int_{0}^{\infty}\int_{0}^{\lambda_{b}^{*}}f_{\lambda_{min}}(\lambda_{b}^{*})f_{\lambda_{max}}(\lambda_{c}^{*})d\lambda_{c}^{*}d\lambda_{b}^{*}.$
(45)

Substituting (35) and (41) into (45), $\Pr(\mathcal{C}_{s,m}>0)$ can be easily
derived. But PNSMC can also be derived from the SOPM expression as the
following.

$\displaystyle\Pr(\mathcal{C}_{s,m}>0)=1-P_{out}(\xi_{s})|_{\xi_{s}=0}.$ (46)

Hence we first substitute $P_{out}(\xi_{s})$ from (44) into (46), then set
$\xi_{s}=0$, and finally obtain the expression of PNSMC.

###  3.3 ESMC Analysis

$\displaystyle\langle\mathcal{C}_{s,m}\rangle$
$\displaystyle=\sum_{e_{8}=0}^{Q-1}\sum_{e_{9}=0}^{P+Pe_{8}-1}\sum_{\varpi_{e_{9}}}\Biggl{[}\sum_{s_{2}=0}^{\infty}\sum_{e_{4}=0}^{\infty}\sum_{e_{10}=0}^{G_{Q}\mu_{b}+e_{4}-1}\biggl{[}\sum_{d_{2}=0}^{\mathcal{J}_{3}}\frac{D_{5}}{ln(2)}\frac{(-1)^{D_{1}-1}Ei(-\mathcal{J}_{5})}{(\frac{1}{\mathcal{J}_{5}})^{D_{1}}e^{-\mathcal{J}_{5}}}+\sum_{d_{3}=1}^{D_{1}}\frac{(d_{3}-1)!}{(-\frac{1}{\mathcal{J}_{5}})^{D_{1}-d_{3}}}\biggl{]}$
$\displaystyle+\sum_{s_{3}=0}^{\infty}\sum_{e_{5}=0}^{\infty}\sum_{e_{11}=0}^{\mu_{a}+s_{3}-1}\biggl{[}\sum_{d_{4}=0}^{\mathcal{J}_{4}}\frac{D_{6}}{ln(2)}\frac{(-1)^{D_{2}-1}Ei(-\mathcal{J}_{5})}{(\frac{1}{\mathcal{J}_{5}})^{D_{2}}e^{-\mathcal{J}_{5}}}+\sum_{d_{5}=1}^{D_{2}}\frac{(d_{5}-1)!}{(-\frac{1}{\mathcal{J}_{5}})^{D_{2}-d_{5}}}\biggl{]}\Biggl{]}$
$\displaystyle-\sum_{e_{14}=0}^{PW-1}\sum_{\varpi_{e_{14}}}\Biggl{[}\sum_{s_{4}=0}^{\infty}\sum_{e_{6}=0}^{\infty}\sum_{e_{15}=0}^{G_{W}\mu_{c}+e_{6}-1}\biggl{[}\sum_{d_{6}=0}^{\mathcal{J}_{8}}\frac{D_{7}}{ln(2)}\frac{(-1)^{D_{3}-1}Ei(-\mathcal{J}_{10})}{(\frac{1}{\mathcal{J}_{10}})^{D_{3}}e^{-\mathcal{J}_{10}}}+\sum_{d_{7}=1}^{D_{3}}\frac{(d_{7}-1)!}{(-\frac{1}{\mathcal{J}_{10}})^{D_{3}-d_{7}}}\biggl{]}$
$\displaystyle-\sum_{s_{5}=0}^{\infty}\sum_{e_{7}=0}^{\infty}\sum_{e_{16}=0}^{\mu_{a}+s_{5}-1}\biggl{[}\sum_{d_{8}=0}^{\mathcal{J}_{9}}\frac{D_{8}}{ln(2)}\frac{(-1)^{D_{4}-1}Ei(-\mathcal{J}_{10})}{(\frac{1}{\mathcal{J}_{10}})^{D_{4}}e^{-\mathcal{J}_{10}}}+\sum_{d_{9}=1}^{D_{4}}\frac{(d_{9}-1)!}{(-\frac{1}{\mathcal{J}_{10}})^{D_{4}-d_{9}}}\biggl{]}\Biggl{]}.$
(48)

The ESMC can be defined as

$\displaystyle\langle\mathcal{C}_{s,m}\rangle$
$\displaystyle=\int_{0}^{\infty}\log_{2}(1+\lambda_{b}^{*})f_{\lambda_{min}}(\lambda_{b}^{*})d\lambda_{b}^{*}$
$\displaystyle-\int_{0}^{\infty}\log_{2}(1+\lambda_{c}^{*})f_{\lambda_{max}}(\lambda_{c}^{*})d\lambda_{c}^{*}.$
(47)

Replacing (35) and (41) into (47) and integrating by making use of [41, eq.
4.222.8], the novel expression for the ESMC is exhibited in (48), where
$D_{1}=\mathcal{J}_{3}-d_{2}$, $D_{2}=\mathcal{J}_{4}-d_{4}$,
$D_{3}=\mathcal{J}_{8}-d_{6}$, $D_{4}=\mathcal{J}_{9}-d_{8}$,
$D_{5}=\frac{\mathcal{J}_{1}\mathcal{J}_{3}!}{D_{1}!\mathcal{J}_{5}^{\mathcal{J}_{3}+1}}$,
$D_{6}=\frac{\mathcal{J}_{2}\mathcal{J}_{4}!}{D_{2}!\mathcal{J}_{5}^{\mathcal{J}_{4}+1}}$,
$D_{7}=\frac{\mathcal{J}_{6}\mathcal{J}_{8}!}{D_{3}!\mathcal{J}_{10}^{\mathcal{J}_{8}+1}}$,
and
$D_{8}=\frac{\mathcal{J}_{7}\mathcal{J}_{9}!}{D_{4}!\mathcal{J}_{10}^{\mathcal{J}_{9}+1}}$.

#### Significance of ESMC Expression

It is clear that how do the physical properties of the channels as well as the
system parameters affects the secrecy capacity, can be easily quantified with
(48). Besides, how does the cooperative diversity provided by the relays help
to enhance secrecy capacity in spite of harsh channel conditions can also be
evaluated easily. Since (48) is a generalized expression, it also represents
secrecy analysis over several classical models. It is noted that, as a special
case of Shadowed Rician fading distribution with ($P=0$, $Q=W=1$
$\kappa_{b}=\kappa_{c}=\kappa_{a}=K$, $\mu_{b}=\mu_{c}=\mu_{a}=1$ and
$m_{b}=m_{c}=m_{a}\rightarrow m$), our obtained results with (48) are
equivalent with the analysis in [46, eq. 51]. In parallel, utilizing (48),
another special scenario i.e. Shadowed Rician fading distribution for
satellite links and Rayleigh fading distribution ($Q=W=1$,
$\kappa_{b}=\kappa_{c}=\kappa_{a}=0$, $\mu_{b}=\mu_{c}=\mu_{a}=1$ and
$m_{b}=m_{c}=m_{a}$$\rightarrow$ $\infty$) for terrestrial links can be shown
as a special case of our model [47].

##  4 Numerical Results

In this section, the numerical results concerning the expressions of SOPM,
PNSMC, and ESMC as given in (44), (46), and (48), respectively are
demonstrated graphically in order to gain some useful insights regarding the
enhancement of security by taking the advantages of the physical properties of
the propagation medium. Since the infinite series converges rapidly after a
few terms, we take the first 25 terms to obtain the numerical results.
Additionally, the validity of the analytical expressions as described in the
previous sections are justified by Monte-Carlo (MC) simulations. 111 To
accomplish this task, we generate $\kappa-\mu$ shadowed random variables in
MATLAB and $10^{6}$ realizations of the channels are averaged to calculate
each value of the secrecy parameters. The analytical results are shown using
markers and corresponding simulation results are shown using solid lines. In
each figure, we observe a close agreement between the analytical and
simulation results which clearly justifies the validity of our analysis.

Figure 2: The PNSMC versus $\bar{\lambda}_{ab}$ for selected values of $Q$ and
$W$ when
$\bar{\lambda}_{ac}=-10dB$,$G_{Q}=G_{W}=2$,$\mu_{a}=\mu_{b}=\mu_{c}=2$,$\kappa_{a}=\kappa_{b}=\kappa_{c}=2$,
and $m_{a}=m_{b}=m_{c}=\infty$.

Figure 2 presents the PNSMC against $\bar{\lambda}_{ab}$ in which the impact
of the multicast receivers ($Q$) and eavesdroppers ($W$) are illustrated. We
assume two scenarios considering $Q=$ 5 and 10. Both the scenarios clearly
dictate that the PNSMC decreases with increasing values of $Q$, and $W$. In
the multicast scenario, due to the increase in $Q$, the bandwidth per user is
reduced which causes a reduction in the received SNR at the user node. Hence
the PNSMC performance degrades remarkably which is testified in [26]. On the
contrary, increasing $W$ escalates the strength of the eavesdroppers by
ensuring the maximum SNR at the eavesdropper node (since increasing $W$
indicates an increasing probability of obtaining maximum SNR at the
eavesdropper channel) and thus PNSMC is degraded. It is also observed from
Fig. 2 that the MC and analytical simulation are in a good agreement, which
clearly indicates the exactness of our PNSMC expression in (46).

Figure 3: The PNSMC versus $\bar{\lambda}_{ab}$ for selected values of
$\mu_{b}$, $\mu_{a}$, and $\mu_{c}$ when $m_{a}=m_{b}=m_{c}=\infty$,
$\kappa_{a}=\kappa_{b}=\kappa_{c}=1$, and $\bar{\lambda}_{ac}=0dB$.

In Fig. 3, the PNSMC is plotted against $\bar{\lambda}_{ab}$ in which the
effect of different number of clusters of relay channel ($\mu_{a}$), multicast
channels ($\mu_{b}$) as well as eavesdropper channel ($\mu_{c}$) are shown. It
is observed from the figure that the increment in the number of clusters of
relay ($\mu_{a}$) and multicast channels ($\mu_{b}$) enhances the PNSMC of the
proposed model. Because, with increasing values of $\mu_{a}$ and $\mu_{b}$, we
are increasing the end-to-end SNR indirectly by including more incoming
signals based on various diversity schemes. As a result, the total fading of
the multicast channels reduces which ensures the increment of the PNSMC. Due
to this similar reason, the PNSMC of the system degrades with $\mu_{c}$. It is
noteworthy that the impact of $\mu_{a}$ upgrading the secrecy performance is
much superior than that of $\mu_{b}$. The outcomes from [25] and [48] exhibits
the same characteristics which validate our result convincingly.

Figure 4: The PNSMC versus $\bar{\lambda}_{ab}$ for selected values of
$\bar{\lambda}_{ac}$ when $G_{Q}=G_{W}=2$, $\mu_{a}=\mu_{b}=\mu_{c}=2$,
$\kappa_{a}=\kappa_{b}=\kappa_{c}=1$, and $m_{a}=m_{b}=m_{c}=\infty$.

The MC simulation and analytical results of PNSMC as a function of
$\bar{\lambda}_{ab}$ is shown in Fig. 4, where the outcomes due to the
variation in average SNR of the eavesdropper channel ($\bar{\lambda}_{ac}$)
are depicted. It is noted that the PNSMC is degraded with the increase of
$\bar{\lambda}_{ac}$ because an increase in $\bar{\lambda}_{ac}$ indicates a
higher SNR at the eavesdropper terminals which enhances the strength of the
wiretap channels in terms of secrecy performance as shown in [49].

Figure 5: The SOPM versus $\bar{\lambda}_{ab}$ for selected values of $m_{b}$,
$m_{a}$, and $m_{c}$ when $\bar{\lambda}_{ac}=-5dB$, $G_{Q}=G_{W}=2$,
$\mu_{a}=\mu_{b}=\mu_{c}=1$, and $\kappa_{a}=\kappa_{b}=\kappa_{c}=1$.

Figure 5 demonstrates the SOPM as a function of $\bar{\lambda}_{ab}$, where
the consequences of variation in the shadowing parameter for multicast
channels ($m_{b}$) and relay channel ($m_{a}$) are represented. From the
figure, it is recognized that increment in $m_{b}$ and $m_{a}$ minimize the
shadowing effect of the multicast channel, and thus the SOPM of the model
decrease. So superior secrecy performance can be achieved if the shadowing
present in the multicast network is comparatively lower. Our numerical result
is also confirmed by MC simulation. A similar conclusion (related to
shadowing) was also drawn in [25].

Figure 6: The SOPM versus $\bar{\lambda}_{ab}$ for selected values of
$\xi_{s}$ and $\bar{\lambda}_{ac}$.

Figure 6 depicts SOPM versus $\bar{\lambda}_{ab}$ to represent the effect of
target secrecy rate ($\xi_{s}$) on the secure outage performance. We consider
two cases herein with $\bar{\lambda}_{ac}=$ 0 and -10 dB. It is noted that the
SOPM increases with $\xi_{s}$ for both cases of $\bar{\lambda}_{ac}$ as shown
in [17]. It is also noticeable that the impact of $\xi_{s}$ in case of
$\bar{\lambda}_{ac}$= -10 dB is more significant than that of
$\bar{\lambda}_{ac}$= 0 dB. Moreover, the SOPM performance improves when the
eavesdropper channel becomes worse (i.e. $\bar{\lambda}_{ac}=$ -10 dB). A good
agreement between MC and analytical outcomes proves that the SOPM derived in
(44) is accurate.

Figure 7: The SOPM versus $\bar{\lambda}_{ab}$ for selected values of $P$ when
$\bar{\lambda}_{ac}=-10dB$, $G_{Q}=G_{W}=2$, $\mu_{a}=\mu_{b}=\mu_{c}=1$,
$\kappa_{a}=\kappa_{b}=\kappa_{c}=1$, and $m_{a}=m_{b}=m_{c}=\infty$.

In Fig. 7, the SOPM is varied against $\bar{\lambda}_{ab}$ to illustrate the
consequences of differing the number of relays ($P$). From the figure, it can
be seen that an increment in $P$ offers a remarkable improvements in system’s
SOPM performance. As the number of relays are increased, the $P$ relays
compete among themselves to be the best one. Additionally, the cooperative
diversity provided by multiple relays also play a notable role in reducing the
impacts of fading and shadowing of multicast links.

Figure 8: The ESMC versus $\bar{\lambda}_{ab}$ for selected values of
$\kappa_{b}$, $\kappa_{a}$, and $\kappa_{c}$ when $\mu_{a}=\mu_{b}=\mu_{c}=1$,
$m_{a}=m_{b}=m_{c}=\infty$ and $\bar{\lambda}_{ac}=-5dB$.

Figure. 8 represents ESMC with respect to $\bar{\lambda}_{ab}$ for different
values of $\kappa_{a}$, $\kappa_{b}$ and $\kappa_{c}$. It is evident from the
figure that both $\kappa_{a}$ and $\kappa_{b}$ increases the secrecy capacity
of the proposed system. The increment of $\kappa_{a}$ and $\kappa_{b}$
enhances the dominant component as well as reduces the scattering component of
the multicast channel which results in higher SNR at the receiver and ESMC of
the proposed model. But, the capacity degrades sharply with $\kappa_{c}$ as it
upgrades the SNR of the eavesdropper channel. Same results were obtained in
[25], and [48] which undoubtedly confirms our analysis. The MC simulation also
has a tight agreement with the analytical result.

Figure 9: The ESMC versus $\bar{\lambda}_{ab}$ for selected values of $G_{Q}$
and $G_{W}$ when $\bar{\lambda}_{ac}=-10dB$,
$\kappa_{a}=\kappa_{b}=\kappa_{c}=1$, $\mu_{a}=\mu_{b}=\mu_{c}=1$, and
$m_{a}=m_{b}=m_{c}=\infty$.

In Fig. 9, ESMC is plotted against $\bar{\lambda}_{ab}$ with a view to
observing the impacts due to the variation in number of antennas of each user
($G_{Q}$) and eavesdropper ($G_{W}$). From the figure, it is observed that the
ESMC escalates if $G_{Q}$ increases. This is because an increase in $G_{Q}$
significantly reduce the fading of multicast channels by enhancing the antenna
diversity at the receiver. On the other hand, it is also noted that, ESMC
degrades if $G_{W}$ is increased. In that case, the eavesdroppers are capable
of overhearing more confidential messages from the multicast channels due to
increase in antenna diversity at the eavesdropper terminals. Even ESMC
degrades dramatically when the eavesdroppers are equipped with large number of
antennas (i.e. $G_{W}$=6). Similar outcomes are also presented in [39] which
manifest the exactness of our results.

Figure 10: The ESMC versus $\bar{\lambda}_{ab}$ for comparing performance of
different classical fading channels as a special cases of $\kappa-\mu$
shadowed fading channel when $\bar{\lambda}_{ac}=-10dB$, $G_{Q}=G_{W}=2$, and
$P=Q=W=2$.

Figure 10 exhibits a graphical representation in which the generic
characteristics of the proposed scenario is illustrated by plotting ESMC
against $\bar{\lambda}_{ab}$. Note that the secrecy analysis over generalized
shadowed model using opportunistic relaying is absent in the previous studies.
Moreover, simply reorienting the system parameters, we can generate some
existing models as special cases which is a clear indication of superiority of
our proposed work. Hence, the conclusive remarks based on the aforementioned
discussion is that this novel proposed work is more purposeful than all the
conventional multipath/ shadowed secure models.

### Comparative Analysis with Existing Related Literature

We consider generalized distribution (i.e. $\kappa-\mu$ shadowed fading) at
$S\rightarrow P$, $P\rightarrow Q$, and $P\rightarrow W$ links in the proposed
dual-hop scheme that encompasses a number of well-known fading distributions
which can be acquired as special cases of our model as shown in Table 1.
Hence, it is noteworthy that the derived expressions in (46), (44), and (48)
regarding our proposed scenario are also generalized, and can be utilized to
unify the secrecy performances of the mentioned channels in Table 1.

##  5 Conclusions

This paper considers PLS in the dual-hop secure wireless multicast relay
networks over $\kappa-\mu$ shadow-fading channels with multiple eavesdroppers.
Under this scenario, secrecy enhancement is ensured by choosing the best relay
among multiple relays. The effect of all the system parameters on the secrecy
performance of the proposed model is thoroughly observed by deriving the exact
and analytical expressions of the performance metrics, i.e., PNSMC, SOPM, and
ESMC. Also, these analytical results are numerically verified with Monte-Carlo
simulations. Form such analyses, it is shown that the secrecy performance of
this dual-hop relay communication model mostly affected by the channel
environment of the first hop, i.e., source-to-relay link, than that of the
second hops, i.e., relay-to-destination and relay-to-eavesdropper links.
Shadowing is an important aspect of this study. The security of this proposed
scenario can be improved by increasing the shadowing effect in relay-to-
eavesdropper link. The proposed generalized shadowing model with opportunistic
relay operation can also be employed to improve the security of different
classical fading scenarios irrespective of the harsh channel conditions in the
presence of a large number of multicast receivers and eavesdroppers. The
proposed model can also be extended to non-terrestrial networks where
shadowing is one of the major impairments.

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