5G-NR

    5G New Radio
3GPP

    3rd Generation Partnership Project
AWGN

    Additive White Gaussian Noise
AWGN

    additive white Gaussian noise
AN

    Artificial Noise
ES

    eavesdropper satellite
LS

    legitimate satellite
UAV

    unmanned aerial vehicles
BEP

    bit error probability
BER

    bit error rate
BS

    base station
BW

    BandWidth
CDF

    cumulative distribution function
CDF

    Cumulative Distribution Function
CDF

    cumulative distribution function
CCDF

    complementary cumulative distribution function
CCDF

    complementary CDF
CCDF

    complementary cumulative distribution function
CDMA

    Code Division Multiple Access
ch.f.

    characteristic function
CSI

    channel state information
PLS

    Physical Layer Security
DFT

    discrete Fourier transform
DL

    DownLink
DFT-s-OFDM

    Discrete Fourier Transform-spread-Orthogonal Frequency Division Multiplexing
FFT

    fast Fourier transform
FDMA

    Frequency Division Multiple Access
GA

    Gaussian approximation
GW

    Gateway
HAP

    high altitude platform
HetNet

    Heterogeneous network
IFFT

    inverse fast Fourier transform
i.i.d.

    independent, identically distributed
IoT

    Internet of Things
IR

    impulse radio
IEEE

    Institute of Electrical and Electronics Engineers
LEO

    Low Earth Orbit
LF

    likelihood function
LoS

    Line-of-Sight
LRT

    likelihood ratio test
LPWAN

    Low Power Wide Area Network
LoRaWAN

    Low power long Range Wide Area Network
NLoS

    Non-Line-of-Sight
GRV

    Gamma Random Variable
MGF

    Moment Generating Function
NB

    narrowband
NBI

    narrowband interference
NLA

    nonlinear sparse approximation
NLOS

    Non-Line of Sight
LOS

    Line of Sight
OFDMA

    Orthogonal Frequency-division Multiple Access
PDF

    Probability Density Function
PDF

    probability distribution function
PMF

    probability mass function
PMF

    probability mass function
PP

    Point Process
PPP

    Poisson Point Process
BPP

    Binomial Point Process
PCP

    Poisson Cluster Process
PGFL

    Probability Generating Functional
MCP

    Matern Cluster Process
LT

    Laplace Transform
RV

    Random Variable
RB

    resource block
SINR

    signal-to-interference and noise ratio
SIR

    signal-to-interference ratio
SNR

    signal-to-noise ratio
SR

    Shadowed-Rician
SG

    Stochastic Geometry
TBS

    terrestrial base station
PL

    path-loss
UE

    User Equipment
UL

    UpLink
WLAN

    wireless local area network
WM

    Wishart matrix
WMAN

    wireless metropolitan area network
WPAN

    wireless personal area network
WSN

    wireless sensor network
WSS

    Wide-Sense Stationary
WHO

    World Health Organization
Wi-Fi

    Wireless Fidelity
VLC

    Visible Light Communication
VPN

    Virtual Private Network
RF

    Radio Frequency
FSO

    Free Space Optics
IoST

    Internet of Space Things
GSM

    Global System for Mobile Communications
2G

    Second-generation cellular network
3G

    Third-generation cellular network
4G

    Fourth-generation cellular network
5G

    Fifth-generation cellular network
gNB

    next-generation Node-B Base Station
NR

    New Radio
UMTS

    Universal Mobile Telecommunications Service
LTE

    Long Term Evolution
QoS

    Quality of Service

# Enhancing Physical Layer Security in
LEO Satellite-Enabled IoT Network Communications

Anna Talgat, Student Member, IEEE, Ruibo Wang,
Mustafa A. Kishk, Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE Anna
Talgat, Ruibo Wang, and Mohamed-Slim Alouini are with KAUST, CEMSE division,
Thuwal 23955-6900, Saudi Arabia. Mustafa Kishk is with the Department of
Electronic Engineering, Maynooth University, W23 F2H6, Ireland. (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Corresponding author: Ruibo Wang.)

###### Abstract

The extensive deployment of Low Earth Orbit (LEO) satellites introduces
significant security challenges for communication security issues in Internet
of Things (IoT) networks. With the rising number of satellites potentially
acting as eavesdroppers, integrating Physical Layer Security (PLS) into
satellite communications has become increasingly critical. However, these
studies are facing challenges such as dealing with dynamic topology
difficulties, limitations in interference analysis, and the high complexity of
performance evaluation. To address these challenges, for the first time, we
investigate PLS strategies in satellite communications using the Stochastic
Geometry (SG) analytical framework. We consider the uplink communication
scenario in an LEO-enabled IoT network, where multi-tier satellites from
different operators respectively serve as legitimate receivers and
eavesdroppers. In this scenario, we derive low-complexity analytical
expressions for the security performance metrics, namely availability
probability, successful communication probability, and secure communication
probability. By introducing the power allocation parameters, we incorporate
the Artificial Noise (AN) technique, which is an important PLS strategy, into
this analytical framework and evaluate the gains it brings to secure
transmission. In addition to the AN technique, we also analyze the impact of
constellation configuration, physical layer parameters, and network layer
parameters on the aforementioned metrics.

###### Index Terms:

Physical layer security, stochastic geometry, LEO satellite-based IoT network,
artificial noise techniques, uplink transmission.

##  I Introduction

###  I-A Motivation

Low Earth Orbit (LEO) satellites play a crucial role in enhancing the
connectivity of the Internet of Things (IoT) devices, especially in remote and
underserved areas where traditional terrestrial networks struggle to provide
coverage [1, 2, 3]. The rapid expansion of satellite launches, which has been
growing by 30% annually since 2012, is due to advancements in rocket launch
platforms [4]. This brings enormous opportunities for the connectivity of IoT
devices but also poses new challenges for communication security.

The deployment of mega-constellations makes more satellites act as potential
eavesdropping threats, especially for uplink transmission from IoT devices to
satellites. Furthermore, due to the dynamic nature and high speed of LEO
satellite constellations relative to the Earth, there are frequent changes in
the signal quality received by both the primary and potential eavesdropping
channels. Atmospheric conditions, multipath fading, and interference further
impact the signal. At certain times, the main channel may experience poor
conditions, resulting in a weaker received signal. Simultaneously, an
eavesdropper may be in a location with better channel conditions, leading to
it experiencing a stronger signal. For example, rain fade can weaken the main
channel signal, while an eavesdropper in a clear sky region may experience
less attenuation and, therefore, better signal quality. This variability can
lead to scenarios where the eavesdropper receives a higher quality signal than
the main communication link, thus posing a significant security threat.
Therefore, the urgent need for enhancing uplink security strategies should be
emphasized.

To address these challenges, our research integrates the Stochastic Geometry
(SG) analytical framework with advanced Physical Layer Security (PLS)
techniques. This novel combination enables comprehensive analysis of dynamic
topology and interference, providing low-complexity solutions for evaluating
security performance in IoT networks based on multi-tier LEO satellite
constellations.

###  I-B PLS Techniques in Satellite Networks

Recently, PLS techniques have been increasingly recognized as potential
solutions for enhancing security within satellite networks. The idea of PLS
originates from Shannon’s foundational concept about perfect secrecy.
Subsequently, authors in [5] proposed the eavesdropper channel model,
considering that the information received by the eavesdropper is a degraded
version of the legitimate received information.

The core idea of applying PLS in satellite networks is to use the randomness
of wireless channels, such as interference, fading, and noise, to achieve
secure transmission [6, 7, 8, 9]. A notable example is the Artificial Noise
(AN) technique, which was first proposed in [10]. The principle of AN
technology is that the IoT device generates an additional interference signal
while transmitting useful signals [11]. This interference signal can be
filtered out by legitimate receiving satellites during demodulation, thereby
affecting only the eavesdropping satellites [12]. As a result, IoT devices
reduce the signal quality of the eavesdropping link by sacrificing some of
their own power to transmit AN.

Several pioneering studies have advanced the integration of PLS techniques
within satellite communications. Authors in [13] provided foundational
frameworks by optimizing resources for secure satellite communications across
different receiver scenarios, including fixed and mobile units, while [14]
provided a detailed theoretical analysis on securing non-geostationary orbit
satellite downlinks. Further, research like [15] introduced threshold-based
user scheduling to improve secrecy in multiuser systems, and [16] proposed
cooperative secrecy tactics in LEO-integrated networks through jamming signals
and optimized power allocation. Moreover, [17] explored the role of unmanned
aerial vehicles (UAV)s in strengthening satellite-vehicular communication
security, and [18] leveraged time-packing techniques to safeguard IoT devices’
direct satellite links. Additionally, recent advancements in PLS techniques
include secrecy-energy efficient hybrid beamforming for satellite-terrestrial
integrated networks to improve security and energy efficiency [19], signal-to-
leakage-plus-noise ratio (SLNR)-based secure energy-efficient beamforming
applied in multibeam satellite systems [20], and refracting reconfigurable
intelligent surface (RIS)-aided hybrid satellite-terrestrial relay networks
developed for joint beamforming design and optimization [21]. Furthermore,
rate-splitting multiple access, which indirectly supports PLS by improving
interference management and signal reliability, has been used to support IoT
in satellite and aerial-integrated networks [22]. These studies demonstrate
the wide range of PLS applications in safeguarding satellite communications
against potential security threats, highlighting the most recent developments
in improving the security and efficiency of satellite and IoT networks.

Although the issue of PLS has gained wide attention in the field of satellite
communications, there are still some challenges that remain unresolved. The
first challenge is that the satellite network’s topology is dynamic; thus, the
number of eavesdropping and legitimate satellites follow a specific stochastic
distribution. Furthermore, the spatial distribution of satellites also greatly
influences the probability of eavesdropping or successful transmission [23].
The second challenge is that the deployment of a large number of devices makes
the interference hard to ignore [24]. Finally, as the number of satellites and
IoT devices increases, evaluating the effectiveness of PLS strategies becomes
increasingly challenging. This is because more device locations need to be
generated, and more IoT-satellite links need to be simulated [13, 16].
Therefore, the third challenge lies in evaluating the performance of PLS
strategies with low computational complexity.

###  I-C Related Works for Stochastic Geometry

SG is a mathematical tool that can perfectly address the above challenges. It
is one of the most suitable methods for analyzing large-scale random network
topologies [25], and the SG-based analytical framework can account for
stochastic variations in interference [26]. The analytical expressions derived
within the SG framework can be represented as functions of system parameters
such as the number of satellites [27]. Therefore, we can achieve a low-
complexity mapping from system parameters to performance metrics, and the
complexity does not increase with the scale of the constellation [28]. Next,
we will introduce the research related to satellites and PLS in the field of
SG.

Modeling satellites through the SG model is an emerging research method.
Binomial Point Process (BPP) is one of the most widely used satellite modeling
methods [29, 30]. The topological performance (such as the distribution of
distances between communication parties [31]) and the network-level
performance (such as coverage probability and data rate [32, 33]), derived
from BPP, align well with actual satellite constellation outcomes. Compared to
BPP, Poisson Point Process (PPP) is more suitable for modeling networks in
non-closed areas [34, 35], such as ground-based IoT networks. Among SG-based
satellite works, studies most closely related to our paper are [36, 37] and
our previous work [38]. Our past research [38] involved addressing coverage
issues in multi-tier satellite networks, where contact distance and
association strategies laid the groundwork for this paper. However, [38] did
not involve IoT devices or PLS issues. While [36] studied PLS in the downlink
LEO satellite networks, focusing on the spatial distribution of satellites,
users, and potential eavesdroppers modeled using homogeneous spherical PPP
with eavesdroppers located on the ground. As for [37], authors explored the
effect of single-tier satellite eavesdroppers on uplink transmission. The
study provided support and inspiration for the research scenario of this
paper. Unlike our paper, however, authors in [37] did not consider PLS
techniques, nor did it involve modeling of legitimate constellations.
Therefore, our study significantly extends the scope beyond [37].

Additionally, some literature has utilized the SG framework to analyze the
secure communication performance of planar networks [39, 40]. Authors in [41]
initiated PLS methodology within cellular networks, and authors in [42]
further explored this concept within millimeter-wave relaying networks.
Authors in [40] expanded the SG framework to IoT networks, where sensors,
access points, and eavesdroppers are modeled by PPPs. Furthermore, authors in
[11] and [43] introduced AN technique into secure transmission in terrestrial
networks, highlighting the significance of utilizing spatial randomness and
strategic AN allocation. Although these studies have laid a solid foundation
for researching PLS strategies in the SG field within ground networks,
investigating PLS issues in satellite networks still faces many challenges.
The spherical topology, channel models, and performance metrics, such as
availability in satellite networks, are significantly different from those in
ground networks.

###  I-D Contribution

As the first study for PLS in satellite networks under SG, the specific
contributions are as follows:

  * •

We introduce a novel multi-tier satellite network topology for uplink
transmission, where one tier consists of legitimate satellites while other
tiers consist of potential eavesdroppers. This topology, which includes
satellites at various altitudes serving distinct roles, accurately reflects
the complexity of real-world LEO satellite constellations.

  * •

We develop an analytical framework using SG to model the spatial distribution
and interaction of satellites and IoT devices. This framework allows us to
derive low-complexity analytical expressions for key performance metrics:
satellite availability probability, coverage probability, secrecy outage
probability, successful communication probability, and secure communication
probability.

  * •

By incorporating the AN technique within our SG-based analytical framework, we
enhance physical layer security. This technique, with the strategic allocation
of power to AN, degrades the signal quality perceived by potential
eavesdroppers while maintaining the communication quality for legitimate
receivers, thereby improving secure transmission performance.

  * •

We conduct Monte Carlo simulations to validate the accuracy of our analytical
results. These simulations confirm that the derived expressions can serve as
accurate and low-complexity alternatives to more computationally intensive
simulations for performance evaluation.

  * •

We reveal useful system-level insights regarding the impact of constellation
configuration (such as the number of satellites and their altitude), physical
layer parameters (such as beamwidth), and network layer parameters (such as
IoT device density) on availability probability and secure communication
probability. These insights offer practical guidelines for optimizing secure
LEO satellite-enabled IoT networks.

Figure 1: Illustration of the multi-tier architecture of the network,
depicting LS and ES at various altitudes. This configuration enables detailed
analysis of signal interactions and security measures across d different
altitude levels.

##  II System Model

This section presents the system model for our investigation into enhancing
the security of IoT communications through LEO satellites using PLS
techniques. We consider uplink transmission in a multi-tier LEO satellite
network with terrestrial IoT devices. In addition, we employ AN as a
countermeasure against potential eavesdropping threats. The symbols, key
abbreviations, and their meanings are listed in TABLE I.

###  II-A Spatial Distribution Model

The IoT devices are distributed across the Earth’s surface, following a
homogeneous PPP model, which closely reflects real-world conditions. The PPP
is denoted as $\Phi_{u}\equiv\\{\mathbf{u}_{i}\\}\in\mathbb{R}^{3}$ with
density $\lambda_{u}$. IoT devices transmit their data to the designated LEO
satellites. Without loss of generality, the typical IoT device is assumed to
be located at $\mathbf{u_{0}}=\left(0,0,R_{\oplus}\right)$ in the spherical
system, where the $R_{\oplus}$ is the radius of the Earth ($\approx 6371$ km).
Apart from the typical device, other IoT devices act as potential sources of
interference in the uplink transmission.

We consider a $K$-tier LEO satellite constellation with different altitudes
$a_{k}$, where $k=1,\dots,K$. Satellites from the $k$-th tier follow a
homogeneous BPP on the surface of a sphere $S_{k}$ with radius
$R_{k}=a_{k}+R_{\oplus}$, where the positions of $N_{k}$ satellites in the
$k$-th tier are independent, identically distributed (i.i.d.). The sphere
$S_{k}$ can be expressed with spherical coordinates as
$S_{k}=\\{R_{k},0\leq\theta\leq\pi,0\leq\varphi\leq 2\pi\\}$, where $\theta$
and $\varphi$ are polar angle and azimuthal angle, respectively. Finally, we
denote the location of satellites on $k$-th sphere, as
$\Phi_{S_{k}}\equiv\\{\mathbf{x}_{kj}\\}$ where
$\mathbf{x}_{kj}\in\mathbb{R}^{3}$ indicates the location of $j$-th satellite
on $k$-th tier. As shown in Fig. 1, the satellites in the constellation
serving the IoT devices are legitimate satellite (LS)s, while other satellites
potentially intercepting communications from IoT devices are known as
eavesdropper satellite (ES)s. The $K$-tier LEO satellite constellation
includes a legitimate tier composed of LSs, as well as $K-1$ tiers consisting
of ESs.

The multi-tier architecture, as illustrated in Fig. 1, represents a realistic
deployment scenario where satellites are positioned at varying altitudes. This
structure allows for an in-depth analysis of signal propagation and
interception across different layers, enhancing the practical relevance of our
study. By considering satellites at multiple altitudes, we can evaluate
complex interaction dynamics and develop more effective PLS strategies.

TABLE I: Notations and Abbreviations

Symbol/Abbr. | Description
---|---
LEO | Low Earth Orbit
PLS | Physical Layer Security
SG | Stochastic Geometry
AN | Artificial Noise
BPP | Binomial Point Process
PPP | Poisson Point Process
LS | Legitimate Satellite
ES | Eavesdropper Satellite
$\Phi_{S_{k}}\equiv\\{\mathbf{x}_{kj}\\}$ |  BPP for satellites in $k$-th tier
$\Phi_{u}\equiv\\{\mathbf{u}_{i}\\}$ |  PPP for IoT devices locations
$\mathcal{A}_{m,\rm vis}$ | Visibility area for the $m$-th tier
$\mathcal{I}_{m}$ | Interference power for the serving LS at tier $m$
$\mathcal{I}_{kj}$ | Interference power for $j$-th ES at tier $k$
$\mathcal{L}_{\mathcal{I}}(\cdot)$ | Laplace transform of the interference $\mathcal{I}$
$\mathcal{P}_{m,\rm av}$ | Availability probability for $m$-th tier
$\mathcal{P}_{m,\rm cov}$ | Coverage probability for $m$-th tier
$\mathcal{P}_{m,\rm suc}$ | Successful communication probability for $m$-th tier
$\mathcal{P}_{\rm out}$ | Secrecy outage probability
$\mathcal{P}_{\rm sec}$ | Secure communication probability

###  II-B Channel Model

In our channel model, we combine the effects of large-scale and small-scale
fading to represent signal propagation accurately. The large-scale fading
component, modeled by the free-space path loss, accounts for the overall
signal attenuation due to the distance between the transmitter and the
receiver. This component is expressed as $\left(\frac{c}{4\pi
f_{c}d_{kj,i}}\right)^{2}$, where $f_{c}$ indicates the carrier frequency, $c$
is the speed of light and $d_{kj,i}$ is the distance between the $i$-th IoT
device and the $j$-th satellite in the $k$-th tier,

$\displaystyle
d_{kj,i}=\sqrt{R_{\oplus}^{2}+R_{k}^{2}-2R_{\oplus}R_{k}\cos{\theta_{kj,i}}},$
(1)

where $\theta_{kj,i}$ is the central angle between the $i$-th IoT device and
the $j$-th satellite. An example of the central angle is shown in Fig. 2.

We adopt the Shadowed-Rician distribution to model small-scale fading, as it
provides a realistic and precise approximation of satellite-terrestrial
channels, accurately capturing small-scale variations and reflecting real-
world conditions [44, 45, 23]. Thus, the small-scale fading component,
represented by $|h_{kj,i}|^{2}$, follows the Shadowed-Rician distribution,
which describes the fading between the $i$-th IoT device and the $j$-th
satellite in the $k$-th tier.

Combining these components, the received signal power $P_{\rm rec}$, at the
satellite during uplink transmission is modeled as

$\displaystyle P_{\rm rec}\left(d_{kj,i}\right)=P_{t}\left(\frac{c}{4\pi
f_{c}d_{kj,i}}\right)^{2}G|h_{kj,i}|^{2},$ (2)

where $P_{t}$ denotes the transmit power of the IoT device, $G$ is the product
of the gains of the transmitting and receiving antennas.

Next, we determine the range for the central angle $\theta$. Firstly, we
consider that IoT devices located within the satellite receiving beam’s main
lobe can establish a stable link with the satellite. The satellite beam is
pointed towards the center of the Earth, and the main lobe angle of the
receiving beam is denoted as $2\theta_{\rm beam}$. As illustrated in Fig. 2,
$\theta_{\rm beam}$ is the half beamwidth angle of the satellite. In addition,
assuming that only satellites with line-of-sight links established with IoT
devices can communicate, the upper bound of the central angle can be
represented as,

$\displaystyle\theta_{k,\rm
max}=\begin{dcases}\sin^{-1}{\left(\frac{R_{k}}{R_{\oplus}}\sin{\theta_{\rm
beam}}\right)}-\theta_{\rm beam},&\theta_{\rm
beam}<\sin^{-1}{\left(\frac{R_{\oplus}}{R_{k}}\right)},\\\
\cos^{-1}{\left(\frac{R_{\oplus}}{R_{k}}\right)},&\theta_{\rm
beam}\geq\sin^{-1}{\left(\frac{R_{\oplus}}{R_{k}}\right)},\end{dcases}$ (3)

and the range for $\theta$ is $0<\theta<\theta_{k,\rm max}$.

As mentioned earlier, the Shadowed-Rician distribution is used to model small-
scale fading. However, due to the complexity of deriving closed-form solutions
from the Shadowed-Rician channel model, we adopt the Gamma fading model as a
practical approximation for the Shadowed-Rician fading [35]. Thus, the
Cumulative Distribution Function (CDF) of small-scale fading $|h|^{2}$ is
tightly bounded by as follows

$F_{|h|^{2}}(x)\leq\left[1-\exp\left({-\frac{(m_{1}!)^{-\frac{1}{m_{1}}}}{m_{2}}x}\right)\right]^{m_{1}}\quad\text{if}\quad
m_{1}\leq 1,\\\ $ (4)

where $m_{1}$ and $m_{2}$ denote shape and scale parameters, respectively.

Figure 2: Illustration of satellite beamwidth $\theta_{\rm beam}$ and its
impact on Earth’s surface coverage.

###  II-C Artificial Noise

In this article, we introduce AN into the transmitted signal to enhance its
secrecy. The transmit power of IoT device, $P_{t}$, is divided into two parts:
a fraction $\gamma P_{t}$ is allocated for transmitting confidential
information, $\left(1-\gamma\right)P_{t}$ is dedicated to transmitting AN.
$\gamma$ is known as the information-bearing ratio, which falls within the
interval $[0,1]$. In scenarios where $\gamma=1$, AN is not applied, and all of
the IoT device’s transmit power is dedicated to signal transmission.

Then, we can present the communication links into the legitimate link (typical
IoT device to LS), eavesdropping link (typical IoT device to ES), and
interfering link (interfering IoT device to the satellite of interest). Due to
LSs knowing in advance that typical IoT devices employ AN strategies, the AN
can be filtered out by LSs. When the IoT device is served by the $m$-th tier,
the signal-to-interference and noise ratio (SINR) attained at the serving
satellite is expressed as

$\displaystyle{\rm SINR}_{\rm LS}=\frac{\gamma P_{\rm
rec}\left(d_{m0,0}\right)}{\mathcal{I}_{m}+\sigma_{\rm S}^{2}},$ (5)

where $P_{\rm rec}(d_{m0,i})$ represents the received power at the serving
satellite from both typical ($i=0$) and interfering ($i\neq 0$) devices, with
the distance function $d_{m0,i}$ defined in (1). For notation simplicity, and
because all IoT devices (typical or interfering) communicate only with the
serving satellite in the legitimate tier $m$, the subscript $0$ used for the
serving satellite is omitted from $d_{m0,i}$ in favor of $d_{m,i}$, $\forall
i$. The noise power at the satellite is denoted by $\sigma_{\rm
S}^{2}=N_{0}\times B$, where $N_{0}$ is the noise spectral density and $B$ is
the bandwidth. Furthermore,

$\mathcal{I}_{m}=\sum\limits_{i\in\Phi_{u}\setminus\\{i=0\\}}P_{\rm
rec}(d_{m,i})$ (6)

denotes the cumulative interference from other IoT devices ($i\neq 0$) at the
serving satellite.

ESs, in contrast, are incapable of filtering out the AN. For the $j$-th ES
located in the $k$-th tier ($\forall k\neq m$), the SINR of the eavesdropping
link is given by

$\displaystyle\text{SINR}_{kj}$ $\displaystyle=\frac{\gamma P_{\rm
rec}\left(d_{kj,0}\right)}{(1-\gamma)P_{\rm
rec}\left(d_{kj,0}\right)+\mathcal{I}_{kj}+\sigma_{\rm S}^{2}},$ (7)

where $P_{\rm rec}(d_{kj,i})$ indicates the received power at the eavesdropper
from both typical ($i=0$) and interfering ($i\neq 0$) IoT devices.

$\mathcal{I}_{kj}=\sum\limits_{i\in\Phi_{u}\setminus\\{i=0\\}}P_{\rm
rec}(d_{kj,i})$ (8)

represents the total interference from other IoT devices ($i\neq 0$) at the
ES.

In conclusion, the addition of AN considerably impairs the eavesdropper’s
ability to intercept the communication while leaving the reception for the
intended receiver unaffected. By carefully managing the power of the AN, it is
feasible to markedly improve the confidentiality of uplink transmissions from
IoT devices to LEO satellites, thereby enhancing security at the physical
layer.

##  III Performance Analysis

In secure satellite communications, evaluating the effectiveness of PLS
techniques requires a comprehensive understanding of how well the system can
maintain connectivity, protect against eavesdropping, and ensure reliable and
secure data transmission. The performance metrics, availability probability,
coverage probability, and secure connection probability are integral to
assessing these aspects. The multi-tier architecture is central to our
analysis, enabling us to model various orbital configurations and assess their
impacts on key performance metrics. This approach allows evaluation under
different conditions, including potential eavesdropping scenarios,
highlighting its significance in optimizing secure communication in LEO
satellite-enabled IoT networks. In this section, we evaluate the security of
the considered system network using the above performance metrics or their
combination and provide analytical expressions for these performance metrics.

To facilitate analysis, we begin by characterizing the distribution of the
central angle from the typical IoT device to the serving satellite, which is
known as the contact angle [46]. Given that $N_{m}$ satellites are modeled
uniformly as BPP, it follows that any randomly chosen points share an
identical distribution of the contact angle. Therefore, the CDF and
Probability Density Function (PDF) of the contact distance can be articulated
in the following lemma.

######  Lemma 1 (Contact Angle Distribution).

Given the typical IoT device and the closest satellite within the $m$-th tier,
the CDF of the contact angle $\theta_{m,0}$ is given by

$F_{\theta_{m,0}}(\theta)=1-\left(\frac{1+\cos{\theta}}{2}\right)^{N_{m}},\quad
0<\theta<\theta_{m,\mathrm{max}},$ (9)

and the corresponding PDF is

$f_{\theta_{m,0}}(\theta)=\frac{N_{m}\sin{\theta}}{2}\left(\frac{1+\cos{\theta}}{2}\right)^{N_{m}-1},\quad
0<\theta<\theta_{m,\mathrm{max}},$ (10)

where $\theta_{m,\mathrm{max}}$ is defined in (3).

###### Proof:

See appendix A. ∎

Next, we derive the availability probability, which is directly related to the
contact angle distribution. The availability probability stands as a
fundamental metric, establishing the foundational viability of any
communication attempt. Without satellite availability, neither data
transmission nor secure communication can occur, as defined and mathematically
presented as follows:

######  Definition 1.

Availability probability is the probability that, within the visible area of a
typical IoT device, at least one satellite from the $m$-th tier is available
for communication. It is mathematically presented as follows:

$\mathcal{P}_{m,\rm
av}\triangleq\mathbb{P}\left[\mathcal{N}\left(\mathcal{A}_{m,\rm
vis}\right)>0\right],$ (11)

where $\mathcal{A}_{m,\rm vis}$ is the surface area from which satellites at
tier $m$ are visible to the typical IoT device, defined by the set
$\mathcal{A}_{m,\rm vis}=\\{R_{m},0\leq\theta\leq\theta_{m,\rm
max},0\leq\varphi\leq 2\pi\\}$. $\mathcal{N}(\mathcal{A}_{m,\rm vis})$ counts
the number of satellites in the surface area $\mathcal{A}_{m,\rm vis}$.

Next, we derive the availability probability in the following lemma as defined
in Definition 1, which is directly related to the contact angle distribution.

######  Lemma 2 (Availability Probability).

The probability of availability $\mathcal{P}_{m,\mathrm{av}}$ of a satellite
in the $m$-th tier is given as

$\displaystyle\mathcal{P}_{m,\rm av}=1-\left(\frac{1+\cos{\theta_{m,\rm
max}}}{2}\right)^{N_{m}},$ (12)

where $\theta_{m,\rm max}$ is defined in (3) and $N_{m}$ is the total number
of satellites within tier $m$.

###### Proof:

Essentially, the CDF of contact angle distribution, which is defined as
$\mathbb{P}\left[\theta_{m,0}<\theta\right]$ is equivalent to the availability
probability $\mathbb{P}\left[\mathcal{N}\left(\mathcal{A}_{m,\rm
vis}\right)=0\right]$, when $\theta_{m,0}=\theta_{m,\rm max}$, where
$\mathcal{A}_{m,\rm vis}$ describes the maximum visibility area. Following the
void probability, the lemma is proved. ∎

Once the availability of satellites is determined, the analysis progresses to
the coverage probability. Coverage probability assesses the likelihood that a
communication link, specifically between a typical IoT device and its serving
satellite, can be established with sufficient signal quality. This quality is
necessary to decode transmitted signals with minimal errors accurately. It is
conditional upon the SINR exceeding a predetermined threshold, $\beta_{\rm
LS}$ and is defined as follows:

######  Definition 2.

The coverage probability is the probability that the SINR for the legitimate
link is greater than $\beta_{\rm LS}$, thereby ensuring reliable signal
decoding. Formally, it is expressed as:

$\displaystyle\mathcal{P}_{m,\rm cov}$
$\displaystyle\triangleq\mathbb{P}\left[{\rm SINR}_{\rm LS}>\beta_{\rm
LS}\right],$ (13)

where the ${\rm SINR}_{\rm LS}$ is defined in (5).

Prior to the analysis of coverage probability, we need to understand the
impact of interference on the communication link. Given the spatial
distribution of IoT devices follows PPP, we employ the Laplace Transform (LT)
to obtain more manageable and closed-form expressions for interference
analysis.

######  Lemma 3 (Laplace Transform of Interference).

For a given satellite located in tier $m$, the LT of the interference
$\mathcal{L}_{\mathcal{I}_{m}}(s)$ experienced by this satellite is:

$\displaystyle\mathcal{L}_{\mathcal{I}_{m}}\left(s\right)=\exp$
$\displaystyle\left(-\lambda_{u}\int\limits_{0}^{2\pi}\int\limits_{0}^{\theta_{m,\rm
max}}\left[1-\left(1+m_{2}sP_{t}G\left(\frac{c}{4\pi
fd}\right)^{2}\right)^{-m_{1}}\right]\right.\left.\vphantom{\int\limits_{0}^{2\pi}}R_{\oplus}^{2}\sin{\theta}d\theta
d\varphi\right),$

where $\theta_{m,\rm max}$ is defined in (3).

###### Proof:

See appendix B. ∎

Building upon the interference analysis, the derivation of the coverage
probability proceeds as follows.

######  Lemma 4 (Coverage Probability).

The coverage probability for the uplink communication between a typical IoT
device and its nearest LS in the $m$-th tier with SINR threshold $\beta_{\rm
LS}$ is derived as follows:

$\displaystyle\mathcal{P}_{m,\rm
cov}=\sum\limits_{q=1}^{m_{1}}\binom{m_{1}}{q}(-1)^{q+1}\int\limits_{0}^{\theta_{m,\rm
max}}\exp{\left(-qs_{\rm LS}\sigma_{S}^{2}\right)}$
$\displaystyle\mathcal{L}_{\mathcal{I}_{m}}\left(qs_{\rm
LS}\right)f_{\theta_{m,0}}(\theta)d\theta,$

where $s_{\rm LS}=\frac{(m_{1}!)^{-\frac{1}{m_{1}}}}{m_{2}}\frac{\beta_{\rm
LS}}{\gamma P_{t}G}\left(\frac{4\pi fd_{m,0}}{c}\right)^{2}$, and
$\theta_{m,\rm max}$ is as defined in (3). The functions
$f_{\theta_{m,0}}(\cdot)$, $\mathcal{P}_{m,\rm av}$, and
$\mathcal{L}_{\mathcal{I}_{m}}(\cdot)$ are provided in Lemma 1, Lemma 2 and
Lemma 3, respectively.

###### Proof:

See appendix C. ∎

We are ready to introduce the concept of successful communication probability,
one of the main metrics, which involves the insights gained from the
availability and coverage probabilities. The successful communication
probability denoted as $\mathcal{P}_{m,\rm suc}$, is defined as the
probability that an uplink connection made by the typical IoT device to its
nearest satellite in the $m$-th tier successfully meets both availability and
coverage criteria. Mathematically, this probability is expressed as the
product of the availability probability and the coverage probability:

$\displaystyle\mathcal{P}_{m,\rm suc}=\mathcal{P}_{m,\rm
av}\times\mathcal{P}_{m,\rm cov},$ (14)

where $\mathcal{P}_{m,\rm av}$ and $\mathcal{P}_{m,\rm cov}$ are derived in
Lemma 2 and Lemma 4, respectively.

Moving beyond the feasibility of communication, the definition of secrecy
outage probability addresses the security aspect, which investigates the
scenario where the SINR experienced by any potential ES falls below a certain
predefined threshold, $\beta_{\rm ES}$. Hence, within a multi-tiered satellite
constellation framework, we define and derive the secrecy outage probability
as follows.

######  Definition 3.

Secrecy outage probability is the probability that the highest SINR
experienced by any ES across a multi-tiered satellite constellation falls
below a predefined threshold $\beta_{\rm ES}$. Mathematically,
$\mathcal{P}_{\rm out}$ is defined as:

$\displaystyle\mathcal{P}_{\rm out}$
$\displaystyle\triangleq\mathbb{P}\left[\max_{k\neq m}\max_{j\in\Phi_{\rm
S_{k}}}\left\\{\text{\rm SINR}_{kj}\right\\}<\beta_{\rm ES}\right],$ (15)

where ${\rm SINR}_{kj}$ represents the SINR experienced by $j$-th ES within
the $k$-th tier, excluding the legitimate $m$-th tier , where the expression
is given in (7).

Upon the definition of the secrecy outage probability, we can derive the
expression of the $\mathcal{P}_{\rm out}$ in the following lemma.

######  Theorem 1 (Secrecy Outage Probability).

For a given value of $\beta_{\rm ES}$, the probability of secrecy outage, as
defined in Definition 3, is derived as:

$\displaystyle\mathcal{P}_{\rm out}=\prod_{\begin{subarray}{c}k=1\\\ k\neq
m\end{subarray}}^{K}$
$\displaystyle\left[\sum\limits_{q=0}^{m_{1}}\binom{m_{1}}{q}(-1)^{q}\int\limits_{0}^{\theta_{k,\rm
max}}\exp{\left(-qs_{\rm
ES}\sigma_{S}^{2}\right)}\mathcal{L}_{\mathcal{I}_{kj}}\left(qs_{\rm
ES}\right)\right.\left.\vphantom{\int\limits_{0}^{2\pi}}\times
f_{\theta_{k,0}}(\theta)d\theta+\frac{1+\cos{\theta_{k,\rm
max}}}{2}\right]^{N_{k}},$

with $s_{\rm ES}=\frac{(m_{1}!)^{-\frac{1}{m_{1}}}}{m_{2}}\frac{\beta_{\rm
ES}}{\left[\gamma-\beta_{\rm ES}(1-\gamma)\right]P_{t}G}\left(\frac{4\pi
fd_{kj,0}}{c}\right)^{2}$ and
$f_{\theta_{k,0}}(\theta)=\frac{\sin{\theta}}{2}$ The LT of interference,
$\mathcal{L}_{\mathcal{I}_{kj}}(\cdot)$, is derived in Lemma 3.

###### Proof:

See appendix D. ∎

Finally, our contribution goes further by not only considering the successful
connection but also ensuring the data transmitted is secure from
eavesdropping. The secure communication probability, denoted as
$\mathcal{P}_{\rm sec}$, is calculated as the product of two key factors: the
probability of the typical IoT device achieving successful communication
within the legitimate $m$-th tier and the secrecy outage probability
considering the influence of all ES. Mathematically, the secure communication
probability is formulated as:

$\displaystyle\mathcal{P}_{\rm sec}=\mathcal{P}_{m,\rm
suc}\times\mathcal{P}_{\rm out},$ (16)

where $\mathcal{P}_{\rm out}$, the secrecy outage probability is elaborated on
in Theorem 1, and $\mathcal{P}_{m,\rm suc}$, the probability of achieving
successful communication, is described in (14).

(a) $\mathcal{P}_{m,\rm av}$ against beamwidth angle and the altitude of
satellites

(b) $\mathcal{P}_{m,\rm av}$ against number of satellites and the altitude of
satellites

Figure 3: Heat map representing the $\mathcal{P}_{m,\rm av}$ for different
parameters of satellite network. TABLE II: Simulation Parameters

Notation | Parameter | Default Value
---|---|---
$f_{c}$ | Center frequency | 2 $\mathrm{GHz}$
$R_{\oplus}$ | Earth radius | 6371 km
$N_{k}$ | Number of satellites in tier $k$, $\forall k$ | 500
$a_{k}$ | Altitude of satellites for all tier, $k=1,2,3$ | [500, 1000, 1500] km
$\theta_{\rm beam}$ | Half beamwidth angle of satellite | ${\pi}/{3}$
$P_{t}$ | Uplink transmit power of IoT devices | 23 $\mathrm{dBm}$
$\lambda_{u}$ | Density of users (IoT devices) |  $10^{-6}$ devices$/$km-2
$N_{0}$ | Noise spectral density | -174 $\mathrm{dBm/Hz}$
$B$ | Bandwidth | 180 $\mathrm{KHz}$
$\Gamma(m_{1},m_{2})$ | Gamma shape and scale parameters | $[m_{1},m_{2}]=[1,0.1269]$
$\gamma$ | AN power allocation coefficient | 0.1
$G$ | Antenna bea, gain | 41.9 $\mathrm{dBi}$
$\beta_{\rm ES}$ |  SINR threshold at ES | -10 $\mathrm{dB}$
$\beta_{\rm LS}$ |  SINR threshold at LS | -30 $\mathrm{dB}$

##  IV Numerical Results

In this section, we present numerical results for the derived performance
metrics. We employed Monte Carlo simulations with 10,000 iterations. In each
iteration, the positions of the satellites and IoT devices were varied to
simulate the dynamic changes in the topology. This approach ensures that the
performance metrics derived from our analytical expressions correspond to the
average performance of the network under dynamic conditions rather than a
static snapshot. We use lines to present analytical results and markers to
present Monte Carlo simulation results. The matching between them in the
figures provided in this section aims to affirm the accuracy of our analysis
further. The parameters employed in these simulations are detailed in Table
II.

###  IV-A System Design on Network Availability

This subsection provides how the design of satellite networks influences a
network’s ability to establish communication links between IoT devices and
satellites. As shown in (12), the availability probability $\mathcal{P}_{m,\rm
av}$ is influenced by the beamwidth, number, and altitude of satellites. In
Fig. 3, we highlight the optimal parameters design with regard to
$\mathcal{P}_{m,\rm av}$.

In Fig. 3(a), as the beamwidth angle increases, the area covered by each
satellite’s communication beam expands, increasing the availability
probability $\mathcal{P}_{m,\rm av}$. Furthermore, with the same beam angle,
satellites at higher altitudes cover a larger area of the Earth’s surface,
thus increasing $\mathcal{P}_{m,\rm av}$. With a fixed constellation size
(e.g., $N_{m}=500$), we observe that we can reach $\mathcal{P}_{m,\rm
av}\approx 1$ for satellite altitudes higher than 500 km when the beamwidth
angle is set to $\theta_{\rm beam}=\frac{\pi}{3}$.

Fig. 3(b) demonstrates how $\mathcal{P}_{m,\rm av}$ is influenced by
constellation configurations. Increasing the number and altitude of satellites
can enhance $\mathcal{P}_{m,\rm av}$. The analysis identifies a constellation
size of $N_{m}=500$ satellites as particularly effective in securing near-
universal availability across all network tiers with $\mathcal{P}_{m,\rm
av}\approx 1$. Further increasing the number of satellites offers limited
benefits to availability.

(a) $\theta_{\rm beam}=\pi/3$

(b) $\theta_{\rm beam}=\pi/3$

(c) $\theta_{\rm beam}=\pi/4$

Figure 4: The effects of power allocation across three configurations on
$\mathcal{P}_{\rm sec}$. Figure 5: Impact of satellite numbers on
$\mathcal{P}_{\rm sec}$ against beam angle.

###  IV-B Enhancing PLS: Techniques and Trade-offs

This subsection discusses how different strategies can influence the secure
communication probability $\mathcal{P}_{\rm sec}$. Unless otherwise stated, we
adopt a 3-tier constellation, with altitudes set at $a_{\rm
S}=[500,1000,1500]$ km. The number of satellites for all tiers is fixed as
$N_{k}=500$, $\forall k=1,2,3$.

Fig. 4 shows how different choices of the legitimate tier and the power
allocation parameter $\gamma$ influence $\mathcal{P}_{\rm sec}$. We observe a
consistent trend across all subfigures: $\mathcal{P}_{\rm sec}$ decreases as
$\gamma$ increases. This is because less power is allocated to AN for a higher
$\gamma$, making it easier for eavesdroppers to decode the signal. More
precisely, the transition from Fig. 4(a) to Fig. 4(b) and Fig. 4(c)
illustrates changes in satellite number and beam angle. When the satellite
count is increased, the chance of having a satellite available for
communication ($\mathcal{P}_{m,\rm av}$) is high, leading to Tier-1 having the
highest $\mathcal{P}_{\rm sec}$ as depicted in Fig. 4(a). However, more
satellites can also create more interference, particularly at higher
altitudes, thus lowering $\mathcal{P}_{\rm sec}$ in Tiers 2 and 3. Conversely,
in Fig. 4(b), reducing the number of satellites significantly impacts Tier-1,
making it less secure due to a lower $\mathcal{P}_{m,\rm av}$; thus, Tier-2
becomes the preferred choice for the legitimate tier due to its balanced
performance. Fig. 4(c) shows the effects of reducing beam angle from
$\theta_{\rm beam}=\frac{\pi}{3}$ to $\theta_{\rm beam}=\frac{\pi}{4}$ while
maintaining $N_{k}=500$, $\forall k$. In this configuration, Tiers 2 and 3
outperform Tier 1 across the entire range of the $\gamma$. The narrower beam
angle implies a need for more satellites to maintain high $\mathcal{P}_{m,\rm
av}$. Overall, we can draw the same conclusion because whether reducing
$N_{k}$ $\forall k$, or $\theta_{\rm beam}$ decreases the $\mathcal{P}_{m,\rm
av}$.

The adjustments illustrate a trade-off between availability and signal
quality, indicating that Tier-2, with an altitude of $a_{m}=1000$ km, is the
optimal choice for the legitimate tier due to its consistently stable maximum
$\mathcal{P}_{\rm sec}$ as depicted in Fig. 4. Therefore, unless otherwise
stated, subsequent analyses of system performance in Figs. 5, 6, and 7 focus
on Tier-2 as the legitimate tier.

Fig. 5 highlights the nuanced impact of varying satellite numbers and beam
angles on $\mathcal{P}_{\rm sec}$. A key trade-off emerges in the number of
satellites: more satellites can improve coverage and security up to a point,
but beyond that, they might introduce excessive interference or potential
eavesdroppers. In this case, we should emphasize the need for an optimized
satellite constellation design.

Moving to Fig. 6, as $\gamma$ increases, the behaviour of $\mathcal{P}_{\rm
sec}$ is totally differently for the three densities of IoT devices. For the
sparsest IoT network ($\lambda_{u}=10^{-6}$), $\mathcal{P}_{\rm sec}$ exhibits
a peak at lower $\gamma$ values. When interference is negligible, allocating
more power to AN can effectively protect the PLS. In contrast, in the high-
density scenario ($\lambda_{u}=10^{-4}$), an inverse trend is observed where
$\mathcal{P}_{\rm sec}$ is the lowest at $\gamma=0.1$ but increases steadily.
This trend reflects that ensuring the information can effectively reach LSs in
an interference-limited scenario is more important than preventing
eavesdropping. When $\lambda_{u}=10^{-5}$, the trend of $\mathcal{P}_{\rm
sec}$ initially rising and then declining suggests that balancing between
ensuring successful, legitimate communication and preventing eavesdropping is
crucial. In summary, Fig. 6 highlights a critical density-dependent
optimization strategy for allocating power between messaging and AN to
maximize the $\mathcal{P}_{\rm sec}$.

Figure 6: $\mathcal{P}_{\rm sec}$ as a function of power allocation parameter
across different densities of IoT devices.

(a) $\lambda_{u}=10^{-6}$

(b) $\lambda_{u}=10^{-5}$

Figure 7: Optimal beam angle and AN power allocation for maximizing
$\mathcal{P}_{\rm sec}$ in different IoT devices.

Figure 8: Impact of the LS altitude on $\mathcal{P}_{m,\rm suc}$ and
$\mathcal{P}_{\rm sec}$ across different IoT device densities.

###  IV-C Security Performance Optimization

In this subsection, we explore adjustments to the network configuration and
security outcomes to maximize or achieve optimal security.

Fig. 7 visually represents the optimization of $\mathcal{P}_{\rm sec}$ in
sparse ($\lambda_{u}=10^{-6}$) versus denser ($\lambda_{u}=10^{-5}$) IoT
devices deployment. There is a clear distinction in how $\theta_{\rm beam}$
and $\gamma$ influence $\mathcal{P}_{\rm sec}$ at different device densities.
As shown in Fig. 7(a), for lower density ($\lambda_{u}=10^{-6}$),
$\mathcal{P}_{\rm sec}$ is more sensitive to the configuration of $\theta_{\rm
beam}$ and $\gamma$, favoring wider beam angles and a large share of power
dedicated to AN. This indicates that securing communications relies more on AN
in environments with fewer devices, with a wider signal spread proving
beneficial due to the lower interference risk. Conversely, Fig. 7(b)
corresponds to a denser network ($\lambda_{u}=10^{-5}$), shows a different
trend, where $\mathcal{P}_{\rm sec}$ demonstrates reduced sensitivity to
variations in $\gamma$, especially at lower beam angles. The optimal
$\mathcal{P}_{\rm sec}$ levels, as indicated by a yellow gradient, are
observed across a broader range of beam angles compared to Fig. 7(a). This
pattern suggests that high-security levels are attainable with a variety of
$\gamma$ settings, with the yellow regions extending throughout the full range
of $\gamma$ values from low to high.

Overall, lower-density environments show the highest secrecy with significant
AN, while higher-density environments maintain high secrecy across varying
levels of AN power.

In Fig. 8, we explore the effects of varying altitudes in the legitimate tier
while keeping the altitudes of the potential eavesdropping tiers fixed at
$500$ and $1500$ km. The optimal altitude exists for both $\mathcal{P}_{m,\rm
suc}$ and $\mathcal{P}_{\rm sec}$ across varying densities of IoT devices.
Specifically, the optimal legitimate altitude decreases as the density
increases. At very high densities, the optimal altitude is found to be 600 km.
As the density decreases slightly, this optimal altitude increases to 1000 km,
and with a further reduction in density, it reaches 1200 km. This is the
result of a trade-off between availability and received signal strength. In
most cases, as the density of IoT devices increases, $\mathcal{P}_{m,\rm suc}$
and $\mathcal{P}_{\rm sec}$ tends to decrease. When $\lambda_{u}=10^{-5}$,
$\mathcal{P}_{\rm sec}$ presents more nuanced results. We interpret this
interesting phenomenon as interference protecting signal security. This
protective mechanism is similar to AN technology, where the received SINR of
LSs is sacrificed to introduce more interference to eavesdroppers. However,
this trade-off has its limits, as demonstrated by the highest density
($\lambda_{u}=10^{-4}$), which does not yield any beneficial impact on
$\mathcal{P}_{\rm sec}$. This suggests a threshold beyond which additional
interference no longer benefits securing communications.

##  V Conclusion

In conclusion, this paper represents a significant advancement in securing IoT
communications via LEO satellites by implementing PLS strategies enhanced by
SG tools and AN techniques. The primary contribution of this study lies in its
innovative approach to modeling the spatial distribution of legitimate and ESs
as well as IoT devices. This enables a more accurate analysis of security
threats and the effectiveness of PLS measures within a LEO satellite-enabled
IoT network. By integrating AN into the PLS framework and utilizing SG for
network analysis, this research not only addresses the substantial security
challenges presented by the extensive coverage and accessibility of LEO
satellites but also provides a scalable solution capable of adapting to the
dynamic nature of satellite positions and fluctuating channel conditions.
However, the trade-off analysis suggests that while the proposed model
significantly enhances security, careful management of AN is required to
prevent compromising communication integrity. Furthermore, the model operates
under the assumption of known eavesdropper altitudes, which may not always be
applicable in practical scenarios. Nonetheless, these findings offer valuable
insights into ongoing efforts to secure satellite-based IoT networks,
proposing avenues for future research to enhance the robustness and
applicability of PLS solutions in such complex and evolving communication
environments.

##  Appendix A Proof of Lemma 1

We consider the spatial distribution of satellites within a homogeneous BPP
across the surface of a sphere; we focus on the $m$-th tier (i.e., the sphere
$S_{m}$) to ascertain the distribution characteristics of the nearest
satellite. Within this framework, the probability
$\mathbb{P}[\theta_{m,0}\leq\theta]$ that a randomly chosen satellite falls
within a spherical cap defined by a central angle $\theta$ is proportional to
the area of this cap relative to the sphere’s total surface area.
Mathematically, this probability is given by:

$\displaystyle\mathbb{P}[\theta_{m,0}\leq\theta]$
$\displaystyle=\frac{\mathcal{S}\left(\mathcal{A}_{m,\rm
vis}\right)}{\mathcal{S}\left(\mathcal{A}_{S_{m}}\right)}=\frac{2\pi
R_{m}^{2}(1-\cos\theta)}{4\pi R_{m}^{2}}.$

Simplifying, we find:

$\mathbb{P}[\theta_{m,0}\leq\theta]=\frac{1-\cos\theta}{2}.$

The CDF for the contact angle $F_{\theta_{m,0}}(\theta)$, considers the case
where the serving satellite is the nearest among all $N_{m}$ i.i.d. satellites
and derived easily as follows

$\displaystyle F_{\theta_{m,0}}(\theta)$
$\displaystyle=1-\prod\limits_{1}^{N_{m}}\mathbb{P}[\theta_{m,0}>\theta]=1-\left[1-\frac{1-\cos\theta}{2}\right]^{N_{m}}.$

This equation captures the probability that at least one satellite is within
the spherical cap, effectively accounting for the presence of $N_{m}$
satellites in the calculation. The PDF can be obtained by differentiating the
CDF with respect to $\theta$.

##  Appendix B Proof of Lemma 3

Recall that the interference is expressed as

$\mathcal{I}_{m}=\sum\limits_{i\in\Phi_{u}\setminus\\{i=0\\}}P_{t}\left(\frac{c}{4\pi
fd_{m,i}}\right)^{2}G|h_{m,i}|^{2},$ (17)

where the $d_{m,i}$ is the distance between the serving LS with the
interfering $i$-th IoT device. We proceed with the derivation of the LT of
interference by definition:

$\displaystyle\mathcal{L}_{\mathcal{I}_{m}}$
$\displaystyle(s)\triangleq\mathbbm{E}\left\\{\exp{\left(-s\mathcal{I}_{m}\right)}\Bigg{|}\theta,\mathcal{I}_{m}\right\\}$
$\displaystyle=\mathbbm{E}_{\Phi_{u},\left\\{|h_{m,i}|^{2}\right\\}}\left\\{\exp{\left(-s\sum\limits_{i\in\Phi_{u}\setminus\\{i=0\\}}P_{t}\left(\frac{c}{4\pi
fd_{m,i}}\right)^{2}G|h_{m,i}|^{2}\right)}\right\\}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbbm{E}_{\Phi_{u}}\left\\{\prod\limits_{i\in\Phi_{u}\setminus\\{i=0\\}}\mathbbm{E}_{|h_{m}|^{2}}\left[\exp{\left(-sP_{t}\left(\frac{c}{4\pi
fd_{m,i}}\right)^{2}G|h_{m}|^{2}\right)}\right]\right\\}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\mathbbm{E}_{\Phi_{u}}\left\\{\prod\limits_{i\in\Phi_{u}\setminus\\{i=0\\}}\left(1+m_{2}sP_{t}G\left(\frac{c}{4\pi
fd_{m,i}}\right)^{2}\right)^{-m_{1}}\right\\}$
$\displaystyle\begin{multlined}\stackrel{{\scriptstyle(c)}}{{=}}\exp\left(-\lambda_{u}\int\limits_{0}^{2\pi}\int\limits_{0}^{\theta_{m,\rm
max}}\left[1-\left(1+m_{2}sP_{t}G\left(\frac{c}{4\pi
fd}\right)^{2}\right)^{-m_{1}}\right]\right.\left.\vphantom{\int\limits_{0}^{2\pi}}\times
R_{\oplus}^{2}\sin{\theta}d\phi
d\theta\right),\end{multlined}\stackrel{{\scriptstyle(c)}}{{=}}\exp\left(-\lambda_{u}\int\limits_{0}^{2\pi}\int\limits_{0}^{\theta_{m,\rm
max}}\left[1-\left(1+m_{2}sP_{t}G\left(\frac{c}{4\pi
fd}\right)^{2}\right)^{-m_{1}}\right]\right.\left.\vphantom{\int\limits_{0}^{2\pi}}\times
R_{\oplus}^{2}\sin{\theta}d\phi d\theta\right),$

where step $(a)$ stands since the fading distribution $|h_{m,i}|^{2}$ for each
IoT device is i.i.d. and is independent of the interfering point process
$\Phi_{u}$. Step $(b)$ follows the Moment Generating Function (MGF) of the
Gamma distribution. Finally, step $(c)$ follows the Probability Generating
Functional (PGFL) of the PPP, considering that the interference signal
originates from devices within the satellite’s coverage area.

##  Appendix C Proof of Lemma 4

Having already derived the expression for availability probability, we now
turn our attention to finalizing the successful communication probability by
deriving the coverage probability. As previously noted, under the assumption
that tier $m$ represents the legitimate tier, it suffices to derive the
expression for $\mathcal{P}_{m,\rm cov}$, the probability that the predefined
threshold is lower than the SINR of the LS. The derivation proceeds are as
follows:

$\displaystyle\mathcal{P}$ ${}_{m,\rm
cov}\triangleq\mathbb{P}\left[\text{SINR}_{\rm LS}>\beta_{\rm LS}\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\mathbb{P}\left[\frac{\gamma
P_{t}\left(\frac{c}{4\pi
fd_{m,0}}\right)^{2}G|h_{m,0}|^{2}}{\mathcal{I}_{m}+\sigma_{\rm
S}^{2}}>\beta_{\rm LS}\right]$
$\displaystyle=\mathbb{E}_{\theta,\mathcal{I}_{m}}\left\\{1-F_{|h_{m,0}|^{2}}\left(\frac{\beta_{\rm
LS}(\mathcal{I}_{m}+\sigma_{\rm S}^{2})}{\gamma P_{t}G}\left(\frac{4\pi
fd_{m,0}}{c}\right)^{2}\right)\right\\}$
$\displaystyle\begin{multlined}\stackrel{{\scriptstyle(b)}}{{\approx}}\mathbb{E}_{\theta,\mathcal{I}_{m}}\left\\{1-\left[1-\exp\left(-\frac{(m_{1}!)^{-\frac{1}{m_{1}}}}{m_{2}}\frac{\beta_{\rm
LS}}{\gamma P_{t}G}\left(\frac{4\pi
fd_{m,0}}{c}\right)^{2}\right.\right.\right.\\\
\left.\left.\left.\vphantom{\frac{(m_{1}!)^{-\frac{1}{m_{1}}}}{m_{2}}}\times\left(\mathcal{I}_{m}+\sigma_{\rm
S}^{2}\right)\right)\right]^{m_{1}}\right\\}\end{multlined}\stackrel{{\scriptstyle(b)}}{{\approx}}\mathbb{E}_{\theta,\mathcal{I}_{m}}\left\\{1-\left[1-\exp\left(-\frac{(m_{1}!)^{-\frac{1}{m_{1}}}}{m_{2}}\frac{\beta_{\rm
LS}}{\gamma P_{t}G}\left(\frac{4\pi
fd_{m,0}}{c}\right)^{2}\right.\right.\right.\\\
\left.\left.\left.\vphantom{\frac{(m_{1}!)^{-\frac{1}{m_{1}}}}{m_{2}}}\times\left(\mathcal{I}_{m}+\sigma_{\rm
S}^{2}\right)\right)\right]^{m_{1}}\right\\}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\mathbb{E}_{\theta,\mathcal{I}_{m}}\left\\{1-\sum\limits_{q=0}^{m_{1}}\binom{m_{1}}{q}(-1)^{q}\exp{\left(-s_{\rm
LS}q\left(\mathcal{I}_{m}+\sigma_{\rm S}^{2}\right)\right)}\right\\}$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{=}}\sum\limits_{q=1}^{m_{1}}\binom{m_{1}}{q}(-1)^{q+1}\int\limits_{0}^{\theta_{m,\rm
max}}\exp{\left(-qs_{\rm LS}\sigma_{\rm
S}^{2}\right)}\mathcal{L}_{\mathcal{I}_{m}}\left(qs_{\rm
LS}\right)f_{\theta_{m,0}}(\theta)d\theta,$

where step $(a)$ is derived by direct expansion of the definition in (5), step
$(b)$ follows the upper bound for the CDF of the Gamma function detailed in
(4). This approximation is crucial for simplifying the expression involving
the channel gain’s distribution, facilitating a more tractable analysis of the
coverage probability. Step $(c)$ utilizes Newton’s generalized binomial
theorem to expand the expression. The scaling factor $s$ in step $(c)$ is
defined as:

$\displaystyle s_{\rm
LS}=\frac{(m_{1}!)^{-\frac{1}{m_{1}}}}{m_{2}}\frac{\beta_{\rm LS}}{\gamma
P_{t}G}\left(\frac{4\pi fd_{m,0}}{c}\right)^{2}$

Finally, step $(d)$ integrates the expectation over the distribution of the
contact angle, $\theta_{m,0}$, and the interference model, taking into account
the spatial geometry of satellite coverage and the statistical properties of
the channel. This concludes the proof.

##  Appendix D Proof of Theorem 1

To state the final expression of the secure communication probability, it’s
sufficient to outline the derivation secrecy outage probability. Here, we
consider the influence of potential ESs across all tiers, with the exception
of the legitimate tier. The objective is to evaluate the maximum SINR
experienced by these ESs. Therefore, the secrecy probability of a typical IoT
device is defined as the probability that the SINR by all ESs for all tiers is
less than threshold $\beta_{\rm ES}$. First, we will find the strongest SINR
of any tier $k\neq m$ such that

$\displaystyle\mathcal{P}_{\rm out}$
$\displaystyle\triangleq\mathbb{P}\left[\max_{k\neq m}\max_{j\in\Phi_{\rm
S_{k}}}\left\\{\text{SINR}_{kj}\right\\}<\beta_{\rm ES}\right]$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\prod_{\begin{subarray}{c}k=1\\\
k\neq
m\end{subarray}}^{K}\prod\limits_{j=1}^{N_{k}}\mathbb{P}\left[\text{SINR}_{kj}<\beta_{\rm
ES}\right]$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{=}}\prod_{\begin{subarray}{c}k=1\\\
k\neq m\end{subarray}}^{K}\left[\int\limits_{0}^{\theta_{k,\rm
max}}\mathbb{P}\left[\text{SINR}_{k}<\beta_{\rm
ES}\right]f_{\theta_{k,0}}(\theta)d\theta+\int\limits_{\theta_{k,\rm
max}}^{\pi}f_{\theta_{k,0}}(\theta)d\theta\right]^{N_{k}}$

where step $(a)$ follows from the independence of satellites’ positions. Step
$(b)$ holds because of the i.i.d. characteristic of the channel fading.

Diving into the probability calculation for a single ES, we manipulate the
SINR condition under the constraint $d<d_{\text{max}}$, leading to a form that
involves expected values over the contact angle distribution and the
interference model. The contact angle distribution defined for each satellite
can be derived from Lemma 1 by assuming $N_{k}=1$:

$\displaystyle f_{\theta_{k,0}}(\theta)$
$\displaystyle=\frac{\sin{\theta}}{2}.$ (18)

Following similar steps to the coverage probability proof, we arrive at the
formulation for the integral part of step $(b)$:

$\displaystyle\int\limits_{0}^{\theta_{k,\rm
max}}\mathbb{P}\left[\text{SINR}_{S_{k}}<\beta_{\rm
ES}\right]f_{\theta_{k,0}}(\theta)d\theta$
$\displaystyle=\sum\limits_{q=0}^{m_{1}}\binom{m_{1}}{q}(-1)^{q}\int\limits_{0}^{\theta_{k,\rm
max}}\exp{\left(-qs_{\rm
ES}\sigma_{S}^{2}\right)}\mathcal{L}_{\mathcal{I}_{kj}}\left(qs_{\rm
ES}\right)f_{\theta_{k,0}}(\theta)d\theta$

with $s_{\rm ES}$ defined as:

$\displaystyle s_{\rm
ES}=\frac{(m_{1}!)^{-\frac{1}{m_{1}}}}{m_{2}}\frac{\beta_{\rm
ES}}{\left[\gamma-\beta_{\rm
ES}\left(1-\gamma\right)\right]P_{t}G}\left(\frac{4\pi
fd_{kj,0}}{c}\right)^{2}.$

The second integral part in step $(b)$ can be calculated as:

$\displaystyle\int\limits_{\theta_{k,\rm
max}}^{\pi}f_{\theta_{k,0}}(\theta)d\theta=\frac{1+\cos{\theta_{k,\rm
max}}}{2}.$

Combining the above results, we articulate the secrecy outage probability (1)
as

$\displaystyle\mathcal{P}_{\rm out}=\prod_{\begin{subarray}{c}k=1\\\ k\neq
m\end{subarray}}^{K}$
$\displaystyle\left[\sum\limits_{q=0}^{m_{1}}\binom{m_{1}}{q}(-1)^{q}\int\limits_{0}^{\theta_{k,\rm
max}}\exp{\left(-qs_{\rm
ES}\sigma_{S}^{2}\right)}\mathcal{L}_{\mathcal{I}_{kj}}\left(qs_{\rm
ES}\right)\right.$ $\displaystyle\hskip
85.35826pt\left.\vphantom{\int\limits_{0}^{2\pi}}f_{\theta_{k,0}}(\theta)d\theta+\frac{1+\cos{\theta_{k,\rm
max}}}{2}\right]^{N_{k}}.$

In summary, this proof thoroughly derives the secrecy probability by
considering the strongest eavesdropper’s SINR across all non-legitimate tiers,
effectively encapsulating the multifaceted dynamics of satellite communication
systems where secrecy against ESs is paramount.

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  *[LEO]: Low Earth Orbit
  *[IoT]: Internet of Things
  *[SG]: Stochastic Geometry
  *[PLS]: Physical Layer Security
  *[AN]: Artificial Noise
  *[UAV]: unmanned aerial vehicles
  *[BPP]: Binomial Point Process
  *[PPP]: Poisson Point Process
  *[i.i.d.]: independent, identically distributed
  *[LS]: legitimate satellite
  *[ES]: eavesdropper satellite
  *[CDF]: Cumulative Distribution Function
  *[SINR]: signal-to-interference and noise ratio
  *[PDF]: Probability Density Function
  *[LT]: Laplace Transform
  *[MGF]: Moment Generating Function
  *[PGFL]: Probability Generating Functional