# Novel Outage-Aware NOMA Protocol for Secrecy Fairness Maximization Among
Untrusted Users

Sapna Thapar, Deepak Mishra,
and Ravikant Saini Copyright (c) 2015 IEEE. Personal use of this material is
permitted. However, permission to use this material for any other purposes
must be obtained from the IEEE by sending a request to pubs-
<EMAIL_ADDRESS>Thapar and R. Saini are with the Department of
Electrical Engineering, Indian Institute of Technology Jammu, Jammu, Jammu
$\&$ Kashmir 181 221, India (e-mail<EMAIL_ADDRESS>ravikant.saini@iitjammu.ac.in). D. Mishra is with the School of Electrical
Engineering and Telecommunications, University of New South Wales Sydney, NSW
2052, Australia (e-mail: d.mishra@unsw.edu.au).This research work was
supported by the Tata Consultancy Services Research Scholar Program (TCS RSP)
Fellowship.A preliminary version [1] of this work has been published at IEEE
GLOBECOM, Hawaii, USA, Dec. 2019.

###### Abstract

Observing the significance of spectrally-efficient secure non-orthogonal
multiple access (NOMA), this paper proposes a novel quality of service (QoS)
aware secure NOMA protocol that maximizes secrecy fairness among untrusted
users. Considering a base station (BS) and two users, a novel decoding order
is designed that provides security to both users. With the objective of
ensuring secrecy fairness between users, while satisfying their QoS
requirements under BS transmit power budget constraint, we explore the problem
of minimizing the maximum secrecy outage probability (SOP). Closed-form
expression of pair outage probability (POP) and optimal power allocation (PA)
minimizing POP are obtained. To analyze secrecy performance, analytical
expressions of SOP for both users are derived, and individual SOP minimization
problems are solved using concept of generalized-convexity. High signal-to-
noise ratio approximation of SOP and asymptotically optimized solution
minimizing this approximation is also found. Furthermore, global-optimal
solution from secrecy fairness standpoint is obtained at low computational
complexity, and tight approximation is derived to get analytical insights.
Numerical results present useful insights on globally optimized PA which
ensure secrecy fairness and provide performance gain of about $55.12\%$,
$69.30\%$, and $19.11\%$, respectively, compared to fixed PA and individual
users’ optimal PAs. Finally, a tradeoff between secrecy fairness performance
and QoS demands is presented.

Index Terms$-$ 5G communications, non-orthogonal multiple access, physical
layer security, power allocation, optimization.

##  I Introduction

Exponentially increasing network traffic, imposed by internet-enabled
applications, poses severe challenges of supporting massive connections loaded
with high data rate demands for upcoming fifth generation (5G) communication
networks. Non-orthogonal multiple access (NOMA) has been recognized as a
potential breakthrough because of the possibility of accommodating a number of
users within the same subcarrier allocation [2]. This exhibits a significant
spectral efficiency enhancement compared to traditional orthogonal multiple
access techniques. But, the broadcast nature of communication at the
transmitter makes NOMA vulnerable to the eavesdropping. Therefore, research on
security issues in NOMA networks is of great significance. The amalgamation of
NOMA and physical layer security (PLS) has been observed as a promising
solution to provide spectrally-efficient secure wireless communication [3].
Despite merits, the design process has secrecy challenge of wiretapping in the
presence of untrusted NOMA users.

###  I-A Related Art

To address the demand for massive connections, modern wireless networks strive
for better spectral efficiency. Motivated by the potential of NOMA in
providing spectrally-efficient networks, survey papers [2], [4] have
summarized recent research contributions in power-domain NOMA. Besides, the
broadcast nature of wireless transmission leads to security issues on the
information-carrying signal. Triggered by these security concerns, researchers
have considered the PLS technique which was first recommended by Wyner [5] as
a complement to cryptographic approaches. To highlight the capabilities of
PLS, [6] has provided the latest survey of PLS research on diverse 5G
technologies, such as multiple input multiple output, full-duplex, and
millimeter wave.

Even though a great amount of research has been done in NOMA and PLS
independently, researchers have recently focused attention on using PLS as a
security measure in NOMA. An investigation of PLS in large-scale networks by
invoking stochastic geometry has been done in [3], where a base station (BS)
communicates with randomly distributed NOMA users and eavesdroppers. The
authors have proposed a protected zone around BS to maintain an eavesdropper
free zone. Secure NOMA transmission for multiple users against eavesdropper in
single-input single-output network has been studied in [7], where the authors
have derived a closed-form expression of optimal power allocation (PA) that
maximizes sum secure rate of NOMA users. In [8], the authors have investigated
the optimal decoding order, individual transmission rates, and PA to each user
to design secure NOMA system against external eavesdropper under secrecy
outage constraint. The authors have proved that optimal decoding order for
secure NOMA scheme against external eavesdropper is same as that for
conventional NOMA. Secrecy of a cooperative NOMA system with a decode-and-
forward and an amplify-and-forward relay against external eavesdropper has
been analyzed in [9]. The authors in [10] have proposed a NOMA-inspired
jamming and relaying scheme for enhancing the PLS of untrusted relay networks.
In [11], the authors considered a multiple-input single-output NOMA cognitive
relay radio network in the presence of multiple eavesdroppers and proposed an
artificial noise (AN) aided cooperative jamming scheme for secure transmission
in considered network.

It is noteworthy that the aforementioned works have focused on securing NOMA
against external eavesdropper only. Besides external eavesdroppers, secure
data transmission among NOMA users is much more challenging due to the
underlying SIC based decoding at receivers. Actually, due to the sharing of
same resource blocks among users in NOMA systems, users may not trust each
other in real scenario, and hence at least some level of secrecy must be
provided against internal eavesdropping. Based on the idea, assuming far users
as untrusted, sum secrecy rate of trusted near users is investigated in [12]
for a multiple-input single-output NOMA system. In [13] also, a NOMA system is
considered where near user is assumed to be trusted, whereas far user as
untrusted, and SOP of trusted user against untrusted user has been
investigated. Note that in [12] and [13], it is assumed that the untrusted far
user first decodes its own data and then can access the message of the trusted
near user with the aid of the SIC technique.

###  I-B Research Gap and Motivation

Note that existing works [3], [7]$-$[11] have considered potential PLS
techniques such as AN aided strategy [3], [11], optimal PA [7], [8], and
cooperative relaying [9], [10] to enhance secrecy of NOMA transmission against
external eavesdroppers assuming all NOMA users as trusted. Since NOMA users
share same resource block and have access to decode the message of other
paired user to perform SIC, there exists an inherent security issue in NOMA in
the case of untrusted users. Therefore, at least some level of secrecy must be
provided against internal eavesdropping. Note that the untrusted scenario is a
hostile situation, where users have no mutual trust, and each user focuses on
achieving secure communication from BS in the presence of other untrusted
users, which leads to more complex and constrained resource allocations [14],
[15]. It is noteworthy that the existing works [12] and [13] only considered
the security problem of trusted near user in a NOMA system against untrusted
far untrusted user. However, near user also decodes the data of far user,
which causes a serious security risk for far user in the case of untrusted
near user. Therefore, from security point of view, it is more desirable to
design a secure NOMA system for worst-case scenario, where both near and far
users are assumed to be untrusted. Thus, with the objective of designing a
smart secure communication network, our work focuses on investigating whether
secure communication is possible in NOMA in the presence of untrusted users.

In this direction, [16] considered symbol-level SIC (SLSIC) receiver where
other user’s signal is demodulated but not decoded to perform SIC. In
contrast, codeword-level SIC (CLSIC) receiver exists in literature [17], where
other user’s signal is demodulated and decoded. Since channel decoding is
conducted during signal detection in CLSIC receiver, the probability of
successful recovery of other user’s signal increases in comparison to SLSIC,
and hence, the impact of error propagation can be reduced drastically [17].
Nevertheless, securing all users’ data in an untrusted scenario from a
decoding perspective, which is a better system design aspect, has not been
considered yet in literature.

It also needs to noted that when both users are assumed to be untrusted in
NOMA, one might think that if any of these users is capable of decoding other
user’s signal then security and privacy of decoded users are compromised. But,
decoding of users’ signals can even happen in orthogonal multiple access
because of the broadcast nature of wireless channels [18]. Actually, we focus
on achieving secure communication by utilizing the concept of PLS. According
to the definition of PLS, information leakage of the legitimate user may occur
only when data rate over the desired link is lesser than that of the potential
eavesdropper’s link because the secrecy rate for a legitimate user is defined
as the difference of the rates when a legitimate user decodes itself, and the
rate that another user achieves while decoding data of legitimate user [5],
[6]. _Therefore, considering the definition of PLS into account for an
untrusted NOMA system, our work is focused on designing such a secure NOMA
transmission protocol that can be feasible in providing positive secrecy rate
for all NOMA users._

Taking this challenge into account, a decoding order has been proposed for a
two-user untrusted NOMA scenario in the preliminary version [1] of this work
that is feasible in providing positive secrecy rate for both near and far NOMA
users. As an extension of [1], in this work, we study all possible decoding
orders for a two user NOMA system including conventional one and investigate
the best decoding order under secrecy constraint for both users. Besides, in
[1], we have considered ideal SIC receivers in which interference from the
decoded user is completely removed while decoding the later user. Though
better spectral efficiency can be achieved through perfect SIC, but it is not
realistic due to various implementation issues such as decoding error and
complexity scaling [4], [19]. Therefore, a practical scenario would be, when
residual interference from the inaccurately decoded user is considered while
decoding later user. To this end, we study secure NOMA to provide positive
secrecy rates at all untrusted users with consideration of imperfect SIC based
decoding which, to the best of our knowledge, is an open problem.

It should be emphasized that, with the vision of ensuring secure communication
to all users, secrecy performance of the far user against the untrusted near
user, and rather secrecy fairness between these users cannot be ignored.
Fulfilling users’ quality of service (QoS) requirement is also an essential
parameter for spectrally-efficient communication. Also, PA to users may play
deciding role in obtaining optimal pair outage and secrecy outage performance
of a system. Inspired by these solid observations, this paper focuses on
investigating numerical and analytical global-optimal PA solutions for a
secure NOMA system with untrusted users from a secrecy fairness maximization
point of view while satisfying users’ QoS requirements.

###  I-C Key Contributions

The contributions of this work are summarized as follows:

  * •

Considering a NOMA system with one BS, two untrusted users and imperfect SIC
model, a novel optimal decoding order is proposed that provides security to
both users.

  * •

Pair outage probability (POP), as a QoS measure, has been derived
analytically. Optimal PA minimizing POP has been obtained using generalized-
convexity of POP.

  * •

Analytical expressions of SOPs have been derived for both users. SOP
minimization problems have been solved using pseudoconvexity of SOP at both
users and optimal PAs are obtained numerically. Asymptotic approximations of
SOP, and optimal PA for both users are also derived to get analytical
insights.

  * •

To maximize the secrecy fairness between users while satisfying users’ QoS
demands, we formulate the problem of minimizing the maximum SOP between users
under the POP and PA constraints. Global-optimal PA solution is obtained using
a low complexity algorithm. Tight analytical approximation for optimal PA is
also derived.

  * •

Extensive simulations are conducted that validate the accuracy of analysis,
present insights on optimal performance, and evaluate performance gains by the
proposed solution. Tradeoff between secrecy fairness performance of system and
users’ QoS demands is also investigated.

###  I-D Organization

The system model, proposed NOMA protocol and investigation on optimal decoding
order have been presented in Section II. In Section III, we provide analysis
of pair outage performance of the proposed system and derive an optimal PA
policy to minimize POP. In section IV, we present SOP analysis for both users,
along with their closed-form asymptotic expressions, and also investigate
their optimality. Analysis of optimal PA for secrecy fairness maximization
between untrusted users under POP constraint is described in Section V.
Performance evaluation of proposed analytical model via simulation results is
discussed in Section VI, and Section VII presents concluding remarks and
future directions.

##  II Secure NOMA Protocol for Untrusted Users

In this section, we first present the system model, which is followed by the
NOMA transmission protocol. Then, an optimal decoding order to secure
untrusted NOMA is investigated.

Figure 1: Illustration of downlink secure NOMA protocol with proposed optimal
decoding order for two untrusted users where decoding order is changed for the
far user in comparison to the conventional approach.

###  II-A System Model

We consider the downlink NOMA system with one BS and two untrusted users (Fig.
1). The near and far users are denoted as U$1$ and U$2$, respectively. Our
consideration for two-user model is motivated by two reasons: firstly, NOMA
system is an interference limited system because co-channel interference is
strong in NOMA due to sharing of same resource by multiple users [20], [21];
and secondly, the implementation complexity at transmitter and receiver sides
increases with an increase in number of users because the users perform SIC to
remove messages of other users before decoding their own messages [22]. Note
that due to these reasons, asking all the users in the system to perform NOMA
jointly is not preferable in practice [21] and therefore, the users can be
divided into multiple groups, where NOMA is implemented within each group. The
users are ordered according to their distance to BS. As a result, U$1$ and
U$2$ are considered as strong and weak users, respectively [23], [24]. The
Rayleigh fading channel gain coefficients from BS to U$i$ is denoted by
$h_{i}$ for $i\in\\{1,2\\}$. Channels between BS and users are assumed to be
independent and suffer both small scale fading and path loss effects, such
that channel power gains $|h_{i}|^{2}$ follows exponential distribution with
mean parameter $\lambda_{i}=L_{p}d_{i}^{-n}$. Here $L_{p}$ is path loss
constant, $n$ is path loss exponent, and $d_{i}$ is the distance from BS to
U$i$. The channel power gains have been sorted such that
$|h_{1}|^{2}>|h_{2}|^{2}$ [23], [24]. All three nodes are assumed to be
equipped with a single antenna [13]. $P_{t}$ denotes total power transmitted
by BS to users, and $\alpha$ represents the PA coefficient denoting the
fraction of $P_{t}$ allocated to U$1$. Remaining $(1-\alpha)P_{t}$ is
allocated to U$2$.

###  II-B Proposed Secure NOMA Protocol

In NOMA, the message signals dedicated to the users are superimposed at the BS
and then transmitted to the users. The signal transmitted by BS is
$\sqrt{\alpha P_{t}}x_{1}+\sqrt{(1-\alpha)P_{t}}x_{2}$ [13] where $x_{1}$ and
$x_{2}$ are unit power signals which contains the message required by U$1$ and
U$2$, respectively. The signals received by U$1$ and U$2$, respectively, from
BS are given as

$y_{1}=h_{1}(\sqrt{\alpha P_{t}}x_{1}+\sqrt{(1-\alpha)P_{t}}x_{2})+n_{1},$ (1)
$y_{2}=h_{2}(\sqrt{\alpha P_{t}}x_{1}+\sqrt{(1-\alpha)P_{t}}x_{2})+n_{2},$ (2)

where $n_{1}$ and $n_{2}$ denote zero mean additive white Gaussian noise with
variance $\sigma^{2}$ at U$1$ and U$2$, respectively.
$\rho_{t}\stackrel{{\scriptstyle\Delta}}{{=}}P_{t}$/$\sigma^{2}$ denotes BS
transmit signal-to-noise ratio (SNR).

After obtaining received signals, two steps are followed at receivers in
conventional decoding order: (1) far user decodes its own information signal
first and after obtaining its own signal it may decode the information signal
of the near user using SIC in case of the far user is an untrusted user [12],
[13]; (2) near user first decodes the signal intended for far user to perform
SIC, and then decodes its own message. _As a result, near and far users both
have access to the signals of far and near user, respectively, which is a
critical issue in secure communication with untrusted users._ Note that the
SIC can be performed by all users in a system because SIC on the receivers’
side is actually a physical layer capability enabling the receiver to decode
packets that arrive collectively. Thus, far user may also decode the
information signal of near user using SIC which has been extensively studied
in literature also [12], [13], [25], [26]. In this work, we consider imperfect
SIC model, where decoding error may occur at both users, resulting in residual
interference from the incompletely decoded user after SIC.

Since all users focus to safe their own data from others in untrusted
scenario, next we investigate a decoding order that can provide security at
both users. Even the conventional decoding order has also not yet been
investigated in the literature to ensure positive secrecy for both users in an
untrusted NOMA system. Hence, we discuss all possible decoding orders
including conventional one to investigate the best decoding order under
secrecy constraint.

###  II-C Optimal Decoding Order from Secrecy Perspective

In secure NOMA, U$1$ needs to be protected from U$2$ and, vice-versa. Decoding
order identifies whether any user decodes its own data first or other user’s
data. Note that both U$1$ and U$2$ are the multiplexed NOMA users which
themselves are the part of the system. Now since BS decides the decoding order
for both the users of the system, decoding order can also be changed [25].
Thus, for two users’ case, four decoding orders are possible. Let us denote
the decoding order as $(i,j)$, where $i$ and $j$, respectively, denotes users,
i.e., U$i$ and U$j$ (for $i\in\\{1,2\\},j\in\\{1,2\\}$) whose data will be
decoded first by U$1$ and U$2$. For example, $(2,1)$ means U$1$ and U$2$ will
decode other user’s data first, and then decode its own data. Specifically,
the four possible cases are: (a) $(1,1)$, (b) $(1,2)$, (c) $(2,1)$ and (d)
$(2,2)$. Under secrecy considerations, the idea is to study whether it is
possible to achieve positive secrecy rate at both users. In this regard,
investigation of feasible decoding orders out of total possible decoding
orders is presented by Theorem $1$.

###### Theorem 1

Positive secrecy rate for both near and far users in untrusted NOMA can be
obtained in decoding orders $(2,1)$, $(1,2)$, and $(1,1)$.

###### Proof:

We consider each case of decoding orders one by one and investigate the
feasibility under positive secrecy constraint. We first consider the
conventional decoding order $(2,2)$, where each user first decodes U$2$’s
data. Then, the proof continues with a consideration of remaining cases, i.e.,
$(2,1)$, $(1,2)$ and $(1,1)$. Adopting the imperfect SIC model, we use
$\beta_{ij}$ to denote the residual interference from the imperfectly decoded
U$i$ when data of U$i$ is decoded by U$j$.

Case 1: Considering conventional decoding order $(2,2)$ [13], U$2$ first
decodes its own signal by considering other signal as noise. After decoding
its own signal, it may decode the signal of near user using SIC. Then, U$1$
decodes signal associated to U$2$, applies SIC, and then decode its own signal
from the remaining signal. Let $\Gamma_{ij}$ denote the received signal-to-
interference-plus-noise-ratio at U$i$ when data of U$i$ is decoded by U$j$.
Various $\Gamma_{ij}$ can be given as [13]

$\Gamma_{21}=\frac{(1-\alpha)|h_{1}|^{2}}{\alpha|h_{1}|^{2}+\frac{1}{\rho_{t}}},\quad\Gamma_{11}=\frac{\alpha|h_{1}|^{2}}{\beta_{21}+\frac{1}{\rho_{t}}},$
(3)
$\Gamma_{22}=\frac{(1-\alpha)|h_{2}|^{2}}{\alpha|h_{2}|^{2}+\frac{1}{\rho_{t}}},\quad\Gamma_{12}=\frac{\alpha|h_{2}|^{2}}{\beta_{22}+\frac{1}{\rho_{t}}}.$
(4)

The achievable secrecy rate $R_{s1}$ of U$1$ can be given as

$R_{s1}=R_{11}-R_{12},$ (5)

where $R_{11}$ and $R_{12}$ are given by Shannon’s Theorem [27] as

$R_{11}=\log_{2}(1+\Gamma_{11}),\quad R_{12}=\log_{2}(1+\Gamma_{12}).$ (6)

Positive secrecy rate condition $R_{11}>R_{12}$ (5), simplified as
$\Gamma_{11}>\Gamma_{12}$ (6) for U$1$ gives a condition
$(\rho_{t}\beta_{22}+1)|h_{1}|^{2}>(\rho_{t}\beta_{21}+1)|h_{2}|^{2}$, which
shows that based on the values of $\rho_{t}$, $\beta_{22}$, $\beta_{21}$,
$|h_{1}|^{2}$ and $|h_{2}|^{2}$, a feasible condition can be obtained. This
proves that positive secrecy rate can be achievable at U$1$. Similarly, the
achievable secrecy rate $R_{s2}$ of U$2$ is

$R_{s2}=R_{22}-R_{21},$ (7)

where

$R_{22}=\log_{2}(1+\Gamma_{22}),\quad R_{21}=\log_{2}(1+\Gamma_{21}).$ (8)

Here the condition $R_{22}>R_{21}$ (7), simplified as
$\Gamma_{22}>\Gamma_{21}$ (8) gives $|h_{2}|^{2}>|h_{1}|^{2}$ which is an
infeasible condition because channel power gains are sorted as
$|h_{1}|^{2}>|h_{2}|^{2}$. Thus, positive secrecy rate cannot be achieved at
U$2$.

Since our goal is to provide positive secrecy rate to both users, the
conventional decoding order cannot be considered for untrusted NOMA because it
cannot provide positive secrecy rate for the far user against the untrusted
near user. Therefore, next we study other possible decoding orders one by one
and investigate the optimal decoding order.

Case 2: Next, we consider decoding order $(2,1)$ where both users first decode
signals associated to other user, and then decode their own signal after
performing SIC. In decoding order (2,1), the order of decoding is different
only for the far user in comparison to conventional decoding order. Note that
the far user can even first decode the signal of near user, and then decodes
its own signal after applying SIC [25], [26]. As a result, we obtain

$\Gamma_{21}=\frac{(1-\alpha)|h_{1}|^{2}}{\alpha|h_{1}|^{2}+\frac{1}{\rho_{t}}},\quad\Gamma_{11}=\frac{\alpha|h_{1}|^{2}}{\beta_{21}+\frac{1}{\rho_{t}}},$
(9)
$\Gamma_{12}=\frac{\alpha|h_{2}|^{2}}{(1-\alpha)|h_{2}|^{2}+\frac{1}{\rho_{t}}},\quad\Gamma_{22}=\frac{(1-\alpha)|h_{2}|^{2}}{\beta_{12}+\frac{1}{\rho_{t}}}.$
(10)

Positive secrecy rate condition $\Gamma_{11}>\Gamma_{12}$, at U$1$ gives

$\alpha<1+\frac{|h_{1}|^{2}-|h_{2}|^{2}-|h_{2}|^{2}\rho_{t}\beta_{21}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}}.$
(11)

Similarly, the feasibility condition $\Gamma_{22}>\Gamma_{21}$ for positive
secrecy rate at U$2$, gives

$\alpha>\frac{|h_{1}|^{2}-|h_{2}|^{2}+|h_{1}|^{2}\rho_{t}\beta_{12}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}}.\\\
$ (12)

Observing (11) and (12), we note that decoding order $(2,1)$ can provide
positive secrecy rate to both the users in untrusted NOMA, provided $\alpha$
is constrained as
$\left(\frac{|h_{1}|^{2}-|h_{2}|^{2}+|h_{1}|^{2}\rho_{t}\beta_{12}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}}<\alpha<1+\frac{|h_{1}|^{2}-|h_{2}|^{2}-|h_{2}|^{2}\rho_{t}\beta_{21}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}}\right)$.

Case 3: In decoding order $(1,2)$, U$1$ and U$2$ first detect their own
signals, apply SIC, and then decode signal of other user. Similar to the
aforementioned analysis, we obtain
$\Gamma_{11}=\frac{\alpha|h_{1}|^{2}}{(1-\alpha)|h_{1}|^{2}+\frac{1}{\rho_{t}}},\Gamma_{21}=\frac{(1-\alpha)|h_{1}|^{2}}{\beta_{11}+\frac{1}{\rho_{t}}},\Gamma_{22}=\frac{(1-\alpha)|h_{2}|^{2}}{\alpha|h_{2}|^{2}+\frac{1}{\rho_{t}}},\Gamma_{12}=\frac{\alpha|h_{2}|^{2}}{\beta_{22}+\frac{1}{\rho_{t}}}$.
In this case, $\Gamma_{11}>\Gamma_{12}$ leads to
$\alpha>1-\frac{|h_{1}|^{2}-|h_{2}|^{2}+|h_{1}|^{2}\rho_{t}\beta_{22}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}}$,
and $\Gamma_{22}>\Gamma_{21}$ provides
$\alpha<\frac{|h_{2}|^{2}-|h_{1}|^{2}+|h_{2}|^{2}\rho_{t}\beta_{11}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}}$
which are feasible. Thus, $(1,2)$ is concluded as feasible decoding order in
providing positive secrecy rate to both users under constraint
$\left(1-\frac{|h_{1}|^{2}-|h_{2}|^{2}+|h_{1}|^{2}\rho_{t}\beta_{22}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}}<\alpha<\frac{|h_{2}|^{2}-|h_{1}|^{2}+|h_{2}|^{2}\rho_{t}\beta_{11}}{|h_{1}|^{2}|h_{2}|^{2}\rho_{t}}\right)$.

Case 4: In $(1,1)$ decoding order, U$1$ first decodes its own signal, applies
SIC and then decodes signal of other user, whereas U$2$ first decodes signal
of other user, subtracts it via SIC and then decodes its own signal. As a
result,
$\Gamma_{11}=\frac{\alpha|h_{1}|^{2}}{(1-\alpha)|h_{1}|^{2}+\frac{1}{\rho_{t}}},\Gamma_{21}=\frac{(1-\alpha)|h_{1}|^{2}}{\beta_{11}+\frac{1}{\rho_{t}}},\Gamma_{12}=\frac{\alpha|h_{2}|^{2}}{(1-\alpha)|h_{2}|^{2}+\frac{1}{\rho_{t}}},\Gamma_{22}=\frac{(1-\alpha)|h_{2}|^{2}}{\beta_{12}+\frac{1}{\rho_{t}}}$.
Here $\Gamma_{11}>\Gamma_{12}$ gives $|h_{1}|^{2}>|h_{2}|^{2}$, and
$\Gamma_{22}>\Gamma_{21}$ required for positive secrecy rate at U$2$ gives
$(\rho_{t}\beta_{11}+1)|h_{2}|^{2}>(\rho_{t}\beta_{12}+1)|h_{1}|^{2}$ which is
also feasible. Thus, $(1,1)$ also can also provide security to both the users,
when obtained feasibility conditions are satisfied. ∎

Thus, it can be concluded from the aforementioned analysis that utilizing the
concept of PLS, three decoding orders, i.e., $(2,1)$, $(1,2)$, $(1,1)$, are
feasible in providing positive secrecy rate for both near and far users, but
with a suitable constraint on PA. Next, the investigation of optimal decoding
order among these three feasible decoding orders is presented through Theorem
$2$.

###### Theorem 2

For untrusted NOMA scenario, the optimal decoding order among the feasible
decoding orders is $(2,1)$ that gives the best secrecy rate in the system.

###### Proof:

See Appendix A. ∎

###### Remark 1

Note that the number of total possible decoding orders is enormous with more
users due to which excessive computational complexity occurs in finding an
optimal secure decoding order. Therefore, we have studied the system with two
users, however, the investigation can be extended for more users easily.

##  III Pair Outage Performance of untrusted NOMA

In order to ensure users’ QoS demands for reliable communication over all the
links, we first derive expression of POP for the optimal decoding order
$(2,1)$. Next, we obtain optimal PA to minimize POP by utilizing the concept
of generalized-convexity.

###  III-A Pair Outage Probability Analysis

Each user in a network has a predefined QoS demand, i.e., each user desires
the transmitter to send data with a minimum information rate guarantees. POP,
denoted as $p_{o}$, ensures minimum rate guarantee to each user. In other
words, $p_{o}$ is defined as the probability that a pair outage happens, i.e.,
the achievable rate at each user falls below a pre-determined threshold rate.
Assuming $R_{i}^{th}$ as the threshold rate for U$i$ and defining
$\pi_{i}\stackrel{{\scriptstyle\Delta}}{{=}}2^{R_{i}^{th}}-1$, $p_{o}$ can be
given as

$\displaystyle p_{o}$
$\displaystyle=1-\text{Pr}\\{\Gamma_{11}>\pi_{1},\Gamma_{21}>\pi_{2},\Gamma_{12}>\pi_{1},\Gamma_{22}>\pi_{2}\\},$
$\displaystyle\stackrel{{\scriptstyle(\mathrm{g})}}{{=}}1-\text{Pr}\\{\Gamma_{11}>\pi_{1},\Gamma_{21}>\pi_{2}\\}\text{Pr}\\{\Gamma_{12}>\pi_{1},\Gamma_{22}>\pi_{2}\\},$
$\displaystyle=1-\text{Pr}\\{|h_{1}|^{2}>\max(\zeta_{1},\zeta_{2})\\}\text{Pr}\\{|h_{2}|^{2}>\max(\zeta_{3},\zeta_{4})\\},$
$\displaystyle=1-(1-F_{|h_{1}|^{2}}(\max(\zeta_{1},\zeta_{2})))(1-F_{|h_{2}|^{2}}(\max(\zeta_{3},\zeta_{4}))),$
$\displaystyle=1-\bar{F}_{|h_{1}|^{2}}(\max(\zeta_{1},\zeta_{2}))\bar{F}_{|h_{2}|^{2}}(\max(\zeta_{3},\zeta_{4})),$
(13)

where $\text{Pr}\\{.\\}$ denotes the probability measure. $(\mathrm{g})$
follows from the property of independent events [28]. Here
$\zeta_{1}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{\pi_{1}\gamma_{21}}{\rho_{t}\alpha},\zeta_{2}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{\pi_{2}}{\rho_{t}(1-\alpha-\alpha\pi_{2})},\zeta_{3}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{\pi_{1}}{\rho_{t}(\alpha-(1-\alpha)\pi_{1})}$,
and
$\zeta_{4}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{\pi_{2}\gamma_{12}}{\rho_{t}(1-\alpha)}$.
Note that
$\gamma_{21}\stackrel{{\scriptstyle\Delta}}{{=}}\rho_{t}\beta_{21}+1$ and
$\gamma_{12}\stackrel{{\scriptstyle\Delta}}{{=}}\rho_{t}\beta_{12}+1$.
$F_{|h_{1}|^{2}}(x)$ and $\bar{F}_{|h_{1}|^{2}}(x)$ are cumulative
distribution function (CDF) and complementary cumulative distribution function
(CCDF), respectively, of channel power gain $|h_{1}|^{2}$. Similarly,
$F_{|h_{2}|^{2}}(x)$ and $\bar{F}_{|h_{2}|^{2}}(x)$ are CDF and CCDF,
respectively, of channel power gain $|h_{2}|^{2}$.

Note that $\bar{F}_{|h_{1}|^{2}}(\max(\zeta_{1},\zeta_{2}))$ in (III-A) can be
rewritten for two cases $\zeta_{1}>\zeta_{2}$ and $\zeta_{1}<\zeta_{2}$. The
first case $\zeta_{1}>\zeta_{2}$, results in a constraint on $\alpha$ as
$\alpha<\frac{\pi_{1}\gamma_{21}}{\pi_{1}\gamma_{21}+\pi_{2}+\pi_{1}\pi_{2}\gamma_{21}}$.
Similarly, the constraint
$\alpha>\frac{\pi_{1}\gamma_{21}}{\pi_{1}\gamma_{21}+\pi_{2}+\pi_{1}\pi_{2}\gamma_{21}}$
results in the second case $\zeta_{1}<\zeta_{2}$. In addition, from the
definition of CDF of exponential distribution, we observe that $\zeta_{1}>0$
and $\zeta_{2}>0$ which leads to $\alpha>0$ and $\alpha<\frac{1}{1+\pi_{2}}$,
respectively. Considering
$\alpha_{1}\\!\stackrel{{\scriptstyle\Delta}}{{=}}\\!\frac{\pi_{1}\gamma_{21}}{\pi_{1}\gamma_{21}+\pi_{2}+\pi_{1}\pi_{2}\gamma_{21}}$
and $\alpha_{2}\\!\stackrel{{\scriptstyle\Delta}}{{=}}\\!\frac{1}{1+\pi_{2}}$,
$\bar{F}_{|h_{1}|^{2}}(\max(\zeta_{1},\zeta_{2}))$ is given as

$\displaystyle\textstyle\bar{F}_{|h_{1}|^{2}}(\max(\zeta_{1},\zeta_{2}))\\!=\\!\begin{cases}\exp\\{-\frac{\zeta_{1}}{\lambda_{1}}\\},\\!&0\\!<\\!\alpha\\!<\\!\alpha_{1}\\\
\exp\\{-\frac{\zeta_{2}}{\lambda_{1}}\\},\\!&\alpha_{1}\\!<\\!\alpha\\!<\\!\alpha_{2}\\\
0,&\text{otherwise}.\end{cases}$ (14)

Similarly, $\bar{F}_{|h_{2}|^{2}}(\max(\zeta_{3},\zeta_{4}))$ can be obtained
as follows

$\displaystyle\textstyle\bar{F}_{|h_{2}|^{2}}(\max(\zeta_{3},\zeta_{4}))\\!$
$\displaystyle=\\!\begin{cases}\exp\\{-\frac{\zeta_{3}}{\lambda_{2}}\\},&\alpha_{3}\\!<\\!\alpha\\!<\\!\alpha_{4}\\\
\exp\\{-\frac{\zeta_{4}}{\lambda_{2}}\\},&\alpha_{4}\\!<\\!\alpha\\!<\\!1\\\
0,&\text{otherwise}\end{cases}$ (15)

where $\alpha_{3}=\frac{\pi_{1}}{1+\pi_{1}}$, and
$\alpha_{4}=\frac{\pi_{1}+\pi_{1}\pi_{2}\gamma_{12}}{\pi_{1}+\pi_{2}\gamma_{12}+\pi_{1}\pi_{2}\gamma_{12}}$.
Using (III-A), (14), (15), the piecewise definition of $p_{o}$ as a function
of $\alpha$ is given at the top of next page in (16)

$p_{o}=\begin{cases}1-\bar{F}_{|h_{1}|^{2}}(\zeta_{1})\bar{F}_{|h_{2}|^{2}}(\zeta_{3}),\quad\quad\alpha_{3}<\alpha<\alpha_{1}\\\
1-\bar{F}_{|h_{1}|^{2}}(\zeta_{2})\bar{F}_{|h_{2}|^{2}}(\zeta_{3}),\quad\quad\begin{cases}[\alpha_{1}<\alpha<\alpha_{2}]\wedge[\alpha_{2}<\alpha_{4}]\wedge[\alpha_{3}<\alpha_{1}]\\\
[\alpha_{1}<\alpha<\alpha_{4}]\wedge[\alpha_{2}>\alpha_{4}]\wedge[\alpha_{3}<\alpha_{1}]\\\
[\alpha_{3}<\alpha<\alpha_{2}]\wedge[\alpha_{2}<\alpha_{4}]\wedge[\alpha_{3}>\alpha_{1}]\\\
[\alpha_{3}<\alpha<\alpha_{4}]\wedge[\alpha_{2}>\alpha_{4}]\wedge[\alpha_{3}>\alpha_{1}]\\\
\end{cases}\\\
1-\bar{F}_{|h_{1}|^{2}}(\zeta_{2})\bar{F}_{|h_{2}|^{2}}(\zeta_{4}),\quad\quad\alpha_{4}<\alpha<\alpha_{2}\\\
1,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\text{otherwise.}\end{cases}$
(16)

where

$\bar{F}_{|h_{1}|^{2}}(\zeta_{1})=\exp\bigg{\\{}-\frac{\pi_{1}\gamma_{21}}{\rho_{t}\alpha\lambda_{1}}\bigg{\\}},$
(17)
$\bar{F}_{|h_{1}|^{2}}(\zeta_{2})=\exp\bigg{\\{}-\frac{\pi_{2}}{\rho_{t}(1-\alpha-\alpha\pi_{2})\lambda_{1}}\bigg{\\}},$
(18)
$\bar{F}_{|h_{2}|^{2}}(\zeta_{3})=\exp\bigg{\\{}-\frac{\pi_{1}}{\rho_{t}(\alpha-(1-\alpha)\pi_{1})\lambda_{2}}\bigg{\\}},$
(19)
$\bar{F}_{|h_{2}|^{2}}(\zeta_{4})=\exp\bigg{\\{}-\frac{\pi_{2}\gamma_{12}}{\rho_{t}(1-\alpha)\lambda_{2}}\bigg{\\}}.$
(20)

###### Remark 2

From the closed-form expression of $p_{o}$ in (16), it can be observed that
$p_{o}$ depends on key system parameters such as threshold rates of users and
SNR. It is noted that POP increases with an increase in threshold rates of
users and decreases with an increase in the SNR.

###  III-B Pair Outage Probability Minimization

Observing $p_{o}$ as a function of $\alpha$, the pair outage optimization
problem can be formulated as

$\displaystyle(J1):\underset{\alpha}{\text{minimize}}$ $\displaystyle p_{o},$
s.t. $\displaystyle(C1):0<\alpha<1,$ (21)

where $(C1)$ indicates PA coefficient bounds.

Note that the closed-form piecewise expression of $p_{o}$ in (16) is a
function of $\alpha$. Therefore, in order to obtain optimal PA solution, next
we need to study each case of piecewise expression one by one and find
candidate optimal points to obtain the global-optimal solution. The global-
optimal solution is given by Lemma $1$.

###### Lemma 1

The unique global-optimal PA solution $\alpha_{p_{o}}^{*}$ is the feasible
optimal point from the set of obtained optimal points that minimizes $p_{o}$,
which is given as

$\displaystyle\\!\alpha_{p_{o}}^{*}\\!\stackrel{{\scriptstyle\Delta}}{{=}}\\!\underset{\alpha\in\\{\alpha_{c1},\alpha_{r1},\alpha_{r2},\alpha_{c2}\\}}{\mathrm{argmin}}\\!p_{o},$
(22)

where $\alpha_{c1}$, $\alpha_{r1}$, $\alpha_{r2}$, and $\alpha_{c2}$ are the
candidate optimal points.

###### Proof:

In order to solve the optimization problem, closed-form piecewise expression
of $p_{o}$ in (16) is considered. Each case is taken one by one to find the
global-optimal solution.

Case 1 : In the first case, where
$p_{o}=1-\bar{F}_{|h_{1}|^{2}}(\zeta_{1})\bar{F}_{|h_{2}|^{2}}(\zeta_{3})$ and
$\alpha_{3}<\alpha<\alpha_{1}$, first-order derivative obtained by
differentiating $p_{o}$ with respect to $\alpha$ is given as

$\displaystyle\frac{\mathrm{d}p_{o}}{\mathrm{d}\alpha}\\!=\\!-\bigg{(}\frac{\pi_{1}\gamma_{21}}{\rho_{t}\alpha^{2}\lambda_{1}}+\frac{\pi_{1}s_{1}}{\rho_{t}(\alpha
s_{1}-\pi_{1})^{2}\lambda_{2}}\bigg{)}$
$\displaystyle\exp{\Big{\\{}\frac{-\pi_{1}\gamma_{21}}{\rho_{t}\alpha\lambda_{1}}+\frac{-\pi_{1}}{\rho_{t}(\alpha
s_{1}-\pi_{1})\lambda_{2}}\Big{\\}}},$ (23)

where $s_{1}\stackrel{{\scriptstyle\Delta}}{{=}}(1+\pi_{1})$. Since
$\frac{\mathrm{d}p_{o}}{\mathrm{d}\alpha}<0$, $p_{o}$ is a monotonically
decreasing function of $\alpha$ in this range. The optimal point is taken as
the corner, given as $\alpha_{c1}=\alpha_{1}$.

Case 2 : For second case, where
$p_{o}=1-\bar{F}_{|h_{1}|^{2}}(\zeta_{2})\bar{F}_{|h_{2}|^{2}}(\zeta_{3})$,
the derivative is given as

$\displaystyle\frac{\mathrm{d}p_{o}}{\mathrm{d}\alpha}=\bigg{(}\frac{\pi_{2}s_{2}}{\rho_{t}(1-\alpha
s_{2})^{2}\lambda_{1}}-\frac{\pi_{1}s_{1}}{\rho_{t}(\alpha
s_{1}-\pi_{1})^{2}\lambda_{2}}\bigg{)}$
$\displaystyle\exp{\Big{\\{}\frac{-\pi_{2}}{\rho_{t}(1-\alpha
s_{2})\lambda_{1}}+\frac{-\pi_{1}}{\rho_{t}(\alpha
s_{1}-\pi_{1})\lambda_{2}}\Big{\\}}},$ (24)

where $s_{2}\stackrel{{\scriptstyle\Delta}}{{=}}(1+\pi_{2})$. Note that
(III-B) does not indicate any monotonic behavior. However, we observe that
$p_{o}$ is monotonically increasing function of $\alpha$ if
$\frac{\pi_{2}s_{2}}{\rho_{t}(1-\alpha
s_{2})^{2}\lambda_{1}}>\frac{\pi_{1}s_{1}}{\rho_{t}(\alpha
s_{1}-\pi_{1})^{2}\lambda_{2}}$, and monotonically decreasing function
otherwise. The point of inflection can be obtained by solving
$\frac{\pi_{2}s_{2}}{\rho_{t}(1-\alpha
s_{2})^{2}\lambda_{1}}=\frac{\pi_{1}s_{1}}{\rho_{t}(\alpha
s_{1}-\pi_{1})^{2}\lambda_{2}}$ which can be simplified as a quadratic
equation $t_{1}\alpha^{2}+t_{2}\alpha+t_{3}=0$ where
$t_{1}\stackrel{{\scriptstyle\Delta}}{{=}}r_{2}s_{1}s_{2}-r_{1}s_{2},t_{2}\stackrel{{\scriptstyle\Delta}}{{=}}2r_{1}-2r_{2}\pi_{1},t_{3}\stackrel{{\scriptstyle\Delta}}{{=}}\pi_{1}^{2}\pi_{2}s_{2}\lambda_{2}-\pi_{1}s_{1}\lambda_{1},r_{1}\stackrel{{\scriptstyle\Delta}}{{=}}\pi_{1}\lambda_{1}s_{1}s_{2},$
and $r_{2}\stackrel{{\scriptstyle\Delta}}{{=}}\pi_{2}\lambda_{2}s_{1}s_{2}$.
Optimal solutions are the roots of the quadratic equation, given as

$\textstyle\alpha_{r1},\alpha_{r2}\\!=\\!\frac{(r_{2}\pi_{1}-r_{1})\pm\sqrt{(s_{2}\pi_{1}-s_{1})^{2}\lambda_{1}\lambda_{2}s_{1}s_{2}\pi_{1}\pi_{2}}}{\lambda_{2}s_{1}^{2}s_{2}\pi_{2}-\lambda_{1}s_{1}s_{2}^{2}\pi_{1}}.$
(25)

Case 3 : In the third case, where
$p_{o}=1-\bar{F}_{|h_{1}|^{2}}(\zeta_{2})\bar{F}_{|h_{2}|^{2}}(\zeta_{4})$,
the derivative is given as

$\displaystyle\frac{\mathrm{d}p_{o}}{\mathrm{d}\alpha}=\bigg{(}\frac{\pi_{2}s_{2}}{\rho_{t}(1-\alpha
s_{2})^{2}\lambda_{1}}+\frac{\pi_{2}\gamma_{12}}{\rho_{t}(1-\alpha)^{2}\lambda_{2}}\bigg{)}$
$\displaystyle\exp{\Big{\\{}\frac{-\pi_{2}}{\rho_{t}(1-\alpha
s_{2})\lambda_{1}}+\frac{-\pi_{2}\gamma_{12}}{\rho_{t}(1-\alpha)\lambda_{2}}\Big{\\}}},$
(26)

which is always greater than zero. Thus, $p_{o}$ is a monotonically increasing
function of $\alpha$. The optimal point, in this case, is given by the corner
point of considered range as $\alpha_{c2}=\alpha_{4}$.

Case 4 : In fourth case, $p_{o}$ is a constant, i.e., $p_{o}=1$, and it is the
maximum feasible value.

From above analysis, we observe that $\alpha_{c1}$ and $\alpha_{c2}$ are two
corner points due to monotonically decreasing and increasing property,
respectively. $\alpha_{r1}$ and $\alpha_{r2}$ are roots as explained in Case
$2$. As a result, POP minimization problem has global-optimal solution
$\alpha_{p_{o}}^{*}$, which is feasible optimal point from set {$\alpha_{c1}$,
$\alpha_{r1}$, $\alpha_{r2}$, $\alpha_{c2}$} at which $p_{o}$ is minimum. ∎

##  IV Secrecy Outage Performance

To analyze secrecy performance, we derive expressions of SOP for both users
and investigate optimal PAs for optimizing SOPs. Next, we provide closed-form
asymptotic expressions of SOPs, and optimal PAs to gain analytical insights.

###  IV-A Secrecy Outage Probability Analysis

For each user, an outage event happens when the received secrecy rate is below
a pre-determined threshold. Particularly, SOP is defined as the probability
that the maximum achievable secrecy rate is less than a target secrecy rate.
Next, we derive analytical expressions of SOPs for both near and far users
which have been denoted by $s_{oi}$ for U$i$.

####  IV-A1 Near user

With $R_{s1}$ and $R_{s1}^{th}$ as achievable and target secrecy rate for
U$1$, $s_{o1}$ is stated as

$\displaystyle s_{o1}$
$\displaystyle=\text{Pr}\\{R_{s1}<R_{s1}^{th}\\}=\text{Pr}\Big{\\{}\frac{1+\Gamma_{11}}{1+\Gamma_{12}}<\Pi_{1}\Big{\\}},$
$\displaystyle=\text{Pr}\Big{\\{}|h_{1}|^{2}<\frac{\gamma_{21}\Pi_{1}|h_{2}|^{2}}{\rho_{t}(1-\alpha)|h_{2}|^{2}+1}+A_{1}\Big{\\}},$
$\displaystyle=\int_{0}^{\infty}F_{\mid
h_{1}\mid^{2}}\bigg{(}\frac{\gamma_{21}\Pi_{1}|h_{2}|^{2}}{\rho_{t}(1-\alpha)|h_{2}|^{2}+1}+A_{1}\bigg{)}f_{|h_{2}|^{2}}(y_{1})dy_{1},$
$\displaystyle=\\!1-\\!\frac{1}{\lambda_{2}}\int_{0}^{\infty}\\!\exp\bigg{\\{}\\!\frac{-\gamma_{21}\Pi_{1}y_{1}}{(\rho_{t}(1-\alpha)y_{1}+1)\lambda_{1}}\\!-\\!\frac{y_{1}}{\lambda_{2}}-\frac{A_{1}}{\lambda_{1}}\bigg{\\}}dy_{1},$
(27)

where $\Pi_{1}\stackrel{{\scriptstyle\Delta}}{{=}}2^{R_{s1}^{th}}$,
$A_{1}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{\gamma_{21}(\Pi_{1}-1)}{\rho_{t}\alpha}$,
and $f_{\mid h_{2}\mid^{2}}(x)$ is probability density function (PDF) of
channel power gain $|h_{2}|^{2}$.

####  IV-A2 Far user

With $R_{s2}$ and $R_{s2}^{th}$ as achievable and target secrecy rate for
U$2$, $s_{o2}$ is given as

$\displaystyle s_{o2}$
$\displaystyle=\text{Pr}\\{R_{s2}<R_{s2}^{th}\\}=\text{Pr}\Big{\\{}\frac{1+\Gamma_{22}}{1+\Gamma_{21}}<\Pi_{2}\Big{\\}},$
$\displaystyle=\text{Pr}\Big{\\{}|h_{2}|^{2}<\frac{\gamma_{12}\Pi_{2}|h_{1}|^{2}}{\rho_{t}\alpha|h_{1}|^{2}+1}+A_{2}\Big{\\}},$
$\displaystyle=\int_{0}^{\infty}F_{\mid
h_{2}\mid^{2}}\bigg{(}\frac{\gamma_{12}\Pi_{2}|h_{1}|^{2}}{\rho_{t}\alpha|h_{1}|^{2}+1}+A_{2}\bigg{)}f_{|h_{1}|^{2}}(y_{2})dy_{2},$
$\displaystyle=\\!1\\!-\\!\frac{1}{\lambda_{1}}\int_{0}^{\infty}\exp\bigg{\\{}\frac{-\gamma_{12}\Pi_{2}y_{2}}{(\rho_{t}\alpha
y_{2}+1)\lambda_{2}}-\frac{y_{2}}{\lambda_{1}}-\frac{A_{2}}{\lambda_{2}}\bigg{\\}}dy_{2},$
(28)

where $\Pi_{2}\stackrel{{\scriptstyle\Delta}}{{=}}2^{R_{s2}^{th}}$,
$A_{2}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{\gamma_{12}(\Pi_{2}-1)}{\rho_{t}(1-\alpha)}$,
and $f_{|h_{1}|^{2}}(x)$ denotes PDF of channel power gain $|h_{1}|^{2}$.

###  IV-B Individual Secrecy Outage Probability Minimization

Next, we investigate optimality of SOPs of both the users.

####  IV-B1 Near User

Considering $s_{o1}$ derived in (IV-A1) as a function of $\alpha$, SOP
minimization problem for U$1$ can be formulated as

$\displaystyle(J2):\underset{\alpha}{\text{minimize}}$ $\displaystyle s_{o1},$
s.t. $\displaystyle(C1).$ (29)

Noting the complexity of derived expression of $s_{o1}$, next we solve SOP
minimization problems using pseudoconvexity of SOP. The feasibility of unique
solution is asserted by Lemma $2$.

###### Lemma 2

$s_{o1}$ is a pseudoconvex function of $\alpha$.

###### Proof:

$s_{o1}$ in (IV-A1) can be rewritten as

$\displaystyle\textstyle
s_{o1}\\!=\\!1\\!-\\!\frac{1}{\lambda_{2}}\int\limits_{0}^{\infty}\exp\bigg{\\{}\\!-\\!\frac{\gamma_{21}\Pi_{1}y_{1}}{(\rho_{t}(1-\alpha)y_{1}+1)\lambda_{1}}\\!-\\!\frac{y_{1}}{\lambda_{2}}\\!-\\!\frac{\gamma_{21}(\Pi_{1}-1)}{\rho_{t}\alpha\lambda_{1}}\bigg{\\}}dy_{1}.$
(30)

Denoting the integrand of $s_{o1}$ as $I_{1}$, it can be defined as

$\textstyle
I_{1}=\exp\bigg{\\{}-\frac{\gamma_{21}\Pi_{1}y_{1}}{(\rho_{t}(1-\alpha)y_{1}+1)\lambda_{1}}-\frac{y_{1}}{\lambda_{2}}-\frac{\gamma_{21}(\Pi_{1}-1)}{\rho_{t}\alpha\lambda_{1}}\bigg{\\}}.$
(31)

Second-order derivative of the logarithm of the integrand function $I_{1}$,
with respect to $\alpha$ is

$\begin{split}\textstyle\frac{\mathrm{d}^{2}\log(I_{1})}{\mathrm{d}\alpha^{2}}=-\Bigg{(}\frac{2\gamma_{21}(\Pi_{1}-1)}{\rho_{t}\lambda_{1}\alpha^{3}}+\frac{2\gamma_{21}\Pi_{1}\rho_{t}^{2}y_{1}^{3}}{\lambda_{1}((1-\alpha)\rho_{t}y_{1}+1)^{3}}\Bigg{)},\end{split}$
(32)

which is non-increasing. This indicates that the integrand $I_{1}$ of the
objective function $s_{o1}$ is a logarithmically concave (log-concave)
function. Since log-concavity is preserved under integration [29], we note
that integral function in (30) is also log-concave function. Observing
pseudoconcave property [30, Lemma 5] of log-concave function, we can state
that integral function of (30) is pseudoconcave, and negative of a
pseudoconcave function is a pseudoconvex function [31]. Hence, $s_{o1}$ is a
pseudoconvex function of $\alpha$. ∎

Due to pseudoconvexity of $s_{o1}$ of U$1$, there exists unique PA that
minimizes $s_{o1}$ [31, Chap. 3.5.9]. We apply a computationally efficient
golden section search algorithm [32] to find $\alpha_{1}^{*}$ (optimal value
of $\alpha$). The algorithm considers pseudoconvex function $s_{o1}$,
$\alpha_{lb}$ and $\alpha_{ub}$ as input, where $\alpha_{lb}$ and
$\alpha_{ub}$, respectively, are the lower and upper bounds of $\alpha$ at
initial stage. It gives optimal solution $\alpha_{1}^{*}$ and corresponding
minimum SOP as the output. Since $(0<\alpha<1)$, we first consider
$\alpha_{lb}=0$ and $\alpha_{ub}=1$ and then the algorithm searches along
$\alpha$ with acceptable tolerance level of $\epsilon<<1$. The algorithm works
by reducing the search space with a fixed ratio of $0.618$ at the end of every
iteration. Finally, the algorithm terminates after $N$ iterations if search
length is less than the given tolerance [32].

####  IV-B2 Far User

SOP minimization problem for U$2$, considering expression of $s_{o2}$ (IV-A2)
is stated as

$\displaystyle(J3):\underset{\alpha}{\text{minimize}}$ $\displaystyle s_{o2},$
s.t. $\displaystyle(C1).$ (33)

In Lemma $3$, we prove the feasibility of unique solution.

###### Lemma 3

$s_{o2}$ is a pseudoconvex function of $\alpha$.

###### Proof:

Similar to the proof of $s_{o1}$ minimization problem, the pseudoconvexity of
SOP for U$2$ can be proved. ∎

Here also, optimal PA solution of $s_{o2}$ minimization problem can be
evaluated numerically using golden section search algorithm. Now, the
algorithm takes pseudoconvex function $s_{o2}$, $\alpha_{lb}=0$,
$\alpha_{ub}=1$ as input. It provides $\alpha_{2}^{*}$, and corresponding
minimum SOP as optimal output.

###  IV-C Asymptotic SOP: Analysis and Optimization

Observing the complexity of derived expressions, and involved computations in
optimizing SOPs, optimal PAs have been obtained numerically. In order to gain
analytical insights, next we derive tight asymptotic expressions of SOPs and
optimal PAs under high SNR scenario.

####  IV-C1 Near User

Asymptotic approximation of SOP for U$1$, $\hat{s}_{o1}$, can be obtained by
setting $\rho_{t}\gg 1$ in (30), which leads to $(\rho_{t}(1-\alpha)y_{1}+1)$
$\approx$ $\rho_{t}(1-\alpha)y_{1}$. Thus, $\hat{s}_{o1}$ is given as

$\displaystyle\textstyle\hat{s}_{o1}\\!$
$\displaystyle=\\!1\\!-\\!\exp\bigg{\\{}\\!\frac{-\gamma_{21}\Pi_{1}}{\rho_{t}(1\\!-\\!\alpha)\lambda_{1}}\\!-\\!\frac{\gamma_{21}(\Pi_{1}\\!-\\!1)}{\rho_{t}\alpha\lambda_{1}}\\!\bigg{\\}}\\!\int_{0}^{\infty}\\!\frac{\exp\\{\frac{-y_{1}}{\lambda_{2}}\\}}{\lambda_{2}}dy_{1},$
$\displaystyle=\\!1\\!-\\!\exp\bigg{\\{}\frac{\gamma_{21}(\Pi_{1}+\alpha-1)}{\rho_{t}\alpha(\alpha-1)\lambda_{1}}\bigg{\\}}.$
(34)

Considering (IV-C1), $\hat{s}_{o1}$ minimization problem is given as

$\displaystyle(J4):\underset{\alpha}{\text{minimize}}$
$\displaystyle\hat{s}_{o1},$ s.t. $\displaystyle(C1).$ (35)

Asymptotic optimal PA for $(J4)$ is given by Lemma $4$.

###### Lemma 4

The optimal PA $\hat{\alpha}_{1}$ minimizing $\hat{s}_{o1}$, is given as

$\hat{\alpha}_{1}=-(\Pi_{1}-1)+\sqrt{(\Pi_{1}(\Pi_{1}-1)}.$ (36)

###### Proof:

See Appendix B. ∎

####  IV-C2 Far User

Similarly, the asymptotic expression $\hat{s}_{o2}$ for U$2$, using
$(\rho_{t}\alpha y_{2}+1)$ $\approx$ $\rho_{t}\alpha y_{2}$ in (IV-A2), can be
given as

$\displaystyle\textstyle\hat{s}_{o2}\\!$
$\displaystyle=\\!1\\!-\\!\exp\bigg{\\{}\frac{-\gamma_{12}\Pi_{2}}{\rho_{t}\alpha\lambda_{2}}-\frac{\gamma_{12}(\Pi_{2}-1)}{\rho_{t}(1-\alpha)\lambda_{2}}\bigg{\\}}\int_{0}^{\infty}\frac{\exp\\{\frac{-y_{2}}{\lambda_{1}}\\}}{\lambda_{1}}dy_{2},$
$\displaystyle=\\!1\\!-\\!\exp\bigg{\\{}\frac{\gamma_{12}(\Pi_{2}-\alpha)}{\rho_{t}\alpha(\alpha-1)\lambda_{2}}\bigg{\\}}.$
(37)

The $\hat{s}_{o2}$ (IV-C2) minimization problem for U$2$ can be stated as

$\displaystyle(J5):\underset{\alpha}{\text{minimize}}$
$\displaystyle\hat{s}_{o2},$ s.t. $\displaystyle(C1).$ (38)

###### Lemma 5

The unique PA $\hat{\alpha}_{2}$ minimizing $\hat{s}_{o2}$ is given as

$\hat{\alpha}_{2}=\Pi_{2}-\sqrt{(\Pi_{2}(\Pi_{2}-1)}.$ (39)

###### Proof:

See Appendix C. ∎

###### Remark 3

It can be observed from the asymptotic approximations of SOPs, i.e.,
$\hat{s}_{o1}$ and $\hat{s}_{o2}$, respectively, for U$1$ and U$2$, that the
SOPs for near user and far user depend on target secrecy rates and SNR. Note
that SOP increases with an increase in target secrecy rate of the user and
decreases with an increase in the SNR.

##  V Secrecy Fairness Maximization

In order to ensure secrecy rate guarantee to both the users while satisfying
users’ QoS requirements, next we formulate secrecy fairness optimization
problem and investigate optimal PA. Also, we present asymptotic approximation
of optimal PA to get deeper insights.

###  V-A Problem Formulation

QoS demands of users, in the form of maximum allowable POP forces an upper
bound $\xi$ on $p_{o}$, i.e., $p_{o}\leq\xi$. Hence, QoS constrained secrecy
fairness maximization problem using (16), (IV-A1) and (IV-A2) that minimizes
the maximum SOP between users under power budget and POP constraints can be
formulated as

$\displaystyle(J6):\underset{\alpha}{\text{minimize }}\max[s_{o1},s_{o2}],$
$\displaystyle\text{s.t.}\quad(C1),(C2):p_{o}\leq\xi.$ (40)

Using $x_{c}\stackrel{{\scriptstyle\Delta}}{{=}}\max[s_{o1},s_{o2}]$, an
equivalent formulation of $(J6)$ can be obtained as follows

$\displaystyle(J7):\underset{\alpha,x_{c}}{\text{minimize}}\quad
x_{c},\quad\text{s.t.}\quad(C1),(C2),$ $\displaystyle(C3):s_{o1}\leq
x_{c},(C4):s_{o2}\leq x_{c},$ (41)

where $(C3)$ and $(C4)$ appears from the definition of max$[\cdot]$.

Due to the presence of nonconvex constraints $(C3)$ and $(C4)$, $(J7)$ is a
nonconvex problem. We solve it by analyzing candidate optimal points that are
characterized by Karush-Kuhn-Tucker (KKT) conditions [33].

###  V-B Power Control for optimizing Min-Max Secrecy Outage

First, we solve min-max SOP optimization problem without consideration of
users’ rate demands. Next, QoS demand is taken into account and optimal NOMA
protocol is analyzed.

####  V-B1 Without Pair Outage Constraint

The secrecy fairness maximization problem without $(C2)$ can be formulated as

$\displaystyle(J8):\underset{\alpha,x_{c}}{\text{minimize }}$ $\displaystyle
x_{c},$ s.t. $\displaystyle(C1),(C3),(C4).$ (42)

The global-optimal solution is given by Lemma $6$.

###### Lemma 6

The global-optimal solution $\widetilde{\alpha_{sop}}$ of the nonconvex
problem $(J8)$, at which maximum SOP between users is minimized, is given as

$\\!\widetilde{\alpha_{sop}}\\!\stackrel{{\scriptstyle\Delta}}{{=}}\\!\underset{\alpha\in\\{\alpha_{1}^{*},\alpha_{2}^{*},\alpha_{3}^{*}\\}}{\mathrm{argmin}}\\!\max[s_{o1},s_{o2}],$
(43)

where $\alpha_{1}^{*},\alpha_{2}^{*},\alpha_{3}^{*}$ are, respectively,
obtained by golden section search algorithm by minimizing $s_{o1},s_{o2}$ (cf.
Section IV-B), and solving $s_{o1}=s_{o2}$.

###### Proof:

Keeping the boundary constraint $(C1)$ implicit, and associating Lagrange
multipliers $\eta_{1}$ with $(C3)$ and $\eta_{2}$ with $(C4)$, the Lagrangian
function $\mathcal{L}$ can be stated as

$\mathcal{L}\stackrel{{\scriptstyle\Delta}}{{=}}x_{c}+\eta_{1}[s_{o1}-x_{c}]+\eta_{2}[s_{o2}-x_{c}].$
(44)

The corresponding primal feasibility KKT conditions are given by the
constraints $(C3)$ and $(C4)$. Following (42) and (44), the dual feasibility
conditions are given as $\eta_{1}\geq 0$ and $\eta_{2}\geq 0$. The subgradient
conditions are obtained as

$\frac{\mathrm{d}\mathcal{L}}{\mathrm{d}x_{c}}=1-\eta_{1}-\eta_{2}=0,\quad\frac{\mathrm{d}\mathcal{L}}{\mathrm{d}\alpha}=\eta_{1}\frac{\mathrm{d}s_{o1}}{\mathrm{d}\alpha}+\eta_{2}\frac{\mathrm{d}s_{o2}}{\mathrm{d}\alpha}=0.$
(45)

The two complementary slackness conditions are given as

$\eta_{1}[s_{o1}-x_{c}]=0,\quad\eta_{2}[s_{o2}-x_{c}]=0.$ (46)

Note that there exists three cases. _Case 1: $\eta_{1}>0$ and $\eta_{2}=0$_,
implies $\frac{\mathrm{d}s_{o1}}{\mathrm{d}\alpha}=0$ which results in the
same solution as SOP minimization problem (29) of U$1$, i.e.,
$\alpha=\alpha_{1}^{*}$. _Case 2: $\eta_{2}>0$ and $\eta_{1}=0$_, implies
$\frac{\mathrm{d}s_{o2}}{\mathrm{d}\alpha}=0$, which results in the solution
of SOP minimization (33) of U$2$, i.e., $\alpha=\alpha_{2}^{*}$. _Case 3:
$\eta_{1}>0$ and $\eta_{2}>0$_, implies $s_{o1}=s_{o2}$ using (46), which
indicates equal SOP for both the users, and this case gives
$\alpha=\alpha_{3}^{*}$. Hence, $(J8)$ has three optimal points, i.e.,
$\alpha_{1}^{*}$ and $\alpha_{2}^{*}$ for minimizing $s_{o1}$ and $s_{o2}$,
respectively, and $\alpha_{3}^{*}$ is obtained from the condition of
$s_{o1}=s_{o2}$. Finally, the global-optimal solution
$\widetilde{\alpha_{sop}}$ to this problem is obtained at the optimal point
for which maximum SOP between users is minimum. ∎

Thus, we conclude that since the secrecy fairness maximization problem is
nonconvex due to the presence of nonconvex constraints, we have successfully
obtained all possible optimal points of this problem by KKT conditions which
are the candidates for the global-optimal solution [33]. Lastly, the optimal
point for which maximum SOP between users is minimum is considered as the
global-optimal solution.

####  V-B2 With Pair Outage Constraint

Now we solve $(J7)$ problem by taking QoS requirements into account. In this
case, the global-optimal solution from secrecy perspective must satisfy the
constraint $(C2)$. Theoretically $\alpha\in[0,1]$, but due to $(C2)$,
acceptable range of $\alpha$ gets limited, as obtained in Lemma $7$.

###### Lemma 7

The lower and upper bound on $\alpha$, required to achieve maximum allowable
POP, are given as

$\alpha_{lb}^{*}=\max[\alpha_{l1}^{*},\alpha_{l2}^{*},\alpha_{l3}^{*}],\quad\alpha_{ub}^{*}=\min[\alpha_{u1}^{*},\alpha_{u2}^{*},\alpha_{u3}^{*}],$
(47)

respectively, where
$\alpha_{l1}^{*},\alpha_{l2}^{*},\alpha_{l3}^{*},\alpha_{u1}^{*},\alpha_{u2}^{*},\alpha_{u3}^{*}$
are the roots of quadratic equations obtained by making $(C2)$ an active
constraint, i.e., $p_{o}=\xi$.

Figure 2: Proposed solution methodology for pair outage constrained secrecy
fairness maximization.

###### Proof:

Due to POP constraint, tight analytical bounds on $\alpha$ are obtained by
using the expression of $p_{o}$ given in (16). Since $p_{o}$ is a piecewise
function of $\alpha$, we first consider each case of (16) one by one,
substitute $p_{o}=\xi$, and find bounds in accordance with the respective
$\alpha$ range. In first case of $p_{o}$, where
$p_{o}=1-\bar{F}_{|h_{1}|^{2}}(\zeta_{1})\times\bar{F}_{|h_{2}|^{2}}(\zeta_{3})$
and $\alpha_{3}<\alpha<\alpha_{1}$, the lower and upper bound can be obtained
by solving $p_{o}=\xi$ which results in a quadratic equation given as
$\log(1-\xi)s_{1}\alpha^{2}-(\pi_{1}\log(1-\xi)-z_{1}s_{1}-z_{2})\alpha-\pi_{1}z_{1}=0$,
where
$z_{1}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{\gamma_{21}\pi_{1}}{\rho_{t}\lambda_{1}}$
and
$z_{2}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{\pi_{1}}{\rho_{t}\lambda_{2}}$.
The roots of the quadratic equation, denoted as $\alpha_{l1}$ and
$\alpha_{u1}$, are given in (48) at next page.

$\alpha_{u1},\alpha_{l1}\\!=\\!\frac{(\pi_{1}\log(1-\xi)-z_{1}s_{1}-z_{2})\\!\pm\\!\sqrt{(\pi_{1}\log(1-\xi)-z_{1}s_{1}-z_{2})^{2}\\!+\\!4\log(1-\xi)s_{1}\pi_{1}z_{1}}}{2\log(1-\xi)s_{1}}.\\\
$ (48)

The lower bound of PA in this case can be obtained as
$\alpha_{l1}^{*}=\alpha_{l1}$ iff $\alpha_{l1}$ exists as a feasible point in
the range else $\alpha_{l1}^{*}=0$. Similarly, the upper bound of PA in this
case can be given as $\alpha_{u1}^{*}=\alpha_{u1}$ iff
$\alpha_{3}<\alpha_{u1}<\alpha_{1}$ otherwise $\alpha_{u1}^{*}=1$. Similar to
the above analysis, lower and upper bounds can be obtained for other cases
also. Let us denote lower and upper bounds, respectively, for the second case
as $\alpha_{l2}^{*}$ and $\alpha_{u2}^{*}$, and for third case as
$\alpha_{l3}^{*}$ and $\alpha_{u3}^{*}$. After identifying lower and upper
bounds of PA in each case, the $\alpha_{lb}^{*}$ in the whole range can be
obtained as
$\alpha_{lb}^{*}=\max[\alpha_{l1}^{*},\alpha_{l2}^{*},\alpha_{l3}^{*}]$ and
similarly, the upper bound $\alpha_{ub}^{*}$ can be obtained as
$\alpha_{ub}^{*}=\min[\alpha_{u1}^{*},\alpha_{u2}^{*},\alpha_{u3}^{*}]$. ∎

Global-optimal solution of $(J7)$ is given by Lemma $8$.

###### Lemma 8

The global-optimal power control solution $\alpha_{sop}^{*}$ at which maximum
SOP between users is minimized, is given as

$\\!\alpha_{sop}^{*}\\!\stackrel{{\scriptstyle\Delta}}{{=}}\\!\underset{\alpha\in\\{\alpha_{1}^{*},\alpha_{2}^{*},\alpha_{3}^{*}\\}}{\mathrm{argmin}}\\!\max[s_{o1},s_{o2}],$
(49)

subject to the feasibility of candidate optimal points as
$\alpha_{1}^{*},\alpha_{2}^{*},\alpha_{3}^{*}\in[\alpha_{lb}^{*},\alpha_{ub}^{*}]$.

###### Proof:

Since $(J7)$ is a nonconvex optimization problem due to the presence of
nonconvex constraints $(C3)$ and $(C4)$, the candidate optimal points
$(\alpha_{1}^{*},\alpha_{2}^{*},\alpha_{3}^{*})$ are obtained similar to the
proof of Lemma $6$. In order to ensure users’ rates necessity, we check if the
obtained candidate optimal points satisfy POP constraint $(C2)$. Knowing
$\alpha_{lb}^{*}$ and $\alpha_{ub}^{*}$ as obtained in the proof of Lemma $7$,
feasibility of optimal points can be ensured. After investigating feasible
optimal points, the global-optimal solution $\alpha_{sop}^{*}$ is obtained at
that feasible optimal point for which maximum SOP between users is minimum. ∎

Fig. $2$ presents the methodology to find optimal PA for pair outage
constrained secrecy fairness maximization problem.

###  V-C Proposed Algorithm and Complexity Analysis

Algorithm 1 Algorithm to find global-optimal PA $\alpha_{sop}^{*}$.

1:$n$, $L_{p}$, $d_{1}$, $d_{2}$, $\xi$, $R_{1}^{th}$, $R_{2}^{th}$,
$R_{s1}^{th}$, $R_{s2}^{th}$ and $\rho_{t}$

2:$\alpha_{sop}^{*}$
(A) Finding potential candidates

3:Obtain $\alpha_{1}^{*}$, $\alpha_{2}^{*}$, $\alpha_{3}^{*}$ for minimizing
$s_{o1}$, $s_{o2}$, $s_{o1}=s_{o2}$, respectively, using golden section search
algorithm
(B) Check Feasibility

4:Obtain $\alpha_{lb}^{*}$ and $\alpha_{ub}^{*}$ by solving the equation
$p_{o}=\xi$ where $p_{o}$ is given in (16)

5:if $\alpha_{lb}^{*}<\alpha_{1}^{*}<\alpha_{ub}^{*}$ then

6: $\alpha_{1}^{*}$ is feasible optimal point

7:else

8: update $\alpha_{1}^{*}=\alpha_{lb}^{*}$

9:end if

10:if $\alpha_{lb}^{*}<\alpha_{2}^{*}<\alpha_{ub}^{*}$ then

11: $\alpha_{2}^{*}$ is feasible optimal point

12:else

13: update $\alpha_{2}^{*}=\alpha_{ub}^{*}$

14:end if

15:if $\alpha_{lb}^{*}<\alpha_{3}^{*}<\alpha_{ub}^{*}$ then

16: $\alpha_{3}^{*}$ is feasible optimal point

17:else

18: update $\alpha_{3}^{*}$ = infeasible point

19:end if
(C) Global-optimal Solution

20:Calculate
$\alpha_{sop}^{*}\stackrel{{\scriptstyle\Delta}}{{=}}\underset{\alpha\in\\{\alpha_{1}^{*},\alpha_{2}^{*},\alpha_{3}^{*}\\}}{\mathrm{argmin}}\max[s_{o1},s_{o2}]$

Next, we present an algorithm that finds global-optimal PA which maximizes
secrecy fairness between users while fulfilling their rate demands. For a
given set of simulation parameters, $\alpha_{lb}^{*}$ and $\alpha_{ub}^{*}$
need to be calculated just once. Also, golden section search is applied to
find three candidate optimal points of nonconvex optimization problem $(J7)$.
Once they are calculated, $\alpha_{sop}^{*}$ is obtained by following the
steps detailed in Algorithm 1. In order to investigate the complexity of
proposed algorithm solving the optimization problem, we calculate total number
of computations. The proposed algorithm is based on golden section search
which terminates after $N$ iterations if
$(\alpha_{ub}^{*}-\alpha_{lb}^{*})\times 0.618^{N}\leq\epsilon$. Hence, the
total number of iterations in the proposed algorithm are $3N$, where
$N=2\ln\Big{(}\frac{\alpha_{ub}^{*}-\alpha_{lb}^{*}}{\epsilon}\Big{)}$.

###  V-D Asymptotic Power Control for min-max Secrecy Outage

In above analysis, we solved the min-max SOP optimization problem numerically.
In order to gain analytical insights, now we derive the asymptotic closed-form
expression of global-optimal PA under high SNR region. Considering the users’
rate requirements, problem can be stated as

$\displaystyle(J9):\underset{\alpha}{\text{minimize }}$
$\displaystyle\max[\hat{s}_{o1},\hat{s}_{o2}],$ s.t. $\displaystyle(C1),(C2).$
(50)

Following
$\hat{x}_{c}\stackrel{{\scriptstyle\Delta}}{{=}}\max[\hat{s}_{o1},\hat{s}_{o2}]$,
an equivalent formulation of $(J9)$ can be given as

$\displaystyle(J10):\underset{\alpha,\hat{x}_{c}}{\text{minimize}}\quad\hat{x}_{c},\quad\text{s.t.}\quad(C1),(C2),$
$\displaystyle(C5):\hat{s}_{o1}\leq\hat{x}_{c},(C6):\hat{s}_{o2}\leq\hat{x}_{c},$
(51)

where $(C5)$ and $(C6)$ appears from the definition of max$[\cdot]$.

We first solve optimization problem without considering QoS constraint, which
can be given as

$\displaystyle(J11):\underset{\alpha,\hat{x}_{c}}{\text{minimize }}$
$\displaystyle\hat{x}_{c},$ s.t. $\displaystyle(C1),(C5),(C6).$ (52)

Global-optimal solution of $(J11)$ is provided by Lemma $9$.

###### Lemma 9

The global-optimal solution $\hat{\alpha}$ of the optimization problem $(J11)$
for maximizing the secrecy fairness is given by

$\dot{\alpha}_{sop}\stackrel{{\scriptstyle\Delta}}{{=}}\underset{\alpha\in\\{\hat{\alpha}_{1},\hat{\alpha}_{2},\hat{\alpha}_{3}\\}}{\mathrm{argmin}}\max[\hat{s}_{o1},\hat{s}_{o2}],$
(53)

where $\hat{\alpha}_{1},\hat{\alpha}_{2}$, respectively, have been obtained by
minimizing $\hat{s}_{o1},\hat{s}_{o2}$ (cf. IV-C), and $\hat{\alpha}_{3}$ is
obtained by solving $\hat{s}_{o1}=\hat{s}_{o2}$.

###### Proof:

We associate Lagrange multipliers $\mu_{1}$ with $(C5)$, $\mu_{2}$ with
$(C6)$, and keep the boundary constraint $(C1)$ implicit. Then, Lagrangian
function $\mathcal{\hat{L}}$ can be written as

$\mathcal{\hat{L}}\stackrel{{\scriptstyle\Delta}}{{=}}\hat{x}_{c}+\mu_{1}[\hat{s}_{o1}-\hat{x}_{c}]+\mu_{2}[\hat{s}_{o2}-\hat{x}_{c}].$
(54)

The corresponding KKT conditions are given by constraints $(C5)$ and $(C6)$.
Dual feasibility conditions are obtained as $\mu_{1}\geq 0$ and $\mu_{2}\geq
0$ using (52) and (54). The subgradient conditions are given as
$\frac{\mathrm{d}\mathcal{\hat{L}}}{\mathrm{d}\hat{x}_{c}}=1-\mu_{1}-\mu_{2}=0,$
and
$\frac{\mathrm{d}\mathcal{\hat{L}}}{\mathrm{d}\alpha}=\mu_{1}\frac{\mathrm{d}\hat{s}_{o1}}{\mathrm{d}\alpha}+\mu_{2}\frac{\mathrm{d}\hat{s}_{o2}}{\mathrm{d}\alpha}=0$.
The complementary slackness conditions are obtained as
$\mu_{1}[\hat{s}_{o1}-\hat{x}_{c}]=0$ and
$\mu_{2}[\hat{s}_{o2}-\hat{x}_{c}]=0$. Here also, three cases exist. _Case 1:
$\mu_{1}>0$ and $\mu_{2}=0$_, implies
$\frac{\mathrm{d}\hat{s}_{o1}}{\mathrm{d}\alpha}=0$ and results
$\alpha=\hat{\alpha}_{1}$ from $\hat{s}_{o1}$ minimization for U$1$ (36).
_Case 2: $\mu_{2}>0$ and $\mu_{1}=0$_, implies
$\frac{\mathrm{d}\hat{s}_{o2}}{\mathrm{d}\alpha}=0$ and provides the same
solution as the optimal solution of $\hat{s}_{o2}$ minimization, i.e.,
$\alpha=\hat{\alpha}_{2}$ from (39). _Case 3: $\mu_{1}>0$ and $\mu_{2}>0$_,
implies $\hat{s}_{o1}=\hat{s}_{o2}$ and it gives $\hat{\alpha}_{3}$ which is
given as

$\hat{\alpha}_{3}=\frac{\gamma_{12}\Pi_{2}\lambda_{1}+(\rho_{t}\beta_{21}-\gamma_{21}\Pi_{1})\lambda_{2}}{\gamma_{12}\lambda_{1}+\gamma_{21}\lambda_{2}}.$
(55)

Observing that the minimization problem has three optimal points, i.e.,
$\hat{\alpha}_{1}$, $\hat{\alpha}_{2}$, and $\hat{\alpha}_{3}$, the global
optimal solution $\dot{\alpha}_{sop}$ is obtained as the point where maximum
SOP is minimum. ∎

Next, while taking the QoS constraint into account, the global-optimal
solution of $(J10)$ is given by Lemma $10$.

###### Lemma 10

Global-optimal solution $\hat{\alpha}_{sop}$ of optimization problem $(J10)$
for minimizing the maximum SOP is given by

$\hat{\alpha}_{sop}\stackrel{{\scriptstyle\Delta}}{{=}}\underset{\alpha\in\\{\hat{\alpha}_{1},\hat{\alpha}_{2},\hat{\alpha}_{3}\\}}{\mathrm{argmin}}\max[\hat{s}_{o1},\hat{s}_{o2}],$
(56)

subject to the feasibility of candidate optimal points as
$\hat{\alpha}_{1},\hat{\alpha}_{2},\hat{\alpha}_{3}\in[\alpha_{lb}^{*},\alpha_{ub}^{*}]$.

###### Proof:

Similar to proof of Lemma $9$, we first find the candidate optimal points
$\hat{\alpha}_{1},\hat{\alpha}_{2},$ and $\hat{\alpha}_{3}$. Next, the
feasibility of these points is investigated between $\alpha_{lb}^{*}$ and
$\alpha_{ub}^{*}$. Finally, the global optimal point is obtained as the
feasible optimal point where maximum SOP is minimized. ∎

##  VI Numerical Investigations

Figure 3: Validation of pair outage probability $p_{o}$ with users’ threshold
rates, (a) $R_{2}^{th}=R_{1}^{th}$, and (b) $\rho_{r}=20$ dB.

Figure 4: Validation of secrecy outage probability, (a) $s_{o1}$ for U$1$, and
(b) $s_{o2}$ for U$2$.

Figure 5: Variation of pair outage probability $p_{o}$ versus U$2$’s distance
$d_{2}$ with $R_{2}^{th}=R_{1}^{th}=0.1$.

Figure 6: Variation of secrecy outage probability versus U$2$’s distance
$d_{2}$, (a) $s_{o1}$ for U$1$, and (b) $s_{o2}$ for U$2$.

Next, numerical results are presented to assess the performance of proposed
protocol. We present simulation setup, and discuss the system performance
under various system settings. Specifically, we validate accuracy of
analytical results, provide insights on effect of users’ distance and optimal
PAs, and identify the performance gains achieved by global-optimal PA.
Finally, tradeoff between QoS parameter POP and secrecy performance SOP is
presented.

###  VI-A Simulation Setup

Downlink of a NOMA system is considered, where BS communicates with two
untrusted users. For the sake of simulation study, near user distance $d_{1}$
and far user distance $d_{2}$ from BS are taken as $50$ meter and $100$ meter,
respectively. $L_{p}$ and $n$ are taken as $1$ and $2.5$, respectively. Noise
signal at both users follow Gaussian distribution with a noise power of $-60$
dBm. Small scale fading affecting both links is assumed to have exponential
distribution with mean value $1$ [13]. Residual interference terms
$\beta_{21}$ and $\beta_{12}$ are assumed to be equal and $-30$ dBm [19].
Simulation results are averaged over $10^{6}$ randomly generated channel
realizations for both U$1$ and U$2$. The correctness of results is observed
using the fact that root mean square error (RMSE) in approximating the derived
results should be less than $0.0003$ [34]. For golden section search,
$\epsilon=0.01$. $\rho_{r}$ is considered as the received SNR. POP and SOP are
considered as performance metrics to evaluate the performance of system.

###  VI-B Validation of Analysis

We first validate analytical expressions of $p_{o}$ in Fig. 4, and $s_{o1}$
and $s_{o2}$ in Fig. 4. For these results, $\alpha$ is considered as $0.5$. A
close match between the simulation and analytical results validates the
analysis of $p_{o}$, $s_{o1}$, and $s_{o2}$ with a RMSE of the order of
$10^{-4}$. Accuracy of asymptotic results, marked as ‘Asy’, is also verified
with simulation results in Fig. 4(b). We observe that $p_{o}$ increases with
increase in the threshold rates $R_{1}^{th}$ or $R_{2}^{th}$. Similarly,
increasing target secrecy rates $R_{s1}^{th}$ and $R_{s2}^{th}$ increase
$s_{o1}$ and $s_{o2}$, respectively. Following the fact that the outage occurs
when the users’ maximum achievable rate is below a threshold rate, it is
natural that increasing threshold rates and target secrecy rates at users,
respectively, increases POP and SOP.

From the results it is also observed that threshold rate pair
($R_{1}^{th},R_{2}^{th}$) with lesser value of $R_{2}^{th}$ compared to
$R_{1}^{th}$ results a performance improvement in POP. The reason is that the
achievable information rate at U$2$ is always less in comparison to U$1$ due
to the poorer channel condition of U$2$. On the other side, outage happens
when the rate falls below a threshold rate. Therefore, in such a case, if more
value of $R_{2}^{th}$ is considered, outage probability at U$2$ will be
increased, which will further increase POP. Therefore, it can be concluded
that the threshold rate pair ($R_{1}^{th},R_{2}^{th}$) with
$R_{2}^{th}<R_{1}^{th}$ has more practical significance to obtain better pair
outage performance.

Further, we observe that increasing $\rho_{r}$ decreases $p_{o}$, $s_{o1}$ and
$s_{o2}$. With increasing SNR, achievable data rates and secrecy rates at
users increase, hence, for a given threshold rate and target secrecy rate, POP
and SOP decreases, respectively. Noting the above observations, we can
conclude that the performance of a system is highly dependent on system
parameters.

Figure 7: Pseudo convexity of POP through $p_{o}$ versus $\alpha$ analysis,
(a) $R_{2}^{th}=R_{1}^{th}=0.1$, and (b) $\rho_{r}=20$ dB.

Figure 8: Pseudo convexity of SOP through variation of SOP with $\alpha$, (a)
$s_{o1}$ at $R_{s1}^{th}=1$, and (b) $s_{o2}$ at $\rho_{r}=20$ dB.

Figure 9: Optimal SOP $\max[s_{o1},s_{o2}]$ and PA coefficient $\alpha$
analysis with $R_{s2}^{th}=R_{s1}^{th}=1$, $R_{2}^{th}=R_{1}^{th}=0.1$, and
POP constraint $\xi=0.5$.

Figure 10: (a) Optimal PA with target secrecy rate $R_{s1}^{th}$, and (b)
secrecy fairness $\min\max[s_{o1},s_{o2}]$ with $R_{s1}^{th}$, $\rho_{r}=30$
dB, $R_{2}^{th}=R_{1}^{th}=0.1$, $\xi=0.5$.

###  VI-C Impact of Relative Distance between Users

Fixing $d_{1}$ as $50$ meter, now we observe the effect of variation of
$d_{2}$ from BS on achievable POP and SOP. In Fig. 6, increasing $d_{2}$
implies decrease in achievable data rate and hence, increasing the outage
probability of U$2$ which results in an increase in POP. On the contrary, the
results presented in Fig. 6(a), show that $s_{o1}$ monotonically decreases
with the increase in $d_{2}$. The reason is, with increasing distance $d_{2}$,
a decrease in achievable information rate at U$2$ improves secrecy rate at
U$1$, and hence, SOP at U$1$ decreases. Also, decrease in information rate at
U$2$ implies decrease in secrecy rate at U$2$ which increases SOP for U$2$ as
shown in Fig. 6(b). Note that increasing the distance between BS to U$2$ has
an opposite effect on $p_{o}$ and $s_{o1}$. Similarly, Fig. 6(a) and Fig. 6(b)
also show a contradicting effect of $d_{2}$ on $s_{o1}$ and $s_{o2}$. Hence,
it can be summarized that since POP and SOPs for both the users are important
system parameters for reliable and secure NOMA communication and fulfilling
them together is highly dependent on the relative distance between BS and
users.

Figure 11: Performance comparison of global-optimal PA, $\alpha_{sop}^{*}$,
with fixed PA $\alpha=0.33$, and individual optimal PA $\alpha_{1}^{*}$ and
$\alpha_{2}^{*}$, $R_{2}^{th}=R_{1}^{th}=0.1$, $\xi=0.5$.

Figure 12: Tradeoff between optimal SOP $\max[s_{o1},s_{o2}]$ and POP
threshold $\xi$ at SNR $\rho_{r}=10$ dB for different secrecy rate targets.

###  VI-D Insights on optimality

Now we investigate optimal pair outage and secrecy outage performance by
presenting the numerical proof of the generalized-convexity of POP and SOP
with respect to $\alpha$. Fig. 8(a) and Fig. 8(b) depict POP performance with
$\alpha$ for different values of SNRs and threshold rates, respectively. The
pseudoconvex nature of POP can be easily observed. It confirms the unique
solution of PA that provides minimum POP. Analytical results have also been
plotted as a validation to the analysis. Similarly, Fig. 8(a) and Fig. 8(b),
respectively, validate pseudoconvex nature of $s_{o1}$ and $s_{o2}$ with
$\alpha$ for different SNRs and threshold rates. The numerical optimal
solution obtained using golden section algorithm has also been indicated. We
observe that numerical results match with asymptotic results, which confirms
the validity of asymptotic analysis. Further, to demonstrate secrecy fairness
maximization under QoS constraint, Fig. 10 presents $\max[s_{o1}$,$s_{o2}]$
with $\alpha$. $\alpha_{lb}^{*}$ and $\alpha_{ub}^{*}$ are shown as lower and
upper bounds on $\alpha$, respectively, for an example case of $\xi=0.5$. A
close match between numerical and asymptotically optimal results can be
observed at $\rho_{r}\geq 20$ dB, which validates high SNR analysis. From the
observations, it is worth noting that PA $\alpha$ decides allocated powers to
users, which effects POP and SOPs. Hence, an appropriate PA for given system
parameters can help to obtain an optimal secure and reliable communication
system.

Next, Fig. 10(a) shows global-optimal $\alpha_{sop}^{*}$ that minimizes the
maximum SOP between users as a function of target secrecy rates $R_{s1}^{th}$
for various values of $R_{s2}^{th}$. Results show that for each target secrecy
rate pair ($R_{s1}^{th}$, $R_{s2}^{th}$), there is one and only one $\alpha$
such that $\alpha P_{t}$ and $(1-\alpha)P_{t}$ are the optimal powers for U$1$
and U$2$, respectively. On increasing $R_{s1}^{th}$, the optimal value of
$\alpha$ decreases, whereas the optimal SOP obtained from min-max optimization
problem, presented in Fig. 10(b) increases. Further, lower value of
$R_{s2}^{th}$ in comparison to $R_{s1}^{th}$ provides improvement in SOP which
results in a higher value of $\alpha$. This is because the achievable data
rate at U$2$ is less compared to the achievable data rate at U$1$ because of
the weaker channel condition of U$2$, due to which the achievable secrecy rate
at U$2$ decreases. Note that according to the definition of secrecy outage,
outage happens when the rate falls below a threshold rate. Therefore, it is
natural if more value of $R_{s2}^{th}$ is considered, SOP for U$2$ will be
increased. Therefore, we can conclude that the threshold secrecy rate pair
($R_{s1}^{th},R_{s2}^{th}$) with $R_{s2}^{th}<R_{s1}^{th}$ is a better choice
to improve system performance.

###  VI-E Performance Comparison

To demonstrate the performance gain achieved by the proposed algorithm for
secrecy fairness maximization, Fig. 12 presents the performance comparison of
global-optimal PA $\alpha_{sop}^{*}$ with fixed PA $\alpha=0.33$, and
individual optimal PAs $\alpha_{1}^{*}$ and $\alpha_{2}^{*}$ obtained by
minimizing $s_{o1}$ and $s_{o2}$, respectively. Percentage gain depicts that
$\alpha_{sop}^{*}$ achieves best SOP performance, because it ensures secrecy
fairness between U$1$ and U$2$. The average percentage improvement by
$\alpha_{sop}^{*}$ over fixed PA, individual optimal PAs $\alpha_{1}^{*}$ and
$\alpha_{2}^{*}$ are respectively around $55.12\%$, $69.30\%$ and $19.11\%$.
Thus, it can be concluded that the global optimal PA allocated to users plays
an important role in providing the best system performance.

###  VI-F Tradeoff

Secrecy performance and users’ data rates demands both are important system
performance parameters for secure and reliable communication. However,
fulfilling them together is a challenge because when QoS is considered, system
sacrifices secrecy rate to improve QoS demands. This interesting tradeoff
between minimized SOP and POP, as shown in Fig. 12, emphasizes the effect of
QoS demands on optimal SOP performance. Decrease in SOP with an increase in
POP highlights that higher rate demands lead to poor secrecy performance.

##  VII Concluding Remarks

This paper has proposed a novel decoding order that is capable of providing
positive secrecy rate for both near and far users in a two-user untrusted NOMA
system. To analyze secrecy performance, SOP for both users are derived, and
closed-form expressions are given for the high SNR regime. Individual PA
optimization to minimize SOPs and asymptotic optimal solutions have also been
presented. In order to ensure users’ QoS demands for reliable communication
over all the links, POP has been derived as a QoS measure, and optimal PA
minimizing POP has also been obtained. Further, with an intention to provide
secrecy fairness between users while satisfying their predefined QoS demands,
optimization of PA to minimize the maximum SOP between users is also
investigated. Numerical results are provided to verify the correctness of
analytical expressions, provide insights on generalized-convexity of POP and
SOP, highlight significant performance gains achieved by global-optimal PA,
and describe the tradeoff between different system performance requirements.
While this work has considered a two user NOMA system, the work can be
extended to the study of secrecy in a multi-user scenario and considering
other variants like code-domain NOMA. Furthermore, for an untrusted users’
scenario, it would be interesting to compare the performance of a NOMA enabled
system with jammer assisted orthogonal multiple access.

##  Appendix A Proof of Theorem 2

Having obtained feasible decoding orders for NOMA system with the positive
secrecy rate consideration at all users, we compare these orders in terms of
users’ secrecy rates to investigate the optimal decoding order. The key idea
here is, to find out that one decoding order which gives best secrecy rates at
users. In this context, we have observed that, for each decoding order,
secrecy rates $R_{s1}$ and $R_{s2}$ at U$1$ and U$2$, respectively, are in the
form of $\log_{2}\frac{1+(A/B)}{1+(C/D)}$. Therefore, on comparing two
decoding orders, we can easily show that a given secrecy rate is higher than
its counterpart if either of $A$ or $D$ is higher, or either of $B$ or $C$ is
lower, with other respective parameters being same. Thus, we can find out the
optimal decoding order having maximum secrecy rate.

Considering Theorem $1$, there exists three feasible decoding orders: $(2,1)$,
$(1,2)$, and $(1,1)$. We first compare $(2,1)$ and $(1,2)$. In $(2,1)$,
$B=\beta_{21}+\frac{1}{\rho_{t}}$ and
$D=(1-\alpha)|h_{2}|^{2}+\frac{1}{\rho_{t}}$ for $R_{s1}$, and,
$B=\beta_{12}+\frac{1}{\rho_{t}}$ and $D=\alpha|h_{1}|^{2}+\frac{1}{\rho_{t}}$
for $R_{s2}$. Whereas, in $(1,2)$,
$B=(1-\alpha)|h_{1}|^{2}+\frac{1}{\rho_{t}}$ and
$D=\beta_{22}+\frac{1}{\rho_{t}}$ for $R_{s1}$, and,
$B=\alpha|h_{2}|^{2}+\frac{1}{\rho_{t}}$ and $D=\beta_{11}+\frac{1}{\rho_{t}}$
for $R_{s2}$. The remaining parameters, i.e., $A$ and $C$ are the same for the
two considered orders for $R_{s1}$. The same holds true for $R_{s2}$.
Observing $R_{s1}$ in $(2,1)$ and $(1,2)$, we note that $B$ in $(2,1)$ is
lower in comparison to $B$ in $(1,2)$, i.e.,
$\beta_{21}<(1-\alpha)|h_{1}|^{2}$. Also, $D$ in $(2,1)$ is higher compared to
$D$ in $(1,2)$, i.e., $(1-\alpha)|h_{2}|^{2}>\beta_{22}$. This is because
$\beta_{ij}$ is the residual interefence from imperfectly decoded U$i$ by
U$j$. Similarly, for $R_{s2}$ also, $B$ is lower and $D$ is higher for $(2,1)$
in comparison to $B$ and $D$, respectively, of $(1,2)$. Therefore, it can be
concluded that secrecy rate in $(2,1)$ is higher than $(1,2)$ for both U$1$
and U$2$. In the similar manner, we compare $(2,1)$ and $(1,1)$. In $(1,1)$,
$B=(1-\alpha)|h_{1}|^{2}+\frac{1}{\rho_{t}}$ for $R_{s1}$ and
$D=\beta_{11}+\frac{1}{\rho_{t}}$ for $R_{s2}$. Here also, for $R_{s1}$ in
$(2,1)$ and $(1,1)$, $B$ is lower in $(2,1)$ compared to the $B$ in $(1,1)$,
and, for $R_{s2}$, $D$ is higher in $(2,1)$ in comparison to the $D$ in
$(1,1)$. Thus, $(2,1)$ will provide better secrecy rate than $(1,1)$ also, at
both users.

Thus, we conclude that out of the three feasible decoding orders that can
provide positive secrecy rate at both users, $(2,1)$ is a decoding order that
ensures maximum secrecy rate at both the users as compared to other two
decoding orders. Therefore, $(2,1)$ is considered as an optimal decoding order
regardless of any PA, which is followed in analysis further.

##  Appendix B Proof of Lemma 4

Note that $\hat{\alpha}_{1}$ is obtained by minimizing $\hat{s}_{o1}$ (IV-C1).
The second-order derivative of $\hat{s}_{o1}$ with respect to $\alpha$ does
not imply monotonicity. By setting
$\frac{\mathrm{d}\hat{s}_{o1}}{\mathrm{d}\alpha}=0$, we obtain
$\hat{\alpha}_{1}=-(\Pi_{1}-1)\pm\sqrt{(\Pi_{1}(\Pi_{1}-1)}$. We observe that
$\hat{\alpha}_{1}=-(\Pi_{1}-1)-\sqrt{(\Pi_{1}(\Pi_{1}-1)}$ is negative, and
hence, infeasible. Thus, asymptotic optimal PA minimizing $\hat{s}_{o1}$ is
given as (36).

##  Appendix C Proof of Lemma 5

Here also by setting, $\frac{\mathrm{d}\hat{s}_{o2}}{\mathrm{d}\alpha}=0$ we
obtain $\hat{\alpha}_{2}=\Pi_{2}\pm\sqrt{(\Pi_{2}(\Pi_{2}-1)}$. We note
$\hat{\alpha}_{2}=\Pi_{2}+\sqrt{(\Pi_{2}(\Pi_{2}-1)}$ is infeasible, because
it forces $R_{s2}^{th}<0$ which is not possible. Hence, asymptotic optimal PA
for U$2$ is given as (39).

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