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

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.

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.

The contributions of this work are summarized as follows:

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  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.

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

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  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.

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  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.

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  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.

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.

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$2$.

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

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.

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 $1$.

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 $2$.

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

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

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

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)

where

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

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 $1$.

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$i$.

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