Secure Transmission in NOMA-enabled Industrial IoT with Resource-Constrained Untrusted Devices

Recently, many works have utilized NOMA to ensure massive connectivity requirements in different scenarios for IIoT [12], [13], [14]. However, because of the broadcast nature of the wireless transmission channel, potential adversaries can cause a security risk to communication in NOMA-enabled IIoT networks. Many existing works have considered PLS techniques to protect NOMA networks in different scenarios. For example, in [15], the authors optimized the resource allocation in terms of decoding order, power allocation, and data rate to protect the information associated with the intended device. In [16], the authors derived the optimal power allocation to maximize the achievable secrecy sum rate at devices for a single-input single-output NOMA network. In [17], a novel beamforming scheme that exploits artificial noise to improve the secrecy performance at devices in a multiple-input single-output network was proposed. In [18], a NOMA-assisted secure computation offloading was investigated under an eavesdropping attack, where a wireless user is paired with an edge-computing user to provide cooperative jamming to the eavesdropper while gaining the opportunity to transmit its data. In addition to eavesdropping, other attack modes like jamming were investigated in [19], where intelligent learning-based algorithms, such as Q-learning algorithms, were used to counter the intelligent attackers. Also, in [20], an artificial noise-aided beamforming approach was proposed to achieve secure communication in a large-scale NOMA network with randomly dispersed devices.

The studies cited above [15]-[20] were limited to the security issue of multiplexed NOMA devices against external eavesdropping devices. However, the multiplexed devices sharing the same resource block also can be potential eavesdroppers intercepting the confidential information of each other [21], [22]. Therefore, we should consider an antagonistic network in which each device is assumed to be untrusted. An untrusted scenario is a hostile but realistic situation in which no device trusts others and wants to safeguard its own confidential information. As a result, it becomes essential to allocate resources in the network in such a way that the secrecy of each device is ensured against the other multiplexed untrusted devices, which is a relatively more complex problem.

Assuming the strong device (with better channel gain) as trusted and the weak device (with poorer channel gain) as untrusted for a two-device NOMA network, the authors derived the secrecy outage probability (SOP) for the strong device in [21]. Similarly, [23] analyzed the sum secrecy rate of the strong devices against weak devices for a multiple-input single-output network. In [24], the SOP was investigated for the strong device against the untrusted weak device for a friendly jammer relay scenario. In contrast, [25] considered the strong device as untrusted, and analyzed the optimal power allocation for a secure NOMA network by adjusting the order of successive interference cancellation (SIC) and utilizing a cooperative jammer. Likewise, in [26], a directional demodulation based method was proposed to secure the data of the weak device against the strong device. Furthermore, to safeguard the data of each NOMA device from the other, the authors proposed a linear precoding approach in [27]. Similarly, to obtain a positive secrecy rate for each device against the other device in a two-device NOMA-enabled network, an optimal decoding order was explored in [22], and SOP and its optimization over power allocation were derived for each device. In [28], the ergodic secrecy rate performance was analyzed for each possible decoding order in an untrusted NOMA network, and then, the optimal decoding order was identified.

Notwithstanding the gainful results in handling secrecy issues among untrusted devices in NOMA-enabled networks, several works, e.g. [21], [23]-[27], over-optimistically considered that the devices could perform perfect SIC. According to this ideal setup, the interference from previously decoded devices is fully subtracted when the signal associated with the later devices is decoded. Thus, the decoded devices do not interfere with other devices. This strong assumption makes the system model simple and might not be realistic. In a practical scenario, the devices are resource-constrained to perform perfect SIC due to various practical implementation issues in IIoT networks, such as hardware limitation, inaccurate calibration, estimation error, multiple types of noise, and complexity scaling [29]. Consequently, imperfect SIC, where the residual interference (RI) from the formerly imperfectly decoded devices inevitably abides while decoding the signals of subsequent devices [29], should be taken at receivers while doing any investigation on NOMA.

In the literature of NOMA, some research works considered the RI as a particular constant value [30], [31], referred to as the Constant RI Model. In contrast, many other works took the RI as a linear function of the power assigned to the interfering signal [29], [32], referred to as the Linear RI Model. Nevertheless, there seem to be fewer studies that have considered the impact of RI while handling the secrecy issue in untrusted NOMA networks. The authors in [22] analyzed the secrecy performance of devices in an untrusted NOMA network with imperfect SIC but considered the constant RI model. However, the RI obtained from decoded devices may not be a constant value in practice. The constant RI is a strong and unrealistic assumption that over-simplifies the model and leads to prediction errors. In contrast, realistic influence of imperfect SIC may be observed with the linear RI model since decoders’ performance significantly depends on the interfering signal’s power. Motivated by this solid observation, in this work, we mainly focus on obtaining a secure NOMA-enabled IIoT network in the presence of untrusted devices, considering the effect of imperfect SIC at receivers with a linear RI model.

Note that [28] explored a secure NOMA network with a linear RI model, but the investigation was carried out for maximizing ergodic secrecy performance for each device. However, this article fills a significant gap in the literature by optimizing resources to maximize secrecy fairness among devices, which has not yet been studied in the literature. Note that fairness is an important performance metric in order to guarantee the achievable rates for weak devices, as considered in many works in the literature [22], [33]. It is because focusing exclusively on the sum rate may result in substantial rate loss for weak devices, as the system tends to allocate most of the communication resources to strong devices when the sum rate is maximized. The weak devices may even be unable to be served in some extreme cases. Thus, in this work, the fundamental basis for studying secrecy fairness is that weak devices may also obtain enough communication resources similar to strong devices so that there is no loss in the achievable secrecy rate performance for weak devices. Therefore, by following [22], [33], we in this work focus on maximizing secrecy fairness between devices, where we maximize the minimum secrecy rate between devices.

To optimize the network’s secrecy fairness performance, the decoding order and power allocation to the multiplexed devices are the two key parameters. In the literature, many research works are limited to the conventional decoding order of NOMA. However, we may change the decoding sequence for each device [22], [28], [34]. The fact is that SIC is a physical layer capability that allows receiving ends to extract the superimposed signal. Thus, any device can decode a signal of itself or others at any stage resulting in various decoding orders. Besides, most existing literature assumes that NOMA is based on more power allocation to the device with weaker channel conditions, which is not true [35]. Therefore, it would be interesting to investigate if the conventional approach of decoding order and power allocation is optimal from the secrecy fairness viewpoint. Encouraged by these substantial observations, in this work, we jointly optimize the resources, such as decoding order and power allocation, for maximizing the secrecy fairness between devices.

The key contributions of this paper are summarized below:

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  We focus on achieving a secure NOMA-enabled IIoT network in the presence of untrusted devices, considering the real effect of imperfect SIC with a linear RI model. In this respect, we first find out the feasible power allocation condition to obtain a positive secrecy rate for each device in all possible decoding orders. This way, we identify the feasible secure decoding orders that can provide a positive secrecy rate for each device.

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  We focus on optimizing the resources, such as secure decoding order and transmission power allocated to devices, from the perspective of secrecy fairness. Under the secrecy fairness criterion, we formulate and solve a joint optimization problem of maximizing the minimum secrecy rate between devices over a set of secure decoding orders and transmission power allocation. The formulated problem is combinatorial and non-convex. Therefore, we first find the optimal secure decoding order and then solve it over power allocation by obtaining candidates of optimal solution with Karush-Kuhn-Tucker (KKT) conditions. Thus, we provide the closed-form global-optimal solution to the formulated problem.

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  Lastly, we present numerical results to confirm the accuracy of the analysis, provide insightful discussion into the impact of network parameters on the optimal performance, and show the performance gains achieved by the optimal results over different benchmarks.

Notations: Bold uppercase and lowercase letters, respectively, are used to refer to matrices and column vectors. We denote the $(u,v)$-th entries of matrix A by $[\textbf{A}]_{u,v}$. The $u$-th entry of vector a is indicated by $[\textbf{a}]_{u}$.

We consider a NOMA-enabled IIoT network, where the base station communicates with two devices, as depicted in Fig. 1. In the network, both devices are assumed to be untrusted. The $n$-th device of the network is symbolized by $U_{n}$, where $n\in\mathbb{N}=\{1,2\}$. All nodes in the network are assumed to have one antenna. All the channels from the base station to devices are assumed to be Rayleigh faded. The channel gain coefficient from the base station to $U_{n}$ is represented by $h_{n}$. As a result, the channel power gain, denoted by $|h_{n}|^{2}$, follows an exponential distribution having mean parameter $\lambda_{n}=L_{p}d_{n}^{-e}$, where $L_{p}$ indicates path loss constant, $d_{n}$ stands for the distance of $U_{n}$ from the base station, and $e$ refer to the path loss exponent. Without any loss of generality, we presume that the channel power gains are arranged as $|h_{1}|^{2}>|h_{2}|^{2}$. Thus, based on the channel power gain conditions, $U_{1}$ and $U_{2}$ could be referred to as strong device and weak device, respectively. The transmission power broadcasted from the base station to both devices is denoted by $P_{t}$. The fraction of transmission power $P_{t}$ allocated to $U_{1}$ is indicated by a power allocation coefficient $\alpha$, where $0\leq\alpha\leq 1$. The remaining fraction $(1-\alpha)$ is allocated to $U_{2}$.

The base station first superposes all the message signals dedicated for devices and then transmits the superposed signal to each device. Thus, the broadcasted signal from the base station to both devices can be expressed as

where $\text{x}_{1}$ and $\text{x}_{2}$, respectively, signify the message signals with unit power dedicated for $U_{1}$ and $U_{2}$. Then, the received signal $\text{y}_{n}$ for $U_{n}$, where $n\in\mathbb{N}$, can be given as

where $w_{n}$ represents the additive white Gaussian noise (AWGN) for $U_{n}$. Without loss of generality, we assume that $w_{n}$ has a mean equal to zero and variance equal to $\sigma^{2}$.

After obtaining the superposed signal, receivers apply SIC to remove the inter-device interference imposed by the superposition and get the desired signal. During SIC, each device decodes its own and other devices’ signals in a particular sequence. The collection of these sequences abide by devices is referred to as a decoding order of the network. According to the conventional decoding order of NOMA, the strong device $U_{1}$ considers $U_{2}$’s signal as interference. Therefore, it decodes $U_{2}$’s signal at the first stage, applies SIC to cancel its interference, and then decodes its own signal at the second stage. Conversely, the weak device $U_{2}$ decodes its own signal at the first stage by considering the signal associated with $U_{1}$ as noise. Thus, a mutual trust is assumed between the devices that they will not intercept each other’s information.

An IIoT network with untrusted devices is an antagonistic but realistic circumstance in which devices do not trust each other and always want to safeguard their data from other devices. Therefore, practically, we should consider the possibility that each device may attempt to decode the signal of itself or other device at any stage [22], [28], [34]. Thus, in a two-device network, each device has two stages of decoding the signal of itself and the other device. As a result, four decoding orders exist based on the concept of permutation. We depict the $o$-th decoding order as a matrix $\mathbf{D}_{o}$ of size $2\times 2$. The $m$-th column of matrix $\mathbf{D}_{o}$ is expressed by a $2\times 1$ column vector $\mathbf{d}_{m}$, which shows the sequence of SIC followed by $U_{m}$, where $m\in\mathbb{N}$. Specifically, $[\mathbf{d}_{m}]_{k}=n$ defines that signal of the device $U_{n}$ is decoded by the device $U_{m}$ at $k$-th stage, where $[\mathbf{d}_{m}]_{1}\neq[\mathbf{d}_{m}]_{2}$ and $n,k\in\mathbb{N}$. Thus, a decoding order of the network can be represented as $\mathbf{D}_{o}=[[\mathbf{d}_{1}]_{1},[\mathbf{d}_{2}]_{1};[\mathbf{d}_{1}]_{2},[\mathbf{d}_{2}]_{2}]$. All decoding orders can be expressed as $\mathbf{D}_{1}=[2,2;1,1]$, $\mathbf{D}_{2}=[2,1;1,2]$, $\mathbf{D}_{3}=[1,2;2,1]$, and $\mathbf{D}_{4}=[1,1;2,2]$. We define the set of these four decoding orders as $\mathbb{D}=\{\mathbf{D}_{o}|1\leq o\leq 4\}$.

For a given decoding order $\mathbf{D}_{o}$, the data rate achieved at $U_{m}$ when $U_{n}$ is decoded by $U_{m}$, for all combinations of $n,m\in\mathbb{N}$ with Shannon’s formula can be expressed as

where $\Gamma^{[o]}_{nm}$ denotes the received signal to interference plus noise ratio (SINR) at $U_{m}$, when $U_{n}$ is decoded by $U_{m}$, and it can be given as

where $\rho_{t}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{P_{t}}{\sigma^{2}}$ is the base station transmit signal-to-noise ratio (SNR). The parameters $a$ and $b$ required to define $\Gamma^{[o]}_{nm}$ for each combination of $m,n\in\mathbb{N}$ in all decoding orders are given in Table I. Note that $n=m$ in Table I indicates the SINR achieved by the legitimate device $U_{n}$ when decoding its own data, namely $\Gamma^{[o]}_{nn}$, resulting which we obtain $R^{[o]}_{nn}$ using (3). In Table I, $\beta_{\widehat{n}m}$ is the RI factor indicating the fraction of the residual error from the previous decoding stage, i.e., when $m$ imperfectly decodes $\widehat{n}$, where $m,\widehat{n}\in\mathbb{N}$ and $\widehat{n}\neq n$. Note that $0\leq\beta_{\widehat{n}m}\leq 1$. Here $\beta_{\widehat{n}m}=0$ and $\beta_{\widehat{n}m}=1$, respectively, indicates perfect SIC and absolutely imperfect SIC [29], [32].

Next, in order to ensure secure communication, we utilize the concept of PLS. According to PLS, the secrecy rate of a legitimate device can be measured by the difference between the rate achieved at the legitimate device when decoding its own data and the rate achieved at another device when decoding the data of the legitimate device. Accordingly, the secrecy rate for $U_{n}$ against $U_{m}$, where $m,n\in\mathbb{N}$ can be expressed as [10], [11]

where $n\neq m$ and $[\Diamond]^{+}=\max\{0,\Diamond\}$. To get a positive secrecy rate for a device, the rate of the main communication link must be greater than the rate of the eavesdropper’s link, i.e., for obtaining $R^{[o]}_{sn}>0$, the condition $R^{[o]}_{nn}>R^{[o]}_{nm}$, simplified to $\Gamma^{[o]}_{nn}>\Gamma^{[o]}_{nm}$ using (3), needs to be satisfied. $[\Diamond]^{+}=\max\{0,\Diamond\}$ indicates that negative secrecy rates are considered to be zero.

With the conventional decoding order in untrusted environment, a weak device $U_{2}$ may try to decode the signal of $U_{1}$ after cancelling the signal of itself through SIC [21]. Thus, the decoding order can be written as $\mathbf{D}_{1}=[2,2;1,1]$. Below in Proposition 1, we first prove that $\mathbf{D}_{1}$ is not efficient to achieve a secure NOMA network.

Thus, it can be concluded that $\mathbf{D}_{1}$ is not a feasible decoding order in achieving a positive secrecy rate to both devices.

Now we check the feasibility of decoding orders $\mathbf{D}_{2}$, $\mathbf{D}_{3}$, and $\mathbf{D}_{4}$, one by one, in achieving secure NOMA transmission.

As we obtained in Section III that there are two secure decoding orders $\mathbf{D}_{2}$ and $\mathbf{D}_{4}$ that can ensure a positive secrecy rate for each device, we will optimize the decoding order over set $\mathbb{S}$ of secure decoding orders (Refer Remark 3). Also, since $R^{[o]}_{s1}$ and $R^{[o]}_{s2}$, is a function of both decoding order and $\alpha$, we formulate a joint optimization problem as maximizing the minimum secrecy rate between devices over a set $\mathbb{S}$ of secure decoding orders and power allocation coefficient $\alpha$ as

where $\mathcal{C}_{1}$ refers to the constraint on power allocation coefficient (Refer Section II-A), and $\mathcal{C}_{2}$ and $\mathcal{C}_{3}$ denote the positive secrecy rate conditions for $U_{1}$ and $U_{2}$, respectively.

We observe that $\mathcal{P}_{1}$ is a combinatorial optimization problem because there are two secure decoding orders, and the secrecy rate depends on $\alpha$ in each secure decoding order. Therefore, to reduce the computational complexity in determining the joint optimal solution of decoding order and power allocation, we solve the joint optimization problem $\mathcal{P}_{1}$ in two steps as described below.

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  In the first step, ignoring the power allocation constraint, we optimize the decoding order over a set $\mathbb{S}$ of secure decoding orders for maximizing the minimum secrecy rate between the devices. This way we find the optimal secure decoding order for maximizing the minimum secrecy rate between devices. (Refer Section IV-C1)

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  In the second step, to complete joint optimization, we obtain the optimal power allocation solution in closed form for only optimal secure decoding order obtained in the first step. (Refer Section IV-C2)

Firstly, Fig. 2 is plotted to validate the accuracy of Lemma 1, stating that the optimal secure decoding order maximizing the minimum secrecy rate between devices is $\mathbf{D}_{2}$. Here, through Fig. 2(a) and Fig. 2(b), the variation in secrecy rates $R_{s1}^{[o]}$ and $R_{s2}^{[o]}$ for $U_{1}$ and $U_{2}$, respectively, with $\alpha$ for all four decoding orders is shown. The results confirm that for all $\alpha$ values, $\mathbf{D}_{2}$ provides more secrecy rate for each device than other decoding orders. Hence, to maximize the minimum secrecy rate, the optimal secure decoding order is $\mathbf{D}_{2}$.

Further, to validate optimal power allocation solution (Refer Lemma 2) for optimal secure decoding order $\mathbf{D}_{2}$, Fig. 3 is plotted. Here, we show the variation of $\min\left[R_{s1}^{[2]},R_{s2}^{[2]}\right]$ with $\alpha$ for different values of $\beta_{21}$ and $\beta_{12}$. Results indicate that there exists a unique global-optimal solution for $\min\left[R_{s1}^{[2]},R_{s2}^{[2]}\right]$ in terms of $\alpha$. The perfect match between the simulation and analytical results confirms the accuracy of the analysis. We also observe from the results that the optimal power allocation can be greater than 0.5. It means providing lesser power to the strong device than the power allocated to the weak device, as presumed in many works in the NOMA literature, is not always necessary. Thus, we conclude that the power allocation associated with devices in a NOMA-enabled IIoT network should be decided based on the given network parameters.

Through the results presented in Fig. 4, we study the impact of $\rho_{t}$ and different values of far device’s distance $d_{2}$ on the average optimal secrecy rate performance of the network. $d_{1}$ is set to as $50$ meters. We observe that average optimal secrecy rate increases by increasing $\rho_{t}$. The reason is that the achievable data rates for devices increase with an increase in SNR. Here we also notice that the average optimal secrecy rate performance decreases with an increase in distance $d_{2}$. The reason is that increasing the distance $d_{2}$ results in a decrease in the achievable data rate of $U_{2}$, and consequently, an increase in secrecy rate for $U_{1}$ and a decrease in secrecy rate for $U_{2}$ is obtained. Through simulations, we notice that less secrecy rate is obtained for $U_{2}$ for most channel realizations. Therefore, while calculating the average max min secrecy rate, the secrecy rate for $U_{2}$ is dominant for given network parameters, which decreases by increasing $d_{2}$. Therefore, the results show that average performance degrades with an increase in $d_{2}$.

Through Fig. 5, we demonstrate that the joint optimal solution, i.e., optimal decoding order and optimal power allocation, is capable of improving the average secrecy rate over different benchmarks. In this regard, the joint optimal solution is compared with four different benchmarks to evaluate its performance in terms of average secrecy rate, and the percentage gain is calculated. Four different benchmarks are considered: (i) ODEP: optimal decoding order and equal power allocation, (ii) ODFP: optimal decoding order and fixed power allocation, (iii) FDEP: fixed decoding order and equal power allocation, and (iv) FDFP: fixed decoding order and fixed power allocation. The optimal and fixed decoding orders are $\mathbf{D}_{2}$ and $\mathbf{D}_{4}$, respectively. For the equal power allocation, $\alpha=0.5$ is assumed, which is considered to examine the case in which both devices are assigned with equal power. However, in a fixed power allocation scheme, $\alpha=0.33$ is taken, which means $\alpha=0.33$ is allocated to $U_{1}$ and the remaining fraction $1-\alpha=0.66$ is allocated to $U_{2}$. Taking $\alpha=0.33$ is intended to examine the situation in which a weak device is assigned more power than a strong one, as considered in many works in the literature. Results show that the joint optimal solution provides an average percentage gain in secrecy fairness performance over benchmarks ODEP, ODFP, FDEP, and FDFP, of around $22.75\%$, $50.58\%$, $94.59\%$, and $98.16\%$, respectively. In this way, we observe that the result achieved by FDFP actually differs greatly from the joint optimal solution obtained.

Through Fig. 6, we demonstrate a variation in $\min\left[R_{s1}^{[o]},R_{s2}^{[o]}\right]$ with $\alpha$ for conventional decoding order and optimal decoding order. Note that the conventional and optimal secure decoding order, respectively, are $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$. Results indicate that for all $\alpha$ values, $\mathbf{D}_{2}$ ensures secrecy fairness between devices and gives a significant secrecy rate. Also, there exists a unique global-optimal solution for $\min\left[R_{s1}^{[2]},R_{s2}^{[2]}\right]$ in terms of $\alpha$ for $\mathbf{D}_{2}$. However, in the case of conventional decoding order $\mathbf{D}_{1}$, maximizing minimum secrecy rates between devices, i.e., $\min\left[R_{s1}^{[1]},R_{s2}^{[1]}\right]$ will always result in zero secrecy rates. The reason is that if conventional decoding order is followed, a positive secrecy rate cannot be achieved for weak devices (Refer proposition 1), resulting in $\min\left[R_{s1}^{[2]},R_{s2}^{[2]}\right]=0$. Thus, from the secrecy fairness viewpoint, we can conclude that $\mathbf{D}_{1}$ is not a feasible decoding order.