Abstract
In this work, we investigate the physical layer security problem of wireless cooperative network, where the communication from a source to a destination is assisted by an untrusted full-duplex amplify-and-forward (AF) relay. In order to realize a positive secrecy rate, cooperative jamming is exploited at the destination. The secrecy outage probability (SOP), connection outage probability (COP), and effective secrecy throughput (EST) are, respectively, derived in closed-form expressions for the fixed gain relaying (FGR) scheme and the variable gain relaying (VGR) scheme. Subsequently, we conduct an asymptotic analysis for both the relaying schemes in the high signal-to-noise ratio (SNR) region to offer valuable insights into practical design. Theory and simulation results demonstrate that the self-interference caused by full-duplex transmission is harmful for the reliability performance, while it is beneficial to the security performance. Moreover, the EST of the considered system increases as the self-interference level decreases, which implies that the self-interference deteriorates an overall performance in terms of the security-reliability tradeoff.
1. Introduction
Due to the broadcast nature of wireless transmission, the communications between legitimate users are particularly vulnerable to be overheard by malicious users [1, 2]. Traditionally, cryptographic mechanism in the upper layers of the network based on computational complexity has been employed to guarantee secure communication in wireless networks [3]. However, as the computing power of malicious users is continuously enhancing (e.g., by adopting the quantum computing technology), cryptographic mechanism is being increasingly challenged. Unlike the cryptographic mechanism, physical layer security, which explores the time-varying properties of physical channels to achieve perfect security, has attracted considerable attention from academia and industry [4, 5].
In recent years, full-duplex relay techniques, which can transmit and receive simultaneously in the same frequency band, have been exploited to improve physical layer security performance of wireless networks. The authors in [6] proposed a newly full-duplex jamming relay scheme, which has better secrecy performance than the half duplex scheme. For full-duplex two-way relay networks, [7] exploited artificial noise to strengthen the sum secrecy rate. In [8], a power allocation strategy between data and jamming was developed to effectively improve the tradeoff between security and reliability, where both the relay and the destination operated in full-duplex mode. Subsequently, full-duplex techniques were extended to secure multirelay networks [9] and nonorthogonal multiple access (NOMA) networks [10].
The works in [6–10] were focused on the security of full-duplex relay networks, where the eavesdroppers are external nodes in addition to legitimate nodes. However, even in the absence of external eavesdroppers, the security caused by legitimate nodes in the network may still be a concern [11–13]. For example, the relay may be untrusted, which means that they can possibly try to eavesdrop the source’s confidential information. The untrusted relay is also a potential eavesdropper, even though it complies with the communication protocols to assist the source forwarding information to the destination [14–16]. In practice, untrusted relay scenario may occur in Internet of Things (IoT), government intelligence networks, and device-to-device (D2D) communications, where not all users do have the same security clearance.
Recently, a few works about untrusted full-duplex relay networks have been considered in the context of physical layer security. Based on the source jamming scheme, [17] derived the secrecy outage probability for untrusted full-duplex relay networks. The impact of outdated channel state information (CSI) on the ergodic secrecy rate of untrusted full-duplex relay networks was investigated in [18]. The authors of [19] proposed an artificial noise-aided secure transmission scheme to confuse the untrusted full-duplex relay and external eavesdroppers. In [20], an optimal beamforming scheme was design to maximize the secrecy sum rate for full-duplex multiple-input multiple-output two-way untrusted relay networks. The aforementioned contributions are mainly concerned on enhancing the secrecy rate of untrusted full-duplex relay networks without paying much attention to the communication reliability. In particular, the impact of self-interference caused by full-duplex transmission on the security-reliability tradeoff is indeed worthy of our attention.
Motivated by the above considerations, we explore the physical layer security of untrusted full-duplex relay networks, where a source operates in half duplex mode and an untrusted relay and a destination operate in a full-duplex mode. In addition, the source equipped with multiple antennas uses a maximal ratio transmission (MRT) scheme to increase reliability. Depending on the availability of the CSI, the relay amplifies and forwards the source’s signal with a fixed-gain or variable-gain factor. Different from the source-based jamming scheme in [17], we employ destination-assisted jamming for secrecy improvement. Compared with [18–20], we consider the outage performance of the untrusted full-duplex relay network, where the secrecy outage probability (SOP) is used to evaluate the security performance and the connection outage probability (COP) is used to characterize the reliability performance. The main contributions of this paper are summarized as follows: (i)We first present exact expressions for the SOP and COP of the fixed gain relaying (FGR) scheme and the variable gain relaying (VGR) scheme and then derive closed-form expressions for the EST of both the relaying schemes, which provide an efficient means to evaluate the reliability and security performance comprehensively(ii)To gain further insights, we conduct an asymptotic analysis in the high signal-to-noise ratio (SNR) region for the SOP, COP, and EST. Based on the asymptotic analysis, we find that there exists a EST floor for both the relaying schemes in the high SNR region and the floor of the VGR scheme is higher than that of the FGR scheme(iii)The results demonstrate the intuition that the self-interference caused by full-duplex transmission is harmful for the reliability performance, while it is beneficial to the security performance. In addition, the EST of the considered system increases as the self-interference level decreases, which implies that the self-interference deteriorates an overall performance in terms of the security-reliability tradeoff
The remainder of this paper is organized as follows. Section 2 gives the system model and describes the key metric in evaluating the performance of physical layer security, including the SOP, COP, and EST. In Section 3, we present the closed form expressions of the SOP, COP, and EST for the VGR scheme and the FGR scheme and analyze the corresponding asymptotic behavior in the high SNR region. Numerical results and conclusions are, respectively, provided in Sections 4 and 5.
2. System Model
We consider an untrusted full-duplex relay network shown in Figure 1, where a source communicates with a destination via an untrusted relay . The source has antennas and operates in the half-duplex mode, while the destination and the relay are equipped with dual antennas and operate in the full-duplex mode. It means that the destination and the relay can receive and send data concurrently. The system considered in this work is applicable to numerous practical full-duplex relay scenarios, for instance, the wireless sensor networks and IoT. Moreover, the direct link between the source and the destination is unavailable, due to severe shadow fading or significant obstacle [21]. Thus, source-destination communication can be performed only through the relay. In addition, all channels are modeled as quasistatic block fading channels, following the Rayleigh distribution. As in [13, 15], we assume that the channel between the relay and the destination is reciprocal.
Because the relay is untrusted, the source wishes to keep the information secret from the relay while simultaneously making use of it to forward the information. To maintain the confidentiality of the source information, the destination sends a jamming signal to the relay when the source transmits information to the relay. Thus, the received signal at the relay can be expressed as where is the channel vector between the source and the relay and its entries follow independent and identically distributed (i.i.d.) complex Gaussian distribution with zero mean and variance . is the transmit precoding vector at the source. is the channel coefficient between the relay and the destination with parameter . is the loopback interference channel coefficient between transmitting and receiving antennas at the relay. is the confidential signal transferred from the source at time slot and satisfies , where represents the expectation operation. is the jamming signal transferred from the destination at time slot and satisfies . Here, is the power allocation factor between the source and the destination. is the loopback interference signal transferred from the relay at time slot and satisfies . is the additive white Gaussian noises (AWGN) at the relay with zero mean and variance . Since is known to the relay, it can apply multiple stage interference cancellation methods to alleviate the loopback interference. Thus, the received signal at the relay after compensation can be expressed as where can be modeled as a Gaussian random variable with mean zero and variance [18, 22]. corresponds to the self-interference cancellation capability at the relay.
Then, the relay amplifies the signal with factor and forwards it to the destination. The two most common factors employed at the relay are so-called fixed gain factor and variable gain factor. In the FGR scheme, the factor is constant based on the statistical CSI, resulting in an output signal with variable power. In the VGR scheme, the factor is variable depended on the instantaneous CSI, resulting in an output signal with fixed power. The factors of both FGR and VGR schemes can be given by
Hence, the received signal at the destination is given by where is the loopback interference channel coefficient between transmitting and receiving antennas at the destination and is the AWGN at the destination with zero mean and variance . Analogously, the received signal at the destination after self-interference cancellation can be written as where can be modeled as a Gaussian random variable with mean zero and variance . corresponds to the self-interference cancellation capability at the destination.
According to (2), the achievable eavesdropping rate with the FGR scheme and the VGR scheme can be obtained as where denotes the transmit SNR. According to (3) and (5), the achievable data rate with the FGR scheme and the VGR scheme can be, respectively, obtained as where , , , and .
As discussed in [23], when the achievable eavesdropping rate is larger than the positive rate difference between the codeword transmission rate and the confidential information rate , there will be a secrecy outage event. When the achievable data rate is less than , there will be a connection outage event. Thus, we can formulate the SOP and COP of our system as
The SOP and COP give practical insights into security and reliability performance of the transmission system, respectively. To establish the direct relationship between security and reliability, we adopt effective secrecy throughput (EST) to measure the overall efficiency of our system, which is given by
3. Secrecy Performance Analysis
In this section, we endeavor to analyze both the reliability and security performance comprehensively for the untrusted full-duplex relay network. Specifically, we first derive closed form expressions of the SOP, COP, and EST for the FGR scheme and the VGR scheme. Then, the asymptotic analysis of the SOP, COP, and EST is provided to reveal additional insights on the secrecy performance.
3.1. Fixed Gain Relaying
3.1.1. Secrecy Outage Probability
Combining (6) and (9), the SOP of the untrusted full-duplex relay network with the FGR scheme can be given by where . Noting that is a central chi-square distribution random variable with degrees of freedom, its cumulative distribution function (CDF) is given by
On the other hand, is an exponential variable with probability density function (PDF) given by
Based on (13) and (14), we can rewrite (12) as
Then, using , we can calculate as
From (16), we find that the SOP increases as the power allocation factor increases from 0 to 1. This is because that increasing results in decreasing transmission power of the jamming signal. Moreover, the SOP decreases as increases, which means that the self-interference caused by full-duplex transmission is beneficial to the security performance.
3.1.2. Connection Outage Probability
Combining (7) and (10), the COP of the untrusted full-duplex relay network with the FGR scheme can be given by where . Based on (13) and (14), we can rewrite (17) as
Then, with the help of the binomial theorem and ([24], 3.471.9), the exact COP of the untrusted full-duplex relay network with the FGR scheme can be derived as where is the Bessel functions of imaginary argument. From (19), we find that the COP decreases as the power allocation factor increases from 0 to 1. The main reason is that when increases, the achievable data rate is larger which makes reliability better. In addition, the COP increases as or increases, which means that the self-interference caused by full-duplex transmission is harmful for the reliability performance.
3.1.3. Effective Secrecy Throughput
Combining (6), (7), and (11), the EST of the untrusted full-duplex relay network with the FGR scheme can be given by
Since and have only one point of intersection when , and the intersection is . Thus, expression (20) can be expressed as where , , , and . Then, using [24, 1.211.1], [24, 3.351.2], and [24, 3.381.3], the exact EST of the untrusted full-duplex relay network with the FGR scheme can be derived as
From (22), we find that the EST of the considered system is small, when the confidential information rate is small. At the other extreme, when the confidential information rate is too large, the secrecy performance of the considered system is poor which also results in small EST. Hence, there exists an optimal to maximize the EST of the considered system. Furthermore, the EST increases as the self-interference level decreases, which means that the self-interference deteriorates an overall performance in terms of the security-reliability tradeoff.
3.1.4. Asymptotic Behavior
To extract additional insights on the FGR scheme, we now investigate the high SNR asymptotic behavior of the SOP, COP, and EST of the considered system. According (16), we have
From (19), we have where . According to (22), we have where . It is interesting to note that when transmit power is sufficiently large, the security and reliability of the untrusted full-duplex relay network are still depend on the power allocation factor. Thus, the power allocation factor enables a tradeoff between reliability and security for the untrusted full-duplex relay network. From the numerical results, we find that there exists an optimal power allocation factor maximizing the EST.
3.2. Variable Gain Relaying
3.2.1. Secrecy Outage Probability
The gain factor employed at the relay determines the quality of signal reception at the destination and has no influence on the eavesdropping ability at the relay. Therefore, the SOP of the VGR scheme is the same as the FGR scheme and it is omitted here.
3.2.2. Connection Outage Probability
Combining (8) and (10), the COP of the untrusted full-duplex relay network with the VGR scheme can be given by
Based on (13) and (14), we can rewrite (26) as where , and . Then, let , and use [24, 1.211.1] and [24, 3.471.9], the exact COP of the untrusted full-duplex relay network with the VGR scheme can be derived as
When , we have , where follows the law of large numbers. Thus, from (26), we clarify that increasing the number of antennas at the source unboundedly is not helpful because the channel between the relay and the destination will finally become the bottleneck and dominate the achievable data rate.
3.2.3. Effective Secrecy Throughput
Combining (6), (8), and (11), the EST of the untrusted full-duplex relay network with the VGR scheme can be given by
Since and have only one point of intersection when , and the intersection is . Thus, the expression (29) can be expressed as where , , , , and . Then, using [24, 1.211.1], [24, 3.351.2] and [24, 3.381.3], the exact EST of the untrusted full-duplex relay network with the VGR scheme can be derived as
The derived exact results in (31) provide an efficient way to holistically examine the security and reliability performance with arbitrary system parameters, i.e., the number of antennas at the source and the self-interference cancellation capability at the relay and destination. Additionally, it is worth noting that the EST of the considered system will tend toward zero when . An intuitive explanation to this phenomenon is that, more antennas at the source would bring us better reliability performance, but at the same time, the grade of the security is severely degraded. Thus, the designers have to carefully take into account the number of antennas at the source.
3.2.4. Asymptotic Behavior
To get more insight, we further study the asymptotic behavior of the COP and EST for the VGR scheme under high SNR case. According to (28), we have where , , , , and . From (31), we have where , , , , and .
From the asymptotic behaviour of the SOP, COP, and EST, we find that the continued boosting of the transmit power will not always improve the security and reliability performance of the considered system. This is due to the fact that the self-interference caused by full-duplex transmission becomes a bottleneck of improving the security and reliability performance at the high SNR region.
4. Numerical Results
In this section, we present simulation and numerical results to illustrate tendencies of the performance of the untrusted full-duplex relay network. Unless otherwise stated, we set the codeword transmission rate, bits/sec/Hz, and the confidential information rate bits/sec/Hz. Also, we set . It is evident from Figures 2–7 that the derived closed form expressions of SOP, COP, and EST agree well with their Monte Carlo simulation points, which indicates the accuracy of our theoretical analysis stated above.
Figure 2 shows the SOP and COP versus the transmit SNR of the FGR scheme as well as the VGR scheme, where , , , dB, and dB. We first observe that the SOP of the FGR and VGR schemes increases as the transmit SNR increases, whereas the COP of the FGR and VGR schemes decrease as the transmit SNR increases. This is due to the fact that an increasing transmit SNR will lead to a higher achievable eavesdropping rate and achievable data rate. Second, we observe that the SOP and COP of the FGR and VGR schemes converge to a constant when the transmit SNR is sufficiently large. Thus, we can conclude that the transmit SNR is not the larger the better. Moreover, we observe that the COP of the VGR scheme outperforms the FGR scheme at the high SNR region.
Figure 3 shows the SOP and COP versus the self-interference cancellation capability of the FGR scheme as well as the VGR scheme, where , , , dB, and dB. We observe that the SOP of the FGR and VGR schemes decreases as the self-interference cancellation capability decreases, whereas the COP of the FGR and VGR schemes increase as the self-interference cancellation capability decreases. The reason is that as the self-interference cancellation capability decreases, the loopback interference at the destination and the relay will be stronger. Thus, the self-interference cancellation capability enables a tradeoff between security and reliability of the untrusted full-duplex relay network.
Figure 4 depicts the EST as a function of the transmit SNR, where , , dB, and dB. We first observe that the EST of the FGR scheme and the VGR scheme converge to a constant when the transmit SNR tends to infinity. This phenomenon is due to the fact that the EST is constrained by the self-interference caused by full-duplex transmission in the high SNR regime. Second, we observe that the EST of the VGR scheme is better than the FGR scheme in the high SNR region. Moreover, we observe that for a given transmit SNR, the EST of both the FGR scheme and the VGR scheme with dB outperform the EST with dB. This is because that the improved quality of the second hop channel is beneficial for secure transmission.
Figure 5 examines the impact of power allocation factor on the EST performance, where , dB, dB, and dB. We observe that when the power allocation factor is either extremely small or large, the EST of the FGR scheme and the VGR scheme approach zero. This is because when the power allocation factor is small, it is hard to establish a reliable link from the source to the destination. On the other hand, when the power allocation factor is large, the power of the jamming signal at the destination is poor and secure transmission is impossible. Thus, there exists an optimal power allocation factor maximizing the EST. Moreover, the optimal power allocation factor gradually increases as increases. It indicates that the destination does not need more power to transmit jamming when the channel between the relay and the destination is better.
Figure 6 illustrates the EST of the FGR scheme and the VGR scheme as a function of the number of antennas at the source, with dB, , dB, and dB. We observe that the EST of the FGR scheme and the VGR scheme first increase to a peak value and then decrease as a function of the number of antennas at the source. This phenomenon is due to the fact that increasing the number of antennas at the source leads to the enhancement of the reliability performance, but the degradation of the security performance. Second, we observe that the performance gain of the VGR scheme is much more pronounced for small , and gradually diminishes when becomes large.
Figure 7 plots the EST of the FGR scheme and the VGR scheme as a function of the self-interference cancellation capability, with , , , and dB. It can be seen from the figure that the EST of the FGR and VGR schemes decrease as the self-interference cancellation capability decreases. From Figure 3, we know that the self-interference cancellation capability has a positive effect on the secure transmission but incurs a negative effect on the reliable transmission. Thus, we can conclude that the self-interference caused by full-duplex transmission does more harm than good for the untrusted full-duplex relay network. When the the level of self-interference is strong, the EST is close to zero, which suggests that both reliable and secure transmission can not be achieved.
5. Conclusion
In this paper, we investigated the security-reliability tradeoff of an untrusted full-duplex relay network over the Rayleigh fading channels. Based on the availability of the CSI, we considered two types of relaying scheme, namely, the FGR scheme and the VGR scheme. The closed form expressions of the SOP, COP, and EST were, respectively, derived, which quantitatively reveal the relationship between the system performances and the number of antennas at the source, power allocation factor, and self-interference cancellation capability, as well as other various parameters. In addition, simple asymptotic results for the high SNR region, i.e., , were also provided to offer valuable insights into practical design. Analytical and numerical results demonstrated that the self-interference caused by full-duplex transmission is harmful for the reliability performance, while it is beneficial to the security performance. Moreover, the EST of the considered system increases as the self-interference level decreases, which implies that the self-interference deteriorates an overall performance in terms of the security-reliability tradeoff.
Data Availability
The simulation data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This work was supported in part by the Doctoral Research Start-up Funding of Nanyang Normal University under Grant 2022ZX017, in part by the Cultivating Fund Project for the National Natural Science Foundation of China of Nanyang Normal University under Grant 2022PY024, in part by the open research fund of Key Lab of Broadband Wireless Communication and Sensor Network Technology (Nanjing University of Posts and Telecommunications), Ministry of Education under Grant JZNY202107, in part by the Key Scientific Research Projects of Colleges and Universities in Henan Province of China under Grant 21A520033, and in part by the Key Scientific and Technological Research Projects in Henan Province under Grant 222102320369.