System Modeling and Deep Learning-Based Security Analysis of Uplink NOMA Relay Networks with IRS and Fountain Codes

1  Introduction

Due to its improved spectrum efficiency, Non-Orthogonal Multiple Access (NOMA) is crucial for sixth-generation (6G) and future wireless networks [1]. NOMA significantly enhances user connection and spectrum efficiency compared to conventional Orthogonal Multiple Access (OMA) methods [2]. In the meanwhile, Intelligent Reflecting Surfaces (IRSs) are becoming a cutting-edge and reasonably priced technology that maximizes energy efficiency (EE) and minimizes transmit power [3,4]. Therefore, combining IRS with NOMA is a viable approach to enhancing the overall performance of wireless communication systems. Furthermore, a downlink IRS-assisted NOMA network functioning on Rician fading channels is examined by the authors in [5–7]. They offer accurate and asymptotic formulas for the throughput, ergodic rate, and outage probability (OP). The findings show that the IRS-NOMA system performs better than the IRS-OMA system, especially as the Rician factor and reflecting component count rise.

Recently, digital contents like games, extended realities (XRs), and movies have been widely and easily distributed over wireless networks. As a result, concerns about unauthorized access, copyright infringement by third parties or eavesdroppers, and cyberattacks have grown significantly. This highlights the urgent need to address content protection and security issues. Consequently, safeguarding against copyright violations and illegal content distribution has become a critical consideration in the design of modern wireless networks [8,9]. Moreover, wireless communication networks are now susceptible to serious information leakage due to eavesdropping attacks, primarily because of the broadcast nature of radio propagation and the inherent instability of transmission links. A potential approach to safeguarding wireless communications is physical layer security (PLS), which seeks to create information-theoretic protection against hackers [1,10]. Historically, techniques such as beamforming, cooperative jamming, and cooperative relaying have been employed to provide secure communication, often in conjunction with the use of artificial noise [11–13]. But there is still a basic problem, especially at higher frequencies. Although these techniques have improved the performance of wireless communication systems’ PLS, they still require the placement of numerous relays, which can be costly and power-hungry. Moreover, to strengthen PLS in Internet of Things (IoT) networks against eavesdropping risks, the authors in [14,15] investigate an IRS-NOMA system. Their study demonstrates that IRS-NOMA outperforms conventional OMA by analyzing key metrics, such as OP, secrecy outage probability (SOP), and secrecy capacity, particularly when power allocation is optimized and IRS placement is properly considered.

In wireless communication, Fountain Codes (FCs) provide a viable option for dependable data transfer. In contrast to conventional codes, FCs dynamically adjust to shifting channel circumstances by producing an infinite number of encoded symbols from a finite set [8,16,17]. Due to their versatility, they work effectively in various settings with different noise and interference levels. Additionally, FCs achieve great performance with minimal computing cost and are robust against burst errors. Their adaptability encompasses a wide range of applications, including distributed storage systems, multimedia streaming, and satellite communication. Because of this, FCs are now widely used as a flexible and effective error-correcting method in wireless communication [18]. Furthermore, FCs are members of a family of linear block codes that are well-known for producing an infinite number of encoded packets [19,20]. Due to their outstanding coding performance, systematic forward error correction (FEC) codes like Raptor-Q codes have gained widespread acceptance as Application-Layer Forward Error Correction (AL-FEC) codes in the 3GPP Multimedia Broadcast and Multicast Services (MBMS) standard [21].

Recently, deep learning has emerged as a powerful data-driven approach for addressing various challenging issues, including wireless communication applications, pattern recognition, and image processing [22,23]. The authors created a deep neural network (DNN) model in [24] to forecast the likelihood of coverage in random wireless networks. Interestingly, the DNN model performs better than conventional mathematical techniques, which are typically limited to network settings that are overly simplistic. Additionally, in [25], the authors outperformed simulation and analytical findings in unmanned aerial vehicle networks by using a DNN model to forecast the SOP, demonstrating the fastest running time for SOP prediction. Furthermore, a NOMA-based downlink system with IRS and a PLS environment is examined by the authors in [8]. While addressing the possibility of a malevolent eavesdropper, their work also incorporates the usage of FCs. This paper’s primary contribution is the derivation of exact closed-form formulas for the suggested system’s OP, EE, intercept probability (IP), and average secrecy rate (ASR). In addition, they create a DNN model to assess the average number of time slots (ATS), OP, IP, and ASR. In [26], this study investigates the security of a downlink IRS-assisted NOMA relay network enhanced with FC and DNN. Closed-form expressions for OP and IP are derived, and Monte Carlo simulations are used for validation. The results confirm the performance gains of IRS-NOMA, especially under varying numbers of reflecting elements, user threshold rates, and encoded packet limits.

1.1 Motivation and Major Contributions

In documents [27] and [28], the authors examined an uplink secure relay network combined with the NOMA technique, considering the case where an eavesdropper (E) intercepts signals from both the relay (R) and direct links between users and the base station (BS). However, these studies did not incorporate the IRS or the FC technique. In the document [1], the authors introduced several improvements over the previous works by integrating IRS into the uplink secure system model. Nevertheless, none of these studies applied the FC technique to enhance system security. Furthermore, our research incorporates the DNN technique to predict system performance and security. Additionally, we analyze system performance through OP and evaluate security performance using the IP.

The main contributions of this study are succinctly summarized as follows:

•   This study investigates an IRS-NOMA network uplink using a new secure communication protocol that makes good use of FC properties. This protocol, in contrast to conventional techniques, guarantees that packets are encrypted securely, making it far more difficult for the E to collect and decode the original data. Additionally, adding an IRS to the system enhances the transmission quality of the uplink signal.

•   In this study, the PLS of the suggested system using FCs is examined. Analytical techniques are used to calculate key performance measures, such as the OP, throughput, diversity evaluation for two users. The study also evaluates the IP necessary for safe communication between users and between authorized users and possible adversaries. In addition to highlighting the benefits of FCs in enhancing communication security, the study provides a comprehensive examination of security and reliability concerns associated with their use.

•   Unlike previous research, our proposed IRS-based uplink NOMA system utilizes FC and DNN techniques to evaluate OP and IP. Moreover, when paired with FC characteristics, the DNN model is a valuable tool for assessing system performance, particularly when closed-form analytical expressions are hard to come by. This method yields precise approximations of system performance measures by streamlining the assessment of highly nonlinear functions. Consequently, the application of DNN and FCs is closely related to performance evaluation. In this study, we have created closed-form formulae for both OP and IP, but we also use the DNN model to forecast more accurate results with real-world applications. Additionally, simulations were conducted to gather useful technical data and validate the theoretical approach. By examining various outcomes, the advantages of integrating IRS, NOMA, and FCs to enhance system performance, particularly in scenarios involving eavesdropping, were further investigated.

1.2 Organization

The structure of this paper is set up as follows: The system model is presented in Section 2, and the OP of the system is discussed in Section 3. Moreover, the IP of the system is discussed in Section 4. The DNN application is examined in Section 5. Lastly, Sections 6 and 7 show the simulation findings and the research conclusion.

2  System Model

Fig. 1 illustrates the uplink NOMA relay network security system supported by an IRS and employing the FC technique. The system model consists of a BS, a R, an IRS with H elements, two users (D1 and D2), and an E. We assume that user D2 is located farther from both the BS and R compared to user D1 and thus requires IRS assistance.

Figure 1: IRS-assisted secure network model for a NOMA relay system

The signal transmission process occurs in two phases:

i) First phase: The signal from user D1 is transmitted directly to R, while D2, being farther away, relies on IRS assistance before forwarding its signal to R. At the same time, E attempts to eavesdrop on the signals from both D1 and D2.

ii) Second phase: R decodes the signals from both users and then transmits them to the BS. Meanwhile, E also eavesdrops on the signals from D1 and D2 as they are relayed by R.

It is important to note that the IRS only reflects signals during transmission and does not participate in signal decoding. At R, the successive interference cancellation (SIC) decoding process is performed to separate the signals from D1 and D2 before forwarding them to the BS, as R receives signals from both users simultaneously. Furthermore, we assume that there is no direct signal transmission from Di, (i=1,2) to BS or from BS to E.

To create the encoded packets, the user divides its original data into C packets and makes sure they are correctly encoded. Di sends each of these encoded packets to the BS separately at each time slot, and E tries to intercept them. Suppose both the BS and E properly receive at least G-encoded packets, where A=C(1+ζ) and ζ is the decoding overhead, which varies depending on the particular code design. In that case, it is assumed that both can successfully retrieve the original data [8,29]. Additionally, the BS signals Di to stop the transmission by sending an acknowledgment (ACK) message when a sufficient number of encoded packets have been gathered. The E cannot obtain Di’s data if it is unable to record the necessary quantity of encrypted packets. On the other hand, it can reassemble the original data if it can intercept enough packets.

2.1 Evaluation of Signal in Phase 1

LetB denote the length of each FC packet. Assume that D1 sends the packets u1[b] (indicating the b-th packet of D1) and D2 sends the packets u2[b] (indicating the b-th packet of D2) to the BS if the Di delivers the signal qi[b],(b=1,…,B) to the BS [8]. The signal that is received at R from Di is supplied by [27,30]

yDR=gDRdDR−ρ2μ1PD1q1[b]+gDIΦgIRdDI−ρ2dIR−ρ2μ2PD2q2[b]+ϑR[b],(1)

where the link between D1 and R, denoted by gDR∼CN(0,σgDR) in the Rayleigh fading model, fades on a small scale. In particular, the complex channel coefficients for D2-IRS and IRS-R are denoted by gDI=[gDI,1,…,gDI,h,…,gDI,H] and gIR=[gIR,1,…,gIR,h,…,gIR,H]T, respectively. With unique fading parameters (mgDI and mgIR, respectively), each element in gDI and gIR adheres to the Nakagami-m fading model. The distance between D1-R link and D2-IRS link and IRS-R link are represented by dDR, dDI, dIR, respectively. The path loss exponents are represented by ρ. Φ=diag[α1ejχ1,…,αhejχh,…,αHejχH],(j=−1) stands for the reflection coefficient matrix related to the D2-IRS-R, whereas αH∈[0,1] represents the amplitude-reflection coefficient, and we assume that α1=⋯=αh=⋯=αH [30]. The IRS can control the h-th element’s adjustable phase shift variables using χh∈[0,2π), which are supposed to be represented by (h=1,…,H). μ1 and μ2 stand for the power allocation coefficients for D1 and D2, respectively. Notably, we assume a fixed power allocation share between the two users and set μ1<μ2 for user fairness, and μ1+μ2=1 [4,30]. The transmitted signals of the b-th symbol of u1[b] and u2[b] are denoted by q1[b] and q2[b], respectively. The transmission power of the Di is denoted by PDi. The additive white Gaussian noise (AWGN) at R is indicated by the symbol ϑR[b]∼CN(0,ω02).

The signal for D1 acts as interference at R, and the signal for D2 is deciphered first. Consequently, the following formula provides the signal to interference plus noise ratio (SINR) for decoding the signal for D2 at R:

γDR(q2)=|gDIΦgIR|2dDI−ρdIR−ρμ2βD2|gDR|2dDR−ρμ1βD1+1,(2)

where βD1=PD1ω02,βD2=PD2ω02.

We then use SIC to decode the D1 signal. The signal-to-noise ratio (SNR) for decoding the signal for D1 at R is now shown as follows:

γDR(q1)=|gDR|dDR−ρ2μ1βD1.(3)

At the same time, the E also intercepts the signals from D1 and D2. At this point, the signal intercepted by E from both users D1 and D2 is given by the following expression [27,31]:

yE1=gDE1dDE1−ρ2μ1PD1q1[b]+gDE2dDE2−ρ2μ2PD2q2[b]+ϑE[b],(4)

where the between D1-E link denoted by gDE1∼CN(0,σgDE1) in the Rayleigh fading model, fades on a small scale; the between D2-E link denoted by gDE2∼CN(0,σgDE2) in the Rayleigh fading model, fades on a small scale. The distance between D1-E link and D2-E link are represented by dDE1, dDE2, respectively. The AWGN at E is indicated by the symbol ϑE[b]∼CN(0,ωE2).

Based on the operating principle of SIC, the E device will attempt to detect and decode the signal for D2 first, as D2 is assumed to have better channel conditions. The quality of the signal for D2 at the E is evaluated through SINR. The SINR at E for detecting the signal for D2 is given by the following expression:

γE1(q2)=|gDE2|2dDE2−ρμ2βE2|gDE1|2dDE1−ρμ1βE1+1,(5)

where βE1=PD1ωE2,βE2=PD2ωE2.

Following the detection of the D2 signal at E, we use SIC to find the D1 signal. The following equation provides the SNR for detecting the signal for D1 at E:

γE1(q1)=|gDE1|dDE1−ρ2μ1βE1.(6)

2.2 Evaluation of Signal in Phase 2

Following the successful decoding of the signals for Di at R, R proceeds to send the signals from Di to the BS. The following equation represents the signal received at the BS at this time [27]:

yRB=gRBdRB−ρ2(μ1PRBq1[b]+μ2PRBq2[b])+ϑBS[b],(7)

where the between R-BS link denoted by gRB∼CN(0,σgRB) in the Rayleigh fading model, fades on a small scale. The distance between R-BS link is represented by dRB. The transmission power of the R is denoted by PRB. The AWGN at BS is indicated by the symbol ϑBS[b]∼CN(0,ω02).

As in (2), the D2 signal is decoded first, and the D1 signal is regarded as interference. At this stage, the following formula provides the SINR for decoding the signal for D2 at the BS:

γRB(q2)=|gRB|2dRB−ρμ2βRB|gRB|2dRB−ρμ1βRB+1,(8)

where βRB=PRBω02.

At the BS, use SIC to decode the D1 signal. At this stage, the following formula provides the SNR at the BS for decoding the signal for D1:

γRB(q1)=|gRB|dRB−ρ2μ1βRB.(9)

The eavesdropping device E concurrently intercepts the signal from R as it is sent to the BS. The following is the expression for the signal that was received at E from R [32,33]:

yRE=gREdRE−ρ2(μ1PRBq1[b]+μ2PRBq2[b])+ϑE[b],(10)

where the between R-E link denoted by gRE∼CN(0,σgRE) in the Rayleigh fading model, fades on a small scale. The distance between R-E link is represented by dRE.

The E device will attempt to detect and decode the signal for D2 first. The SINR at E from R for detecting the signal for D2 is given by the following expression:

γRE(q2)=|gRE|2dRE−ρμ2βRE|gRE|2dRE−ρμ1βRE+1,(11)

where βRE=PRBωE2.

Next, we use SIC to find the D1 signal. The following equation provides the SNR for detecting the signal for D1 at E:

γRE(q1)=|gRE|dRE−ρ2μ1βRE.(12)

3  OP-Based System Reliability

The OP in communication systems is the probability that an outage will occur when a user’s information rate drops below a certain goal rate. A commonly used statistic for assessing system performance in situations involving fixed-rate transmission is OP. In this part, we examine the uplink IRS-NOMA networks’ outage performance.

3.1 D1’s OP Assessment

Since the SIC operation is assumed to be flawless in this study, the OP for D1 of one FC packet may be found as follows: i) D1 fails to decode the signal packet q1[b] during the first transmission phase. ii) D1 cannot interpret the signal packet q1[b] once more during the second transmission phase. Consequently, the OP of one FC packet at D1 may be written as follows [27,30]:

ΩD1=1−Pr(γDR(q1)>γ1(th),γRB(q1)>γ1(th)),(13)

where γi(th)=22RDi−1, and the target date rate for user Di is denoted by RDi.

As stated in [4,30], the probability density function (PDF) and cumulative distribution function (CDF) for J∈{gRB,gDR,gRE,gDE1,gDE2} may be written as

f|J|2(x)=1σJe−xσJ,(14)

F|J|2(x)=1−e−xσJ.(15)

Based on Eq. (13), ΩD1 may be written as follows:

ΩD1=1−Pr(γDR(q1)>γ1(th))Pr(γRB(q1)>γ1(th))=1−Pr(|gDR|2>γ1(th)dDR−ρμ1βD1)Pr(|gRB|2>γ1(th)dRB−ρμ1βRB).(16)

Using the CDF function of (15), ΩD1 is further calculated as follows:

ΩD1=1−[1−F|gDR|2(γ1(th)dDR−ρμ1βD1)][1−F|gRB|2(γ1(th)dRB−ρμ1βRB)]=1−exp⁡(−γ1(th)dDR−ρμ1βD1σgDR)exp⁡(−γ1(th)dRB−ρμ1βRBσgRB).(17)

The OP of D1 in the case of one FC packet is represented by (17). The following is how we analyze the OP of D1 for the case of A FC packets using these equations: The number of time slots given or the number of FC packet transfers by the D1 is restricted because of the latency limitation. We represent ZDi as the number of transmission times utilized by the Di, where Zth≥ZDi≥A, and Zth as the maximum number of transmission times for the Di. This implies that the Di will cease transmission after Zth transmissions, and if BS does not receive at least A-encoded packets, it will either experience an outage or be unable to reconstruct the original data. Additionally, we represent ZBS and ZE as the number of FC packets that nodes BS and E have successfully received, respectively. Di is in the outage if ZBS is smaller than A. Consequently, the OP at D1 for A FC packets may be written as [8,16]

OPD1,A=Pr(ZBS<A|ZD1=Zth).(18)

Next, using [16, Eq. (21)], we may get the precise formula for D1’s OP for A FC packets as follows:

OPD1,A=∑ZBSA−1(ZthZBS)(1−ΩD1)ZBS(ΩD1)Zth−ZBS.(19)

3.2 D2’s OP Assessment

One FC packet’s OP for D2 is determined as follows: i) It is impossible to interpret the q1[b] signal packet during the initial transmission phase. ii) The q1[b] signal packet is also undecodable during the second transmission phase. At this stage, one FC packet’s OP for D2 may be written as follows [8,30]:

ΩD2=1−Pr(γDR(q2)>γ2(th),γRB(q2)>γ2(th))=1−Pr(γDR(q2)>γ2(th))⏟Θ1aPr(γRB(q2)>γ2(th))⏟Θ1b.(20)

Based on [3,30], our goal for the D2-IRS-R connection is to optimize the channel quality for D2 by modifying the IRS’s characteristics1. With gDI,h and gIR,h being the h-th elements of gDI and gIR, respectively, this is done in order to maximize |gDIΦgIR|=|∑h=1HαhgDI,hgIR,hejχh|. To do this, each element’s phase-shift variable χh may be adjusted such that all gDI,hgIR,hejχh phases are set to the same. Since χ~ is an arbitrary constant that ranges in [0,2π), there is more than one solution for {χh}, and χh=χ~−arg⁡(gDI,hgIR,h) provides the generalized solution. Following the adoption of the optimum {χh}, we have

|gDIΦgIR|2=α2(∑h=1H|gDI,h||gIR,h|)2,(21)

are assumed without sacrificing generality.

According to [3,30], the PDF and CDF of Υ=(∑h=1H|gDI,h||gIR,h|)2H(1−κ) are written as follows:

fΥ(w)=e−w+σDIR2∑n=0∞σDIRnwn−12n!22n+12Γ(n+12),(22)

and

FΥ(w)=e−σDIR2∑n=0∞σDIRnγ(i+12,w2)n!2nΓ(n+12),(23)

where σDIR=Hκ1−κ, κ=1mgDImgIR(Γ(mgDI+12)Γ(mgDI))2(Γ(mgIR+12)Γ(mgIR))2. The distribution of Υ tends to be noncentral chi-squared as Υ∼Υ2(σDIR) when the number of reflecting components H is large. Gamma and lower incomplete gamma functions are denoted by Γ(.) and γ(.,.), respectively.

From (20), Θ1a can be rewritten as

Θ1a=Pr(|gDIΦgIR|2>γ2(th)(|gDR|2dDR−ρμ1βD1+1)dDI−ρdIR−ρμ2βD2)=Pr(α(∑h=1H|gDI,h||gIR,h|)2H(1−κ)>γ2(th)(|gDR|2dDR−ρμ1βD1+1)dDI−ρdIR−ρμ2βD2)=Pr((∑h=1H|gDI,h||gIR,h|)2>γ2(th)H(1−κ)(|gDR|2dDR−ρμ1βD1+1)αdDI−ρdIR−ρμ2βD2).(24)

Given (14) and (23), Θ1a may be expressed as follows:

Θ1a=∫0∞[1−F(∑h=1H|gDI,h||gIR,h|)2(γ2(th)H(1−κ)(xdDR−ρμ1βD1+1)αdDI−ρdIR−ρμ2βD2)]f|gDR|2(x)dx=1−exp⁡(−σDIR2)σgDR∑n=0∞σDIRnn!2nΓ(n+12)∫0∞γ(i+12,γ2(th)H(1−κ)(xdDR−ρμ1βD1+1)2αdDI−ρdIR−ρμ2βD2)×exp⁡(−xσgDR)dx.(25)

It is rather challenging to derive the above equation in a closed form for the end outcome. As a result, we estimate the integral between 0 and a big number L [4,34]. Θ1a is now rewritten as:

Θ1a=1−exp⁡(−σDIR2)σgDR∑n=0∞σDIRnn!2nΓ(n+12)∫0Lγ(i+12,γ2(th)H(1−κ)(xdDR−ρμ1βD1+1)2αdDI−ρdIR−ρμ2βD2)×exp⁡(−xσgDR)dx.(26)

Next, we let u=2x−LL⇒x=uL+L2, and at this point, Θ1a is rewritten as follows:

Θ1a=1−Lexp⁡(−σDIR2)2σgDR∑n=0∞σDIRnn!2nΓ(n+12)×∫−11γ(i+12,γ2(th)H(1−κ)(uL+L2dDR−ρμ1βD1+1)2αdDI−ρdIR−ρμ2βD2)exp⁡(−uL+L2σgDR)du.(27)

Gaussian-Chebyshev quadrature is applied [35], and we obtain

Θ1a=1−πLexp⁡(−σDIR2)2VσgDR∑n=0∞∑v=1V1−uv2σDIRnn!2nΓ(n+12)×γ(i+12,γ2(th)H(1−κ)(ψvdDR−ρμ1βD1+1)2αdDI−ρdIR−ρμ2βD2)exp⁡(−ψvσgDR),(28)

where ψv=uvL+L2, uv=cos⁡[(2v−1)π2V], and the accuracy and complexity trade-off parameter is called V.

Using (15) and (20), Θ1b may be expressed as:

Θ1b=Pr(|gRB|2>γ2(th)dRB−ρβRB(μ2–γ2(th)μ1))=1−F|gRB|2(γ2(th)dRB−ρβRB(μ2–γ2(th)μ1))=exp⁡(−γ2(th)dRB−ρβRBσgRB(μ2–γ2(th)μ1)).(29)

By combining (28) and (29) into (20), the closed-form expression of ΩD2 is rewritten as follows:

ΩD2=1−[1−πLexp⁡(−σDIR2)2VσgDR∑n=0∞∑v=1V1−uv2σDIRnn!2nΓ(n+12)×γ(i+12,γ2(th)H(1−κ)(ψvdDR−ρμ1βD1+1)2αdDI−ρdIR−ρμ2βD2)exp⁡(−ψvσgDR)]×exp⁡(−γ2(th)dRB−ρβRBσgRB(μ2–γ2(th)μ1)).(30)

As with (18), the OP at D2 for A FC packets may be expressed exactly as follows:

OPD2,A=∑ZBSA−1(ZthZBS)(1−ΩD2)ZBS(ΩD2)Zth−ZBS.(31)

3.3 Throughput Assessment

One crucial performance indicator for wireless communication systems is system throughput. System throughput research is very important for real-world applications. Information is sent continuously from the BS to Di in the delay-limited transmission mode, which causes outages because of the impact of wireless fading channels. Thus, in the delay-limited transmission mode, the OP determines the system throughput. The expression for Di’s delay-limited throughput is [5]

ℑDi,A=(1−OPDi,A)RDi.(32)

Moreover, by applying (32), the system throughput can be expressed as follows:

ℑSYM,A=ℑD1,A+ℑD2,A.(33)

3.4 Diversity Evaluation

The slope of the OP curve in the high SNR range is the definition of diversity order in communication systems. It is frequently used to evaluate the outage performance of wireless communication networks and demonstrates how quickly the OP drops as the broadcast SNR increases. The following is a mathematical expression for the diversity order [4,5]:

ddi=−limβ→∞⁡log⁡(OPDi,A∞(β))log⁡(β).(34)

where we assume that β = βDi = βRB. In this case, OPDi,A∞ stands for the OP value of the user Di in question. It is important to remember that the system has zero diversity order for every user while limβ→∞⁡exp⁡(−x/β)≈1−(x/β) holds [4,36]. The impact of additional limiting parameters outside the SNR results in the saturation of outage performance.

4  IP-Based Security Evaluation

An important PLS metric is the IP, which measures the probability that a hacker may successfully collect and decode private data sent over a wireless channel. It is the likelihood that a security breach will occur if the E’s data rate is higher than the difference between the source transmission rate and the channel capacity of the authorized user.

4.1 D1’s IP Assessment

Eqs. (6) and (12) may be used to determine user D1’s IP for one FC packet, as shown [8,37]

ΔD1=Pr(γE1(q1)>γ1(th),γRE(q1)>γ1(th))=Pr(|gDE1|2dDE1−ρμ1βE1>γ1(th))Pr(|gRE|2dRE−ρμ1βRE>γ1(th))=Pr(|gDE1|2>γ1(th)dDE1−ρμ1βE1)Pr(|gRE|2>γ1(th)dRE−ρμ1βRE).(35)

Next, based on the CDF function mentioned in (15), ΔD1 is calculated as follows:

ΔD1=[1−F|gDE1|2(γ1(th)dDE1−ρμ1βE1)][1−F|gRE|2(γ1(th)dRE−ρμ1βRE)]=exp⁡(−γ1(th)dDE1−ρμ1βE1σgDE1)exp⁡(−γ1(th)dRE−ρμ1βREσgRE)=exp⁡(−γ1(th)dDE1−ρμ1βE1σgDE1−γ1(th)dRE−ρμ1βREσgRE).(36)

From (36), in the case of one FC packet, represent the IP of D1. We continue with the IP analysis for A FC packets based on these two equations. It should be noted that the data meant for the BS will be intercepted if E gathers enough A-encoded packets before D1’s transmission concludes. Consequently, the IP of D1 for A FC packets may be used as follows [8,38]:

IPD1,A=Pr(ZE=A|ZD1≤Zth).(37)

Continuing, the IP of D1 for A FC packets may be found using the binomial distribution definition [39, Eq. (9)]:

IPD1,A=∑y=AZth(y−1y−A)(ΔD1)A(1−ΔD1)y−A.(38)

4.2 D2’s IP Assessment

Just as in (35), and based on (5) and (11), user D2’s IP is determined as shown in [8,37]:

ΔD2=Pr(γE1(q2)>γ2th,γRE(q2)>γ2th)=Pr(γE1(q2)>γ2th)⏟Θ2aPr(γRE(q2)>γ2th)⏟Θ2b.(39)

Using (39), Θ2a is computed as follows:

Θ2a=Pr(γE1(q2)>γ2th)=Pr(|gDE2|2>γ2th(|gDE1|2dDE1−ρμ1βE1+1)dDE2−ρμ2βE2)=∫0∞(1−F|gDE2|2(γ2th(xdDE1−ρμ1βE1+1)dDE2−ρμ2βE2))f|gDE1|2(x)dx=1σgDE1exp⁡(−γ2thdDE2−ρμ2βE2σgDE2)∫0∞exp⁡(−(γ2thdDE1−ρμ1βE1dDE2−ρμ2βE2σgDE2+1σgDE1)x)dx=dDE2−ρμ2βE2σgDE2γ2thdDE1−ρμ1βE1σgDE1+dDE2−ρμ2βE2σgDE2exp⁡(−γ2thdDE2−ρμ2βE2σgDE2).(40)