Rate Splitting in the Presence of Untrusted Users: Outage and Secrecy Outage Performances

In this contribution, a thorough investigation of the performance of rate splitting is conducted in terms of outage and secrecy outage for the simultaneous service to a near user and far user, where the latter attempts to overhear the message of the former. The source transmits a linear combination of the users’ common stream and private streams. Once the common stream is retrieved, two decoding strategies can be adopted by each user. In the first strategy, the nodes (near or far) treat the far user’s private stream as noise to retrieve the private stream of the near user, then the far user decodes its own stream. In the second strategy, the nodes decode the far user’s private stream by treating the one of the near user as noise, then the near user retrieves its private stream while the far user decodes the stream of the near user in its attempt to overhear it. Considering the four decoding combinations, we obtain exact closed-form expressions for the outage probability, and provide tight approximations for the secrecy outage probability. Comparative results are also provided. In particular, it is shown that to achieve better outage probability, with no concern about secrecy, once the decoding of the common stream is completed, each user should first retrieve the private stream with lower target data rate by treating the other private stream as noise. To improve the secrecy outage probability, once the common stream is decoded, the near user must first decode the far user’s private stream, and the far user should first retrieve the private stream with lower target data rate.


I. INTRODUCTION A. CONTEXT
I NCREASING the capacity of wireless communication systems in terms of the number of users that can be served within a limited spectrum band has always been a fundamental design goal. A promising approach to serve multiple users simultaneously in a shared bandwidth is rate splitting (RS), which is based on the concept of Han-Kobayashi (H-K) signaling [1]. In RS systems, 1 a user's message is split at the source into two parts, known as 1. In this paper, an RS system refers to a system implementing the RS mechanism.
common and private messages. The common parts of the message are combined and encoded to form a single common stream to be decoded by all users. The private messages are separately encoded into private streams. Then, the source broadcasts a linear combination of the users' private streams and the common stream, with a predefined power allocation. After decoding the common stream, each user can obtain its private stream by treating those of the other users as noise [2].
For a system comprised of two-user single-input singleoutput interference channels, implementing H-K signalling is known to yield the best achievable rate region [1]. Also, [3] and [4] demonstrate that the achievable rate region of networks with multiple-input multiple-output interference channels can be improved by utilizing H-K signaling. In particular, applying RS mechanism in multi-user networks can enhance the quality of service [2], [5]. For instance, RS multiple access (RSMA) can achieve equal or larger rate region compared to NOMA (non-orthogonal multiple access) and conventional access schemes such as spacedivision multiple access, and also reduce the computational complexity as compared to NOMA for instance [6]. NOMA and OMA in broadcast channels can be seen as special cases of RS [6], [7]. However, RSMA is superior over the aforementioned conventional multiple access schemes, as has been demonstrated from various aspects, e.g., in terms of spectral efficiency [8]- [11], max-min achievable rates [6], [12], [13], and ergodic sum-rate [14]. In [15], the performance of twouser uplink RSMA was investigated in terms of outage probability (OP) and throughput. Also, the OP of cooperative H-K signalling has been studied in [16], where the near user relays the common message or the far user to improve the corresponding received signal-to-noise ratio (SNR). This is a particular case of RSMA to serve near and far users, which we seek to improve in this paper by considering all possible strategies to decode the common and private messages of the RSMA users. Several studies have also looked at the system performance when RS, or H-K signaling in a more general sense, operates in the presence of eavesdroppers. Indeed, since the signals are transmitted on free medium, eavesdroppers, whether external or internal to the communication system, have the opportunity to overhear the messages of the system's users, [17], [18]. Generally speaking, to ensure secure communication, alongside cryptography protocols, solutions based on physical-layer security (PLS) can leverage the randomness of the fading channels to guarantee message secrecy [19], [20]. A suitable metric to evaluate the secrecy performance of PLS-based schemes is the secrecy outage probability (SOP) [19]. In this context, several works have investigated the SOP of RSMA systems. The secrecy performance of H-K based communication assisted with unmanned aerial vehicle in the presence of external eavesdropper was investigated in [21]. Therein, it was shown that the H-K access outperforms OMA and NOMA in terms of secrecy throughput. In [19], considering a two-user RS system with external eavesdropper, robust beamforming was devised to maximize the secrecy achievable rate, where the common stream was modeled as artificial noise for the eavesdropper. Besides, by enabling cooperation, improvement in the secrecy sum-rate of RSMA in the presence of external eavesdropping was demonstrated in [22], [23]. Cooperative RS was also considered at an aerial base station to protect the secrecy of a two-user system against an external eavesdropper [24]. Deploying RSMA in a system with two legitimate users and a potential eavesdropper was also studied in [25], which addressed the maximization of the minimum achievable secrecy rate.

B. CONTRIBUTION
In this paper, the focus is on the investigation of the robustness of RS technique in terms of SOP and OP. Specifically, the RS system under study is comprised of three nodes: the source, a near user, and a far user; the users being assumed to be scheduled for service. Each user has to first decode the common stream. In addition to retrieving its own private stream, the far user attempts to wiretap the message of the near user. After decoding the common stream, two decoding strategies can be implemented at each node, termed Strategy-1 and Strategy-2. When adopting Strategy-1, the users (near and/or far) first decode the near user's private stream while treating the stream of the far user as noise. In this case, the far user also retrieves its own private stream after decoding the one of the near user. When adopting Strategy-2, the users (near and/or far) retrieve the private stream of the far user while considering the stream of the near user as noise. Here, the near user decodes its own private stream after having access to the far user's private stream. The far user is capable of overhearing the near user's private stream after obtaining its own private stream via this decoding scheme. As each user in the system can individually apply either Strategy-1 or Strategy-2, a total of four decoding strategies are possible. Specifically, Strategy-ij, i, j ∈ {1, 2}, describes the RS system in which the near user and the far user adopt Strategy-i and Strategy-j, respectively. Aiming at the evaluation of the OP and SOP of the RS system in the different decoding scenarios, the focus of this work and its ensuing contributions can be summarized as follows.
We obtain the exact closed-form expressions for the OP in the four decoding strategies for the entire system, and for the near user separately. Monte-Carlo simulations are also presented and confirm the analysis. To compare the decoding strategies, the impacts of the power allocation, power budget, target data rates, and the far user's distance, on the outage performance without security constraints, are investigated.
Further, using the Gauss-Chebyshev quadrature method, we find tractable approximations for the SOP corresponding to the use of each one of the four decoding strategies, and characterize the exact SOP in the system operation with Strategy-22. We also investigate the effects of the power allocation, the source power budget, the target data rates, and the far user's distance, on the SOP of the four RS scenarios. The agreement between the Monte-Carlo simulations and analytical results confirms the accuracy of the analysis.
We also compare the system's OP and SOP in different operation scenarios, and provide guidelines on the decoding mechanisms that lead to enhanced performance. Important findings are revealed. In particular, it is shown that when the allocated powers and the streams' targets rates are equal, then all decoding schemes will yield the same outage performance. Also, when the powers allocated to the private streams of the users are the same, the near user must follow Strategy-2 to achieve the least SOP.
In detailing the above-highlighted contributions and findings, the following content of the paper is organized as follows. Section II elaborates on the RS system and the decoding strategies. The OP and SOP are studied in Sections III and IV, respectively. Numerical results are discussed in Section V, and Section VI concludes the paper.
Notation: For event A, P(A) denotes the probability of occurrence of A, and A is its complementary. F X (x) and f X (x) respectively denote the cumulative distribution function (CDF) and the probability density function (PDF) of random variable X. Operator [y] + returns max(0, y), U(.) is the unit step function, and 1 B is the indicative function corresponding to event B, where 1 B = 1 if B occurs and 1 B = 0 otherwise.

II. THE RATE SPLITTING SYSTEM UNDER EAVESDROPPING
The communication takes place from the source to a near user, U 1 , and a far user, U 2 , using the RS mechanism. Denote the messages intended to U 1 and U 2 by W 1 and W 2 , respectively. As per the RS mechanism, the messages W i , i ∈ {1, 2}, are split into a common part W i,c and private parts W i,p . The common parts are combined together and encoded into a common stream S c . The private parts, W 1,p and W 2,p , are encoded into the private streams S 1 and S 2 , respectively. Here, it is assumed that S c , S 1 and S 2 are i.i.d. zero-mean circularly-symmetric complex Gaussian random variables of unit variance. The source transmits a linear combination of S c , S 1 and S 2 , with a power allocation such that the private stream S 1 is to be kept confidential against the eavesdropping by U 2 . Let the power allocation coefficients pertaining to S c , S 1 and S 2 , be denoted by a c , a 1 and a 2 , respectively, with a 1 + a 2 + a c = 1 and 0 < a 1 , a 2 , a c < 1. Using a power budget P, the source broadcasts the superimposed streams, as Denote the Rayleigh fading channel gains between the source transmitter and the users by h 1 and h 2 , with distribution parameters λ 1 and λ 2 , respectively. The source and node U t , t ∈ {1, 2}, are separated by distance d t , and the path-loss coefficient is α. The signal at node U t , t ∈ {1, 2}, can then be expressed as where n t is the additive white Gaussian noise, with zero mean and variance σ 2 .
To decode its private stream, a user first decodes S c while treating S 1 and S 2 as noise.
For any node U t , the achievable rate to decode the common stream S c is given by, 2 where γ c→t = a c g t (a 1 +a 2 )g t +ρ and ρ = σ 2 P . The four decoding strategies are depicted in Fig. 1. For i, j ∈ {1, 2}, Strategy-ij indicates that U 1 and U 2 follow Strategy-i and Strategy-j, respectively. Strategy-12 corresponds to conventional RS system. As aforementioned, we 2. In this paper, the unit of rates is bit per channel use (BPCU). consider three other decoding mechanisms, and investigate their performance in comparison to the conventional scheme.

R
(1) 2) STRATEGY-12 In this case, U 1 and U 2 use Strategy-1 and Strategy-2, respectively. After decoding S c , U 1 decodes S 1 by assuming S 2 as noise. Therefore, the achievable rate of U 1 for accessing S 1 is as per (4). At the U 2 's side, after having access to S c , U 2 retrieves S 2 with rate where γ (2) 2→2 = a 2 g 2 a 1 g 2 +ρ and S 1 is considered as noise. Since U 2 also attempts to overhear S 1 , the achievable rate of U 2 for decoding S 1 is given by (7), where γ (2) 1→2 = a 1 g 2 ρ .

4) STRATEGY-22
In this approach, U 1 and U 2 utilize Strategy-2 to decode their streams. Therefore, after decoding S c , both users retrieve S 2 by treating S 1 as noise, and lastly obtain S 1 . The achievable rates for U 1 to access S 2 and S 1 are given by (8) and (9), respectively. Also, U 2 achieves the rates (6) and (7) to decode S 2 and overhear S 1 , respectively.

III. OUTAGE PROBABILITY
First, we investigate the OP of each strategy without considering any security constraint. The target date rates associated with S c , S 1 and S 2 are denoted by R c , R 1 and R 2 , respectively. For simplicity, define , define E c→t as the event that node U t is not able to decode S c correctly, i.e., and, for i, j ∈ {1, 2}, define E (j) i→t as the event that U t fails to retrieve S i via Strategy-j, i.e.,

A. PERFORMANCE WITH STRATEGY-11
Here, both users adopt Strategy-1. This strategy faces outage in the following cases: U t , t ∈ {1, 2}, fail to decode S c , i.e., 1→2 ; U 2 is not able to decode S 2 , i.e., E (1) 2→2 . Hence, the OP in Strategy-11 is given by . To avoid the occurrence of outage, the power allocation must satisfy the conditions a c > C c 1+C c and a 1 a 2 > C 1 , and OP (11) is obtained as where superscript (11) refers to Strategy-11. Proof: The proof is provided in Appendix A.

C. PERFORMANCE WITH STRATEGY-21
Outage here occurs in five events: 2→2 . Hence, the outage probability in this case is given by Theorem 3: Let β 2 = max(μ 2 , ρC 1 a 1 ). In order to prevent outage, the power allocation coefficients must satisfy the conditions a c > C c 1+C c and C 1 < a 1 a 2 < 1 C 2 , which yields where the superscript (21) indicates Strategy-21.
Proof: The proof is provided in Appendix B.

D. PERFORMANCE WITH STRATEGY-22
In this system operation, outage is comprised of five events: . As a result, OP (22) is given by Theorem 4: For the system operation with Strategy-22, the power allocation coefficients must satisfy a c > C c 1+C c and a 1 a 2 < 1 C 2 . With this power allocation, OP (22) is obtained as where the superscript (22) refers to Strategy-22.
Proof: The proof is provided in Appendix C.

E. OUTAGE PROBABILITY OF U 1
The OP in Strategy-11 and Strategy-12 can be individually evaluated at node U 1 by considering the cases when U t , t ∈ {1, 2} fail to decode S c , and U 1 cannot retrieve S 1 . Thus, it is obtained as where j ∈ {1, 2} denotes the decoding strategy at U 2 . Similarly, the OP of U 1 in Strategy-21 and Strategy-22 can be evaluated by considering the cases when U t , t ∈ {1, 2}, fail to decode S c , and U 1 fails to decode S 1 and S 2 . That is, where j ∈ {1, 2} denotes the decoding strategy at node U 2 . Calculation of the OP at U 2 can straightforwardly be obtained using the same approach and, hence, not shown here.

F. COMPARISON OF THE DECODING STRATEGIES
The following Lemma compares the OP of the RS systems when the allocated powers to S 1 and S 2 are equal, and the target date rate of S 2 is higher than that of S 1 .
That is, when a 1 = a 2 and R 2 > R 1 , the users must follow Strategy-11 to yield the least OP. Similar to Lemma 1, when a 1 = a 2 and R 1 > R 2 , one can claim that better OP performance can be attained when the users adopt Strategy-22. Indeed, with R 1 > R 2 , decoding S 1 first at U 1 or U 2 will lead to a higher OP. On the other hand, decoding S 2 first at both nodes, trying to achieve a lower data rate R 2 , will lead to a lower OP, thereby justifying the fact that Strategy-22 is the best. Similar inference can also be drawn for the case with R 2 > R 1 , where Strategy-11 performs the best.

IV. SECRECY OUTAGE PROBABILITY
Node U 1 must be able to decode its private stream S 1 securely, while U 2 attempts to overhear S 1 in addition to retrieving its own stream S 2 . Now, we develop closed-form approximations for the SOP when the RS system operates with either one of the four decoding strategies. Let R s be the secrecy target data rate of S 1 , and C s = 2 R s − 1.

A. PERFORMANCE WITH STRATEGY-11
A secrecy outage event occurs when: a node U t , t ∈ {1, 2}, is not able to decode S c , i.e., E c→t ; U 1 is not able to decode its private stream securely, i.e., 3 2→2 . As such, the SOP of the system can be calculated using SOP (11) Since U 2 decodes S 1 by treating S 2 as noise, perfect secrecy is not maintained for the private stream of U 1 , i.e., S 1 .
In this paper, we evaluate the SOP w.r.t. the condition in which the secrecy achievable rate of U 1 to retrieve S 1 , i.e., R

C. PERFORMANCE WITH STRATEGY-21
In this case, a secrecy outage event is the result of the following events: 2→1 ; U 1 fails to retrieve S 1 securely, i.e., 2→2 . As a result, SOP (21) is given by Similar to Section IV-A, perfect secrecy at S 1 cannot be provided with this strategy. In this paper, we investigate the secrecy of S 1 when the secrecy achievable rate of U 1 to decode S 1 , i.e., R (2) 1→1 − R (1) 1→2 , is higher than R s . Next, we provide an approximation for the SOP in Strategy-21.
Theorem 6: The SOP for the system operation with Strategy-21 is obtained as where (21) ), (21) ), (21) Proof: The proof is provided in Appendix F.

D. PERFORMANCE WITH STRATEGY-22
Here, the secrecy outage event occurs in four cases: when users are not able to decode S c , i.e., E c→t , t ∈ {1, 2}; if U 1 fails to decode S 2 , i.e., E (2) 2→1 ; when the near user U 1 is not able to decode S 1 securely, i.e., 2→2 . As such, we have The next theorem characterizes the exact SOP in closedfrom. Theorem 7: The SOP for the system operation with Strategy-22 is obtained as Proof: The proof is provided in Appendix G.
Remark: When U 2 adopts Strategy-1, it first retrieves S 1 to obtain S 2 . This decoding scheme does not provide perfect secrecy at node U 1 . By setting R s as the secrecy target rate of S 1 , some sort of secrecy can be maintained for U 1 . In fact, U 2 is permitted to retrieve S 1 with the achievable rate R (1) 1→2 higher than R 1 , while the secrecy achievable rate of U 1 to decode S 1 must be higher than R s . Clearly, when node U 2 uses Strategy-1, there exists a tradeoff between the ability of U 2 to retrieve S 2 and the secrecy of the private stream of node U 1 . Indeed, on one hand, since U 2 retrieves S 1 before decoding S 2 , then R 1→2 must be sufficiently high to be greater than R 1 . On the other hand, R (1) 1→2 must be as small as possible to make R 1→2 , i ∈ {1, 2}, i.e., the secrecy achievable rate of U 1 to decode S 1 with Strategy-i1, greater than R s .

E. SECRECY OUTAGE PROBABILITY OF U 1
Now, we focus on the secrecy of the near user U 1 only and not on the entire decoding process at the near and far users.
where N 2 is the complexity-vs-accuracy parameter in the Gauss-Chebyshev quadrature rule, ψ l = cos( 2l−1 2N 2 π), and the other parameters are as follows: Strategy-11: M (11) = ρk 0 /2(a 1 a 2 − a 2 k 0 ), (11) Strategy-12: M ; In the above SOP evaluations pertaining to node U 1 , Strategy-11 involves the situation when U 1 is unable to decode S c and fails to retrieve S 1 securely with U 2 performing Strategy-1, i.e.,Ē 11 s , Strategy-12 involves the situation when U 1 is unable to decode S c and fails to retrieve S 1 securely with the user U 2 performing Strategy-2, i.e.,Ē 12 s , and Strategy-21 involves the situation when U 1 is unable to decode S c and S 2 correctly and fails to retrieve S 1 securely, i.e.,Ē 21 s . In the case of Strategy-22, the SOP of U 1 is given by Proof: The proof is provided in Appendix H.

F. COMPARISON OF THE DECODING STRATEGIES
Now, we compare the behavior of the decoding strategies in high-SNR regimes. We assume that the source transmit power, P, is sufficiently large to make the users decode the common stream, and U 2 is able to retrieve its private stream perfectly, i.e., the power allocation satisfy a c >> a 1 and a c >> a 2 . In these cases, the secrecy of the private stream of U 1 has the major impact on the SOP, and the system SOP is almost equal to that of U 1 . This will be confirmed in Section V.  Corollary 1: By adopting the results of Table 1, conventional RS, i.e., when the system operates with Strategy-12, yields the worst SOP, while Strategy-21 excels in SOP compared to the other schemes.
Finally, the next two lemmas investigate the SOP performance for extreme values of P.
Proof: The proof is provided in Appendix M. Lemma 3: When P → ∞, the SOP under Strategy-21 tends to zero, and the SOP under Strategy-22 converges to 1 −

Proof:
The proof is provided in Appendix N.

V. NUMERICAL RESULTS AND DISCUSSIONS
In this part, numerical results corresponding to the obtained OP, SOP and SOP 1 , are presented. These are in full agreement with the Monte-Carlo simulations obtained from 10 5 iterations. The goal is to investigate the effects of the power allocation, the target data rates, the power budget of the source transmitter, and the far user's distance, on performance. Here, the path-loss exponent is set as α = 2.2, the distribution parameters of the channel power gains are such that λ 1 = λ 2 = 1, the noise variance is σ 2 = 1, and the complexity-vs-accuracy parameters of the Gauss-Chebyshev quadrature rule are N 1 = N 2 = 150. Figure 2 illustrates the OP of the four strategies w.r.t. the power allocation coefficients, for constant P = 40dBW and target rates R 1 = R 2 = R c = 0.1BPCU. For different sets of (a 1 , a 2 , a c ), the least OP with Strategy-i, i ∈ {11, 12, 21, 22}, is denoted as OP (i) 0 , while these sets are not necessarily the same for all strategies. When a sufficient portion of P is allocated to each stream such that the users are able to correctly decode the streams, it is demonstrated that OP (22) (21) 0 , which confirms that Strategy-22 is superior. This validates Lemma 1, which illustrates the fact that for equal target rates, by assigning appropriate power allocation coefficients, the users can achieve lower OP when following Strategy-2. The users retrieve the far user's private stream S 2 by treating S 1 as noise. When a 2 is significantly small, Strategy-11 outperforms the other strategies. In fact, with Strategy-2, the users should decode S 2 after obtaining S c , where small values of a 2 indicating low levels of P cannot achieve a high enough rate for the retrieving of S 2 . Hence, when the power allocated to S 2 is much smaller than those allocated to S 1 and S c , then Strategy-11 leads users  1 and a 2 (a c = 1 − (a 1 + a 2 ) 1 and a 2 (a c = 1 − (a 1 + a 2 ) to achieve better outage performance. Similarly, when a 1 is close to zero, Strategy-22 excels in OP, since small levels of a 1 do not allow the users to correctly decode S 1 via Strategy-1. Figure 3 shows the range of SOP of all strategies when R 1 = R 2 = R c = 0.1BPCU, R s = 0.05BPCU, and P = 40dBW. Here, SOP (i) 0 , i ∈ {11, 12, 21, 22}, represents the least achievable SOP with Strategy-i. As illustrated, SOP (21) 0 < SOP (22) 0 < SOP (11) 0 < SOP (12) 0 , which indicates that by adopting suitable power allocation, Strategy-21 has the potential to provide the least SOP. Unlike the OP performance without security constraints in which OP (12) 0 < OP (11) 0 and OP (21) 0 > OP (11) 0 , for the SOP it is found that SOP < SOP (11) 0 . Therefore, with suitable power allocation, the achievable rate of U 1 and U 2 to decode S 1 gets improved with Strategy-2, while the increment in the rate of U 1 to retrieve S 1 reduces the SOP, and the increment in the rate of U 2 to decode S 1 raises it. Similar to the OP performance, when a 1 is significantly smaller than a 2 and a c , then Strategy-22 yields better SOP and, for small a 2 , Strategy-11 outperforms the other strategies. If a 1 + a 2 is close to one, the SOP would be close to unity, since with a low value of a c the users' SINRs are not sufficient for decoding the common stream. Figure 4 compares the OP of the strategies w.r.t. P, for different target rates and a 1 = a 2 = a c = 1 3 . When the rates associated with S 1 and S 2 are equal, e.g., R 1 = R 2 = 0.1BPCU, and R c = 0.1 or 0.3BPCU, the plots of all strategies coincide. While all strategies yield the same performance in these cases, Strategy-12 reduces the decoding complexity at the receiving sides. When R 1 > R 2 or R 2 > R 1 , according to Lemma 1, after decoding S c , all users must decode the private stream for which the target rate is less. For instance, when R 1 = R c = 0.1BPCU and R 2 = 0.3BPCU, OP (11) < OP (21) < OP (12) < OP (22) , i.e., Strategy-11 yields the least OP for the RS system, where each user after retrieving S c decodes S 1 by treating S 2 as noise. An increase in the target rates raises the OP, according to the fact that the decoding ability of the users gets reduced by increasing the rates. For example, the RS system with R 1 = R 2 = 0.1BPCU and R c = 0.3BPCU, and the one with R 1 = R c = 0.1BPCU and R 2 = 0.3BPCU, have higher OP than the system with a 1 = a 2 = a c = 0.1BPCU. Changing the power allocation from a 1 = a 2 = a c = 0.1BPCU to a 1 = a 2 = 0.1BPCU and a c = 0.3BPCU affects the OP more than changing the allocation from a 1 = a 2 = a c = 0.1BPCU to a 1 = a c = 0.1BPCU and a 2 = 0.3BPCU. Therefore, the target rate R c has the major impact on the OP as compared to R 1 and R 2 , since each user must decode S c before having access to its own private stream. The OP performance of the proposed system has also been compared with a benchmark system, namely, a NOMA system where R 1 = R 2 = 0.1BPCU and a 1 = a 2 = 1/2. As observed from Fig. 4, at low transmit power values, the performance with NOMA is slightly better than RSMA because the power budget in the latter is divided into three parts for the common and private streams of the two users, thereby resulting in lower power levels for each stream, whereas the power allocation is only for the messages of the two users in NOMA. On the other hand, for high values of the power budget P, the proposed decoding strategies are able to overcome the allocation of lesser power for the private and common streams as compared to NOMA, thereby yielding improved performance for RSMA as compared to NOMA. Figure 5 illustrates the SOP and SOP 1 w.r.t. the source power, P, when a 1 = a 2 = a c = 1 3 , R 1 = R 2 = R c = 0.1BPCU, and R s = 0.05BPCU. As discussed in Table 1, when the RS strategies operate with high power, e.g., P = 50dBW, SOP 1 ≈ SOP, which confirms the major impact of the U 1 's private stream secrecy on the SOP at high SNRs. If the signals are transmitted with P < 50dBW, then the SOP of Strategy-21 and Strategy-22 decrease with P, due to the increment in the received SNR at U 1 to retrieve S 1 . For P ≥ 50dBW, as per Lemma 3, the SOP of Strategy-21 goes to zero, and SOP (12) tends to the constant value 1 −  Strategy-i. When P > P (i) min , an increase in P raises the SOP. Indeed, in addition to improving the SINR at U 1 to decode S 1 , an increase in P enhances the achievable rate of U 2 to decode S 1 , which degrades the SOP. As a result, the SOP of Strategy-i, i ∈ {11, 12}, converges to unity in high-SNR regimes, as stated in Lemma 2. Comparing the results for large values of P reveals that SOP (21) < SOP (11) < SOP (12) and SOP (21) < SOP (22) < SOP (12) , which is in agreement with Table 1. Hence, although U 1 is not aware of the strategy adopted by U 2 , following Strategy-2 leads to the RS system performing more securely.

B. IMPACTS OF THE POWER BUDGET AND THE TARGET RATES
The SOP of the RS strategies w.r.t. R s is shown in Fig. 6, for P = 40dBW and R c = 0.1BPCU, and different R 1 and R 2 . The SOP increases with R s , due to the fact that the ability of U 1 to decode S 1 securely gets reduced when R s increases. When R 1 > R 2 , e.g., R 1 = 0.3BPCU and R 2 = 0.1BPCU, Strategy-22 leads to better SOP, while for R 2 > R 1 , e.g., R 2 = 0.3BPCU and R 1 = 0.1BPCU, Strategy-21 performs more securely than the other decoding mechanisms. Therefore, to achieve the best performance, U 1 must follow Strategy-2 by decoding S 2 after having access to S c . In order for U 2 to assist the RS system in achieving the least SOP, it must follow a similar approach to Lemma 1. In  R c = 0.1BPCU, R s = 0.05BPCU, a 1 = a 2 = a c = 1  3  and d 1 = 20m Figure 7 illustrates the OP and SOP of the RS strategies w.r.t. d 2 for different values of P when a 1 = a 2 = a c = 1 3 , R 1 = R 2 = R c = 0.1BPCU and R s = 0.05BPCU. As discussed, when a 1 = a 2 and R 1 = R 2 , all strategies yield the same OP. The OP increases with d 2 . Indeed, such increments reduce the received SINR/SNR at U 2 to decode the streams. When it comes to secrecy, if d 2 is such that U 2 receives sufficient SINR/SNR to decode its private stream, an increase in d 2 improves the secrecy and reduces the SOP. In these cases, said increase reduces the rate of U 2 to retrieve S 1 , and raises the secrecy achievable rate of U 1 to decode S 1 , i.e., R

C. EFFECTS OF THE POSITION OF THE FAR USER
. When d 2 is large enough to significantly reduce the rate of U 2 to decode S 1 , U 1 would be able to retrieve S 1 securely. In these cases, the achievable rates of U 2 to decode the streams S c and S 2 have the main effects on the SOP. Hence, increasing d 2 reduces the ability of U 2 to decode S c and S 2 , which raises the SOP. Now, let d (i) m (P) be the distance d 2 for which the SOP is minimum, denoted SOP (i) m (P), when Strategy-i, i ∈ {11, 12, 21, 22}, operates with the source power P. An increase in P decreases the SOP of Strategy-i, while increasing d 2 enhances the SOP when d 2 > d (i) m (P). As observed from Fig. 7, d (21) m (40) < d (22) m (40) < d (11) m (40) < d (12) m (40). On the other hand, SOP (21) m (40) < SOP (22) m (40) < SOP (11) m (40) < SOP (12) m (40). Hence, by comparing the SOP of Strategy-i and Strategy-j, i, j ∈ {11, 12, 21, 22}, we can say that d . In fact, the strategy with a higher SOP provides U 1 with a lower secrecy achievable rate to decode S 1 . To improve the SOP with less secrecy achievable rate, it is necessary to degrade the achievable rate of U 2 to decode S 1 more by increasing d 2 . Furthermore, comparing d (12) m (40) and d (12) m (45) shows that an increase in P raises the value of d 2 for which the SOP is minimum. As a matter of fact, in addition to enhancing the received SINR/SNR at U 1 to decode S 1 , increasing P raises the received SINR/SNR at U 2 to retrieve S 1 . Hence, to improve the secrecy outage and reduce the increment in the achievable rate of U 2 to access S 1 , which is caused by increasing P, it is required that d 2 be increased.

VI. CONCLUSION
We characterized the OP and SOP of a RS communication system, where the service to a near user and far user is such that the latter attempts to overhear the message of the former. The impacts of the power budget, power allocation, target data rates, and far user's distance, on the OP and SOP performance, were thoroughly investigated. In particular, without concerns about secrecy, it was shown that if the allocated power to one user's private stream is significantly smaller than the other, both users, after decoding the common stream, must first decode the private stream for which the power allocation coefficient is higher. Also, it was demonstrated that for high power budget, the system SOP is almost equal to the SOP of the near user. Besides, it was shown that when the powers allocated to the users' private streams are equal, to achieve the best OP and SOP performances, the far user must decode the private stream for which the target rate is less by treating the other private stream as noise. Furthermore, the near user by decoding the private stream whose rate is less than the other makes the RS system attain a better OP, whereas the least SOP is achievable when the near user first retrieves the far user's private stream by considering its own private stream as noise.

APPENDIX A
Proof of Theorem 1: To simplify the calculations, we present this lemma (cf. proof in [16]).
Lemma 4: Consider the event E T = { a n Z a m Z+a p ≥ x}, where Z is a non-negative random variable, and where a n , a m , a p and x are positive deterministic parameters. For E T to occur, it is required that a n a m > x. Under this condition, E T can be simplified to E T = {Z ≥ a p x a n −a m x }. Now, we characterize the OP of Strategy-11. Using (3)-(5), (12), and adopting Lemma 4, it is ensured that a c > C c 1+C c and a 1 a 2 > C 1 . Therefore, OP (11) is obtained as OP (11) = 1 − P({g 1 ≥ μ 1 } ∩ {g 2 ≥ β 1 }), and then as shown in (13).

APPENDIX E
Proof of Theorem 5: We use i ∈ {11, 12} to differentiate between the decoding strategies. First, define f (11) (g 2 ) = C s ρ(a 2 g 2 +ρ)+2 Rs a 1 ρg 2 −C s a 1 a 2 g 2 +a 1 ρ−C s a 2 (a 2 g 2 +ρ) and f (12) (g 2 ) = ρ(C s ρ+2 Rs a 1 g 2 ) a 1 ρ−C s a 2 ρ−2 Rs a 1 a 2 g 2 . With regard to the events shown in (22) and (24), considering non-negative g 1 and g 2 , and following a similar approach as in Lemma 4 along with simple algebraic manipulations, shows that the inequality a 1 a 2 > C s must hold. As such, E (i) s is reformulated as Using Lemma 4,(23), (25), (39), and some calculations, we can simplify SOP (i) as where while it is clear that The first step for obtaining the SOP consists in deriving P(F Applying E (i) d to (41) leads to P(F (i) 2 }). Note that although g 2 > 0, since (i) = max(β (i) , η (i) ) and β (i) is supposed to be greater than zero, the sign of η (i) is not a concern here. If (i) ≥ δ (i) 2 , P(F (i) 1 ) = 0; otherwise, since g 1 and g 2 are independent, and by using the Gauss-Chebyshev quadrature rule, P(F Since g 2 is non-negative, to make the event E (i) d occur, η (i) must be greater than zero, which results in a 1 > C s a 2 + C s ρ τ c . Using (42), then when Substituting (43)-(45) into (40), the SOP under Strategy-11 and Strategy-12 is as shown in (26).

APPENDIX G
Proof of Theorem 7: Using Lemma 4 and (31), the power allocation must satisfy a c > C c 1+C c and a 1 a 2 < 1 C 2 . Under these conditions, we can write SOP (22) , which is equal to +2 Rs x f g 1 (y)dy dx. (56) As a result, SOP (22) is obtained as shown in (32).

APPENDIX H
Proof of Theorem 8: Using (33), we present the proof for each strategy, separately. 1) Strategy-11: According to (33), SOP (11) 1 is given by Since g 1 and g 2 are non-negative random variables, it is necessary to have a 1 − a 2 C s − 2 Rs a 1 a 2 g 2 a 2 g 2 +ρ > 0, which results in a 1 a 2 > C s and g 2 < 2 (11) . Then, (57) is simplified to Using the Gauss-Chebyshev quadrature method and following a similar approach as in (44), (58) is found as shown in (34).
2) Strategy-12: With the aid of (33), SOP (12) 1 is found as Since g 1 and g 2 are non-negative, it is sure that a 1 − a 2 C s − 2 Rs a 1 a 2 g 2 ρ > 0, which results in a 1 a 2 > C s and g 2 < 2 (12) . Hence, (59) is written as which, by using the Gauss-Chebyshev quadrature and following a similar way as (44), can be approximated as per (34).
3) Strategy-21: According to (33), SOP is obtained as Finally, adopting the Gauss-Chebyshev method, SOP respectively. First, we need to derive F a 1 gt According to F g t (x) = 1 − e −λ t d α t x and using a similar approach as Lemma 4, we have Then, taking the derivative of (65) w.r.t. x, we obtain f a 1 gt Next, we derive F a 1 gt ρ (x). To do so, the CDF F a 1 gt ρ (x) is obtained as is obtained in (66). According to (63), (64) and (66), we get As mentioned in Theorem 8, to avoid the occurrence of a secrecy outage in Strategy-11 and Strategy-12, the power allocation condition a 1 a 2 > C s must hold. Accordingly, for a 1 −a 2 (2 −Rs (1+x)−1) . Therefore, using (65)-(67), and comparing (68) and (69), it is guaranteed that (1 − SOP (12) ) < (1 − SOP (11) ). As a result, SOP (12) > SOP (11) , and Strategy-11 outperforms Strategy-12 in terms of SOP for large values of P. Table 1 (Comparing Strategy-21 and  Strategy-22): As mentioned, for high values of P, the system SOP and the SOP of the near user are almost equal. Hence, using (33), the SOP of Strategy-21 and Strategy-22 in high-SNR regimes are approximately equal to SOP (21) ≈ P g 1 < ρC s a 1 + 2 R s ρg 2 a 2 g 2 + ρ

Proof of Case 3 of
Since Z and g 1 are non-negative, ρ(C s +2 Rs x) a 1 −a 2 (C s +2 Rs x) must be greater than zero, which leads the variable x in (72) to satisfy 0 < x < C * . Further, with the aid of (71) and (66), we can write . On the other hand, C * < a 1 a 2 , which according to (73) gives (21) .