Secrecy Outage Probability of Minimum Relay Selection in Multiple Eavesdroppers DF Cognitive Radio Networks † State
∗ School
Jie Ding∗† , Qiqing Yang∗‡ and Jing Yang∗
of Information Engineering, Yangzhou University, Yangzhou, China Key Laboratory of Software Development Environment, Beihang University, Beijing, China Email:
[email protected],
[email protected],
[email protected] ‡ Corresponding author:
[email protected] Abstract—In this paper, we investigate the secrecy outage probability in a dual-hop decode-and-forward (DF) secondary cognitive radio network, which is equipped with one source, one destination, multiple eavesdroppers (EAVs) and multirelay. The direct links from the source to the destination and the EAVs are considered because of the moderate shadowing environment. All channels experience the Rayleigh fading. To analyse the system performance, we consider the minimum relay selection, where the most annoying eavesdropper is selected first according to the maximum signal-to-noise ratio (SNR) of the direct links from the source to the EAVs. Then, the best relay is selected according to the minimum SNR from the selected EAV to relays. The selection combining (SC) technique is also employed in our system. The circumstance that the relays can or not deliver the information is considered. We evaluate the system performance through the closed-form accurate and asymptotic secrecy outage probability. The simulation results are also derived by Monte Carlo Simulations to verify the numerical results. Keywords— decode-and-forward (DF); minimum relay selection; selection combining (SC); secrecy outage probability
I. I NTRODUCTION Secrecy outage analysis is a hot direction of research in wireless network. In [1], the authors pay attention to the multiuser and multirelay network in amplify-and-forward (AF) and decode-and-forward (DF) transmission schemes under the spectrum-sharing constraints. Basically, according to the conventional selection, selecting the best destination and corresponding relay are based on the same scheme. Details of relay selections of the system without power constraints and the direct links are given in [2], and the authors analyse the asymptotic achievable secrecy rate, secrecy outage probability and the probability of non-zero achievable secrecy rate in minimum relay selection, conventional relay selection and optimal relay selection. The research of optimal relay selection scheme is illustrated minutely in [3]. The relays are not always able to decode the messages during the information transmission, and the authors consider the situation, where no relay can deliver the information in [4]. The authors mainly study the interception of eavesdropper (EAV) in physical-layer security in [5] and [6]. With DF strategy, the performance analysis of using selection combining (SC) and maximal ratio combining (MRC) can be seen in [7].
A new efficient scheme for multiple sources and multirelay cooperative networks is studied in [8] and [9], and the proposed scheme can reduce the number of channel estimation and obtain the same diversity order as using the joint selection scheme. Be different from the multiple DF relay network in [10], the authors study the system performance under the AF relay network in [4]. Most researches are focusing on the perfect channel state information (CSI), and the imperfect CSI is studied in [11]. The system with spectrum-sharing constraints is studied in [12], and the information transmission will be limited at the source and the relays because of the primary users interference. So, the authors give the methods to gain the probability density functions (PDF) of SNR under the constraint condition in [13], [14] and [15]. In our work, when the secrecy outage probability is derived, we can gain many effective methods in [16]. In this paper, we study a new system model, which contains one source, 𝐿 EAVs, one destination and 𝐾 relays. We not only study the direct links when no relays could forward the messages correctly, but also research the S-R-D link, in which the relay is selected by the minimum relay selection. However, the annoying EAV is selected first based on the channel quality of direct link in our system model. The SC technique is also used to combine the S-D link SNR and S-R-D link SNR. We give the asymptotic and accurate secrecy outage probabilities. To verify the numerical results, the simulation results are also given. II. S YSTEM M ODEL AND R ELAY S ELECTION S CHEME A. System Model A dual-hop DF underlay cognitive radio network with direct links is made up of one secondary source (S), one secondary destination (D), 𝐿 EAVs (𝑙=1,2,⋅ ⋅ ⋅ ,𝐿 ) and 𝐾 relays (𝑘=1,2,⋅ ⋅ ⋅ ,𝐾), which are distributed in the figure 1 and all of them are equipped with single antenna. From the source to the destination and the EAVs transmit links, there exists direct links because of the moderate shadowing environment. In this paper, we can use 𝛾𝑖𝑗 to represent the signal-to-noise ratio (SNR) between the arbitrarily two nodes 𝑖 and 𝑗, 𝛾𝑖𝑗 is
978-1-5090-1701-0/16/$31.00 ©2016 IEEE
can only rely on the direct link. It means / 2 max 𝛾𝑆𝑘 = max 𝑃𝑆 ∣ℎ𝑆𝑅𝑘 ∣ 𝑁0 < 𝛾𝑡 .
:LUHWDSOLQN
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𝑘
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hRk D
𝑘
'
System model
given by 𝛾𝑖𝑗 =
2
𝑃𝑖 ∣ℎ𝑖𝑗 ∣ , 𝑁0𝑗
(1)
where 𝑃𝑖 is the transmit power at node 𝑖 (𝑖 ∈ {𝑆, 𝑅}), 𝑁0𝑗 is the variance of additive white Gaussian noise at node 𝑗 (𝑗 ∈ {𝑅, 𝐷, 𝐸}), and the channel gain between 𝑖 and 𝑗 is 2 expressed by ∣ℎ𝑖𝑗 ∣ . 𝛾𝑖𝑗 is exponential with mean of 1/𝛽𝑖𝑗 , and 𝛽𝑖𝑗 is the parameter, which can be defined as 𝛽𝑖𝑗 ≜ 1/𝐸 [𝛾𝑖𝑗 ]. Then, the PDF of the SNR between 𝑖 and 𝑗 with parameter 𝛽𝑖𝑗 is (2) 𝑓 (𝑥) = 𝛽𝑖𝑗 𝑒−𝑥𝛽𝑖𝑗 . In our work, all relays close to each other are assumed. To guarantee the same distance from relays to other nodes, all relays should form a cluster. This also applies to EAVs. We suppose that the any receiving nodes have the same variance of additive white Gaussian noise 𝑁0 for the sake of simplicity. We define some parameters which will be used in our Defining 𝛽𝑆𝐷 ≜ / ] any [ generality. ] / [work without 2 2 = 𝜀1 , 𝛽𝑆𝑙 = 𝛽𝑆𝐸𝑙 ≜ 1 𝐸 𝑃𝑆 ∣ℎ𝑆𝐷 ∣ 𝑁0 , 𝐸 ∣ℎ𝑆𝐷 ∣ / ] [ ] / [ 2 2 1 𝐸 𝑃𝑆 ∣ℎ𝑆𝐸𝑙 ∣ 𝑁0 , 𝐸 ∣ℎ𝑆𝐸𝑙 ∣ = 𝜀2 , 𝛽𝑆𝑘 = 𝛽𝑆𝑅𝑘 ≜ / ] [ ] / [ 2 2 1 𝐸 𝑃𝑆 ∣ℎ𝑆𝑅𝑘 ∣ 𝑁0 , 𝐸 ∣ℎ𝑆𝑅𝑘 ∣ = 𝛼, 𝛽𝑘𝐷 = 𝛽𝑅𝑘 𝐷 ≜ / ] [ ] / [ 2 2 1 𝐸 𝑃𝑅 ∣ℎ𝑅𝑘 𝐷 ∣ 𝑁0 , 𝐸 ∣ℎ𝑅𝑘 𝐷 ∣ = 𝜎1 , 𝛽𝑘𝑙 = 𝛽𝑅𝑘 𝐸𝑙 ≜ / ] [ ] / [ 2 2 1 𝐸 𝑃𝑅 ∣ℎ𝑅𝑘 𝐸𝑙 ∣ 𝑁0 , 𝐸 ∣ℎ𝑅𝑘 𝐸𝑙 ∣ = 𝜎2 , where 𝑃𝑆 is the transmit power at source and 𝑃𝑅 is the transmit power at relay. For the sake of simplicity, 𝛽𝑘𝑙 = 𝛽𝑅𝑘 𝐸𝑙 is used to carry out our next work, and this assumption also apply to 𝛽𝑅𝑘 𝐷 , 𝛽𝑆𝐸𝑙 and 𝛽𝑆𝑅𝑘 . As we can observe, not all relays could decode and forward the information. If the SNR of the transmit link from source to one of the relays cannot reach a target date rate, this relay will be silent and cannot deliver the information, that means log2 (1 + 𝛾𝑆𝑘 ) < 𝑅𝑡 ,
𝑅𝑡
In the opposite situation, i.e., when there exists 𝑁 (𝑁 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝐾}) relays, which can decode and forward the information correctly, the probability can be expressed as ( ) 𝐾 ∑ ) ( 𝐾 Pr 𝛾𝑆1 ⩾ 𝛾𝑡 , ..., 𝛾𝑆𝑁 ⩾ 𝛾𝑡 , 𝛾𝑆(𝑁+1) 0, i.e., there exists 𝑁 relays can decode and forward the information correctly. 𝐼2 can be expressed as ( ) ( ) ∑ 𝐾 𝑒𝑛𝑑 𝐾 1 + 𝛾𝑆𝐷 𝐼2 = Pr < 𝛾𝑡ℎ × 𝑒𝑛𝑑 1 + 𝛾𝑆𝐸 𝑁
𝑁 =1 Pr2 ( ) Pr 𝛾𝑆1 ≥ 𝛾𝑡 , ..., 𝛾𝑆𝑁 ≥ 𝛾𝑡 , 𝛾𝑆(𝑁 +1) < 𝛾𝑡 , ..., 𝛾𝑆𝐾 < 𝛾𝑡 ) 𝐾 ( ∑ )𝐾−𝑁 𝐾 −𝑁 𝛾𝑡 𝛽𝑆𝑘 ( 1 − 𝑒−𝛾𝑡 𝛽𝑆𝑘 = Pr 2 × . (18) 𝑒 𝑁 𝑁 =1
𝜔
Then, Pr2 can be rewritten as ( ) ) ( 𝑒𝑛𝑑 𝑒𝑛𝑑 Pr 2 = Pr 𝛾𝑆𝐷 −1 < 𝛾𝑡ℎ 1 + 𝛾𝑆𝐸 𝑓𝛾𝑆𝐸 𝑒𝑛𝑑 (𝑥)
=
𝑓𝛾𝑆𝐷 𝑒𝑛𝑑 (𝑦) 𝑑𝑥𝑑𝑦.
0
(19)
0
𝑒𝑛𝑑 𝑒𝑛𝑑 Now, we must obtain the PDFs of 𝛾𝑆𝐷 and 𝛾𝑆𝐸 . Because of the use of SC technique, they can be expressed as 𝑒𝑛𝑑 𝛾𝑆𝐷 𝑒𝑛𝑑 𝛾𝑆𝐸
= max (𝛾𝑆𝐷 , 𝛾𝑘∗ 𝐷 ) , ( ) = max max 𝛾𝑆𝑙 , min 𝛾𝑘𝑙∗ . 𝑙
𝑘
𝑙=0
𝑙
𝐿 ( ) ∑ 𝐿
𝑙
𝑙−1
𝛽𝑆𝑙 𝑙 × 𝑒−𝛽𝑆𝑙 𝑙𝑥
𝑙−1
(𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 ) 𝑒−(𝛽𝑆𝑙 𝑙+𝛽𝑘𝑙 𝑁 )𝑥 .
(−1)
(−1)
(21)
Proof: See in Appendix. By substituting (20) and (21) into (19), we can derive Pr2 as shown in (22). We can combine the results of Pr 1 with ∣Ω∣ = 0 and Pr 2 with ∣Ω∣ > 0, and the exact secrecy outage probability can be presented as 𝑃𝑠𝑜𝑝 = Pr 1 × 𝛿 + Pr 2 × 𝜔.
(23)
B. Asymptotic expression With high main-to-eavesdropper ratio (MER) 𝜆, which is defined as 𝜆 = 𝜀𝜀12 = 𝜎𝜎12 , we can obtain the asymptotic secrecy outage probability. With high transmit power, we first obtain 𝐾 the asymptotic expression of I1 with 𝛿 ≃ (𝛾𝑡 𝛽𝑆𝑘 ) , and the asymptotic cumulative distribution function (CDF) of 𝛾𝑆𝐷 is expressed as 𝐹𝛾𝑆𝐷 (𝑥) ≃ 𝛽𝑆𝑘 𝑥.
(24)
Substituting (24) and (16) into (14), we can approximate Pr1 as 𝐿 ( ) ∑ 𝐿 𝑙−1 Pr 1 ≃ 𝛽𝑆𝐷 (𝛾𝑡ℎ −1)+ (−1) 𝛽𝑆𝐷 𝛾𝑡ℎ /(𝛽𝑆𝑙 𝑙). (25) 𝑙 𝑙=0
Then, we extend ( ) to obtain the asymptotic expression of I2 with 𝐾 ∑ 𝐾 𝐾−𝑁 𝑒𝑛𝑑 𝜔 ≃ , the asymptotic CDF of 𝛾𝑆𝐷 is (𝛾𝑡 𝛽𝑆𝑘 ) 𝑁 =1 𝑁 given by 𝐹𝛾𝑆𝐷 𝑒𝑛𝑑 (𝑥) ≃ 𝛽𝑆𝑘 𝑥 × 𝛽𝑘𝐷 𝑥.
𝛾𝑡ℎ (1+𝑥)−1 ∫
∫∞
𝐿 ( ) ∑ 𝐿
𝑙=0
Pr 1 = Pr (𝛾𝑆𝐷 < 𝛾𝑡ℎ (1 + 𝛾𝑆𝑙∗ ) − 1) =
𝑓𝛾𝑆𝐸 𝑒𝑛𝑑 (𝑥) = −
In this equation, Pr1 can be expressed as
(20)
𝑒𝑛𝑑 while the distribution of 𝛾𝑆𝐸 is given in the following lemma. 𝑒𝑛𝑑 is expressed as Lemma 1: When ∣Ω∣ > 0, the PDF of 𝛾𝑆,𝐸
(13)
𝛿
∫∞
𝑒𝑛𝑑 can be expressed as When ∣Ω∣ > 0, the PDF of 𝛾𝑆𝐷 ( ) −𝛽𝑆𝐷 𝑥 𝑓𝛾𝑆𝐷 1 − 𝑒−𝛽𝑘𝐷 𝑥 𝑒𝑛𝑑 (𝑥) = 𝛽𝑆𝐷 𝑒 ( ) + 𝛽𝑘𝐷 𝑒−𝛽𝑘𝐷 𝑥 1 − 𝑒−𝛽𝑆𝐷 𝑥 ,
(26)
Substituting (26) and (21) into (19), Pr2 can be approximated as shown in (27). Thus, we can derive the asymptotic secrecy outage probability 𝑃𝑠𝑜𝑝 ≃ Pr 1 × 𝛿 + Pr 2 × 𝜔.
(28)
We could observe that the diversity order of the system is constant, and the system performance will not change with the changes of numbers of relays and EAVs from the asymptotic results, because of the minimum relay selection and the use of SC technique.
) 𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 𝛽𝑆𝑙 𝑙 − 𝑙 𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 + 𝛾𝑡ℎ 𝛽𝑆𝐷 𝛽𝑆𝑙 𝑙 + 𝛾𝑡ℎ 𝛽𝑆𝐷 𝑙=0 ( ) ( ) 𝐿 ∑ 𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 𝛽𝑆𝑙 𝑙 𝐿 𝑙−1 −(𝛾𝑡ℎ −1)𝛽𝑘𝐷 − − (−1) 𝑒 𝑙 𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 + 𝛾𝑡ℎ 𝛽𝑘𝐷 𝛽𝑆𝑙 𝑙 + 𝛾𝑡ℎ 𝛽𝑘𝐷 𝑙=0 ( ) 𝐿 ( ) ∑ 𝛽𝑆𝑙 𝑙 𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 𝐿 𝑙−1 − − (−1) 𝑒−(𝛾𝑡ℎ −1)(𝛽𝑆𝐷 +𝛽𝑘𝐷 ) . 𝑙 𝛽𝑆𝑙 𝑙 + 𝛾𝑡ℎ (𝛽𝑆𝐷 + 𝛽𝑘𝐷 ) 𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 + 𝛾𝑡ℎ (𝛽𝑆𝐷 + 𝛽𝑘𝐷 )
Pr 2 = 1 −
𝐿 ( ) ∑ 𝐿
(−1)
𝑙−1 −(𝛾𝑡ℎ −1)𝛽𝑆𝐷
(
𝑒
(22)
𝑙=0
) 2 2 (𝛾𝑡ℎ − 1) 𝛾𝑡ℎ 2𝛾𝑡ℎ 2 (𝛾𝑡ℎ − 1) + + Pr 2 ≃ (−1) 𝛽𝑆𝑙 𝑙𝛽𝑆𝐷 𝛽𝑘𝐷 2 3 𝑙 𝛽𝑆𝑙 𝑙 (𝛽𝑆𝑙 𝑙) (𝛽𝑆𝑙 𝑙) 𝑙=0 ( ) 𝐿 ( ) 2 ∑ 2 (𝛾𝑡ℎ − 1) 𝛾𝑡ℎ 2𝛾𝑡ℎ 2 (𝛾𝑡ℎ − 1) 𝐿 𝑙−1 + − + . (−1) (𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 ) 𝛽𝑆𝐷 𝛽𝑘𝐷 3 𝑙 (𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 ) (𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 )2 (𝛽𝑆𝑙 𝑙 + 𝛽𝑘𝑙 𝑁 ) 𝑙=0 𝐿 ( ) ∑ 𝐿
(
𝑙−1
IV. S IMULATION AND N UMERICAL R ESULTS
(27)
0
10
−0.8
10 −1
Secrecy Outage Probability
In this section, to verify the accuracy, some numerical and simulation results are given. All of the channels we considered are Rayleigh fading channels. We set the target SNR threshold 𝛾𝑡 is 1dB, and all the variance of additive white Gaussian noise 𝑁0 = 1. The MER 𝜆 is 𝜆 = 𝜀𝜀12 = 𝜎𝜎12 , and 𝑃𝑆 = 𝑃𝜂𝑅 = 𝜆. The average channel gains of the wiretap channels are set to 𝜀2 = 1,𝜎2 = 1, and the S-R link 𝛼 = 1. Fig.2 depictes the influence on the secrecy outage probability with the change of number of relays, where 𝜂 is set to 0.8 and 𝛾𝑡ℎ is set to 3dB. We can find that the analytical and asymptotic results match well with the simulation results. When the number of EAVs is 3, 𝐾 varies from 1 to 3, the lights are tight and the changes of the system secrecy performance are not very obvious. That is because we use the minimum relay selection and the SC technique. Although when 𝐾 is becoming larger, the EAVs may intercept the less information from the relays because of the minimum SNR from the relays to the EAVs. However, EAVs can also wiretap much information from the source based on the maxim SNR. In the proposed system model with direct links and the use of SC technique, the minimum relay selection is not much useful. Fig.3 illustrates the effect on the secrecy outage probability with the change of number of EAVs, where 𝜂 is set to 0.8 and 𝛾𝑡ℎ is set to 3dB. When the number of relays is 3, 𝐿 varies from 1 to 3, the security of the system is becoming worse, and the information will be wiretapped more easily. The secrecy performance of the system can not change much with the changes of the number of EAVs. Fig.4 indicates the influence of 𝛾𝑡ℎ on the secrecy outage probability. We set 𝑁 = 𝐿 = 3, 𝜂 = 0.8. we can clearly observe that the system security performance is becoming better with the decrease of 𝛾𝑡ℎ . To our knowledge, the given secrecy SNR threshold is larger, the secrecy outage event is more likely occur.
10
−0.9
10
10
10.5
11
−2
10
K=1,L=3(Anal.) K=1,L=3(Sim.) K=1,L=3(Asy.) K=2,L=3(Anal.) K=2,L=3(Sim.) K=2,L=3(Asy.) K=3,L=3(Anal.) K=3,L=3(Sim.) K=3,L=3(Asy.)
−3
10
−4
10
0
Fig. 2.
5
10 15 MER λ (dB)
20
25
Effects of 𝐾 on the secrecy outage probability.
V. C ONCLUSION In our study, we investigated a downlink multiple EAVs and multirelay DF network of minimum relay selection with SC technique, which exists the direct transmit links from the source to the EAVs and relays. We derived the exact and asymptotic secrecy outage probabilities to measure the secrecy performance of the system, and the Monte Carlo Simulation results were also given. With the minimum relay selection, traditional approach that increasing the number of relays was not much useful, because the EAVs could wiretap much information according to the maximum SNR from the source to the EAVs transmit links and less information according to the minimum SNR from the relays to the EAVs transmit links. In addition, we could observe that the diversity order of our system remains unchanged with the change of number of relays and EAVs from the asymptotic results. A PPENDIX
𝑒𝑛𝑑 can be expressed as Thus, the PDF of 𝛾𝑆𝐸
0
10
𝑓𝛾𝑆𝐸 𝑒𝑛𝑑 (𝛾) =
Secrecy Outage Probability
−1
10
which is shown in Lemma1. K=3,L=1(Anal.) K=3,L=1(Sim.) K=3,L=1(Asy.) K=3,L=2(Anal.) K=3,L=2(Sim.) K=3,L=2(Asy.) K=3,L=3(Anal.) K=3,L=3(Sim.) K=3,L=3(Asy.)
−3
10
0
5
ACKNOWLEDGMENT
10 15 MER λ (dB)
20
25
0
10
Secrecy Outage Probability
−1
10
γth=3(Anal.) γth=3(Sim.)
−2
10
γth=3(Asy.) γth=4(Anal.) γ =4(Sim.) th
γth=4(Asy.)
−3
10
γth=5(Anal.) γth=5(Sim.) γ =5(Asy.) th
−4
10
0
5
Fig. 4.
10 15 MER λ (dB)
20
25
Effect of 𝛾𝑡ℎ on the secrecy outage probability.
According to
P ROOF OF LEMMA 1 = max(max 𝛾𝑆𝑙 , min 𝛾𝑘𝑙∗ ), the cumula-
𝑒𝑛𝑑 𝛾𝑆𝐸
𝑙
𝑘
𝑒𝑛𝑑 tive distribution function of 𝛾𝑆𝐸 can be obtained as ( ) ∗ max(max 𝛾𝑆𝑙 , min 𝛾𝑘𝑙 ) < 𝛾 𝐹𝛾𝑆𝐸 𝑒𝑛𝑑 (𝛾) = Pr 𝑙
(
)
= Pr max 𝛾𝑆𝑙 < 𝛾
𝑘
(
)
× Pr min 𝛾𝑘𝑙∗ < 𝛾
𝑙
𝑘
( ( )) Pr (𝛾𝑆𝑙 < 𝛾) × 1 − Pr min 𝛾𝑘𝑙∗ > 𝛾 𝑘
𝐼=1
(
= 1 − 𝑒−𝛽𝑆𝑙 (
= 1 − 𝑒−𝛽𝑆𝑙
) 𝑥 𝐿 ) 𝑥 𝐿
( ×
1−
) Pr (𝛾𝑘𝑙∗ > 𝛾)
𝑛=1
( ×
𝑁 ∏
1−
𝑁 ∏
) (1 − Pr (𝛾𝑘𝑙∗ < 𝛾))
𝑛=1 𝐿 ( ) ∑ 𝐿 𝑙=0
𝑙
( ) 𝑙 (−1) 𝑒−𝛽𝑆𝑙 𝑙𝑥 1 − 𝑒−𝛽𝑘𝑙 𝑁 𝑥 .
The authors acknowledge the financial support by the National Natural Science Foundation of China under Grant No. 61472343, 61301111 and 61402395. R EFERENCES
Effect of 𝐿 on the secrecy outage probability.
Fig. 3.
=
,
−2
−4
𝐿 ∏
𝑑𝛾
10
10
=
𝑑𝐹𝛾𝑆𝐸 𝑒𝑛𝑑 (𝛾)
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