Abstract
We study the physical-layer security of multi-hop secondary network under spectrum sharing constraint caused by the primary, which consists of many simultaneously independent direct transceiver pairs. The existence of a secondary eavesdropper, which attempts to hear confidential information, results in secure communication to be needful. The PUs’ interference forces low secondary SIRs. The secondary power constraint according to PUs causes the QoS to diminish, especially when the PUs transmission simultaneously operates. In this paper, we derive the closed-form of end-to-end multi-hop secrecy outage probability expression with independent non-identically distributed (i.n.i.d) Rayleigh fading. Furthermore, the secondary intercept probability is derived. Combining the outage probability, we present the security-reliability of the system. Besides, the impact of different numbers of primary transceivers on end-to-end multi-hop outage probability is investigated. Finally, our theoretical results verified via extensive Monte Carlo simulations.
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This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2017.317.
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Appendices
Appendices
1.1 Appendix A
Proof of Proposition 1 in (15)
From (3), denote \({\vartheta _i}\left( \upsilon \right) {=}\Pr \! \left( \!{\frac{{{M_i}}}{{\sum \nolimits _{j {=} 1,j \ne i}^{L^{\phantom {\frac{.}{.}}}} {\left( {{M_j}} \right) }{+}{M_{L {+} 1}} {+} 1}} {\le } \upsilon }\!\right) ,\) where \({M_i} \buildrel \varDelta \over ={\varOmega _i}{\kappa _{i,i}}\), \({M_j} \buildrel \varDelta \over ={\varOmega _i}{\kappa _{j,i}}\), \({M_{L + 1}} \buildrel \varDelta \over ={\varDelta _{k - 1}}{\varpi _{k - 1,i}}\), \(M \buildrel \varDelta \over =\sum \nolimits _{j = 1,j \ne i}^L{{M_j}}+{M_{L + 1}} = \sum \nolimits _{a = 1,a \ne i}^{L + 1}{{M_a}}.\) Using the law of total probability in [37, Lemma 1], the \({f_{M,i}}\left( x \right) \) is given by
where \({\beta _{{M_i}}} = {\delta _{i,i}}\), \({\beta _{{M_{L +1}}}} = {\varpi _{k - 1,i}}\), \({\beta _{{M_a}}} ={\delta _{a,i}}\). With the assistance of [41, 4.20], we easily find \({f_{M + 1,i}}\left( x \right) \). Next, the \({F_{M +1,i}} \left( x \right) \) can be derived by integration the \({f_{M+1,i}} \left( x \right) \).
After some algebra and changing variables, we have
Applying the complementary probability theory, \(1-{\vartheta _i}\left( \upsilon \right) \) is the success probability on ith pair. Therefore, \(1-\prod \nolimits _{i = 1}^L \left( {1-{\vartheta _i}\left( \upsilon \right) } \right) \) is the primary outage, on (15).
1.2 Appendix B
Proof of Proposition 2 in (18)
Substituting (5), (6), (7), the (13) is equivalent to
where \(\xi = {2^{K{C_{th}}}}\). We can rewrite (B.1) as
where \(X = {\varDelta _{k - 1}}{\varphi _k}\), \(\hat{X} = {\varDelta _{k - 1}}{\chi _k}\) have the mean \({\varepsilon _k},{\eta _k}\). \(Y = \sum \nolimits _{i = 1}^L {{Y_i}} ,\hat{Y} = \sum \nolimits _{i = 1}^L {{{\hat{Y}}_i}}\), with \({Y_i} = {\varOmega _i}{\omega _{i,k}}\), \({\hat{Y}_i} = {\varOmega _i}{\theta _i}\), are \({\tau _{i,k}},{\sigma _i}\), respectively. We derive the representative \(\Pr \left( {U = 1+X/\left( {Y+1} \right) <u} \right) \) in details. Based on [37, 12], [41, 4.20], we obtain the \(\Pr \left( {U \le u} \right) \) expression as follows (B.3), (B.4)
Let’s denote \(T = \frac{{\hat{X}}}{{\hat{Y}+1}}\), \(V = T+1\). We identified \({f_T}\left( t \right) \) by exploiting [42, 21] and solved the integration by [40, 3.381.4]. Afterwards, we apply [41, 4.20] to exchange between T and V random variables. We obtain
We exploit \({F_{\xi ,k}}\left( \xi \right) \!= \!{F_{U/V}}\left( \xi \right) \!=\! \int \nolimits _1^\infty {{f_V}\left( v \right) .} {F_U} \left( {v\xi } \right) dv,\) derived from [43, 6.42, 6.43]. The result shows in (B.6). Next, we can find the results of the \({Q_i},\;(i = 1,2,3,4)\), in (B.7) by [40, 3.351.4, 3.353.1, 3.352.2]. Substituting (B.7) to (B.6), we obtain (19), and substituting (19) to (18), we have \({\mathrm{SO}}{{\mathrm{P}}_{{\mathrm{e2}}e}}\).
1.3 Appendix C
Proof of Proposition 3 in (22)
Next, from (3), we consider the \({\mathrm{O}}{{\mathrm{P}}_k} =\Pr \left( {\frac{I}{{H+1}} < {\gamma _{th}}} \right) \), where \(I={\varDelta _{k - 1}}{\varphi _k},H = {H_1}+{H_2}...+{H_L} =\sum \nolimits _{i = 1}^L {{\varOmega _i}{\omega _{i,k}}}\),
Using the same methodology with the first proposition, we receive the \({\mathrm{O}}{{\mathrm{P}}_k}\) as follows
Afterwards, the \({\mathrm{O}}{{\mathrm{P}}_{e2e}}\) in (24) is derived following the substitution of \({\mathrm{O}}{{\mathrm{P}}_k}\) with (C.2) in (23).
1.4 Appendix D
From (4), let us denote
where \(P = {\varDelta _{k - 1}}{\chi _k},\;Q = {Q_1}+{Q_2}+\cdots +{Q_L} =\sum \nolimits _{i = 1}^L {{\varOmega _i}{\theta _i}}\). Likewise, the \({f_Q}\left( y \right) \) can be written as
Applying similar way above, we obtain the expression in (26).
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Pham, M.N. On the secrecy outage probability and performance trade-off of the multi-hop cognitive relay networks. Telecommun Syst 73, 349–358 (2020). https://doi.org/10.1007/s11235-019-00608-1
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DOI: https://doi.org/10.1007/s11235-019-00608-1