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Improving Secrecy Outage and Throughput Performance in Two-Way Energy-Constraint Relaying Networks Under Physical Layer Security

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Abstract

In this paper, we propose three secrecy cooperative transmission protocols for a two-way energy-constrained relaying network in which two sources wish to exchange information with the help of multiple intermediate relays being subjected to wiretapping by multiple eavesdroppers. In the secure two-way communication (STW protocol), an energy-constrained relay is preselected via one of three investigated relay-selection strategies, which harvest the energy from the radio-frequency signals of one source and decode-and-forward the signals to another source. In secure two-way communication with network coding (STWNC protocol), the network coding technique is applied at a relay preselected via one of two investigated relay-selection strategies. In secure two-way communication with cooperative jamming and network coding (STWJNC protocol), under cooperative jamming, the network coding technique is applied at two sources and a preselected relay where a jammer-relay pair is preselected via one of two investigated selection strategies. The power-splitting receiver is applied at the energy-constrained relay for all proposed protocols. To evaluate performance, we derive new closed-form expressions for the secrecy outage probability and the throughput performance of the three protocols with the different relay and jammer-selection strategies. Our analysis is verified using Monte Carlo simulations.

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Author information

Authors and Affiliations

  1. Duy Tan University, Da Nang, Vietnam

    Sang Quang Nguyen

  2. University of Ulsan, Ulsan, Republic of Korea

    Hyung Yun Kong

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  1. Sang Quang Nguyen
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  2. Hyung Yun Kong
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Correspondence to Sang Quang Nguyen.

Appendices

Appendix 1: The PDF of RVs \({\varvec{g_{{R_s}E\max }}}\), \({\varvec{g_{S2E\max }}}\), and \({\varvec{g_{S1E\max }}}\) When Using the Relay Selection Strategy in (19a)

The CDFs of RVs \({g_{{R_s}E\max }}\), \({g_{S2E\max }}\), and \({g_{S1E\max }}\) can be given respectively as

$$\begin{aligned} {F_{{g_{{R_s}E\max }}}}\left( x \right)& = \Pr \left[ {\mathop {\min }\limits _{m = 1,2,\ldots ,M} \left( {\mathop {\max }\limits _{l = 1,2,\ldots ,L} {g_{{R_m}{E_l}}}} \right)< x} \right] \nonumber \\& = 1 - \prod \limits _{m = 1}^M {\left\{ {1 - \prod \limits _{l = 1}^L {\Pr \left[ {{g_{{R_m}{E_l}}} < x} \right] } } \right\} } = 1 - {\left[ {1 - {{\left( {1 - {e^{ - {\lambda _{RE}}x}}} \right) }^L}} \right] ^M} \end{aligned}$$
(66)
$$\begin{aligned} {F_{{g_{S2E\max }}}}\left( x \right)& = \Pr \left[ {\mathop {\max }\limits _{m = 1,2,\ldots ,L} {g_{S2E{\mathop {\mathrm {m}}\nolimits } }} < x} \right] = {\left( {1 - {e^{ - {\lambda _{S2E}}x}}} \right) ^L} \end{aligned}$$
(67)
$$\begin{aligned} {F_{{g_{S1E\max }}}}( x )& = \Pr \left[ {{g_{S1E\max }} < x} \right] = {\left( {1 - {e^{ - {\lambda _{S1E}}x}}} \right) ^L} \end{aligned}$$
(68)

Then, by differentiating (66), (67), and (68), we obtain the PDFs of RVs \({g_{{R_s}E\max }}\), \({g_{S2E\max }}\), and \({g_{S1E\max }}\), respectively, as follows:

$$\begin{aligned} {f_{{g_{{R_s} E\max }}}} \left( x \right)& = M L{\lambda _{RE}}{e^{ - {\lambda _{RE}}x}}{\left( {1 - {e^{ - {\lambda _{RE}}x}}} \right) ^{L - 1}}{\left[ {1 - {{\left( {1 - {e^{ - {\lambda _{RE}}x}}} \right) }^L}} \right] ^{M- 1}}\nonumber \\& = M L{\lambda _{RE}} \sum \limits _{k = 0}^{L - 1} {C_{L - 1}^k} {\left( { - 1} \right) ^k} \sum \limits _{u = 0}^{M - 1} {C_{M - 1}^u} {\left( { - 1} \right) ^u}\sum \limits _{v = 0}^{Lu} {C_{Lu}^v} {\left( { - 1} \right) ^v}{e^{ - \left( {1 + k + v} \right) {\lambda _{RE}}x}} \end{aligned}$$
(69)
$$\begin{aligned} {f_{{g_{S2E\max }}}}\left( x \right)& = L{\lambda _{S2E}}{e^{ - {\lambda _{S2E}}x}}{\left( {1 - {e^{ - {\lambda _{S2E}}x}}} \right) ^{L - 1}}\nonumber \\& = L{\lambda _{S2E}}\sum \limits _{w = 0}^{L - 1} {C_{L - 1}^w} {\left( { - 1} \right) ^w}{e^{ - \left( {1 + k} \right) {\lambda _{S2E}}x}} \end{aligned}$$
(70)
$$\begin{aligned} {f_{{g_{S1E\max }}}}\left( x \right)& = L{\lambda _{S1E}}{e^{ - {\lambda _{S1E}}x}}{\left( {1 - {e^{ - {\lambda _{S1E}}x}}} \right) ^{L - 1}}\nonumber \\& = L{\lambda _{S1E}}\sum \limits _{t = 0}^{L - 1} {C_{L - 1}^t} {\left( { - 1} \right) ^t}{e^{ - \left( {1 + t} \right) {\lambda _{S1E}}x}}. \end{aligned}$$
(71)

Appendix 2: Proof of Lemma 1

At first, the integral \({I_{{\mathrm{1,MAS1R}}}}\) can be expressed as

$$\begin{aligned}&{I_{1,\mathrm{MAS1R}}}\mathop = \limits ^{\left( {{\mathrm{72}}{\mathrm{.1}}} \right) } {e^{ \frac{{ - \left( {\varphi - 1} \right) {\lambda _{S2R}}}}{{{\omega _2}\psi }}}}\int _0^\infty {L{\lambda _{S1E}}\sum \limits _{t = 0}^{L - 1} {C_{L - 1}^t} {{\left( { - 1} \right) }^t}{e^{ - \left( {1 + t} \right) {\lambda _{S1E}}{x_1}}}} \nonumber \\&\quad \int _0^\infty {L{\lambda _{S2E}}\sum \limits _{w = 0}^{L - 1} {C_{L - 1}^w} {{\left( { - 1} \right) }^w}{e^{ - \left[ {\frac{{\left( {1 + w} \right) {\omega _2}{\lambda _{S2E}} + \varphi {\omega _1}{\lambda _{S2R}}}}{{{\omega _2}}}} \right] {x_2}}}} \nonumber \\&\quad \int _{\frac{{\varphi - 1}}{{{\omega _2}\psi }} + \frac{{\varphi {\omega _1}{x_1}}}{{{\omega _2}}}}^\infty {M {\lambda _{S1 R}} \sum \limits _{k = 0}^{M - 1} {C_{M - 1}^k} {{\left( { - 1} \right) }^k}{e^{ - \left( {1 + k} \right) {\lambda _{S1R}}{x_3}}}{e^{\frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi {x_3}}}}}} d{x_3}d{x_2}d{x_1}\nonumber \\&\qquad \mathop \approx \limits ^{\left( {{\mathrm{72}}{\mathrm{.2}}} \right) } {e^{\frac{{ - \left( {\varphi - 1} \right) {\lambda _{S2R}}}}{{{\omega _2}\psi }}}}\left( {{I_3} + {I_4}} \right) \end{aligned}$$
(72)

where (72.2) is obtained by approximating \({e^{\frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi {x_3}}}}} \approx 1 + \frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi {x_3}}}\).

The term \({I_3}\) and \({I_4}\) in (72) are denoted and derived as in (73) and (74) as follows

$$\begin{aligned}&{I_3} \buildrel \varDelta \over = \int _0^\infty {L{\lambda _{S1E}}\sum \limits _{t = 0}^{L - 1} {C_{L - 1}^t} {{\left( { - 1} \right) }^t}{e^{ - \left( {1 + t} \right) {\lambda _{S1E}}{x_1}}}} \nonumber \\&\qquad \int _0^\infty {L{\lambda _{S2E}}\sum \limits _{w = 0}^{L - 1} {C_{L - 1}^w} {{\left( { - 1} \right) }^w}{e^{ - \left[ {\frac{{\left( {1 + w} \right) {\omega _2}{\lambda _{S2E}} + \varphi {\omega _1}{\lambda _{S2R}}}}{{{\omega _2}}}} \right] {x_2}}}} \nonumber \\&\qquad \int _{\frac{{\varphi - 1}}{{{\omega _2}\psi }} + \frac{{\varphi {\omega _1}{x_1}}}{{{\omega _2}}}}^\infty {M{\lambda _{S1R}}\sum \limits _{k = 0}^{M - 1} {C_{M - 1}^k} {{\left( { - 1} \right) }^k}{e^{ - \left( {1 + k} \right) {\lambda _{S1R}}{x_3}}}} d{x_3}d{x_2}d{x_1}\nonumber \\&\quad = L{\lambda _{S2E}}\sum \limits _{w = 0}^{L - 1} {\frac{{C_{L - 1}^w{{\left( { - 1} \right) }^w}{\omega _2}}}{{\left( {1 + w} \right) {\omega _2}{\lambda _{S2E}} + \varphi {\omega _1}{\lambda _{S2R}}}}} \nonumber \\&\qquad LM{\lambda _{S1E}}\sum \limits _{t = 0}^{L - 1} {C_{L - 1}^t} {\left( { - 1} \right) ^t}\sum \limits _{k = 0}^{M - 1} {\frac{{C_{M - 1}^k{{\left( { - 1} \right) }^k}{\omega _2}{e^{\frac{{ - \left( {1 + k} \right) \left( {\varphi - 1} \right) {\lambda _{S1R}}}}{{{\omega _2}\psi }}}}}}{{\left( {1 + k} \right) \left[ {\left( {1 + t} \right) {\omega _2}{\lambda _{S1E}} + \left( {1 + k} \right) \varphi {\omega _1}{\lambda _{S1R}}} \right] }}} \nonumber \\&\quad \buildrel \varDelta \over = {\varOmega _3}\left( {L,\varphi ,{\omega _1},{\omega _2},{\lambda _{S2R}},{\lambda _{S2E}}} \right) {\varOmega _4}\left( {L,M,\varphi ,{\omega _1},{\omega _2},\psi ,{\lambda _{S1R}},{\lambda _{S1E}}} \right) \end{aligned}$$
(73)
$$\begin{aligned}&{I_4} \buildrel \varDelta \over = \frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi }}\int _0^\infty {L{\lambda _{S1E}}\sum \limits _{t = 0}^{L - 1} {C_{L - 1}^t} {{\left( { - 1} \right) }^t}{e^{ - \left( {1 + t} \right) {\lambda _{S1E}}{x_1}}}} \nonumber \\&\qquad \int _0^\infty {L{\lambda _{S2E}}\sum \limits _{w = 0}^{L - 1} {C_{L - 1}^w} {{\left( { - 1} \right) }^w}{e^{ - \left[ {\frac{{\left( {1 + w} \right) {\omega _2}{\lambda _{S2E}} + \varphi {\omega _1}{\lambda _{S2R}}}}{{{\omega _2}}}} \right] {x_2}}}}\nonumber \\&\qquad \int _{\frac{{\varphi - 1}}{{{\omega _2}\psi }} + \frac{{\varphi {\omega _1}{x_1}}}{{{\omega _2}}}}^\infty {M{\lambda _{S1R}}\sum \limits _{k = 0}^{M - 1} {C_{M - 1}^k} {{\left( { - 1} \right) }^k}\frac{{{e^{ - \left( {1 + k} \right) {\lambda _{S1R}}{x_3}}}}}{{{x_3}}}} d{x_3}d{x_2}d{x_1}\nonumber \\& \mathop = \limits ^{\left( {{\mathrm{74}}{\mathrm{.1}}} \right) } \frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi }}{\varOmega _3}\left( {L,\varphi ,{\omega _1},{\omega _2},{\lambda _{S2R}},{\lambda _{S2E}}} \right) \nonumber \\&\qquad \int _0^\infty {L{\lambda _{S1E}}\sum \limits _{t = 0}^{L - 1} {C_{L - 1}^t} {{\left( { - 1} \right) }^t}{e^{ - \left( {1 + t} \right) {\lambda _{S1E}}{x_1}}}}\nonumber \\&\qquad M{\lambda _{S1R}}\sum \limits _{k = 0}^{M - 1} {C_{M - 1}^k} {{\left( { - 1} \right) }^k}\varGamma \left[ {0,\left( {1 + k} \right) {\lambda _{S1R}}\left( {\frac{{\varphi - 1}}{{{\omega _2}\psi }} + \frac{{\varphi {\omega _1}{x_1}}}{{{\omega _2}}}} \right) } \right] d{x_1}\nonumber \\& \mathop \approx \limits ^{\left( {{\mathrm{74}}{\mathrm{.2}}} \right) } \frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi }}{\varOmega _3}\left( {L,\varphi ,{\omega _1},{\omega _2},{\lambda _{S2R}},{\lambda _{S2E}}} \right) \nonumber \\&\qquad \int _0^\infty {LM{\lambda _{S1E}}{\lambda _{S1R}}\sum \limits _{t = 0}^{L - 1} {C_{L - 1}^t} {{\left( { - 1} \right) }^t}\sum \limits _{k = 0}^{M - 1} {C_{M - 1}^k} {{\left( { - 1} \right) }^k}} {e^{ - \left( {1 + t} \right) {\lambda _{S1E}}{x_1}}}\nonumber \\&\qquad \varGamma \left[ {0,\frac{{\left( {1 + k} \right) \varphi {\omega _1}{\lambda _{S1R}}{x_1}}}{{{\omega _2}}}} \right] d{x_1}\nonumber \\& \mathop = \limits ^{\left( {{\mathrm{74}}{\mathrm{.3}}} \right) } \frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi }}{\varOmega _3}\left( {L,\varphi ,{\omega _1},{\omega _2},{\lambda _{S2R}},{\lambda _{S2E}}} \right) \nonumber \\&\qquad LM{\lambda _{S1E}}{\lambda _{S1R}}\sum \limits _{t = 0}^{L - 1} {C_{L - 1}^t{{\left( { - 1} \right) }^t}\sum \limits _{k = 0}^{M - 1} {C_{M - 1}^k} {{\left( { - 1} \right) }^k}} \nonumber \\&\qquad \frac{{{\omega _2}}}{{\left( {1 + t} \right) {\omega _2}{\lambda _{S1E}} + \left( {1 + k} \right) \varphi {\omega _1}{\lambda _{S1R}}}}_2{F_1}\left( {1,1;2;\frac{{\left( {1 + t} \right) {\omega _2}{\lambda _{S1E}}}}{{\left( {1 + t} \right) {\omega _2}{\lambda _{S1E}} + \left( {1 + k} \right) \varphi {\omega _1}{\lambda _{S1R}}}}} \right) \nonumber \\& \buildrel \varDelta \over = \frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi }}{\varOmega _3}\left( {L,\varphi ,{\omega _1},{\omega _2},{\lambda _{S2R}},{\lambda _{S2E}}} \right) \nonumber \\&\qquad {\varOmega _5}\left( {L,M,\varphi ,{\omega _1},{\omega _2},{\lambda _{S1R}},{\lambda _{S1E}}} \right) \end{aligned}$$
(74)

where (74.1) is obtained from by using \(\int _0^\infty {L{\lambda _{S2 E}} \sum \limits _{w = 0}^{L - 1} {C_{L - 1}^w} {{\left( { - 1} \right) }^w}{e^{ - \left[ {\frac{{\left( {1 + w} \right) {\omega _2}{\lambda _{S2E}} + \varphi {\omega _1}{\lambda _{S2R}}}}{{{\omega _2}}}} \right] {x_2}}}} = {\varOmega _3} \left( { L, \varphi , {\omega _1}, {\omega _2},{\lambda _{S2 R}}, {\lambda _{S2 E}}} \right)\) and \(\int _{\frac{{\varphi - 1}}{{{\omega _2}\psi }} + \frac{{\varphi {\omega _1}{x_1}}}{{{\omega _2}}}}^\infty {\frac{{{e^{ - \left( {1 + k} \right) {\lambda _{S1R}}{x_3}}}}}{{{x_3}}}} d{x_3} = \varGamma \left[ {0,\left( {1 + k} \right) {\lambda _{S1R}}\left( {\frac{{\varphi - 1}}{{{\omega _2}\psi }} + \frac{{\varphi {\omega _1}{x_1}}}{{{\omega _2}}}} \right) } \right]\) (see [32, Eq. (3.381.3)]); (74.2) is obtained from (74.1) by approximating \(\frac{{\varphi - 1}}{{{\omega _2}\psi }}\mathop \approx \limits ^{{\mathrm{high}}\,\,\psi \,} 0\); (73.3) is obtained from (74.2) by using the Eq. (6.455.1) of [32] in the case of \(\mu = 1\) and \(v = 0\), as \(\int _0^\infty {{e^{ - \beta x}}} \varGamma \left( {0,\alpha x} \right) dx = \frac{1}{{\alpha + \beta }}_2{F_1}\left( {1,\,1\,;\,2\,;\frac{\beta }{{\alpha + \beta }}} \right).\)

We finish the proof by combining (72), (73), and (74).

Appendix 3: Proof of Lemma 2

The integral \({I_{2,\mathrm{MAS1R}}}\) can be obtained after some steps with using [32, Eq. (3.381.3)] and approximating \(\frac{{\varphi - 1}}{{{\omega _2}\psi }} \approx 0\), and \(\frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi {x_3}}} \approx 0\) when \(\psi\) is high, as follows

$$\begin{aligned} {I_{2,maS1R}} \approx \frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS1}}}}{{{\omega _3}\psi }}\left( {{I_5} + {I_6}} \right) \end{aligned}$$
(75)

where \({I_5}\) and \({I_6}\) are denoted and derived as in (76) and (77), respectively

$$\begin{aligned}&{I_5} \buildrel \varDelta \over = \int _0^\infty {L{\lambda _{S1E}}\sum \limits _{u = 0}^{L - 1} {C_{L - 1}^u} {{\left( { - 1} \right) }^u}{e^{ - \left( {1 + u} \right) {\lambda _{S1E}}{x_1}}}} \nonumber \\&\qquad \int _0^\infty {L{\lambda _{S2E}}{\lambda _{S2R}}\sum \limits _{k = 0}^{L - 1} {C_{L - 1}^k} {{\left( { - 1} \right) }^k}{e^{ - \left( {1 + k} \right) {\lambda _{S2E}}{x_2}}}\varGamma \left[ {0,\frac{{\varphi {\omega _1}{\lambda _{S2R}}{x_2}}}{{{\omega _2}}}} \right] } \nonumber \\&\qquad \int _{\frac{{\varphi - 1}}{{{\omega _2}\psi }} + \frac{{\varphi {\omega _1}{x_1}}}{{{\omega _2}}}}^\infty {{\lambda _{S1R}}{e^{ - {\lambda _{S1R}}{x_3}}}} d{x_3}d{x_2}d{x_1}\nonumber \\& = {\varOmega _4}\left( {L,M,\varphi ,{\omega _1},{\omega _2},\psi ,{\lambda _{S1R}},{\lambda _{S 1 E}}} \right) {\varOmega _5} \left( {L, 1, \varphi , {\omega _1}, {\omega _2},{\lambda _{S 2R}},{\lambda _{S 2E}}} \right) \end{aligned}$$
(76)
$$\begin{aligned}&{I_6} \buildrel \varDelta \over = \frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi }}\int _0^\infty {L{\lambda _{S1E}}\sum \limits _{t = 0}^{L - 1} {C_{L - 1}^t} {{\left( { - 1} \right) }^t}{e^{ - \left( {1 + t} \right) {\lambda _{S1E}}{x_1}}}} \nonumber \\&\qquad \int _0^\infty {L{\lambda _{S 2 E}}{\lambda _{S 2 R}} \sum \limits _{w = 0}^{L - 1} {C_{L - 1}^w} {{\left( { - 1} \right) }^w}{e^{ - \left( {1 + w} \right) {\lambda _{S2E}}{x_2}}}\varGamma \left[ {0,\frac{{\varphi {\omega _1}{\lambda _{S2R}}{x_2}}}{{{\omega _2}}}} \right] } \nonumber \\&\qquad \int _{\frac{{\varphi - 1}}{{{\omega _2}\psi }} + \frac{{\varphi {\omega _1}{x_1}}}{{{\omega _2}}}}^\infty {M{\lambda _{S1R}}\sum \limits _{k = 0}^{M - 1} {C_{M - 1}^k} {{\left( { - 1} \right) }^k}\frac{{{e^{ - \left( {1 + k} \right) {\lambda _{S1R}}{x_3}}}}}{{{x_3}}}} d{x_3}d{x_2}d{x_1}\nonumber \\& = \frac{{ - \left( {\varphi - 1} \right) {\lambda _{RS2}}}}{{{\omega _3}\psi }}{\varOmega _5}\left( {L,M,\varphi ,{\omega _1},{\omega _2},{\lambda _{S1R}},{\lambda _{S1E}}} \right) \nonumber \\&\qquad {\varOmega _5}\left( {L,1,\varphi ,{\omega _1},{\omega _2},{\lambda _{S2R}},{\lambda _{S2E}}} \right) \end{aligned}$$
(77)

We finish the proof by combining (75), (76), and (77).

Appendix 4: Proof of Equation (59.2)

The expression for the probability term \(\Pr \left[ {\min \left( {{C_{{J_1}S1}},{C_{{J_1}S2}}} \right) \ge {C_t}} \right]\) can be obtained as follows:

$$\begin{aligned}&\Pr \left[ {\min \left( {{C_{{J_s}S1}},{C_{{J_s}S2}}} \right) \ge {C_t}} \right] \nonumber \\&\quad = \Pr \left[ {{g_{{J_s}S1}} \ge \frac{{\varphi - 1}}{{{\omega _1}\psi }} + \varphi {g_{{J_s}E\max }}} \right] \Pr \left[ {{g_{{J_s}S2}} \ge \frac{{\varphi - 1}}{{{\omega _1}\psi }} + \varphi {g_{{J_s}E\max }}} \right] \nonumber \\&\quad = {e^{\frac{{ - \left( {\varphi - 1} \right) \left( {{\lambda _{JS1}} + {\lambda _{JS2}}} \right) }}{{{\omega _1}\psi }}}}L{\lambda _{JE}}\sum \limits _{k = 0}^{L - 1} {\frac{{C_{L - 1}^k{{\left( { - 1} \right) }^k}}}{{\left( {1 + k} \right) \left( {{\lambda _{JE}} + {\lambda _{JE}}} \right) + \varphi \left( {{\lambda _{JS1}} + {\lambda _{JS2}}} \right) }}} \nonumber \\&\quad = {e^{\frac{{ - \left( {\varphi - 1} \right) \left( {{\lambda _{JS1}} + {\lambda _{JS2}}} \right) }}{{{\omega _1}\psi }}}}{\varOmega _2}\left( {L,\varphi ,{\lambda _{JS1}},{\lambda _{JS2}},{\lambda _{JE}}} \right) \end{aligned}$$
(78)

By substituting (78) into (59.1), we finish the proof.

Appendix 5: Proof of Lemma 3

By substituting (49), (50), (51), and (5) into (61), we obtain

$$\begin{aligned}&P_{out{\mathrm{,OPT}}}^{{\mathrm{STWJNC,1}}} = 1 - \Pr \left[ \begin{array}{l} {g_{S1{R_s}}} \ge \frac{{\varphi - 1}}{{{\omega _1}\psi }},{g_{S2{R_s}}} \ge \frac{{\varphi - 1}}{{{\omega _1}\psi }},\\ {g_{{R_s}S1}} \ge \frac{{\varphi - 1}}{{{\omega _4}\psi {g_{{J_s}{R_s}}}}},{g_{{R_s}S2}} \ge \frac{{\varphi - 1}}{{{\omega _4}\psi {g_{{J_s}{R_s}}}}} \end{array} \right] \nonumber \\&\quad = 1 - \int _{\frac{{\varphi - 1}}{{{\omega _1}\psi }}}^\infty {{f_{{g_{S1{R_s}}}}}\left( {{x_1}} \right) } \int _{\frac{{\varphi - 1}}{{{\omega _1}\psi }}}^\infty {{f_{{g_{S2{R_s}}}}}\left( {{x_2}} \right) } \int _0^\infty {{f_{{g_{{J_s}{R_s}}}}}\left( {{x_3}} \right) } \nonumber \\&\quad \int _{\frac{{\varphi - 1}}{{{\omega _4}\psi {x_3}}}}^\infty {{f_{{g_{{R_s}S1}}}}\left( {{x_4}} \right) } \int _{\frac{{\varphi - 1}}{{{\omega _4}\psi {x_3}}}}^\infty {{f_{{g_{{R_s}S2}}}}\left( {{x_5}} \right) } d{x_5}d{x_4}d{x_3}d{x_2}d{x_1} \end{aligned}$$
(79)

From the optimal jammer and relay selection strategy, in (48a), the CDF of RV \({g_{{J_s}{R_s}}}\) is expressed as

$$\begin{aligned} {F_{{g_{{J_s}{R_s}}}}}\left( {{x_3}} \right) = \Pr \left[ {\mathop {\mathop {\max }\limits _{\scriptstyle \,k = 1,2,\ldots ,K}}\limits _{\, m= 1,2,\ldots ,M} } \left( {{g_{ {J_k} {R_m}}}} \right) < {x_3} \right] = {\left( {1 - {e^{ - {\lambda _{JR}}{x_3}}}} \right) ^{KM}} \end{aligned}$$
(80)

By substituting (80), \({f_{{g_{S1{R_s}}}}}( {{x_1}} ) = {\lambda _{S1R}}{e^{ - {\lambda _{S1R}}{x_1}}}\), \({f_{{g_{S2{R_s}}}}}\left( {{x_2}} \right) = {\lambda _{S2R}}{e^{ - {\lambda _{S2R}}{x_2}}}\), \({f_{{g_{{J_s}{R_s}}}}}\left( {{x_3}} \right) = KM{\lambda _{JR}}\sum \nolimits _{k = 0}^{KM - 1} {C_{KM - 1}^k{{\left( { - 1} \right) }^k}{e^{ - \left( {1 + k} \right) {\lambda _{JR}}{x_3}}}}\), \({f_{{g_{{R_s}S1}}}}\left( {{x_4}} \right) = {\lambda _{RS1}}{e^{ - {\lambda _{RS1}}{x_4}}}\), and \({f_{{g_{{R_s}S2}}}}\left( {{x_5}} \right) = {\lambda _{RS2}}{e^{ - {\lambda _{RS2}}{x_5}}}\) into (79), we can obtain

$$\begin{aligned} P_{out{\mathrm{,opt}}}^{{\mathrm{STWJNC,1}}}& = 1 - {e^{\frac{{ - \left( {\varphi - 1} \right) \left( {{\lambda _{S1R}} + {\lambda _{S2R}}} \right) }}{{{\omega _1}\psi }}}}\nonumber \\&\quad \int _0^\infty {KM{\lambda _{JR}} \sum \limits _{k = 0}^{KM - 1} {C_{KM - 1}^k{{\left( { - 1} \right) }^k}{e^{ - \left( {1 + k} \right) {\lambda _{JR}}{x_3}}}{e^{\frac{{ - \left( {\varphi - 1} \right) \left( {{\lambda _{RS1}} + {\lambda _{RS2}}} \right) }}{{{\omega _4}\psi {x_3}}}}}} } d{x_3} \end{aligned}$$
(81)

From [32, Eq. (3.381.1)], \(\int _0^\infty {{e^{ - \beta /4x - \gamma x}}dx} = \sqrt{\beta /\gamma } {K_1}\left( {\sqrt{\beta \gamma } } \right)\), we obtain

$$\begin{aligned}&\int _0^\infty {{e^{ - \left( {1 + k} \right) {\lambda _{JR}}{x_3}}}{e^{\frac{{ - \left( {\varphi - 1} \right) \left( {{\lambda _{RS1}} + {\lambda _{RS2}}} \right) }}{{{\omega _4}\psi {x_3}}}}}} d{x_3} = \sqrt{\frac{{4\left( {\varphi - 1} \right) \left( {{\lambda _{RS1}} + {\lambda _{RS2}}} \right) }}{{\left( {1 + k} \right) {\omega _4}\psi {\lambda _{JR}}}}} \nonumber \\&\quad {K_1}\left( {\sqrt{\frac{{4\left( {1 + k} \right) \left( {\varphi - 1} \right) \left( {{\lambda _{RS1}} + {\lambda _{RS2}}} \right) {\lambda _{JR}}}}{{{\omega _4}\psi }}} } \right) \end{aligned}$$
(82)

By substituting (82) into (81), we complete the proof.

Appendix 6: Proof of Lemma 4

By substituting (54), (55), (56), and (57) into (62), we obtain

$$\begin{aligned} P_{out{\mathrm{,opt}}}^{{\mathrm{S T W J NC,2}}}& = 1 - \Pr \left[ \begin{array}{l} {g_{S1{R_s}}} \ge \frac{{\varphi - 1}}{{{\omega _1}\psi }} + \varphi {g_{S1E\max }},{g_{S2{R_s}}} \ge \frac{{\varphi - 1}}{{{\omega _1}\psi }} + \varphi {g_{S2E\max }},\\ {g_{{R_s}S 1}} \ge \frac{{\varphi - 1}}{{{\omega _4}\psi {g_{{J_s}{R_s}}}}} + \varphi {g_{{R_s} E \max }},{g_{{R_s}S 2}} \ge \frac{{\varphi - 1}}{{{\omega _4}\psi {g_{{J_s}{R_s}}}}} + \varphi {g_{{R_s} E \max }} \end{array} \right] \nonumber \\& = 1 - \int _0^\infty {{f_{{g_{S1 E \max }}}} \left( {{x_1}} \right) \int _0^\infty {{f_{{g_{S2E \max }}}} \left( {{x_2}} \right) } \int _{\frac{{\varphi - 1}}{{{\omega _1}\psi }} + \varphi {x_1}}^\infty {{f_{{g_{S1{R_s}}}}} \left( {{x_3}} \right) \int _{\frac{{\varphi - 1}}{{{\omega _1}\psi }} + \varphi {x_2}}^\infty {{f_{{g_{S2{R_s}}}}} \left( {{x_4}} \right) } } } \nonumber \\&\quad \int _0^\infty {{f_{{g_{{J_s}{R_s}}}}} \left( {{x_5}} \right) \int _0^\infty {{f_{{g_{{R_s}E \max }}}} \left( {{x_6}} \right) \int _{\frac{{\varphi - 1}}{{{\omega _4}\psi {x_5}}} + \varphi {x_6}}^\infty {{f_{{g_{{R_s}S1}}}} \left( {{x_7}} \right) \int _{\frac{{\varphi - 1}}{{{\omega _4}\psi {x_5}}} + \varphi {x_6}}^\infty {{f_{{g_{{R_s} S 2}}}} \left( {{x_8}} \right) } } } } \nonumber \\&\quad {d{x_8}d{x_7}d{x_6}d{x_5}d{x_4}d{x_3}d{x_2}d{x_1}} \end{aligned}$$
(83)

By substituting the PDFs of the eight RVs \({{g_{S1E\max }}}\), \({S2E\max }\), \({S1{R_s}}\), \({S2{R_s}}\), \({{g_{{J_s}{R_s}}}}\), \({{g_{{R_s}E\max }}}\), \({{g_{{R_s}S1}}}\), and \({{g_{{R_s}S2}}}\), into (83) and after some manipulations of (83), the Eq. (64) in Lemma 4 is obtained. This completes the proof.

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Nguyen, S.Q., Kong, H.Y. Improving Secrecy Outage and Throughput Performance in Two-Way Energy-Constraint Relaying Networks Under Physical Layer Security. Wireless Pers Commun 96, 6425–6457 (2017). https://doi.org/10.1007/s11277-017-4485-8

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