Combining Binary Jamming and Network Coding to Improve Outage Performance in Two-Way Relaying Networks under Physical Layer Security

Abstract

By generating a jamming message at the jammer node, a system can reduce wiretapping in the physical layer because this message can refuse the illegal eavesdropper node. The network coding technique of operating an XOR between two binary messages improves the performance. In this paper, we propose three protocols that use network coding at the two source nodes and/or the binary jamming technique at a relay in the two-way relaying network under physical layer security, compared to a conventional secrecy transmission protocol. The main idea to improve the system performance is that, if the data is transmitted securely, the next transmission time slots using the digital network coding will not consider the presence of the eavesdropper node, because the eavesdropper node cannot obtain the data. The system performance is analyzed and evaluated in terms of the exact closed-form outage probability over Rayleigh fading channels. The simulation results using a Monte-Carlo simulation are in complete agreement with the theoretical results.

Appendices

Appendix 1: Solving the probability $\Pr [ {AS{R_{SR}} > {R_t}} ]$ in (32)

From (5), we can calculate $\Pr [ {AS{R_{SR}} > {R_t}} ]$ as follows

$$\begin{aligned} \Pr \left[ {AS{R_{SR}} > {R_t}} \right]= \Pr \left[ {\frac{1}{3}{{\log }_2}\left( {\frac{{1 + {\gamma _S}{{\left| {{h_1}} \right| }^2}}}{{1 + {\gamma _S}{{\left| {{h_2}} \right| }^2}}}} \right) > {R_t}} \right] \nonumber \\= \Pr \left[ {{{\left| {{h_1}} \right| }^2} > \frac{{{2^{3{R_t}}} - 1}}{{{\gamma _S}}} + {2^{3{R_t}}}{{\left| {{h_2}} \right| }^2}} \right] \end{aligned}$$

(48)

Let ${X_l}$ be i.i.d. exponential RVs ${| {{h_l}} |^2}$, $l \in { {1,2,3,4,5} }$. Then, the probability density function (PDF) of ${X_l}$ is given by

$$\begin{aligned} {f_{{X_l}}}({x_l}) = {\lambda _l}{e^{ - {\lambda _l}{x_l}}} \end{aligned}$$

(49)

Substituting (49) into (48), we obtain

$$\begin{aligned} \Pr \left[ {AS{R_{SR}} > {R_t}} \right]= & {} \int _0^\infty {{f_{{X_2}}}} ({x_2})\int _{\frac{{{2^{3{R_t}}} - 1}}{{{\gamma _S}}} + {2^{3{R_t}}}{x_2}}^\infty {{f_{{X_1}}}} ({x_1})d{x_1}d{x_2}\nonumber \\= & {} \int _0^\infty {{\lambda _2}{e^{ - {\lambda _2}{x_2}}}} \int _{\frac{{{2^{3{R_t}}} - 1}}{{{\gamma _S}}} + {2^{3{R_t}}}{x_2}}^\infty {{\lambda _1}{e^{ - {\lambda _1}{x_1}}}} d{x_1}d{x_2}\nonumber \\= & {} \int _0^\infty {{\lambda _2}{e^{ - {\lambda _2}{x_2}}}} {e^{ - {\lambda _1}(\frac{{{2^{3{R_t}}} - 1}}{{{\gamma _S}}} + {2^{3{R_t}}}{x_2})}}d{x_1}\nonumber \\= & {} \frac{{{\lambda _2}{e^{ - \frac{{\left( {{2^{3{R_t}}} - 1} \right) {\lambda _1}}}{{{\gamma _S}}}}}}}{{{\lambda _2} + {2^{3{R_t}}}{\lambda _1}}} \end{aligned}$$

(50)

Appendix 2: Solving the Probability $\Pr [ {{C_{RS}} > {R_t}} ]$ in (34)

From (14), we can calculate $\Pr [ {{C_{RS}} > {R_t}} ]$ as follows

$$\begin{aligned} \Pr \left[ {{C_{RS}} > {R_t}} \right]= & {} \Pr \left[ {{{\left| {{h_1}} \right| }^2} > \frac{{{2^{3{R_t}}} - 1}}{{{\gamma _R}}}} \right] \nonumber \\= & {} \int _{\frac{{{2^{3{R_t}}} - 1}}{{{\gamma _R}}}}^\infty {{\lambda _1}{e^{ - {\lambda _1}{x_1}}}} d{x_1}\nonumber \\= &\, {e^{ - \frac{{\left( {{2^{3{R_t}}} - 1} \right) {\lambda _1}}}{{{\gamma _R}}}}} \end{aligned}$$

(51)

Appendix 3: Solving the Probability $\Pr [ {AS{R_{RS}} > {R_t},AS{R_{RD}} > {R_t}} ]$ in (36)

From (19) and (20), we can express $\Pr [ {AS{R_{RS}} > {R_t},AS{R_{RD}} > {R_t}} ]$ as follows

$$\begin{aligned} \Pr \left[ {AS{R_{RS}} > {R_t},AS{R_{RD}} > {R_t}} \right]= & {} \Pr \left[ \begin{array}{l} \frac{1}{5}{\log _2}\left( {\frac{{1 + {\gamma _R}{{\left| {{h_1}} \right| }^2}}}{{1 + {\gamma _R}{{\left| {{h_5}} \right| }^2}}}} \right) > {R_t},\\ \left[ {\frac{1}{5}{{\log }_2}\left( {\frac{{1 + {\gamma _R}{{\left| {{h_3}} \right| }^2}}}{{1 + {\gamma _R}{{\left| {{h_5}} \right| }^2}}}} \right) } \right] > {R_t} \end{array} \right] \nonumber \\= & {} \Pr \left[ \begin{array}{l} {\left| {{h_1}} \right| ^2} > \frac{{{2^{5{R_t}}} - 1}}{{{\gamma _R}}} + {2^{5{R_t}}}{\left| {{h_5}} \right| ^2},\\ {\left| {{h_3}} \right| ^2} > \frac{{{2^{5{R_t}}} - 1}}{{{\gamma _R}}} + {2^{5{R_t}}}{\left| {{h_5}} \right| ^2} \end{array} \right] \end{aligned}$$

(52)

Substituting (49) into (52), we obtain

$$\begin{aligned} \Pr \left[ {AS{R_{RS}} > {R_t},AS{R_{RD}} > {R_t}} \right]= & {} \int _0^\infty {{f_{{X_5}}}} ({x_5})\int _{\frac{{{2^{5{R_t}}} - 1}}{{{\gamma _R}}} + {2^{5{R_t}}}{x_5}}^\infty {{f_{{X_3}}}} ({x_3})\int _{\frac{{{2^{5{R_t}}} - 1}}{{{\gamma _R}}} + {2^{5{R_t}}}{x_5}}^\infty {{f_{{X_1}}}} ({x_1})d{x_1}d{x_3}d{x_5}\nonumber \\= & {} \int _0^\infty {{\lambda _5}{e^{ - {\lambda _5}{x_5}}}} \int _{\frac{{{2^{5{R_t}}} - 1}}{{{\gamma _R}}} + {2^{5{R_t}}}{x_5}}^\infty {{\lambda _3}{e^{ - {\lambda _3}{x_3}}}} \int _{\frac{{{2^{5{R_t}}} - 1}}{{{\gamma _R}}} + {2^{5{R_t}}}{x_5}}^\infty {{\lambda _1}{e^{ - {\lambda _1}{x_1}}}} d{x_1}d{x_3}d{x_5}\nonumber \\= & {} \int _0^\infty {{\lambda _5}{e^{ - {\lambda _5}{x_5}}}} {e^{ - {\lambda _3}\left( {\frac{{{2^{5{R_t}}} - 1}}{{{\gamma _R}}} + {2^{5{R_t}}}{x_5}} \right) }}{e^{ - {\lambda _1}\left( {\frac{{{2^{5{R_t}}} - 1}}{{{\gamma _R}}} + {2^{5{R_t}}}{x_5}} \right) }}d{x_5}\nonumber \\= & {} \frac{{{\lambda _5}{e^{ - \left( {{2^{5{R_t}}} - 1} \right) \left( {\frac{{{\lambda _3} + {\lambda _1}}}{{{\gamma _R}}}} \right) }}}}{{{\lambda _5} + {2^{5{R_t}}}\left( {{\lambda _3} + {\lambda _1}} \right) }} \end{aligned}$$

(53)