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Physical Layer Security of Ground-to-UAV Communication in the Presence of an Aerial Eavesdropper Outside the Guard Zone

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Abstract

In this work, we determine the physical layer security (PLS) metrics for ground-to-unmanned aerial vehicle (UAV) link in the presence of a suspicious UAV, which intends to overhear confidential transmission. The ground node sends data up to the UAV, which is positioned exactly above the transmitter and is able to spot an aerial eavesdropper according to the predefined horizontal and/or vertical guard distance. However, the attacker tries to intercept the channel transmission outside of marked zone. The main and wiretap channel are both assumed to be subjected to Fisher-Snedecor fading process. Under such system/channel scenario, the average secrecy capacity, lower bound of secrecy outage probability, intercept probability and non-zero secrecy capacity are characterized in terms of mathematical tractable forms. In addition, numerical and simulation results are presented to verify the correctness of theoretical ones. The impact of different positions of UAVs (their heights and mutual distances) and various conditions over channels on the secrecy transmission is analysed and discussed in details. The proposed PLS scenario can be utilized in Internet of Things environments, with UAV as a data collector, to enhance the security of energy-aware or disaster-stricken transmissions. All obtained results can be helpful in prediction of positioning UAV to achive low probability of interception of confidential communication.

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Availability of data and materials

The datasets generated during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work was supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia (Contract No. 451-03-65/2024-03/200102).

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The authors declare that no funds or grants were received during the preparation of this manuscript.

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Contributions

All authors contributed to the study conceptualization and design. Material preparation, data collection and analysis were performed by Jelena Anastasov, Aleksandra Cvetković and Aleksandra Panajotović. Jelena Anastasov and Aleksandra Cvetković did the analytical derivations, Daniela Milović and Aleksandra Panajotović prepared numerical results and figures; simulations were performed by Dejan Milić and Nenad Milošević, as well as validation. The first draft of the manuscript was written by Jelena Anastasov and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Jelena Anastasov.

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Appendix A: Derivation of Asymptotic ASC

Appendix A: Derivation of Asymptotic ASC

In order to present solution of (11) in more tractable way, we derive asymptotic version of ASC at high SNR regime, i.e. when \(\lambda \rightarrow \infty\).

More specific, at high SNR region, (11) can be rewritten as [25, eq.(20)]

$$\begin{aligned} {\hat{\bar{C}}_s} = \hat{\mathfrak {I}}_1+\hat{\mathfrak {I}}_2 -\mathfrak {I}_3, \end{aligned}$$
(A1)

where \(\hat{\mathfrak {I}}_1\) and \(\hat{\mathfrak {I}}_2\) denotes asymptotic versions of \(\mathfrak {I}_1\) and \(\mathfrak {I}_2\), respectively. Integrals \(\hat{\mathfrak {I}}_1\) and \(\hat{\mathfrak {I}}_2\) can be obtained with the help of asymptotic expressions of cdfs for instantaneous SNRs of main and wiretap channel. Namely, we recall the series expansion of Meijer’s G function when its argument tends to 0 [27, eq. (07.34.06.0001.01)], which corresponds to high SNR regime, and we take only the first term in expansion to obtain asymptotic behavior of (8) and (3), respectively, in the following way

$$\begin{aligned} \hat{F}_{\gamma _E} (\gamma )&= \dfrac{2}{\Gamma (m_E) \Gamma (k_E) (R_2^2 - R_1^2) \xi _E} \dfrac{\Gamma (k_E + m_E)}{(\frac{2}{\xi _E} + m_E)m_E} \nonumber \\&\Bigg [ (H_E^2 + R_2^2) \bigg ({\dfrac{m_E (H_E^2 + R_2^2)^{\xi _E / 2}}{k_E \bar{\gamma }_E} \gamma }\bigg )^{m_E} - (H_E^2 + R_1^2) \bigg ({\dfrac{m_E (H_E^2 + R_1^2)^{\xi _E / 2} }{k_E \bar{\gamma }_E} \gamma }\bigg )^{m_E} \Bigg ] \end{aligned}$$
(A2)

and

$$\begin{aligned} \hat{F}_{\gamma _M} (\gamma ) = \dfrac{\Gamma (k_M + m_M)}{\Gamma (m_M) \Gamma (k_M) m_M} \bigg ({\dfrac{m_M H_M^{\xi _M} }{k_M \bar{\gamma }_M} \gamma }\bigg )^{m_M}. \end{aligned}$$
(A3)

Then, by substituting (A2) and (2) in \(\mathfrak {I}_1\), and with the help of [27, eqs. (01.05.26.0002.01) and (07.34.21.0013.01)], after some mathematical manipulations, we get the following form of \(\hat{\mathfrak {I}}_1\)

$$\begin{aligned} \hat{\mathfrak {I}}_1 =&\dfrac{2}{\Gamma (m_E) \Gamma (k_E) \Gamma (m_M) \Gamma (k_M) (R_2^2 - R_1^2) \xi _E \log (2)} \dfrac{\Gamma (k_E + m_E)}{(\frac{2}{\xi _E} + m_E)m_E} \nonumber \\&\times G^{\,3,2}_{3,3}\left( \, \dfrac{m_M H_M^{\xi _E / 2}}{k_M \bar{\gamma }_M}\left| \begin{array}{c} 1 - k_M, - m_E, 1 - m_E\\ m_M, - m_E, - m_E\end{array}\,\right. \right) \nonumber \\&\times \Bigg [ (H_E^2 + R_2^2)\bigg ({\dfrac{m_E (H_E^2 + R_2^2)^{\xi _E / 2} }{k_E \bar{\gamma }_E}}\bigg )^{m_E} - (H_E^2 + R_1^2) \bigg ({\dfrac{m_E (H_E^2 + R_1^2)^{\xi _E / 2} }{k_E \bar{\gamma }_E}}\bigg )^{m_E} \Bigg ]. \end{aligned}$$
(A4)

One can notice the solution of this integral in a form of Meijer’s G function. These functions are built-in for most of the common mathematical packages. Unfortunately, this is not the case for the bivariate Meijer’s G functions that are more complex. Also, by substituting (A3) and (7) in \(\mathfrak {I}_2\), and again recalling [27, eq. (01.05.26.0002.01)] and [27, eq. (07.34.21.0013.01)], we get

$$\begin{aligned} \hat{\mathfrak {I}}_2 =&\dfrac{2 \Gamma (k_M + m_M)}{\Gamma (m_E) \Gamma (k_E) \Gamma (m_M) \Gamma (k_M) m_M (R_2^2 - R_1^2) \xi _E \log (2)} \bigg ({\dfrac{m_M H_M^{\xi _M} }{k_M \bar{\gamma }_M} }\bigg )^{m_M} \nonumber \\&\times \Bigg [ (H_E^2 + R_2^2) G^{\,3,2}_{3,3}\left( \, {\dfrac{m_E (H_E^2 + R_2^2)^{\xi _E / 2} }{k_E \bar{\gamma }_E}}\left| \begin{array}{c} 1 - k_E, 1-\frac{2}{\xi _E},- m_M, 1 - m_M\\ m_E, - m_M, - m_M, -\frac{2}{\xi _E}\end{array}\,\right. \right) \nonumber \\&- (H_E^2 + R_1^2) G^{\,3,2}_{3,3}\left( \, {\dfrac{m_E (H_E^2 + R1^2)^{\xi _E / 2} }{k_E \bar{\gamma }_E}}\left| \begin{array}{c} 1 - k_E, 1-\frac{2}{\xi _E},- m_M, 1 - m_M\\ m_E, - m_M, - m_M, -\frac{2}{\xi _E}\end{array}\,\right. \right) \Bigg ]. \end{aligned}$$
(A5)

Thus, the final asymptotic form of ASC, in high SNR region, can be obtained by using derived equations (A4), (A5) and (14).

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Anastasov, J., Cvetković, A., Panajotović, A. et al. Physical Layer Security of Ground-to-UAV Communication in the Presence of an Aerial Eavesdropper Outside the Guard Zone. Wireless Pers Commun 138, 1597–1614 (2024). https://doi.org/10.1007/s11277-024-11562-w

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