On Physical Layer Security Over Fox's $H$-Function Wiretap Fading Channels

Most of the well-known fading distributions, if not all of them, could be encompassed by the Fox's <inline-formula><tex-math notation="LaTeX">$H$</tex-math></inline-formula>-function fading. Consequently, we investigate the physical layer security over Fox's <inline-formula><tex-math notation="LaTeX">$H$</tex-math></inline-formula>-function fading wiretap channels, in the presence of non-colluding and colluding eavesdroppers. In particular, for the non-colluding scenario, closed-form expressions are derived for the secrecy outage probability (SOP), the probability of non-zero secrecy capacity, and the average secrecy capacity. These expressions are given in terms of either univariate or bivariate Fox's <inline-formula><tex-math notation="LaTeX">$H$</tex-math></inline-formula>-function. In order to show the effectiveness of our derivations, three metrics are respectively listed over the following frequently used fading channels, including Rayleigh, Weibull, Nakagami-<inline-formula><tex-math notation="LaTeX">$m$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$\alpha -\mu$</tex-math></inline-formula>, Fisher-Snedecor (F-S) <inline-formula><tex-math notation="LaTeX">$\mathcal {F}$</tex-math></inline-formula>, and extended generalized-<inline-formula><tex-math notation="LaTeX">$\mathcal {K}$</tex-math></inline-formula>. Our tractable results are not only straightforward and general, but also feasible and applicable, especially the SOP, which is usually limited to the lower bound in the literature due to the difficulty of deriving closed-from analytical expressions. For the colluding scenario, a super eavesdropper equipped with maximal ratio combining (MRC) or selection-combining (SC) schemes is characterized. The lower bound of SOP and exact PNZ are thereafter derived with closed-form expressions in terms of the multivariate Fox's <inline-formula><tex-math notation="LaTeX">$H$</tex-math></inline-formula>-function. In order to validate the accuracy of our analytical results, Monte Carlo simulations are subsequently conducted for the aforementioned fading channels. One can observe that for the former non-colluding scenario, we have perfect agreement between the exact analytical and simulation results, and highly accurate approximations between the exact and asymptotic analytical results. On the contrary, the SOP and PNZ of colluding eavesdropper is greatly degraded with the increase of the number of eavesdroppers. Also, the so-called super eavesdropper with MRC is much powerful to wiretap the main channel than the one with SC.

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