# Physical-Layer Security Analysis over M-Distributed Fading Channels
## 3 Secrecy Performance Analysis
### Probability of SPSC Analysis  
In physical layer secure communications, the probability of SPSC is often used as a fundamental benchmark, which is used to emphasize the probability of the existence of the secrecy capacity. In this paper, the probability of SPSC is given by setting \(\gamma_{\rm th}=0\) in (12). Therefore, the following theorem is obtained.  
**Theorem 3**.: _For the M-distributed fading channels (i.e., \(H_{n}\sim\mathcal{M}(\alpha_{n},\beta_{n},\rho_{n},\Omega_{n},\xi_{n},\phi_{1,n },\phi_{2,n})\), the closed-form expression of the probability of SPSC is given by_  
\[P_{\rm SPSC}\!\!=\!\!1\!-\  
By employing (21) in [24], (30) can be further written as  
\[C_{3}=\frac{A_{\rm E}}{8\pi}\sum_{k=1}^{\beta_{\rm E}}a_{k,{\rm E}}\bigg{(}\frac{ \sigma_{\rm E}}{\sqrt{P}}\bigg{)}^{\frac{\alpha_{\rm E}+k}{2}}G_{2,6}^{6,1} \bigg{[}\frac{\left(\alpha_{\rm E}\beta_{\rm E}\sigma_{\rm E}\right)^{2}}{16P \big{(}\xi_{g,{\rm E}}\beta_{\rm E}+\Omega_{\rm E}^{\prime}\big{)}^{2}}\bigg{|} \frac{\alpha_{\rm E}-k}{4},\frac{\alpha_{\rm E}-k+2}{4},\frac{k-\alpha_{\rm E }}{4},\frac{2-\alpha_{\rm E}+k}{4},-\frac{\alpha_{\rm E}+k}{4},-\frac{\alpha_ {\rm E}+k}{4}\bigg{]} \tag{31}\]  
Then, substituting (27), (28) and (31), (25) can be derived.  
It can be seen from _Theorem 4_ that the exact expression of the ESC (25) is a very complex integral expression. It is very hard to obtain a closed-form expression for the ESC. Alternatively, a lower bound of the ESC is derived in the following theorem.  
**Theorem 5**: _For the M-distributed fading channels (i.e., \(H_{n}\sim\mathcal{M}(\alpha_{n},\beta_{n},\rho_{n},\Omega_{n},\xi_{n},\phi_{1,n},\phi_{2,n})\), a closed-form expression for the lower bound of the ESC is given by_  
\[C_{\rm Erg}^{\rm L} =\Bigg{\{}\frac{A_{\rm B}}{8\pi}\sum_{j=1}^{\beta_{\rm B}}a_{j,{ \rm B}}\bigg{(}\frac{\sigma_{\rm B}}{\sqrt{P}}\bigg{)}^{\frac{\alpha_{\rm E}+ j}{2}}G_{2,6}^{6,1}\bigg{[}\frac{\left(\alpha_{\rm B}\beta_{\rm B}\sigma_{\rm B }\right)^{2}}{16P\big{(}\xi_{g,{\rm B}}\beta_{\rm B}+\Omega_{\rm E}^{\prime} \big{)}^{2}}\bigg{|}\frac{\alpha_{\rm B}-j}{4},\frac{\alpha_{\rm B}-j+2}{4}, \frac{j-4}{4},\frac{2-\alpha_{\rm B}+j}{4},-\frac{\alpha_{\rm B}+j}{4},-\frac {\alpha_{\rm B}+j}{4}\bigg{]}\] \[-\frac{A_{\rm E}}{8\pi}\sum_{k=1}^{\beta_{\rm E}}a_{k,{\rm E}} \bigg{(}\frac{\sigma_{\rm E}}{\sqrt{P}}\bigg{)}^{\frac{\alpha_{\rm E}+k}{2}}G_ {2,6}^{6,1}\bigg{[}\frac{\left(\alpha_{\rm E}\beta_{\rm E}\sigma_{\rm E} \right)^{2}}{16P\big{(}\xi_{g,{\rm E}}\beta_{\rm E}+\Omega_{\rm E}^{\prime} \big{)}^{2}}\bigg{|}\frac{\alpha_{\rm E}-k}{4},\frac{\alpha_{\rm E}-k+2}{4}, \frac{k-\alpha_{\rm E}}{4},\frac{2-\alpha_{\rm E}+k}{4},-\frac{\alpha_{\rm E} +k}{4},-\frac{\alpha_{\rm E}+k}{4}\bigg{]}\Bigg{\}}^{+} \tag{32}\]  
According to (31), the ergodic capacity for Alice-Bob link is given by  
\[C_{1,{\rm Erg}}=\frac{A_{\rm B}}{8\pi}\sum_{j=1}^{\beta_{\rm B}}a_{j,{\rm B}} \bigg{(}\frac{\sigma_{\rm B}}{\sqrt{P}}\bigg{)}^{\frac{\alpha_{\rm B}+j}{2}}G_ {2,6}^{6,1}\bigg{[}\frac{\left(\alpha_{\rm B}\beta_{\rm B}\sigma_{\rm B} \right)^{2}}{16P\big{(}\xi_{g,{\rm B}}\beta_{\rm B}+\Omega_{\rm E}^{\prime} \big{)}^{2}}\bigg{|}\frac{\alpha_{\rm B}-j}{4},\frac{\alpha_{\rm B}+j+2}{4}, \frac{\frac{j-4}{4},\frac{2-\alpha_{\rm B}+j}{4},-\frac{\alpha_{\rm B}+j}{4}, \frac{\alpha_{\rm B}+j}{4}}\bigg{]} \tag{33}\]  
Similarly, the ergodic capacity for Alice-Eve link is \(C_{3}\) in (31).  
Referring to [25], a lower bound of the ESC is given by  
\[C_{\rm Erg}\geq\big{\{}C_{1,{\rm Erg}}-C_{2,{\rm Erg}}\big{\}}^{+} \tag{34}\]  
Substituting (33) and (31) into (34), (32) can be derived.