## 4 Reliable and Secure Probability Analyses
### Overall Reliable-and-Security of Probability  
Given that the second-phrase transmission can be modeled as the CEO problem, let \(P_{n}\) (where \(n\in\{1,2,3,4\}\)) denote the probability of \((R_{\text{UR}_{1}}^{\mathscr{D}},R_{\text{UR}_{2}}^{\mathscr{D}})\) residing in region \(n\), as depicted in Figure 5. The probability \(P_{\text{out}}^{\text{D}}\) can be determined by due to the block fading assumption leading to varying \(e_{1}^{\prime}\) and \(e_{2}^{\prime}\) values from one block to another (the Berger-Tung bound is not employed as it necessitates a specified distortion level). \[P_{\text{out}}^{\text{D}}=1-\sum_{i=1}^{4}P_{i}. \tag{28}\]  
It should be highlighted that an outage occurs only if the resulting distortion \(\delta\) surpasses \(\min\{e_{1}^{\prime},e_{2}^{\prime}\}\). Regions 3 and 4 in Figure 5 can be regarded as permissible regions, following the definition of outage probability given by (27), especially when \(R_{\text{UR}_{1}}^{\mathscr{D}}\geq H(x_{\text{UR}_{1}})\) and \(R_{\text{UR}_{2}}^{\mathscr{D}}\geq H(x_{\text{UR}_{2}})\). Factoring in the link between rate \(R_{\text{UR}_{k}}^{\mathscr{D}}\) and the associated real-time channel SNR \(\gamma_{\text{UR}\text{-D}}\), as well as the impact from the fluctuations of each link, the outage probability \(P_{n}\) (with \(n\in\{1,2,3,4\}\)) is determined by averaging across all second-stage transmissions. This gives us the following:  
\[P_{1}= \Pr\Bigl{\{}H(x_{\text{UR}_{1}}|x_{\text{UR}_{2}})<R_{\text{UR} _{1}}^{\mathscr{D}}<H(x_{\text{UR}_{1}}),R_{\text{UR}_{1}}^{\mathscr{D}}+R_{ \text{UR}_{2}}^{\mathscr{D}}>H(x_{\text{UR}_{1}},x_{\text{UR}_{2}})\Bigr{\}}\] \[= \int_{2\kappa_{\text{UR}_{1}\text{-D}}^{2}R_{\text{UR}_{1}\text{ -D}}}^{2\kappa_{\text{UR}_{1}\text{-D}-1}}f\bigl{(}\gamma_{\text{UR}_{1}\text{ -D}}\bigr{)}d\gamma_{\text{UR}_{1}\text{-D}}\] \[\int_{2\kappa_{\text{UR}_{2}\text{-D}}^{\infty}\Bigl{[}1+H(x_{ \text{UR}_{1}}|x_{\text{UR}_{2}})^{-\frac{\log(1+\gamma_{\text{UR}_{1}\text{ -D}})}{R_{\text{UR}_{1}\text{-D}}}\Bigr{]}}_{-1}}^{\infty}f\bigl{(}\gamma_{ \text{UR}_{2}\text{-D}}\bigr{)}d\gamma_{\text{UR}_{2}\text{-D}}\] \[= \int_{2\kappa_{\text{UR}_{1}\text{-D}}^{2}R_{\text{UR}_{1}\text{ -D}}}^{2\kappa_{\text{UR}_{1}\text{-D}}^{2}}\underbrace{\frac{m_{\text{UR}_{1} \text{-D}}\Bigl{[}2^{\kappa_{\text{UR}_{2}\text{-D}}\Bigl{(}1+H(x_{\text{UR}_ {1}}|x_{\text{UR}_{2}})^{-\frac{\log(1+\gamma_{\text{UR}_{1}\text{-D}})}{R_{ \text{UR}_{1}\text{-D}}}\Bigr{)}}_{-1}\Bigr{]}}{\Gamma(m_{\text{US-UR}_{2}})}}\] \[\frac{m_{\text{UR}_{1}\text{-D}}^{m_{\text{UR}_{1}\text{-D}}}}{ \Gamma(m_{\text{UR}_{1}\text{-D}})\gamma_{\text{UR}_{1}\text{-D}}^{m_{\text{UR }_{1}\text{-D}}}}^{m_{\text{UR}_{1}\text{-D}}}\exp\left(-\frac{m_{\text{UR}_{1} \text{-D}}\gamma_{\text{UR}_{1}\text{-D}}}{\overline{\gamma}_{\text{UR}_{1} \text{-D}}}\right)d\gamma_{\text{UR}_{1}\text{-D}}, \tag{29}\]  
\((1-a)bE_{T}\), and \((1-a)(1-b)E_{T}\), respectively. \(a\) ( \(0<a\leq 1\)) and \(b\) ( \(0\leq b\leq 1\)) are the power allocation ratios. Note that due to the symmetry of the two untrusted relays, we consider (UR\({}_{1}\) and UR\({}_{2}\)) as a whole in the power allocation investigation.