## 4 Reliable and Secure Probability Analyses
### Analysis of Error Probability in Source-Relay Links  
Given that transmission errors are permissible within the links from the source to untrusted relays, the distortion errors (defined as \(\epsilon_{1}\) and \(\epsilon_{2}\)) in the links between S and UR\({}_{1}\) and the link between S and UR\({}_{2}\), respectively, are acknowledged based on the theory of lossy source-channel separation, as in [39]. These distortions can be expressed as follows:  
\[R_{S_{1}}^{\text{D}}(\epsilon_{1})\dot{R}_{S_{1}} \leq C(\gamma_{\text{S-UR}_{1}}), \tag{9}\] \[R_{S_{2}}^{\text{D}}(\epsilon_{2})\dot{R}_{S_{1}} \leq C(\gamma_{\text{S-UR}_{2}}). \tag{10}\]  
Here, \(R_{S_{k}}^{\text{D}}(\epsilon_{k})\) and \(\dot{R}_{S_{k}}\) for \(k\in(1,2)\) stand for the function of rate-distortion at \(\epsilon_{k}\) (associated with the least distortion level considering the Hamming measure), and the cumulative rate for source-channel joint coding for the associated link, in order. The specific of \(\dot{R}_{S_{k}}\) is introduced in Appendix A. The capacity, which is equivalent to the maximal transmission rate, based on [PERSON]'s Gaussian codebook, is given by  
\[C(x)=\log_{2}(1+x). \tag{11}\]  
With the Hamming measure, the minimum distortion for a provided \(\gamma_{\text{S-UR}_{k}}\) value is tantamount to the error probability \(\epsilon_{\text{S-UR}_{k}}\) within the respective links. For a binary source following a Bernoulli(\(p\)) distribution, the rate-distortion function \(R_{S_{k}}^{\text{D}}\) is given by the following:  
\[R_{S_{k}}^{\text{D}}(\epsilon_{k})= \begin{cases}1-H(\epsilon_{k}),&0\leq \epsilon_{k}\leq\min(p_{k},1-p_{k})\\ 0,&\epsilon_{k}>\min(p_{k},1-p_{k}),\end{cases} . \tag{12}\]  
where \(H(\beta)=-\beta\log_{2}(\beta)-(1-\beta)\log_{2}(1-\beta)\) represents the function of binary entropy. The relationship between the distortion \(\epsilon_{k}\) and the rate \(R_{S_{k}}^{\text{D}}(\epsilon_{k})\) with respect to distortion \(\epsilon_{k}\) is depicted in Figure 2.  
For a Gaussian source with a \(N(0,\sigma^{2})\) distribution, the rate-distortion function \(R_{S_{k}}^{\text{D}}\) is expressed as follows:  
\[R_{S_{k}}^{\text{D}}(\epsilon_{k})= \begin{cases}\frac{1}{2}\log_{2}\frac{ \sigma^{2}}{\epsilon_{k}},&0\leq\epsilon_{k}\leq\sigma^{2}\\ 0,&\epsilon_{k}>\sigma^{2}.\end{cases}  \tag{13}\]  
The connection between \(\epsilon_{k}\) and the rate \(R_{S_{k}}^{\text{D}}(\epsilon_{k})\) with squared-error distortion is illustrated in Figure 3.  
With the measure of hamming distortion, the minimum level of the distortion \(\epsilon_{1}\) and \(\epsilon_{2}\) correspond to the error probabilities in the transmission from the source to UR\({}_{1}\) and from the source to UR\({}_{2}\), respectively, given a specific instantaneous SNR \(\gamma\). Figure 3: Intro-link distortion level \(\mathcal{D}\) versus rate-distortion function with a Gaussian source, where \(\sigma=1\). Figure 2: Intro-link distortion level \(\mathcal{D}\) versus rate-distortion function with a Bernoulli(\(p\)) source, where \(p=\frac{1}{2}\). Based on (11)-(13), the instantaneous SNR \(\gamma_{ij}\) of channel \(i-j\) and \(R_{\text{S}_{1}}^{\text{D}}(\epsilon_{k})\) can be related as follows:  
\[R_{\text{S}_{1}}^{\text{D}}(\epsilon_{1})\hat{R}_{\text{S}_{1}} =(1-H(\mathscr{D}))\hat{R}_{\text{S}_{1}}\leq C(\gamma_{\text{S-UR}_{1}})= \log_{2}(1+\gamma_{\text{S-UR}_{1}}), \tag{14}\] \[R_{\text{S}_{2}}^{\text{D}}(\epsilon_{2})\hat{R}_{\text{S}_{2}} =(1-H(\mathscr{D}))\hat{R}_{\text{S}_{2}}\leq C(\gamma_{\text{S-UR}_{2}})= \log_{2}(1+\gamma_{\text{S-UR}_{2}}), \tag{15}\]  
for a Bernoulli(\(\frac{1}{2}\)) source.