## 4 Reliable and Secure Probability Analyses
### Acceptable Rate Region for the CEO Problem in the Slepian-Wolf Framework  
Let \(\mathcal{U}\) represent the original message transmitted by the source, while \(\mathcal{U}_{3}\) and \(\mathcal{U}_{4}\) denote the sequences output by \(\text{UR}_{1}\) and \(\text{UR}_{2}\), respectively, characterized by \(R_{3}^{s}\) and \(R_{4}^{s}\). Because of the lossy DF scheme, even if the sequences that are taken over by \(\text{UR}_{1}\) and \(\text{UR}_{2}\) have faults (i.e., \(\mathcal{U}\
eq\mathcal{U}_{3}\) and \(\mathcal{U}\
eq\mathcal{U}_{4}\) with a certain probability), \(\text{UR}_{1}\) and \(\text{UR}_{2}\) still forward these erroneous sequences to the destination. As a result, the analysis of the transmissions from \(\text{UR}_{1}\) to D and \(\text{UR}_{2}\) to D falls under the CEO problem category in network information theory [41]. The conceptual framework of the CEO problem is illustrated in Figure 4. Note that \(\mathcal{U}_{1}\) and \(\mathcal{U}_{2}\) are not utilized in this paper to maintain consistency in notation. Under the block fading assumption, the errors occurring in the S-\(\text{UR}_{1}\) and S-\(\text{UR}_{2}\) links have fixed probabilities within a single transmission block.  
Since \(\mathcal{U}_{3}\) and \(\mathcal{U}_{4}\) originate from the same source, they are correlated. This correlation is described using a bit-flipping model: \(\mathcal{U}_{3}=\mathcal{U}_{4}\oplus\varepsilon_{0}\), where \(\varepsilon_{0}\) is a stochastic variable with \(\text{Pr}(\varepsilon_{0}=1)=1-\text{Pr}(\varepsilon_{0}=0)=\rho_{0}\). Here, \(\rho_{0}\) represents the probability of bit-flipping between \(\mathcal{U}_{3}\) and \(\mathcal{U}_{4}\).  
Since \(\mathcal{U}_{3}\) and \(\mathcal{U}_{4}\) are correlated, the Slepian-Wolf theorem states that successful retrieval of \(\mathcal{U}_{3}\) and \(\mathcal{U}_{4}\) through joint decoding at the destination can be achieved if the source rate pair \((R_{3}^{s},R_{4}^{s})\) for \(\mathcal{U}_{3}\) and \(\mathcal{U}_{4}\) satisfies the following conditions:  
\[\begin{cases}R_{3}^{s}&\geq H(\mathcal{U}_{3}|\hat{\mathcal{U}}_{4}),\\ R_{4}^{s}&\geq H(\mathcal{U}_{4}|\hat{\mathcal{U}}_{3}),\\ R_{3}^{s}+R_{4}^{s}&\geq H(\mathcal{U}_{3},\mathcal{U}_{4}),\end{cases} \tag{24}\]  
where \(\hat{\mathcal{U}}_{3}\) and \(\hat{\mathcal{U}}_{4}\) are the estimates of \(\mathcal{U}_{3}\) and \(\mathcal{U}_{4}\) from the destination's final output. Here, \(H(\mathcal{U}_{3}|\hat{\mathcal{U}}_{4})\) and \(H(\mathcal{U}_{4}|\hat{\mathcal{U}}_{3})\) represent conditional entropies. The relationships between \(\mathcal{U}_{3}\) and \(\hat{\mathcal{U}}_{3}\) and between \(\mathcal{U}_{4}\) and \(\hat{\mathcal{U}}_{4}\) are modeled using bit-flipping probabilities \(\rho_{3}\) and \(\rho_{4}\), respectively. With the consideration of block fading, the possibility of errors occurring in the \(\text{UR}_{1}\)-D and \(\text{UR}_{2}\)-D links remain steady within a single communication block, making \(\varepsilon_{3}\) and \(\varepsilon_{4}\) fixed parameters for each stage. Given that the source is i.i.d. in this study, we have \(H(\mathcal{U}_{3}|\hat{\mathcal{U}}_{4})=H(\varepsilon_{0}*\varepsilon_{3})\) and \(H(\mathcal{U}_{4}|\hat{\mathcal{U}}_{3})=H(\varepsilon_{0}*\varepsilon_{4})\), where \(x*y=(1-y)x+(1-x)y\).  
Consider two boundary scenarios. In the first scenario, where both \(\mathcal{U}_{3}\) and \(\mathcal{U}_{4}\) are fully recovered at the destination simultaneously, we have \(\mathcal{U}_{3}=\hat{\mathcal{U}}_{3}\) and \(\mathcal{U}_{4}=\hat{\mathcal{U}}_{4}\), with \(\varepsilon_{3}=0\) and \(\varepsilon_{4}=0\). This relates to the scenario where \((R_{3}^{s},R_{4}^{s})\) falls within regions 1 and 2 in  
Figure 4: Abstract framework of a CEO problem for binary sources.