# An On-Demand Fault-Tolerant Routing Strategy for Secure Key Distribution Network
## 4 Proposed Solution
### Information Aggregation and Decision Making  
The mass functions \(m_{i}\) and \(m_{j}\) of different observational evidence can be combined through Dempster's rule to obtain the combined mass function \(m^{{}^{\prime}}\), as shown in the following formula. \(m^{{}^{\prime}}\) also satisfies the condition of Equation (5) and is the same as the power set \(2^{\Omega}\) mapped by \(m_{i}\) and \(m_{j}\):  
\[m^{{}^{\prime}}(A)=\frac{1}{1-k}\sum_{B\cap C=A}m_{i}(B)m_{j}(C) \tag{16}\]  
\[k=\sum_{B\cap C=\varnothing}m_{i}(B)m_{j}(C) \tag{17}\]  
Therefore, different information can be considered comprehensively to further analyze and reason about the elements in the set. However, the normalized conjunctive rule of combination that defines [PERSON]'s rule assumes the same degree of belief for different evidence, and lacks the distinction of evidence importance through prior knowledge. For this reason, we calculate the combined mass function by using the weighted Dempster's rule. After assigning a positive integer importance factor (IF) to each piece of evidence, for any \(A\in 2^{\Omega}\), Equations (16) and (17) will be transformed into:  
\[m^{{}^{\prime}}(A)=\frac{1}{K}\sum_{B\cap C=A}m_{i}(B)^{\frac{f_{i}}{f_{j}}}m _{j}(C)^{\frac{f_{j}}{f_{i}}} \tag{18}\]  
\[k^{{}^{\prime}}=\sum_{B\cap C\
eq\varnothing}m_{i}(B)^{\frac{f_{i}}{f_{j}}}m _{j}(C)^{\frac{f_{j}}{f_{i}}} \tag{19}\]  
where \(f_{i}\) and \(f_{j}\) are the IFs of \(m_{i}\) and \(m_{j}\), respectively. The IF of the combined mass function \(m^{{}^{\prime}}\) is \((f_{i}+f_{j})/2\). It can be verified that the weighted Dempster's rule satisfies the condition of Equation (5); weighting is more general, and due to the special case \(f_{i}=f_{j}\), it degenerates into Dempster's rule. More importantly, Equations (18) and (19) shows that the mass function with high IF contributes more to the combined mass function, effectively utilizing prior knowledge obtained from key transmission. For the obtained combined mass function \(m^{{}^{\prime}}\), select the element \(\theta_{result}\in\Omega\) that can maximize the Pignistic transform of \(m^{{}^{\prime}}\) as the cause of fault:  
\[\theta_{result}=\operatorname*{arg\,max}_{\theta}(\mathit{BetP}(\theta)) \tag{20}\]  
where Pignistic transform \(\mathit{BetP}(\theta)\) is as follows:  
\[\mathit{BetP}(\theta)=\sum_{A\subseteq\Omega,\theta\in A}\frac{m(A)}{\left|A \right|\left(1-m^{{}^{\prime}}(\varnothing)\right)} \tag{21}\]  
When the detection is triggered, we collect evidence from \(e_{1}\) to \(e_{3+2\log\left[N-1\right]}\), calculate the combined mass function \(m^{{}^{\prime}}\), and calculate \(\theta_{result}\) through Equation (20). When \(\theta_{result}\) is benign, path discovery is performed, and convert the communication key sent from the detection trigger to available, return to the service layer. Otherwise, the communication key in the buffer is cleared, and the detected link will be added to the route request message in the next path discovery, and other nodes will ignore the link during the route discovery phase. Algorithm 1 for combined mass function \(m^{\prime}\) is summarized as follows. where \(f_{i}\) is the IF of each \(e_{i}\), \(m_{i}\) is calculated from Equations (6)-(8). The time complexity of combined mass function calculation is \(\mathcal{O}(\log(N))\), where \(N\) is the number of nodes on the path. This shows that the computational overhead increases slowly as the network size increases, indicating the practical value of the algorithm.