## 4 Proposed Methodology
### Using A Single Route for Each Transmission  
The idea of this case is that each time a message is transmitted, a route will be selected from the route between the sender and the receiver.  
We further propose the addition of a parameter \(P_{nk\ j}\)--the probability of violating the secrecy of the message transmitted through the \(j\)-th route, which will be determined by the secrecy index of the message transmitted through the \(j\)-th route.  
\[P_{nk\ j}=\alpha_{j}(1-P_{con\ j})=\alpha_{j}(1-\prod_{i=1}^{n_{j}}P_{nk\ i,j})\]  
Here, the coefficient \(\alpha_{j}\) is the utilization rate of the \(j\)-th route for transmitting messages, \(0\leq\alpha_{j}\leq 1\). \(\alpha_{j}=\frac{b_{j}}{a}\), where \(a\) is the total number of messages transmitted between the sender and the receiver, \(b_{j}\) is the number of messages transmitted through \(j\)-the route. The coefficient \(\alpha_{j}\) indicates that the more \(j\)-th route is used to transmit messages, the higher the probability of violating the secrecy of the message transmitted through \(j\)-th route.  
\(P_{con}\) is the secrecy index of the message transmitted between the receiver and the sender.  
\(P=1-P\) is the probability of violating the confidentiality of the message transmitted between the sender and the receiver.  
If in the study of author [49], only a single dedicated route is used for transmitting the message every time, meaning \(\alpha_{base}=1\).  
The probability of violating the confidentiality of the message transmitted between the sender and the receiver, using only the primary route for message transmission, i.e., employing a single fixed dedicated route (corresponding to \(n_{base}\) stego channel and having \(\alpha_{base}=1\)), will be determined by the expression:  
\[P_{nk\ base}=1\times(1-P_{con\ base})=1-\prod_{i=1}^{n_{base}}P_{con\ i,base} \tag{2}\]  
An example of a steganographic system illustrating expression (2) is depicted in Figure 5.  
Figure 4: Example of a route with \(n_{j}\) hidden channels.  
Equation (2) represents the probability of violating the security of the case when only a single fixed route is used.  
If we consider the scenario where there is an alternation between using the main route and the backup routes, and each time a message is transmitted, a different route is utilized, the value of \(\alpha_{base}+\sum\limits_{j=1}^{k}\alpha_{j}=1\) will change accordingly (Case 1).  
The probability of violating the security of a message transmitted between the sender and receiver, considering both the main route and k backup routes (corresponding to the stego channels \(n_{base},n_{1},n_{2},\ldots,n_{k}\) and \(\alpha_{base},\alpha_{1},\alpha_{2},\ldots,\alpha_{k}\), respectively, where \(\alpha_{base}+\sum\limits_{j=1}^{k}\alpha_{j}=1\) will be determined using the following expression:  
\[P_{nk\;base,k}=P_{nk\;base}\times\prod\limits_{j=1}^{k}P_{nk\;j}=[\alpha_{ base}(1-\prod\limits_{i=1}^{n_{base}}P_{con\;i,base})]\times\prod\limits_{j=1}^{k} [\alpha_{j}(1-\prod\limits_{i=1}^{n_{j}}P_{con\;i,j})] \tag{3}\]  
An example of a steganographic system illustrating expression (3) is depicted in Figure 6.  
Figure 5: Example of a steganographic system (illustrating for Equation (2)).  
Figure 6: Example of a steganographic system (illustration of Equation (2)).  
Because for any signal transmission channel \"\(i\)-th\" of route \"\(j\)-th\", \(P_{con\;i\;j}\leq 1\), and for any route \"\(j\)-th\", \(\alpha_{j}\leq 1\), therefore \(P_{nk\;base,k}\leq P_{nk\;base}\). The equal sign in this expression will occur when \(k=0\), meaning that no backup route is used.  
From the result that \(P_{nk\;base,k}\leq P_{nk\;base}\), it has been demonstrated that the alternation of multiple communication routes helps decrease the probability of violating the security of transmitted messages between the sender and the receiver. This also implies an increase in the security index of the transmitted messages between the sender and the receiver.