# Model for Quantifying the Quality of Secure Service
## 4 The Quality of Secure Service Model  
Although a communication network typically uses only one network channel between two given nodes, the possibility exists to utilize multiple paths between nodes, as shown in Figure 1. This figure shows an arbitrary network with eight individual channels, any of which may be used to transport data. A message sent through the network in Figure 1 may travel across one of the channels influenced by the set of adversarial listeners, \(A_{L}\), or the set of adversarial disruptors, \(A_{D}\). The relationship between listeners, disruptors, and the total number of needed channels is described by PSMT, which \"abstract[s] away the network entirely and concentrate[s] on solving the Secure Message Transmission Problem\" for a single transmitter and receiver pair [11]. Additional articles explore multi-channel architectures [29, 30], while others strive to prove the general case and optimize the statistical reliability and secrecy [31, 32]. In our model, \(\sigma\) represents the number of wires (channels) between the transmitter and receiver available to the adversarial listener set, \(A_{L}\), and \(\rho\) is the number of channels between the transmitter and receiver available to the adversarial disruptor set, \(A_{D}\)[10]. Communication is two-way between the transmitter and receiver and, following PSMT, the number of channels that must exist between transmitter and receiver is given by  
\[n\geq max\{\sigma+\rho+1;\;2\rho+1\}. \tag{8}\]  
With this equation, we know how many channels must be used to maintain secure and reliable communication. If a channel is unavailable, then it must not be counted as part of \(n\). If we assume the number of channels accessible to a listener or disruptor, then we can arrive at a specific quantification of \(n\). For example, when \(n=8\) and \(\sigma=3\), the probability that any one channel of the eight could be listened to is 0.375. The probability of leakage for each channel within a multi-channel architecture becomes  
\[P(I)=\frac{P(int)\cdot P(dx)\cdot P(dx)\cdot\sigma}{n}. \tag{9}\]  
Similarly, the probability of corruption for each channel within multi-channel architecture becomes  
\[P(c)=\frac{\left( \begin{pmatrix}1-P(n)\end{pmatrix} P(inj)+ \begin{pmatrix}P(n)+ P(s)\end{pmatrix} - \begin{pmatrix}P(n)P(s)\\ -\begin{pmatrix}1-P(n)\end{pmatrix}\begin{pmatrix}P(n)+P(s)\end{pmatrix}P(inj) \end{pmatrix} \rho\right.}{n}. \tag{10}\]  
Therefore, the more channels there are in a network, the lower the probability of adversarial interference of the data [11]. This, then, follows the premise of PSMT: to have more channels than the combined set of listeners and disruptors \(A_{L}\cup A_{D}\). In the same manner that multiple channels may thwart adversarial interference, message fragmentation may also thwart eavesdropping. Message fragmentation is the splitting of data across the available channels, effectively parallelizing the data. Fragmentation describes how many portions the original message is divided into. Various methods of fragmentation are possible, including uniform or non-uniform fragmentation from 1-bit to the total \(m\)-bits in message \(M\). Research has been published on particular approaches to fragmentation [28, 33]; however, in this paper,we focus on the security effects and apply the assumption that fragments are of equal size across the network. If \(C_{n}\) is the set of \(n\) channels, and \(F_{M}\) is the set of \(k\) fragments of \(1\leq|f_{i}|\leq m\)-bits of the message \(M\), then  
\[F_{M}=\{f_{(M,1)},f_{(M,2)},f_{(M,3)},\ldots,f_{(M,k)}\} \tag{11}\]  
Figure 1: A network configuration with multiple possible channels. \[f_{(M,i)}\subseteq M\ for\ 1\leq i\leq k \tag{12}\]  
where each fragment is unique.