## 4 The Development of the MRFCO-ZTS Approach
### MRFCO-ZTS Approach  
Satellite continuous authentication is a significant task in zero-trust security where it verifies user identity, device posture, and session behavior. However, combining continuous checking with zero-trust formulates several problems that impact security, user experience, and system performance in the satellite communication environment. Continuous checking of all users in every session requires the processing of contextual and behavioral data in real-time, which slows down the communication system, increases network strain, and affects overall performance when considering a larger environment. Scaling continuous verification for thousands of users and devices needs robust infrastructure and constant tuning. Hence, the efficiency of zero-trust security still needed to be improved with a greater number of input satellite users. To address the issue of existing Zero-Trust Security, a novel cost function \(C(M)\) has been developed based on attack, risk, and computational delay when considering a growing number of users in a satellite network. To optimize ZTS, our developed MRFCO-ZTS approach aims to minimize multi-objective cost function per user/session \(i\), using  
\[C(M)=a\cdot a_{i}\cdot r_{i}+\beta\cdot d_{i}+\gamma\cdot m_{i} \tag{1}\]  
In Equation (1), \(a_{i}\) refers to attack likelihood, \(r_{i}\) indicates cumulative user risk, \(d_{i}\) represents computational delay/overhead (due to continuous authentication), and \(m\) denotes the misclassification penalty (Table 1). Figure 2: The architecture of the MRFCO-ZTS approach for a communication network with secure inter-satellite links. Let us consider that attack probability increases with the number of users and is higher for non-potential users, which can be expressed mathematically as  
\[a_{i}=a_{1}\cdot log\big{(}M_{p}+1\big{)}+a_{2}\cdot log\big{(}M_{np}1\big{)} \tag{2}\]  
Here, \(a_{1}<a_{2}\) indicates attacks are more expected from non-potential users. \[r_{i}=\frac{1}{M}\bigg{(}\sum_{i=1}^{M_{p}}r_{p,i}+\sum_{i=1}^{M_{np}}r_{np,i} \bigg{)} \tag{3}\]  
In Equation (3), \(r_{p,i}\) and \(r_{np,j}\) show risk scores for individual users. To define a cost function for computational delay that balances reduced latency and improved security in a satellite network using a ZTS concept for both potential and non-potential users, the following key terms are considered: \(M_{p}\) refers to potential (trusted) users; \(M_{np}\) represents non-potential (un-trusted) users. Then, the total number of users \(M\) can be calculated as \(M=M_{p}+M_{np}\). \[d_{i}=w1\cdot CD(M)+w2\cdot L(M)-w3\cdot S(M) \tag{4}\]  
In Equation (4), \(CD(M)\) depicts total computational delay, \(L(M)\) defines latency as a function of users, \(S(M)\) indicates security strength (e.g., number/intensity of ZTS checks), \(D_{i}\) refers to computational delay, and \(w1\), \(w2\), and \(w3\) represent weights balancing security, latency, and user importance. Next, the computational delay of ZTS has been defined based on user types as  
\[CD(M)=M_{p}\cdot d_{p}+M_{np}\cdot d_{np} \tag{5}\]  
In Equation (5), \(M_{p}\) and \(M_{np}\) describe the number of possible non-potential users, \(d_{p}\) represents the average delay per potential user (lower), and \(d_{np}\) refers to the average delay per non-potential user (higher). Latency occurs due to satellite links and ZTS verification, which is represented mathematically as  
\[L(M)=\omega\big{(}M_{p}\cdot l_{p}+M_{np}\cdot l_{np}\big{)} \tag{6}\]  
In Equation (6), \(l_{p}<l_{np}\) and \(\omega\) describes a system-specific constant. \[S(M)=\sigma\big{(}M_{p}\cdot s_{p}+M_{np}\cdot s_{np}\big{)} \tag{7}\]  
In Equation (7), \(s_{p}<s_{np}\) (more secure verifications for non-potential users); \(\sigma\) is a security effectiveness constant. The objective function J(.) of the optimization algorithm is defined mathematically as  
\[\text{J}(.)=\min_{\theta}\sum_{i=1}^{M}C_{i} \tag{8}\]