## 5 Security and Privacy Analysis
### Security  
We begin by analyzing the security properties of our architecture, focusing on availability guarantees in the presence of adversaries attempting to compromise executors. **Theorem 1** (Heterogeneity Security Enhancement). : _Consider a system with \(k\) heterogeneous executors where executor \(P_{i}\) has individual attack success probability \(p_{i}\) and the correlation coefficient between executors \(P_{i}\) and \(P_{j}\) is \(\rho_{i,j}\in[0,1]\). The probability of successful common-mode attack against all executors is approximately \(P_{\text{h}etero}\approx\prod_{i=1}^{k}p_{i}(1+\sum_{1\leq i<j\leq k}\frac{p_{ i,j}\sqrt{(1-p_{i})(1-p_{j})}}{\sqrt{p!p!}})+O(\rho^{2})\). For the symmetric case where \(p_{i}=p\) and \(\rho_{i,j}=\rho\), this simplifies to \(P_{\text{h}etero}\approx p^{k}(1+(\frac{k}{2})\frac{\rho(1-p)}{p})+O(\rho^{2})\). Compared to homogeneous redundancy where \(P_{\text{h}omo}=p\) due to identical vulnerabilities, heterogeneous architecture provides exponential security enhancement._  
Proof. : Let \(X_{i}\) denote the indicator random variable for a successful attack on executor \(P_{i}\) with \(P(X_{i})=p_{i}\). For any pair of executors \((P_{i},P_{j})\), the correlation coefficient \(\rho_{i,j}\) determines their common vulnerability:  
\[P(X_{i}\cap X_{j})=p_{i}p_{j}+\rho_{i,j}\sqrt{p_{i}(1-p_{i})p_{j}(1-p_{j})}, \tag{21}\]  
where \(P(X_{i}\cap X_{j})\) is the success probability of a common-mode attack against executors \(i\) and \(j\). For the general case with \(k\) executors having potentially different attack success probabilities, we apply the inclusion-exclusion principle. The covariance between executors \(P_{i}\) and \(P_{j}\) is \(\text{Cov}(X_{i},X_{j})=\rho_{i,j}\sqrt{p_{i}(1-p_{i})p_{j}(1-p_{j})}\). Using first-order approximation:  
\[P\left(\bigcap_{i=1}^{k}X_{i}\right) \approx\prod_{i=1}^{k}p_{i}+\sum_{1\leq i<j\leq k}\text{Cov}(X_{i },X_{j})+O(\rho^{2}) \tag{22}\] \[=\prod_{i=1}^{k}p_{i}\left(1+\sum_{1\leq i<j\leq k}\frac{\rho_{i,j}\sqrt{(1-p_{i})(1-p_{j})}}{\sqrt{p_{i}p_{j}}}\right)+O(\rho^{2}). \tag{23}\]  
We now consider the symmetric case where all executors have identical individual vulnerability levels (\(p_{i}=p\)) and uniform correlation (\(\rho_{i,j}=\rho\)). Under these conditions, the expression simplifies significantly:  
\[P_{\text{h}etero}\approx p^{k}\left(1+\binom{k}{2}\frac{\rho(1-p)}{p}\right)+ O(\rho^{2}). \tag{24}\]  
In contrast, a homogeneous redundant system employs \(k\) identical executors with identical vulnerabilities. When facing a common-mode attack, success against the common vulnerability immediately compromises all executors simultaneously. Therefore, \(P_{\text{h}omo}=p\) regardless of the number of redundant executors. The security enhancement factor demonstrates the exponential advantage of heterogeneity:  
\[\text{SEF}=\frac{P_{\text{h}omo}}{P_{\text{h}etero}}\approx\frac{p}{p^{k}(1+ \binom{k}{2}\frac{\rho(1-p)}{p})}=\frac{1}{p^{k-1}(1+\binom{k}{2}\frac{\rho(1- p)}{p})}. \tag{25}\]Even with moderate correlation (\(\rho<1\)), the heterogeneous architecture transforms the attack complexity from \(O(1)\) for homogeneous systems to \(O(p^{k-1})\), providing substantial security enhancement that scales exponentially with the number of diverse executors. **Theorem 2** (Dynamic Resilience). : _The proposed architecture with credit-heterogeneity-based scheduling and dynamic credibility updates achieves asymptotic security improvement over static architectures, with compromise probability \(P_{comp}(t)\leq e^{-at}\) for some \(\alpha>0\) as \(t\to\infty\), where \(t\) is the system operation time._  
Proof. : Consider an attacker attempting to compromise the system over time. In a static architecture, the probability of system compromise increases monotonically as attackers discover and exploit vulnerabilities:  
\[P_{static}(t)=1-e^{-\lambda t}, \tag{26}\]  
where \(\lambda\) is the rate of vulnerability discovery. In our dynamic architecture, each executor rotation reduces the attacker's accumulated knowledge.