## 4 Privacy and Security Enhanced Methods
### Privacy-Preserving Arbitration Mechanism
#### 4.1.4 Adaptive Privacy Budget Scheme  
To balance privacy protection with operational efficiency, we introduce an adaptive privacy budget mechanism that allows the arbitration and scheduling module to make preliminary arbitration decisions locally, adjusting privacy exposure based on perceived threat levels.  
We define a privacy budget parameter \(\epsilon\in[0,1]\) that determines the proportion of output bits protected from the arbitration and scheduling module. For an \(n\)-bit output, the module gains access to the first \((1-\epsilon)\times n\) bits for local consistency checking, while the remaining \(\epsilon\times n\) bits remain fully encrypted.  
The local consistency check operates as follows:  
1. Each executor \(e_{i}\) computes its output \(O_{i}\) and separates it into two parts: \(O_{i}^{visible}\) (first \((1-\epsilon)\times n\) bits) and \(O_{i}^{protected}\) (remaining \(\epsilon\times n\) bits).
2. The executor computes a hash of the visible portion, \(H(O_{i}^{visible})\), and sends both this hash and the encrypted complete output \(Enc_{pk}(O_{i})\) to the arbitration and scheduling module.
3. The module compares hashes of visible portions to make preliminary consistency determinations: \[\Phi_{local}(O_{i},O_{j})=\begin{cases}1,&\text{if }H(O_{i}^{visible})=H(O_{j}^{visible})\\ 0,&\text{otherwise}\end{cases}.\] (10)
4. Based on local consistency checks, the module initiates the homomorphic encryption-based scheme for final verification.  
The privacy budget \(\epsilon\) is dynamically adjusted based on observed attack patterns and arbitration error rates:  
\[\epsilon_{t+1}=\begin{cases}\min(\epsilon_{t}+\delta,\epsilon_{max}),&\text{if }r_{t}<\tau_{low}\\ \max(\epsilon_{t}-\delta,\epsilon_{min}),&\text{if }r_{t}>\tau_{high},\\ \epsilon_{t},&\text{otherwise}\end{cases}, \tag{11}\]  
where \(r_{t}\) represents the error rate of local arbitration (discrepancy between local and cloud-verified consistency determinations), \(\tau_{low}\) and \(\tau_{high}\) are threshold parameters, \(\delta\) is the adjustment step size, and \(\epsilon_{min}\) and \(\epsilon_{max}\) define the allowed range for the privacy budget.  
During periods of low attack activity (indicated by low error rates), the system increases the privacy budget to enhance privacy protection. Conversely, when attacks are detected or suspected (indicated by high error rates), the system reduces the privacy budget to prioritize accurate and timely arbitration, thereby maintaining system availability.  
This adaptive approach leverages the natural advantages of cloud-edge collaboration, allowing edge-level preliminary decisions with cloud-level verification, creating a balanced trade-off between privacy protection and security enforcement.  
The selection of thresholds \(\tau_{low}\) and \(\tau_{high}\), as well as step size \(\delta\), is inherently application-dependent and requires careful consideration of domain-specific requirements. The lower threshold \(\tau_{low}\) should be determined based on the minimum acceptable level of privacy leakage risk considering regulatory requirements and organizational privacy policies, while the upper threshold \(\tau_{high}\) should be established based on the maximum tolerable performance degradation that maintains operational requirements--varying significantly across different industrial applications from millisecond-critical manufacturing processesto minute-level process control systems. The step size \(\delta\) represents a fundamental trade-off between convergence speed and system stability, requiring consideration of real-time constraints inherent in ICS environments. In practice, these parameters should be calibrated through pre-deployment testing that evaluates privacy budget performance under both attack and benign scenarios, measuring corresponding performance degradation against application-specific tolerance levels, with domain experts providing guidance based on operational requirements, threat landscape, and privacy protection needs of the target industrial environment.