## 6 Experimental Evaluation and Results
### Performance Evaluation of Privacy-Preserving Arbitration  
While privacy preservation provides essential security guarantees in cloud-edge collaboration scenarios, it inevitably introduces computational and communication overhead that affects system performance. To comprehensively evaluate the practical feasibility of our privacy-preserving arbitration mechanism, we conducted a detailed performance analysis focusing on the arbitration latency introduced by different privacy-preserving schemes.  
Figure 6 presents the arbitration latency comparison across different schemes under both best-case and worst-case scenarios. The experimental results reveal that encryption and decryption operations of executor outputs constitute the primary contributors to arbitration latency.  
Figure 4: Executor outputs under different defense strategies.  
Figure 5: Executor outputs for the proposed method.  
tration delay, while homomorphic operations performed by the arbiter (i.e., the arbitration and scheduling module) and message transmission delays have relatively minimal impact on overall performance. This finding indicates that the computational complexity of cryptographic operations, rather than network communication, represents the primary performance bottleneck in privacy-preserving arbitration.  
Under best-case conditions, the adaptive privacy budget scheme demonstrates superior performance with minimal additional arbitration latency. This efficiency stems from the arbiter's capability to leverage deliberately leaked partial information for local decision-making, thereby reducing the need for extensive homomorphic computations and cloud-based processing. In worst-case scenarios, where maximum privacy protection is required and minimal information leakage is permitted, the adaptive privacy budget scheme maintains performance parity with the communication-optimized scheme.  
Notably, even under worst-case conditions, the adaptive privacy budget scheme introduces arbitration latency of less than 80 milliseconds. This performance level satisfies the timing requirements of time-insensitive industrial control systems, particularly those involving gradual process variations such as natural gas pipeline pressure control systems where pressure adjustments occur over extended time horizons. However, for control systems with stringent real-time requirements, the current implementation using the Paillier cryptosystem may introduce unacceptable delays. In such scenarios, lightweight homomorphic encryption alternatives present viable solutions to achieve acceptable privacy-performance trade-offs. Specifically, symmetric homomorphic encryption schemes offer significant computational advantages over asymmetric approaches. For instance, the provably secure additive and multiplicative privacy homomorphism scheme [53] provides secure arithmetic operations on encrypted data while maintaining efficiency against chosen-ciphertext attacks. More recently, symmetric additive homomorphic encryption implementations [54] have demonstrated the ability to reduce encryption/decryption operations to the microsecond level, which fully satisfies the requirements of millisecond-level control loops. These lightweight alternatives can be integrated into our framework as configurable privacy-preserving mechanisms, allowing system operators to dynamically balance security requirements with real-time performance constraints based on specific application needs. Future work will focus on implementing and evaluating these hybrid approaches to extend the applicability of our architecture to time-critical industrial environments.  
Figure 6: Arbitration latency comparison across different schemes under both best-case and worst-case scenarios.