Privacy-preserving Distributed Optimization Algorithm Based on Elliptic Curve ELGamal
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摘要: 研究一类考虑节点隐私保护的分布式优化问题, 目标为保护各智能体的隐私信息不被泄露, 并最小化所有智能体局部成本函数之和. 首先, 针对无向连通图, 提出一种基于椭圆曲线密码机制的分布式凸优化算法. 通过设计底层权重矩阵, 将基于椭圆曲线的ElGamal同态加密和数字签名与分布式次梯度算法相结合, 克服了椭圆曲线密码机制与分布式一致性策略无法结合的难点. 在无第三方或聚合器的场景下, 该算法实现了系统的隐私保护. 理论分析表明, 该算法能够渐近收敛至全局最优解, 并适用于时变通信拓扑的动态环境. 此外, 算法有效保护智能体的状态和成本函数不受来自诚实但好奇攻击者、外部窃听者和篡改攻击者的威胁. 最后, 通过数值仿真验证了算法的有效性.Abstract: In this paper, we investigate a class of distributed optimization problems considering node privacy protection, aiming to prevent the disclosure of private information of each agent while minimizing the sum of local cost functions across all agents. Firstly, for an undirected connected graph, a distributed convex optimization algorithm based on elliptic curve cryptography is proposed. By designing the underlying weight matrix, elliptic curve-based ElGamal homomorphic encryption and digital signatures are combined with the distributed subgradient algorithm, overcoming the difficulty of integrating elliptic curve cryptosystems with distributed consensus strategies. Without the need for a third party or aggregator, our algorithm achieves system privacy protection. Theoretical analysis shows that the algorithm can asymptotically converge to the global optimal solution and is suitable for dynamic environments with time-varying communication topologies. Moreover, the algorithm effectively safeguards the state and cost functions of agents from threats posed by curious but honest attackers, external eavesdroppers, and adversaries attempting tampering attacks. Finally, the effectiveness of the algorithm is validated through numerical simulations.
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Key words:
- Distributed optimization /
- privacy protection /
- homomorphic encryption /
- elliptic curve
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图 3 节点2接收到的加密信息$ c_1$的演化
Fig. 3 Evolution of the encrypted messages $ c_1$ received by node 2
图 4 节点2接收到的加密信息$ c_2$的演化
Fig. 4 Evolution of the encrypted messages $ c_2$ received by node 2
图 5 不同$\eta$值下$\mathrm{ERR}_{\mathrm{RMSE}}$的演化
Fig. 5 Evolution of $\mathrm{ERR_{\mathrm RMSE}}$ with different $\eta$ values
图 6 $\eta = 0.010$时诚实且好奇敌手对节点1状态的估计
Fig. 6 Honest and curious adversary＇s estimation of the state of node 1 with $\eta = 0.010$
图 8 $\eta = 0.001$时诚实且好奇敌手对节点1状态的估计
Fig. 8 Honest and curious adversary＇s estimation of the state of node 1 with $\eta = 0.001$
图 7 $\eta = 0.005$时诚实且好奇敌手对节点1状态的估计
Fig. 7 Honest and curious adversary＇s estimation of the state of node 1 with $\eta = 0.005$
表 1 不同$\eta$值下算法1的均方根误差
Table 1 Root mean square error of algorithm 1 under different $\eta$ values
$\eta$ $\mathrm{ERR_{\mathrm{RMSE}}}$ $0.100$ $5.47 \times 10^{-4}$ $0.010$ $1.16 \times 10^{-3}$ $0.005$ $1.42 \times 10^{-3}$ $0.001$ $1.47 \times 10^{-3}$ 
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表 2 $N=5$时各种同态加密方案的计算时间 (单位: s)
Table 2 Computation time of various homomorphic encryption schemes for $N=5$ (unit: s)
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表 3 $N=10$时各种同态加密方案的计算时间 (单位: s)
Table 3 Computation time of various homomorphic encryption schemes for $N=10$ (unit: s)
算法 准备时间 加密时间 解密时间 同态运算时间 算法1 $2.350 \times {10^{-3}}$ $5.710 \times {10^{-6}}$ $2.343 \times {10^{-3}}$ $2.510 \times {10^{-6}}$ Paillier 26.891 2.121 0.599 0.718 BFV 0.361 0.716 0.361 0.004 CKKS 4.392 4.811 4.522 0.021
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