In this paper, we present a secure distributed estimation strategy in networked systems. In particular, we consider distributed Kalman filtering as the estimation method and Paillier encryption, which is a partially homomorphic encryption scheme. The proposed strategy protects the confidentiality of the transmitted data within a network. Moreover, it also secures the state estimation computation process. To this end, all the algebraic calculations needed for state estimation in a distributed Kalman filter are performed over the encrypted data. As Paillier encryption only deals with integer data, in general, this, in turn, provides significant quantization error in the computation process associated with the Kalman filter. However, the proposed estimation approach handles quantized data in an efficient way. We provide an optimality and convergence analysis of our proposed method. It is shown that state estimation and a covariance matrix associated with the proposed method remain with a certain small radius of those of a conventional centralized Kalman filter. Simulation results are given to further demonstrate the effectiveness of the proposed scheme.

在本文中，我们提出了一种在网络系统中的安全分布式估计策略。特别地，我们将分布式卡尔曼滤波作为估计方法和Paillier加密，这是一种部分同态的加密方案。所提出的策略保护网络内传输的数据的机密性。而且，还确保了状态估计计算处理。为此，对加密数据执行分布式卡尔曼滤波器中状态估计所需的所有代数计算。由于Paillier加密仅处理整数数据，因此，这反过来又在与卡尔曼滤波器相关的计算过程中提供了很大的量化误差。然而，提出的估计方法以有效的方式处理量化的数据。我们提供了我们提出的方法的最优性和收敛性分析。结果表明，与所提出的方法相关的状态估计和协方差矩阵保持了常规集中式卡尔曼滤波器的半径和一定的小半径。仿真结果进一步证明了该方案的有效性。