Double-Private Distributed Estimation Algorithm Using Differential Privacy and a Key-Like Proportionate Matrix with Its Performance Analysis

In this brief, we present an enhanced privacy-preserving distributed estimation algorithm, referred to as the ``Double-Private Algorithm," which combines the principles of both differential privacy (DP) and cryptography. The proposed algorithm enhances privacy by introducing DP noise into the intermediate estimations of neighboring nodes. Additionally, we employ an inverse of a closed-form reproducible proportionate gain matrix as the cryptographic key matrix to fortify the privacy protection within the proposed double private algorithm. \textcolor{blue}{We improve the algorithm by transmitting alternative variable vectors instead of raw measurements, resulting in enhanced key matrix reconstruction performance. This innovative approach mitigate noise impact, enhancing overall algorithm effectiveness.} We also establish an upper bound for the norm of the error between the non-private Diffusion Least Mean Square (DLMS) algorithm and our double private algorithm. Further, we determine a sufficient condition for the step-size to ensure the mean convergence of the proposed algorithm. Simulation results demonstrate the effectiveness of the proposed algorithm, particularly its ability to attain the final Mean Square Deviation (MSD) comparable to that of the non-private DLMS.