Privacy-preserving Average Consensus: Privacy Analysis and Optimal Algorithm Design

The goal of the privacy-preserving average consensus (PPAC) is to guarantee the privacy of initial states and asymptotic consensus on the exact average of the initial value. This goal is achieved by an existing PPAC algorithm by adding and subtracting variance decaying and zero-sum random noises to the consensus process. However, there is lack of theoretical analysis to quantify the degree of the privacy protection. In this paper, we analyze the privacy of the PPAC algorithm in the sense of the maximum disclosure probability that the other nodes can infer one node’s initial state within a given small interval. We first introduce a privacy definition, named (ǫ, δ)-data-privacy, to depict the maximum disclosure probability. We prove that PPAC provides (ǫ, δ)-data-privacy, and obtain the closed-form expression of the relationship between ǫ and δ. We also prove that the added noise with uniform distribution is optimal in terms of achieving the highest (ǫ, δ)-data-privacy. Then, we prove that the disclosure probability will converge to one when all information used in the consensus process is available, i.e., the privacy is compromised. Finally, we propose an optimal privacy-preserving average consensus (OPAC) algorithm to achieve the highest (ǫ, δ)data-privacy. Simulations are conducted to verify the results.