Differential Private Noise Adding Mechanism: Fundamental Theory and its Application.

Differential privacy is a formal mathematical standard for quantifying the degree of that individual privacy in a statistical database is preserved. To guarantee differential privacy, a typical method is adding random noise to the original data for data release. In this paper, we investigate the fundamental theory of differential privacy considering the general random noise adding mechanism, and then apply this framework for privacy analysis of the privacy-preserving consensus algorithm. Specifically, we obtain a necessary and sufficient condition of $\epsilon$-differential privacy, and the sufficient conditions of $(\epsilon, \delta)$-differential privacy. This theoretical framework provides a useful and efficient criterion of achieving differential privacy. We utilize them to analyze the privacy of some common random noises and the theory matches with the existing literature for special cases. Applying the theory, differential privacy property of a privacy-preserving consensus algorithm is investigated based on the framework. We obtain the necessary condition and the sufficient condition for the privacy-preserving consensus algorithm, under which differential privacy is achieved, respectively. In addition, it is proved that the average consensus and $\epsilon$-differential privacy cannot be guaranteed simultaneously by any privacy-preserving consensus algorithm.