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Differential privacy is a formal mathematical {stand-ard} for quantifying thedegree of that individual privacy in a statistical database is preserved. Toguarantee differential privacy, a typical method is adding random noise to theoriginal data for data release. In this paper, we investigate the conditions ofdifferential privacy considering the general random noise adding mechanism, andthen apply the obtained results for privacy analysis of the privacy-preservingconsensus algorithm. Specifically, we obtain a necessary and sufficientcondition of $\epsilon$-differential privacy, and the sufficient conditions of$(\epsilon, \delta)$-differential privacy. We apply them to analyze variousrandom noises. For the special cases with known results, our theory matcheswith the literature; for other cases that are unknown, our approach provides asimple and effective tool for differential privacy analysis. Applying theobtained theory, on privacy-preserving consensus algorithms, it is proved thatthe average consensus and $\epsilon$-differential privacy cannot be guaranteedsimultaneously by any privacy-preserving consensus algorithm.

Differential Private Noise Adding Mechanism and Its Application on Consensus