Differential Private Discrete Noise Adding Mechanism: Conditions and Properties

Differential privacy is a golden standard to quantify the degree of privacy protection. Many application scenarios of differential privacy in daily life are based on discretely distributed datasets, e.g., traffic data, health records. However, the adoption of differential privacy under discrete distributions is rare. This paper focuses on the discrete random noise adding mechanisms. We first propose basic differential privacy conditions for the general discrete noise adding mechanisms. Then, we theoretically analyze the differential privacy properties for the proposed mechanisms. More concretely, we derive a sufficient and necessary condition for discrete ϵ-differential privacy and a sufficient condition for discrete (ϵ, δ) -differential privacy, with the numerical estimation of differential privacy parameters. These conditions can be applied to analyze the differential privacy properties for the discrete random mechanisms with various noises, e.g., the discrete Gaussian and the discrete Laplacian noise distributions. Further, we design a discrete noise adding mechanism with the Staircase-shaped probability distribution to guarantee arbitrary differential privacy level.