Heavy-Tailed Privacy: The Symmetric alpha-Stable Privacy Mechanism

With the rapid growth of digital platforms, there is increasing apprehension about how personal data is collected, stored, and used by various entities. These concerns arise from the increasing frequency of data breaches, cyber-attacks, and misuse of personal information for targeted advertising and surveillance. To address these matters, Differential Privacy (DP) has emerged as a prominent tool for quantifying a digital system's level of protection. The Gaussian mechanism is commonly used because the Gaussian density is closed under convolution, and is a common method utilized when aggregating datasets. However, the Gaussian mechanism only satisfies an approximate form of Differential Privacy. In this work, we present and analyze of the Symmetric alpha-Stable (SaS) mechanism. We prove that the mechanism achieves pure differential privacy while remaining closed under convolution. Additionally, we study the nuanced relationship between the level of privacy achieved and the parameters of the density. Lastly, we compare the expected error introduced to dataset queries by the Gaussian and SaS mechanisms. From our analysis, we believe the SaS Mechanism is an appealing choice for privacy-focused applications.