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.

翻译：随着数字平台的快速发展，人们对各类实体如何收集、存储和使用个人数据的担忧日益加剧。这些担忧源于数据泄露、网络攻击以及将个人信息用于定向广告和监控等滥用行为的日益频繁。为解决这些问题，差分隐私已成为量化数字系统保护水平的重要工具。高斯机制因其密度函数在卷积下具有封闭性而被广泛使用，是在聚合数据集时的常用方法。然而，高斯机制仅满足近似形式的差分隐私。本文提出并分析了对称α稳定机制。我们证明该机制在保持卷积封闭性的同时实现了纯差分隐私。此外，我们深入研究了所实现的隐私水平与密度参数之间的微妙关系。最后，我们比较了高斯机制与对称α稳定机制对数据集查询引入的期望误差。通过分析，我们认为对称α稳定机制是注重隐私保护应用的一个有吸引力的选择。