Adding random noise to database query results is an important tool for achieving privacy. A challenge is to minimize this noise while still meeting privacy requirements. Recently, a sufficient and necessary condition for $(\epsilon, \delta)$-differential privacy for Gaussian noise was published. This condition allows the computation of the minimum privacy-preserving scale for this distribution. We extend this work and provide a sufficient and necessary condition for $(\epsilon, \delta)$-differential privacy for all symmetric and log-concave noise densities. Our results allow fine-grained tailoring of the noise distribution to the dimensionality of the query result. We demonstrate that this can yield significantly lower mean squared errors than those incurred by the currently used Laplace and Gaussian mechanisms for the same $\epsilon$ and $\delta$.

翻译：在数据库查询结果中添加随机噪音是实现隐私的一个重要工具。 一项挑战是如何在满足隐私要求的同时尽量减少这种噪音。 最近公布了高斯噪音的充足且必要的( epslon,\ delta)$差异隐私条件。 这个条件允许计算用于此分布的最小隐私保护比例。 我们延长了这项工作,为所有对称和对调噪音密度提供了充足且必要的( epslon,\ delta)$差异隐私条件。 我们的结果允许根据查询结果的维度对噪音分布进行细微裁剪。 我们证明,这可以产生大大低于目前使用的拉普和高斯机制在相同 $\epslon$ 和 $\delta$ 上发生的平均平方差 。