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 $(ε, δ)$-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 $(ε, δ)$-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 $ε$ and $δ$.

翻译：向数据库查询结果添加随机噪声是实现隐私保护的重要工具。一个挑战在于如何在满足隐私要求的同时最小化噪声量。近期，高斯噪声满足$(ε, δ)$-差分隐私的充分必要条件被提出，该条件使得计算该分布所需的最小隐私保护尺度成为可能。本文扩展了此项工作，为所有对称且对数凹的噪声密度函数提供了$(ε, δ)$-差分隐私的充分必要条件。我们的研究结果允许根据查询结果的维度对噪声分布进行细粒度定制。实验表明，在相同$ε$和$δ$条件下，该方法相较于当前广泛使用的拉普拉斯机制与高斯机制能够显著降低均方误差。