We introduce the multi-dimensional Skellam mechanism, a discrete differential
privacy mechanism based on the difference of two independent Poisson random
variables. To quantify its privacy guarantees, we analyze the privacy loss
distribution via a numerical evaluation and provide a sharp bound on the
R\'enyi divergence between two shifted Skellam distributions. While useful in
both centralized and distributed privacy applications, we investigate how it
can be applied in the context of federated learning with secure aggregation
under communication constraints. Our theoretical findings and extensive
experimental evaluations demonstrate that the Skellam mechanism provides the
same privacy-accuracy trade-offs as the continuous Gaussian mechanism, even
when the precision is low. More importantly, Skellam is closed under summation
and sampling from it only requires sampling from a Poisson distribution -- an
efficient routine that ships with all machine learning and data analysis
software packages. These features, along with its discrete nature and
competitive privacy-accuracy trade-offs, make it an attractive practical
alternative to the newly introduced discrete Gaussian mechanism.

本研究介绍了基于两个独立的泊松随机变量之差的多维 Skellam 机制，可以用于分散式隐私应用及联邦学习下的安全聚合。该机制提供了与连续高斯机制相同的隐私 - 准确性折衷，即使精度较低也可以实现，而实现该机制只需要从泊松分布中进行取样。