We consider the problem of private membership aggregation (PMA), in which a user counts the number of times a certain element is stored in a system of independent parties that store arbitrary sets of elements from a universal alphabet. The parties are not allowed to learn which element is being counted by the user. Further, neither the user nor the other parties are allowed to learn the stored elements of each party involved in the process. PMA is a generalization of the recently introduced problem of $K$ private set intersection ($K$-PSI). The $K$-PSI problem considers a set of $M$ parties storing arbitrary sets of elements, and a user who wants to determine if a certain element is repeated at least at $K$ parties out of the $M$ parties without learning which party has the required element and which party does not. To solve the general problem of PMA, we dissect it into four categories based on the privacy requirement and the collusions among databases/parties. We map these problems into equivalent private information retrieval (PIR) problems. We propose achievable schemes for each of the four variants of the problem based on the concept of cross-subspace alignment (CSA). The proposed schemes achieve \emph{linear} communication complexity as opposed to the state-of-the-art $K$-PSI scheme that requires \emph{exponential} complexity even though our PMA problems contain more security and privacy constraints.

翻译：我们考虑私有成员聚合（PMA）问题，其中用户计算某个特定元素在由独立参与方组成的系统中被存储的次数，这些参与方存储来自通用字母表的任意元素集。参与方不允许获知用户正在计数的元素。此外，用户和其他参与方均不允许了解过程中各参与方存储的元素。PMA是近期提出的$K$隐私集合交集（$K$-PSI）问题的泛化。$K$-PSI问题涉及$M$个参与方存储任意元素集，用户希望确定某个特定元素是否在$M$个参与方中至少被重复存储于$K$个参与方，同时不泄露哪些参与方拥有该元素、哪些没有。为解决PMA的通用问题，我们根据隐私要求及数据库/参与方之间的共谋程度将其划分为四类。我们将这些问题映射为等价的私有信息检索（PIR）问题。基于跨子空间对齐（CSA）概念，我们针对该问题的四种变体提出了可达方案。与现有$K$-PSI方案需要指数级复杂度不同，即便我们的PMA问题包含更多安全与隐私约束，所提方案仍实现了线性通信复杂度。