In a typical formulation of the private information retrieval (PIR) problem, a single user wishes to retrieve one out of $ K$ datasets from $N$ servers without revealing the demanded message index to any server. This paper formulates an extended model of PIR, referred to as multi-message private computation (MM-PC), where instead of retrieving a single message, the user wishes to retrieve $P>1$ linear combinations of datasets while preserving the privacy of the demand information. The MM-PC problem is a generalization of the private computation (PC) problem (where the user requests one linear combination of the datasets), and the multi-message private information retrieval (MM-PIR) problem (where the user requests $P>1$ datasets). A direct achievable scheme, referred to as baseline scheme, repeats the optimal PC scheme by Sun and Jafar $P$ times, or treats each possible demanded linear combination as an independent dataset and then uses the near optimal MM-PIR scheme by Banawan and Ulukus. However, a direct combination of the PC and the MM-PIR schemes does not result in an achievable scheme. Our main contribution to this new problem is to propose an achievable MM-PC scheme by smartly leveraging the above two schemes with some additional highly non-trivial steps.

翻译：在私有信息检索（PIR）问题的典型表述中，单个用户希望从$N$个服务器中检索$K$个数据集中的一个，而不向任何服务器泄露所请求消息的索引。本文提出了一种扩展的PIR模型，称为多消息私有计算（MM-PC），其中用户不再检索单个消息，而是希望检索$P>1$个数据集的线性组合，同时保护需求信息的隐私。MM-PC问题是私有计算（PC）问题（用户请求数据集的一个线性组合）和多消息私有信息检索（MM-PIR）问题（用户请求$P>1$个数据集）的泛化。一种直接的可行方案（称为基线方案）将Sun和Jafar的最优PC方案重复$P$次，或者将每个可能的请求线性组合视为独立数据集，然后使用Banawan和Ulukus的近最优MM-PIR方案。然而，PC方案与MM-PIR方案的直接组合并不能得到可行的方案。我们对该新问题的主要贡献是，通过巧妙结合上述两种方案并附加一些高度非平凡的步骤，提出了一种可行的MM-PC方案。