In a typical formulation of the private information retrieval (PIR) problem, a single user wishes to retrieve one out of $ K$ files from $N$ servers without revealing the demanded file 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 file, the user wishes to retrieve $P>1$ linear combinations of files 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 files), and the multi-message private information retrieval (MM-PIR) problem (where the user requests $P>1$ files). A baseline achievable scheme repeats the optimal PC scheme by Sun and Jafar $P$ times, or treats each possible demanded linear combination as an independent file and then uses the near optimal MM-PIR scheme by Banawan and Ulukus. In this paper, we propose an achievable MM-PC scheme that significantly improves upon the baseline scheme. Doing so, we design the queries inspiring from the structure in the cache-aided scalar linear function retrieval scheme by Wan et al., where they leverage the dependency between messages to reduce the amount of communication. To ensure the decodability of our scheme, we propose a new method to benefit from the existing dependency, referred to as the sign assignment step. In the end, we use Maximum Distance Separable matrices to code the queries, which allows the reduction of download from the servers, while preserving privacy.

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 $K$ 个文件中的一个，且不向任何服务器泄露所请求文件的索引。本文提出了PIR的扩展模型，称为多消息私有计算（MM-PC），其中用户不是检索单个文件，而是希望检索 $P>1$ 个文件的线性组合，同时保护需求信息的隐私。MM-PC问题是私有计算（PC）问题（用户请求一个文件的线性组合）和多消息私有信息检索（MM-PIR）问题（用户请求 $P>1$ 个文件）的泛化。一种基准可实现方案将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文件，然后使用Banawan和Ulukus的近似最优MM-PIR方案。在本文中，我们提出了一种显著优于基准方案的可实现MM-PC方案。为此，我们借鉴Wan等人缓存辅助的标量线性函数检索方案中的结构设计查询，该方案利用消息之间的依赖性减少通信量。为确保方案的可解码性，我们提出了一种利用现有依赖性的新方法，称为符号分配步骤。最后，我们使用最大距离可分矩阵对查询进行编码，从而在保护隐私的同时减少服务器下载量。