On Multi-Message Private Computation

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 (MMPC), 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 inspired from the structure in the cache-aided scalar linear function retrieval scheme, 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.