Fundamental Limits of 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 (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 a new MM-PC scheme that significantly improves upon the baseline schemes. In doing so, we design the queries inspired by the structure in the cache-aided scalar linear function retrieval scheme by Wan {\it et al.}, which leverages the dependency between linear functions to reduce the amount of communications. 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. By the proposed schemes, we characterize the capacity within a multiplicative factor of $2$.