Private Linear Computation for Noncolluding Coded Databases

Private computation in a distributed storage system (DSS) is a generalization of the private information retrieval (PIR) problem. In such a setting, a user wishes to compute a function of f messages stored in n noncolluding coded databases, i.e., databases storing data encoded with an [n,k] linear storage code, while revealing no information about the desired function to the databases. We consider the problem of private linear computation (PLC) for coded databases. In PLC, a user wishes to compute a linear combination over the f messages while keeping the coefficients of the desired linear combination hidden from the databases. For a DSS setup where data is stored using a code from a particular family of linear storage codes, we derive an outer bound on the PLC rate, which is defined as the ratio of the desired amount of information and the total amount of downloaded information. In particular, the proposed converse is valid for any number of messages and linear combinations, and depends on the rank of the coefficient matrix obtained from all linear combinations. Further, we present a PLC scheme with rate equal to the outer bound and hence settle the PLC capacity for the considered class of linear storage codes. Interestingly, the PLC capacity matches the maximum distance separable coded capacity of PIR for the considered class of linear storage codes.

Automated summary

Private computation is a generalization of the private information retrieval (PIR) problem, where a user wishes to compute a function of messages stored in coded databases without revealing any information. This paper focuses on the problem of private linear computation (PLC) in a distributed storage system using linear storage codes. An outer bound on the PLC rate is derived, and a matching PLC scheme is presented for a specific class of linear storage codes.