Capacity of Private Linear Computation for Coded Databases

TL;DR

This paper investigates the maximum efficiency of private linear computation in distributed coded storage systems, showing that the capacity matches that of private information retrieval for MDS codes, depending on the linear coefficient matrix.

Contribution

It extends previous work by deriving the PLC capacity for a broad class of linear codes, including MDS codes, and establishes a capacity expression based on the rank of the coefficient matrix.

Findings

PLC capacity matches MDS-coded PIR capacity for many codes.

Capacity depends on the rank of the coefficient matrix.

The converse proof is valid for any number of messages and linear combinations.

Abstract

We consider the problem of private linear computation (PLC) in a distributed storage system. In PLC, a user wishes to compute a linear combination of $f$ messages stored in noncolluding databases while revealing no information about the coefficients of the desired linear combination to the databases. In extension of our previous work we employ linear codes to encode the information on the databases. We show that the PLC capacity, which is the ratio of the desired linear function size and the total amount of downloaded information, matches the maximum distance separable (MDS) coded capacity of private information retrieval for a large class of linear codes that includes MDS codes. In particular, the proposed converse is valid for any number of messages and linear combinations, and the capacity expression depends on the rank of the coefficient matrix obtained from all linear combinations.