Private Linear Transformation: The Joint Privacy Case

TL;DR

This paper introduces the Private Linear Transformation problem, focusing on the single-server case, and characterizes the capacity for MDS coefficient matrices while providing bounds for other cases.

Contribution

It formalizes the PLT problem, extends PIR and PLC concepts, and derives capacity results for MDS matrices and bounds for non-MDS cases.

Findings

Capacity characterized for MDS coefficient matrices.

Bounds provided for non-MDS and side information scenarios.

Generalizes PIR and PLC frameworks.

Abstract

We introduce the problem of Private Linear Transformation (PLT). This problem includes a single (or multiple) remote server(s) storing (identical copies of) $K$ messages and a user who wants to compute $L$ linear combinations of a $D$-subset of these messages by downloading the minimum amount of information from the server(s) while protecting the privacy of the entire set of $D$ messages. This problem generalizes the Private Information Retrieval and Private Linear Computation problems. In this work, we focus on the single-server case. For the setting in which the coefficient matrix of the required $L$ linear combinations generates a Maximum Distance Separable (MDS) code, we characterize the capacity -- defined as the supremum of all achievable download rates, for all parameters $K, D, L$. In addition, we present lower and/or upper bounds on the capacity for the settings with non-MDS coefficient matrices and the settings with a prior side information.