Private Linear Transformation: The Individual Privacy Case

TL;DR

This paper investigates the private linear transformation problem with individual privacy on a single server, establishing capacity bounds when the user requests linear combinations of dataset messages while keeping message identities private.

Contribution

It introduces capacity bounds for private linear transformations with individual privacy, especially for MDS code coefficient matrices, and considers side information scenarios.

Findings

Established tight capacity bounds under certain conditions.

Derived lower bounds for scenarios with user side information.

Analyzed the impact of MDS code structures on privacy capacity.

Abstract

This paper considers the single-server Private Linear Transformation (PLT) problem when individual privacy is required. In this problem, there is a user that wishes to obtain $L$ linear combinations of a $D$-subset of messages belonging to a dataset of $K$ messages stored on a single server. The goal is to minimize the download cost while keeping the identity of every message required for the computation individually private. The individual privacy requirement implies that, from the perspective of the server, every message is equally likely to belong to the $D$-subset of messages that constitute the support set of the required linear combinations. We focus on the setting in which the matrix of coefficients pertaining to the required linear combinations is the generator matrix of a Maximum Distance Separable code. We establish lower and upper bounds on the capacity of PLT with individual privacy, where the capacity is defined as the supremum of all achievable download rates. We show that our bounds are tight under certain divisibility conditions. In addition, we present lower bounds on the capacity of the settings in which the user has a prior side information about a subset of messages.