Multi-Server Private Linear Transformation with Joint Privacy

TL;DR

This paper characterizes the maximum achievable data download rate for private linear transformations in multi-server settings, especially when the desired linear combinations form an MDS code, and provides bounds for various parameter regimes.

Contribution

It derives the capacity of the Private Linear Transformation problem for the case of a single linear combination and establishes tight bounds for multiple parameters, advancing understanding of multi-server privacy.

Findings

Capacity characterized for L=1 case.

Upper bounds established for all parameters.

Tight bounds shown for specific regimes.

Abstract

This paper focuses on the Private Linear Transformation (PLT) problem in the multi-server scenario. In this problem, there are $N$ servers, each of which stores an identical copy of a database consisting of $K$ independent messages, and there is a user who wishes to compute $L$ independent linear combinations of a subset of $D$ messages in the database while leaking no information to the servers about the identity of the entire set of these $D$ messages required for the computation. We focus on the setting in which the coefficient matrix of the desired $L$ linear combinations generates a Maximum Distance Separable (MDS) code. We characterize the capacity of the PLT problem, defined as the supremum of all achievable download rates, for all parameters $N, K, D \geq 1$ and $L=1$, i.e., when the user wishes to compute one linear combination of $D$ messages. Moreover, we establish an upper bound on the capacity of PLT problem for all parameters $N, K, D, L \geq 1$, and leveraging some known capacity results, we show the tightness of this bound in the following regimes: (i) the case when there is a single server (i.e., $N=1$), (ii) the case when $L=1$, and (iii) the case when $L=D$.