The Distributed Multi-User Point Function

TL;DR

This paper introduces a novel distributed multi-user point function scheme that ensures correct recovery and privacy, characterizing its capacity bounds in an information-theoretic setting with multiple users and server nodes.

Contribution

It proposes the first multi-user scheme satisfying correctness and privacy constraints, with new capacity bounds derived through a novel converse argument.

Findings

Scheme guarantees correct point function recovery for all users.

Scheme maintains information-theoretic privacy against server responses.

Capacity bounds are characterized with tight inner and outer bounds.

Abstract

In this paper, we study the problem of information-theoretic distributed multi-user point function, involving a trusted master node, $N \in \mathbb{N}$ server nodes, and $K\in \mathbb{N}$ users, where each user has access to the contents of a subset of the storages of server nodes. Each user is associated with an independent point function $f_{X_k,Z_k}: \{1,2,\hdots,T\} \rightarrow{GF(q^{m R_k})},T,mR_k \in \mathbb{N}$. Using these point functions, the trusted master node encodes and places functional shares $G_1,G_2,\hdots,G_N \in GF(q^{M}), M \in \mathbb{N}$ in the storage nodes such that each user can correctly recover its point function result from the response transmitted to itself and gains no information about the point functions of any other user, even with knowledge of all responses transmitted from its connected servers. For the first time, we propose a multi-user scheme that satisfies the correctness and information-theoretic privacy constraints, ensuring recovery for all point functions. We also characterize the inner and outer bounds on the capacity -- the maximum achievable rate defined as the size of the range of each point function $mR_k$ relative to the storage size of the servers $M$ -- of the distributed multi-user point function scheme by presenting a novel converse argument.