An introduction to local differential privacy protocols using block designs

TL;DR

This paper provides a comprehensive introduction to local differential privacy protocols, highlighting their connection with block designs and demonstrating how certain designs lead to optimal estimators.

Contribution

It establishes the equivalence between pure LDP protocols and $(r,\lambda)$-designs, and shows that optimal estimators are derived from the Moore-Penrose inverse of TPMs.

Findings

Pure LDP protocols are equivalent to $(r,\lambda)$-designs.

Optimal estimators are obtained from the Moore-Penrose inverse of TPMs.

BIBD-based LDP protocols provide optimal estimators.

Abstract

The design of protocols for local differential privacy (or LDP) has been a topic of considerable research interest in recent years. LDP protocols utilise the randomised encoding of outcomes of an experiment using a transition probability matrix (TPM). Several authors have observed that balanced incomplete block designs (BIBDs) provide nice examples of TPMs for LDP protocols. Indeed, it has been shown that such BIBD-based LDP protocols provide optimal estimators. In this primarily expository paper, we give a detailed introduction to LDP protocols and their connections with block designs. We prove that a subclass of LDP protocols known as pure LDP protocols are equivalent to $(r,\lambda)$-designs (which contain balanced incomplete block designs as a special case). An unbiased estimator for an LDP scheme is a left inverse of the transition probability matrix. We show that the optimal estimators for BIBD-based TPMs are precisely those obtained from the Moore-Penrose inverse of the corresponding TPM. We also review some existing work on optimal LDP protocols in the context of pure protocols.