An introduction to local differential privacy protocols using block designs

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 $(r,λ)$-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.

💡 Research Summary

The paper provides a comprehensive, largely expository treatment of local differential privacy (LDP) protocols through the lens of combinatorial block designs, especially balanced incomplete block designs (BIBDs) and their generalizations. It begins by formalizing the LDP setting: a finite input domain X of size n and an output (perturbed) domain Y of size m ≥ n. For each input x∈X a “high‑probability” subset Yₓ⊆Y and its complement Y\Yₓ are specified. A local randomizer f maps x to an element of Yₓ with probability θ·|Yₓ|⁻¹ (uniformly within Yₓ) and to an element of the complement with probability (1‑θ)·|Y\Yₓ|⁻¹. The collection of these probabilities forms the transition probability matrix (TPM) Q∈ℝ^{m×n}, where Q_{y,x}=Pr