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.

翻译：近年来，本地差分隐私（LDP）协议的设计已成为一个备受关注的研究课题。LDP协议利用转移概率矩阵对实验结果的随机编码实现隐私保护。多位学者指出，平衡不完全区组设计为LDP协议提供了优质的转移概率矩阵范例。事实上，研究已证明基于此类BIBD的LDP协议能够产生最优估计量。在这篇以综述为主的论文中，我们系统介绍了LDP协议及其与区组设计的关联。我们证明了被称为纯LDP协议的子类等价于$(r,λ)$-设计（其特例包含平衡不完全区组设计）。对于LDP方案的无偏估计量是转移概率矩阵的左逆矩阵。我们证明基于BIBD的转移概率矩阵的最优估计量恰好可通过相应转移概率矩阵的Moore-Penrose逆矩阵获得。此外，我们在纯协议框架下回顾了现有关于最优LDP协议的研究成果。