We consider a discrete distribution estimation problem under a local differential privacy (LDP) constraint in the presence of shared randomness. By exploiting the shared randomness, we suggest a new method for constructing LDP schemes which achieve the exactly optimal privacy-utility trade-off (PUT) with the communication cost of less than or equal to the input data size for any privacy regime. The main idea is to decompose a block design scheme by Park et al. (2023), based on the combinatorial concept called resolution. The LDP scheme decomposed from a block design scheme is called a resolution of the block design scheme, and it achieves the same PUT as the original block design scheme while requiring a less communication cost. We provide two resolutions of an exactly PUT-optimal block design scheme, called the Baranyai's resolution and the cyclic shift resolution, both requiring the communication cost of less than or equal to the input data size. In particular, we show that the Baranyai's resolution achieves the minimum communication cost among all the PUT-optimal resolutions of block design schemes. One drawback of the Baranyai's resolution is that it can be obtained through a recursive algorithm in general. In contrast, the cyclic shift resolution has an explicit structure, but its communication cost can be larger than that of Baranyai's resolution. To complement this, we also suggest resolutions of other block design schemes achieving the optimal PUT for some privacy budgets, which require the minimum communication cost as the Baranyai's resolution and have explicit structures as the cyclic shift resolution.

翻译：本文考虑在局部差分隐私（LDP）约束下，利用共享随机性进行离散分布估计的问题。通过利用共享随机性，我们提出一种新的LDP方案构造方法，该方法能在任意隐私预算下，以不超过输入数据大小的通信成本实现精确最优的隐私-效用权衡（PUT）。核心思想是对Park等人（2023）基于组合学概念“分辨率”提出的块设计方案进行分解。从块设计方案分解得到的LDP方案称为块设计方案的分辨率，其能够实现与原块设计方案相同的PUT，同时所需通信成本更低。我们提供了两种精确PUT最优块设计方案的分辨率——Baranyai分辨率和循环移位分辨率，二者所需通信成本均不超过输入数据大小。特别地，我们证明Baranyai分辨率在所有块设计方案的PUT最优分辨率中实现了最小通信成本。Baranyai分辨率的一个缺点在于其通常需通过递归算法获得。相比之下，循环移位分辨率具有显式结构，但其通信成本可能高于Baranyai分辨率。为弥补这一不足，我们还提出了其他块设计方案的分辨率，这些方案能在特定隐私预算下实现最优PUT，同时兼具Baranyai分辨率的最小通信成本与循环移位分辨率的显式结构。