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We consider a discrete distribution estimation problem under a local differential privacy (LDP) constraint in the presence of shared randomness. For this problem, we propose a new class of LDP schemes achieving the exactly optimal privacy-utility trade-off (PUT), with the communication cost less than or equal to the size of the input data. Moreover, it is shown as a simple corollary that one-bit communication is sufficient for achieving the exactly optimal PUT for a high privacy regime if the input data size is an even number. The main idea is to decompose a block design scheme proposed by Park et al. (2023), based on the combinatorial concept called resolution. We call the resultant decomposed LDP scheme with shared randomness as a resolution of the original block design scheme. A resolution of a block design scheme has a communication cost less than or equal to that of the original block design scheme. Also, the resolution of a block design scheme is exactly optimal whenever the original block design scheme is exactly optimal. Accordingly, we provide two resolutions of the exactly optimal subset selection scheme proposed by Ye and Barg (2018), called the Baranyai’s resolution and the cyclic shift resolution. We show that the Baranyai’s resolution achieves the minimum communication cost among all exactly optimal resolutions of block design schemes. One drawback of the Baranyai’s resolution is that its explicit structure is unknown in general. In contrast, the cyclic shift resolution has an explicit structure, but its communication cost can be larger than that of the Baranyai’s resolution. To complement this, we also suggest resolutions of other block design schemes achieving the exactly optimal PUT for some input data size and privacy budget. Those require the minimum communication cost as the Baranyai’s resolution and have explicit structures as the cyclic shift resolution.