  In this paper, we propose a new class of local differential privacy (LDP)
schemes based on combinatorial block designs for discrete distribution
estimation. This class not only recovers many known LDP schemes in a unified
framework of combinatorial block design, but also suggests a novel way of
finding new schemes achieving the exactly optimal (or near-optimal)
privacy-utility trade-off with lower communication costs. Indeed, we find many
new LDP schemes that achieve the exactly optimal privacy-utility trade-off,
with the minimum communication cost among all the unbiased or consistent
schemes, for a certain set of input data size and LDP constraint. Furthermore,
to partially solve the sparse existence issue of block design schemes, we
consider a broader class of LDP schemes based on regular and pairwise-balanced
designs, called RPBD schemes, which relax one of the symmetry requirements on
block designs. By considering this broader class of RPBD schemes, we can find
LDP schemes achieving near-optimal privacy-utility trade-off with reasonably
low communication costs for a much larger set of input data size and LDP
constraint.
