In this paper, we propose a new class of local differential privacy (LDP)
schemes based on combinatorial block designs for a 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 optimal (or near-optimal) privacy-utility
trade-off with lower communication costs. Indeed, we find many new LDP schemes
that achieve both the optimal privacy-utility trade-off and the minimum
communication cost among all the unbiased 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.Comment: 18 pages, 3 figures, and 1 table. This manuscript was submitted to
IEEE Transactions on Information Theory and a short version of this
manuscript will be presented at 2023 IEEE International Symposium on
Information Theor