We study mechanisms for differential privacy on finite datasets. By deriving \emph{sufficient sets} for differential privacy we obtain necessary and sufficient conditions for differential privacy, a tight lower bound on the maximal expected error of a discrete mechanism and a characterisation of the optimal mechanism which minimises the maximal expected error within the class of mechanisms considered.
@article{arxiv.1505.07254,
title = {Differentially Private Response Mechanisms on Categorical Data},
author = {Naoise Holohan and Doug Leith and Oliver Mason},
journal= {arXiv preprint arXiv:1505.07254},
year = {2015}
}