  The purpose of this paper is to develop a mathematical analysis theory to
solve differential privacy problems. The heart of our approaches is to use
analytic tools to characterize the correlations among the outputs of different
datasets, which makes it feasible to represent a differentially private
mechanism with minimal number of parameters. These results are then used to
construct differentially private mechanisms analytically. Furthermore, our
approaches are universal to almost all query functions. We believe that the
approaches and results of this paper are indispensable complements to the
current studies of differential privacy that are ruled by the ad hoc and
algorithmic approaches.
