  Differential privacy is a notion that has emerged in the community of
statistical databases, as a response to the problem of protecting the privacy
of the database's participants when performing statistical queries. The idea is
that a randomized query satisfies differential privacy if the likelihood of
obtaining a certain answer for a database $x$ is not too different from the
likelihood of obtaining the same answer on adjacent databases, i.e. databases
which differ from $x$ for only one individual. Information flow is an area of
Security concerned with the problem of controlling the leakage of confidential
information in programs and protocols. Nowadays, one of the most established
approaches to quantify and to reason about leakage is based on the R\'enyi min
entropy version of information theory. In this paper, we analyze critically the
notion of differential privacy in light of the conceptual framework provided by
the R\'enyi min information theory. We show that there is a close relation
between differential privacy and leakage, due to the graph symmetries induced
by the adjacency relation. Furthermore, we consider the utility of the
randomized answer, which measures its expected degree of accuracy. We focus on
certain kinds of utility functions called "binary", which have a close
correspondence with the R\'enyi min mutual information. Again, it turns out
that there can be a tight correspondence between differential privacy and
utility, depending on the symmetries induced by the adjacency relation and by
the query. Depending on these symmetries we can also build an optimal-utility
randomization mechanism while preserving the required level of differential
privacy. Our main contribution is a study of the kind of structures that can be
induced by the adjacency relation and the query, and how to use them to derive
bounds on the leakage and achieve the optimal utility.
