  In this paper, we study a stochastic disclosure control problem using
information-theoretic methods. The useful data to be disclosed depend on
private data that should be protected. Thus, we design a privacy mechanism to
produce new data which maximizes the disclosed information about the useful
data under a strong $\chi^2$-privacy criterion. For sufficiently small leakage,
the privacy mechanism design problem can be geometrically studied in the space
of probability distributions by a local approximation of the mutual
information. By using methods from Euclidean information geometry, the original
highly challenging optimization problem can be reduced to a problem of finding
the principal right-singular vector of a matrix, which characterizes the
optimal privacy mechanism. In two extensions we first consider a scenario where
an adversary receives a noisy version of the user's message and then we look
for a mechanism which finds $U$ based on observing $X$, maximizing the mutual
information between $U$ and $Y$ while satisfying the privacy criterion on $U$
and $Z$ under the Markov chain $(Z,Y)-X-U$.
