  The problem of private data disclosure is studied from an information
theoretic perspective. Considering a pair of dependent random variables
$(X,Y)$, where $X$ and $Y$ denote the private and useful data, respectively,
the following problem is addressed: What is the maximum information that can be
revealed about $Y$ (measured by mutual information $I(Y;U)$, in which $U$ is
the revealed data), while disclosing no information about $X$ (captured by the
condition of statistical independence, i.e., $X\independent U$, and henceforth,
called \textit{perfect privacy})? We analyze the supremization of
\textit{utility}, i.e., $I(Y;U)$ under the condition of perfect privacy for two
scenarios: \textit{output perturbation} and \textit{full data observation}
models, which correspond to the cases where a Markov kernel, called
\textit{privacy-preserving mapping}, applies to $Y$ and the pair $(X,Y)$,
respectively. When both $X$ and $Y$ have a finite alphabet, the linear
algebraic analysis involved in the solution provides some interesting results,
such as upper/lower bounds on the size of the released alphabet and the maximum
utility. Afterwards, it is shown that for the jointly Gaussian $(X,Y)$, perfect
privacy is not possible in the output perturbation model in contrast to the
full data observation model. Finally, an asymptotic analysis is provided to
obtain the rate of released information when a sufficiently small leakage is
allowed. In particular, in the context of output perturbation model, it is
shown that this rate is always finite when perfect privacy is not feasible, and
two lower bounds are provided for it; When perfect privacy is feasible, it is
shown that under mild conditions, this rate becomes unbounded.
