  An information theoretic privacy mechanism design problem for two scenarios
is studied where the private data is either hidden or observable. In each
scenario, privacy leakage constraints are considered using two different
measures. In these scenarios the private data is hidden or observable. In the
first scenario, an agent observes useful data $Y$ that is correlated with
private data $X$, and wishes to disclose the useful information to a user. A
privacy mechanism is designed to generate disclosed data $U$ which maximizes
the revealed information about $Y$ while satisfying a per-letter privacy
constraint. In the second scenario, the agent has additionally access to the
private data. First, the Functional Representation Lemma and Strong Functional
Representation Lemma are extended by relaxing the independence condition to
find a lower bound considering the second scenario. Next, lower bounds as well
as upper bounds on privacy-utility trade-off are derived for both scenarios. In
particular, for the case where $X$ is deterministic function of $Y$, we show
that our upper and lower bounds are asymptotically optimal considering the
first scenario.
