  A privacy mechanism design problem is studied through the lens of information
theory. In this work, an agent observes useful data $Y=(Y_1,...,Y_N)$ that is
correlated with private data $X=(X_1,...,X_N)$ which is assumed to be also
accessible by the agent. Here, we consider $K$ users where user $i$ demands a
sub-vector of $Y$, denoted by $C_{i}$. The agent wishes to disclose $C_{i}$ to
user $i$. Since $C_{i}$ is correlated with $X$ it can not be disclosed
directly. A privacy mechanism is designed to generate disclosed data $U$ which
maximizes a linear combinations of the users utilities while satisfying a
bounded privacy constraint in terms of mutual information. In a similar work it
has been assumed that $X_i$ is a deterministic function of $Y_i$, however in
this work we let $X_i$ and $Y_i$ be arbitrarily correlated. First, an upper
bound on the privacy-utility trade-off is obtained by using a specific
transformation, Functional Representation Lemma and Strong Functional
Representaion Lemma, then we show that the upper bound can be decomposed into
$N$ parallel problems. Next, lower bounds on privacy-utility trade-off are
derived using Functional Representation Lemma and Strong Functional
Representaion Lemma. The upper bound is tight within a constant and the lower
bounds assert that the disclosed data is independent of all $\{X_j\}_{i=1}^N$
except one which we allocate the maximum allowed leakage to it. Finally, the
obtained bounds are studied in special cases.
