  A private compression design problem is studied, where an encoder observes
useful data $Y$, wishes to compress it using variable length code and
communicates it through an unsecured channel. Since $Y$ is correlated with
private attribute $X$, the encoder uses a private compression mechanism to
design encoded message $\cal C$ and sends it over the channel. An adversary is
assumed to have access to the output of the encoder, i.e., $\cal C$, and tries
to estimate $X$. Furthermore, it is assumed that both encoder and decoder have
access to a shared secret key $W$. The design goal is to encode message $\cal
C$ with minimum possible average length that satisfies a perfect privacy
constraint. To do so we first consider two different privacy mechanism design
problems and find upper bounds on the entropy of the optimizers by solving a
linear program. We use the obtained optimizers to design $\cal C$. In two cases
we strengthen the existing bounds: 1. $|\mathcal{X}|\geq |\mathcal{Y}|$; 2. The
realization of $(X,Y)$ follows a specific joint distribution. In particular,
considering the second case we use two-part construction coding to achieve the
upper bounds. Furthermore, in a numerical example we study the obtained bounds
and show that they can improve the existing results.
