  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 data $X$, 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 $W$. In this work, we generalize the perfect
privacy (secrecy) assumption and consider a non-zero leakage between the
private data $X$ and encoded message $\cal C$. The design goal is to encode
message $\cal C$ with minimum possible average length that satisfies
non-perfect privacy constraints. We find upper and lower bounds on the average
length of the encoded message using different privacy metrics and study them in
special cases. For the achievability we use two-part construction coding and
extended versions of Functional Representation Lemma. Lastly, in an example we
show that the bounds can be asymptotically tight.
