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 C and sends it over the channel. An adversary is assumed to have access to the output of the encoder, i.e., 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 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 C. In two cases we strengthen the existing bounds: 1. |X| |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.