Private Variable-Length Coding with Zero Leakage

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 $\mathcal{C}$ and sends it over the channel. An adversary is assumed to have access to the output of the encoder, i.e., $\mathcal{C}$, 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 $\mathcal{C}$ 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 $\mathcal{C}$. In two cases we strengthen the existing bounds: 1. $\vert \mathcal{X}\vert \geq\vert \mathcal{Y}\vert; 2$. 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.