Private Variable-Length Coding with Zero Leakage

TL;DR

This work first considers two different privacy mechanism design problems and finds upper bounds on the entropy of the optimizers by solving a linear program and uses the obtained optimizers to design $\mathcal{C}$.

Abstract

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.

Figures (3)

• Figure 1: In this work an encoder wants to compress $Y$ which is correlated with $X$ under a zero privacy leakage constraint and send it over a channel where an eavesdropper has access to the output of the encoder. The encoder and decoder have an advantage of using shared secret key.
• Figure 2: In this work we use two-part construction coding strategy to send codewords over the channels. We hide the information of $X$ using one-time-pad coding and we then use the solution of $g_0(P_{XY})=h_0(P_{XY})$ to construct $U$.
• Figure 3: At the receiver side, we first decode $X$ using the shared key $W$, then by using the fact that $U$ satisfies $H(Y|X,U)=0$, we can decode $Y$ based on $X$ and $U$.

Theorems & Definitions (23)

• Lemma 1
• proof
• Corollary 1
• Lemma 2
• proof
• Remark 1
• Lemma 3
• proof
• Lemma 4
• proof