Can Coding Be Both Lossless and Private?

Table of Links

Abstract and I. Introduction

II. System Model

III. Secure and Private Source Coding Regions

IV. Gaussian Sources and Channels

V. Proof for Theorem 1

Acknowledgment and References

III. SECURE AND PRIVATE SOURCE CODING REGIONS

A. Lossy Source Coding

The lossy secure and and private source coding region RD is characterized below; see Section V for its proof.

We remark that (12) and (13) show that one can simultaneously achieve strong secrecy and strong privacy, i.e., the conditional mutual information terms in (2) and (3), respectively, are negligible, by using a large private key K, which is a result missing in some recent works on secure source coding with private key.

B. Lossless Source Coding

The lossless secure and and private source coding region R is characterized next; see below for a proof sketch.

Denote

IV. GAUSSIAN SOURCES AND CHANNELS

Suppose the following scalar discrete-time Gaussian source and channel model for the lossy source coding problem depicted in Fig.

We next take the liberty to use the lossy rate region in Corollary 1, characterized for discrete memoryless channels, for the model in (29)-(31). This is common in the literature since there is a discretization procedure to extend the achievability proof to well-behaved continuous-alphabet random variables and the converse proof applies to arbitrary random variables; see [2, Remark 3.8]. For Gaussian sources and channels, we use differential entropy and eliminate the cardinality bound on the auxiliary random variable. The lossy source coding region for the model in (29)-(31) without a private key is given below.

One can also show that

Thus, by calculating (25)-(27), the achievability proof follows.

For the converse proof, one can first show that

Next, consider

Combining the distortion constraint given in Corollary 1 with (50) and (51), the converse proof for (35) follows.