The Private and Public Correlation Cost of Three Random Variables with Collaboration

TL;DR

This paper characterizes the trade-off between public and private correlations needed to generate three-party distributions, extending Wyner's common information and exploring collaborative versus adversarial scenarios.

Contribution

It provides a single-letter formula for the correlation cost trade-off in three-party settings, generalizing Wyner's common information to include private correlations.

Findings

Single-letter characterization of correlation trade-off

Private bits generalize Wyner's common information

Differences between collaborative and adversarial scenarios detailed

Abstract

In this paper we consider the problem of generating arbitrary three-party correlations from a combination of public and secret correlations. Two parties -- called Alice and Bob -- share perfectly correlated bits that are secret from a collaborating third party, Charlie. At the same time, all three parties have access to a separate source of correlated bits, and their goal is to convert these two resources into multiple copies of some given tripartite distribution $P_{XYZ}$. We obtain a single-letter characterization of the trade-off between public and private bits that are needed to achieve this task. The rate of private bits is shown to generalize Wyner's classic notion of common information held between a pair of random variables. The problem we consider is also closely related to the task of secrecy formation in which $P_{XYZ}$ is generated using public communication and local randomness but with Charlie functioning as an adversary instead of a collaborator. We describe in detail the differences between the collaborative and adversarial scenarios.