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

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.

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