Multi-User Privacy: The Gray-Wyner System and Generalized Common\n Information

The problem of preserving privacy when a multivariate source is required to\nbe revealed partially to multiple users is modeled as a Gray-Wyner source\ncoding problem with K correlated sources at the encoder and K decoders in which\nthe kth decoder, k = 1, 2, ...,K, losslessly reconstructs the kth source via a\ncommon link and a private link. The privacy requirement of keeping each decoder\noblivious of all sources other than the one intended for it is introduced via\nan equivocation constraint at each decoder such that the total equivocation\nsummed over all decoders is E. The set of achievable rates-equivocation tuples\nis completely characterized. Using this characterization, two different\ndefinitions of common information are presented and are shown to be equivalent.\n