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The problem of preserving privacy when a multivariate source is required tobe revealed partially to multiple users is modeled as a Gray-Wyner sourcecoding problem with K correlated sources at the encoder and K decoders in whichthe kth decoder, k = 1, 2, ...,K, losslessly reconstructs the kth source via acommon link and a private link. The privacy requirement of keeping each decoderoblivious of all sources other than the one intended for it is introduced viaan equivocation constraint at each decoder such that the total equivocationsummed over all decoders is E. The set of achievable rates-equivocation tuplesis completely characterized. Using this characterization, two differentdefinitions of common information are presented and are shown to be equivalent.

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