Multi-user privacy: The Gray-Wyner system and generalized common information

The problem of preserving privacy when a multi-variate source is required to be revealed partially to multiple users is modeled as a Gray-Wyner source coding problem with K correlated sources at the encoder and K decoders in which the k^th decoder, k = 1, 2, ..., K, losslessly reconstructs the k^th source via a common link of rate R₀ and a private link of rate R_k. The privacy requirement of keeping each decoder oblivious of all sources other than the one intended for it is introduced via an equivocation constraint E_k at decoder k such that the total equivocation summed over all decoders E ≥ Δ. The set of achievable ({R_k}^K_k=1,R₀,Δ) rates-equivocation (K + 2)-tuples is completely characterized. Using this characterization, two different definitions of common information are presented and are shown to be equivalent.