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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> decoder, k = 1, 2, ..., K, losslessly reconstructs the k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> source via a common link of rate R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> and a private link of rate R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> . 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> at decoder k such that the total equivocation summed over all decoders E ≥ Δ. The set of achievable ({R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sub> } <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k=1</sub> ,R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ,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.