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In this work, we investigate two source coding models, a \emph{Helper}problem and a \emph{Gray-Wyner} problem, under equivocation constraints.Specifically, in the Helper problem, an encoder communicates with a legitimatereceiver through noise-free rate-limited public and private links; and anexternal passive eavesdropper intercepts every information that is sent on thepublic link. We study two classes of this model: i) when a pair of arbitrarilycorrelated discrete memoryless sources is to be encoded such that one componenthas to be recovered lossily at the legitimate receiver while the equivocationabout both components at the eavesdropper must be maintained no smaller thansome prescribed level; and ii) when the legitimate receiver reproduces bothcomponents, one of which, that is recovered losslessly, has to be concealedfrom the eavesdropper to some equivocation level. For both classes problems, weestablish single-letter characterizations of optimalrate-distortion-equivocation tradeoffs in the discrete memoryless case. Next,we extend our results to the case of two legitimate receivers, i.e., Gray-Wynernetwork with equivocation constraints. Here, two legitimate receivers areconnected to the encoder each through a dedicated error-free private link aswell as a common error-free public link; and an external passive eavesdropperoverhears on the public link. We study two classes of this model that areextensions of the aforementioned instances of Helper problems to the case oftwo receivers. For each of the two classes, we establish a single-lettercharacterization of the optimal rate-distortion-equivocation region. Throughoutthe paper, the analysis sheds light on the role of the private links, and weillustrate the results by computing them for some binary examples. Also, wemake some meaningful connections, e.g., with problems of secret-sharing andencryption.

Secure Lossy Source Coding for Some Classes of Helper and Gray-Wyner Models