Secure Lossy Transmission of Vector Gaussian Sources

We study the secure lossy transmission of a vector Gaussian source to a legitimate user in the presence of an eavesdropper, where both the legitimate user and the eavesdropper have vector Gaussian side information. The aim of the transmitter is to describe the source to the legitimate user in a way that the legitimate user can reconstruct the source within a certain distortion level while the eavesdropper is kept ignorant of the source as much as possible as measured by the equivocation. We obtain an outer bound for the rate, equivocation and distortion region of this secure lossy transmission problem. This outer bound is tight when the transmission rate constraint is removed. In other words, we obtain the maximum equivocation at the eavesdropper when the legitimate user needs to reconstruct the source within a fixed distortion level while there is no constraint on the transmission rate. This characterization of the maximum equivocation involves two auxiliary random variables. We show that a nontrivial selection for both random variables may be necessary in general. The necessity of two auxiliary random variables also implies that, in general, Wyner-Ziv coding is suboptimal in the presence of an eavesdropper. In addition, we show that, even when there is no rate constraint on the legitimate link, uncoded transmission (deterministic or stochastic) is suboptimal; the presence of an eavesdropper necessitates the use of a coded scheme to attain the maximum equivocation.