Secure Lossy Function Computation with Multiple Private Remote Source Observations

We consider that multiple noisy observations of a remote source are used by different nodes in the same network to compute a function of the noisy observations under joint secrecy, joint privacy, and individual storage constraints, as well as a distortion constraint on the function computed. Suppose that an eavesdropper has access to one of the noisy observations in addition to the public messages exchanged between legitimate nodes. This model extends previous models by 1) considering a remote source as the source of dependency between the correlated random variables observed at different nodes; 2) allowing the function computed to be a distorted version of the target function, which allows to reduce the storage rate as compared to a reliable function computation scenario in addition to reducing secrecy and privacy leakages; 3) introducing a privacy metric that measures the information leakage about the remote source to the fusion center in addition to the classic privacy metric that measures the leakage to an eavesdropper; 4) considering two transmitting nodes to compute a function rather than one node. Single-letter inner and outer bounds are provided for the considered lossy function computation problem, and exact lossy rate regions are characterized for two special cases in which either the computed function is partially invertible or the function is invertible and the measurement channel of the eavesdropper is physically degraded with respect to the measurement channel of the fusion center.