We consider a distributed function computation problem in which parties
observing noisy versions of a remote source facilitate the computation of a
function of their observations at a fusion center through public communication.
The distributed function computation is subject to constraints, including not
only reliability and storage but also privacy and secrecy. Specifically, 1) the
remote source should remain private from an eavesdropper and the fusion center,
measured in terms of the information leaked about the remote source; 2) the
function computed should remain secret from the eavesdropper, measured in terms
of the information leaked about the arguments of the function, to ensure
secrecy regardless of the exact function used. We derive the exact rate regions
for lossless and lossy single-function computation and illustrate the lossy
single-function computation rate region for an information bottleneck example,
in which the optimal auxiliary random variables are characterized for
binary-input symmetric-output channels. We extend the approach to lossless and
lossy asynchronous multiple-function computations with joint secrecy and
privacy constraints, in which case inner and outer bounds for the rate regions
differing only in the Markov chain conditions imposed are characterized.

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