Private Computation — k-Connected versus 1-Connected Networks
We study the role of connectivity of communication networks in private computations under information theoretic settings. It will be shown that some functions can be computed by private protocols even if the underlying network is 1-connected but not 2-connected. Then we give a complete characterisation of non-degenerate functions that can be computed on non-2-connected networks.
Furthermore, a general technique for simulating private protocols on arbitrary networks will be presented. Using this technique every private protocol can be simulated on arbitrary k-connected networks using only a small number of additional random bits.
Finally, we give matching lower and upper bounds for the number of random bits needed to compute the parity function on k-connected networks.
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