Private computation: k-connected versus 1-connected networks

We study the role of connectivity of communication networks in private computations under information theoretical settings in the honest-but-curious model. We show that some functions can 1-privately be computed even if the underlying network is 1-connected but not 2-connected. Then we give a complete characterisation of non-degenerate functions that can 1-privately be computed on non-2-connected networks. Furthermore, we present a technique for simulating 1-private protocols that work on arbitrary (complete) networks on $k$-connected networks. For this simulation, at most $\begin{equation*}(1 - \frac{k}{n-1}) \cdot L\end{equation*}$ additional random bits are needed, where $L$ is the number of bits exchanged in the original protocol and $n$ is the number of players. Finally, we give matching lower and upper bounds for the number of random bits needed to 1-privately compute the parity function on $k$-connected networks, namely $\begin{equation*}\lceil \frac{n-2}{k-1}\rceil - 1\end{equation*}$ random bits for networks consisting of $n$ players.