A New Approach to Multi-Party Peer-to-Peer Communication Complexity.

We introduce new models and new information theoretic measures for the study of communication complexity in the natural peer-to-peer, multi-party, number-in-hand setting. We prove a number of properties of our new models and measures, and then, in order to exemplify their effectiveness, we use them to prove two lower bounds. The more elaborate one is a tight lower bound of $\Omega(kn)$ on the multi-party peer-to-peer randomized communication complexity of the $k$-player, $n$-bit Disjointness function. The other one is a tight lower bound of $\Omega(kn)$ on the multi-party peer-to-peer randomized communication complexity of the $k$-player, $n$-bit bitwise parity function. Both lower bounds hold when ${n=\Omega(k)}$. The lower bound for Disjointness improves over the lower bound that can be inferred from the result of Braverman et al.~(FOCS 2013), which was proved in the coordinator model and can yield a lower bound of $\Omega(kn/\log k)$ in the peer-to-peer model. To the best of our knowledge, our lower bounds are the first tight (non-trivial)lower bounds on communication complexity in the natural {\em peer-to-peer} multi-party setting. In addition to the above results for communication complexity, we also prove, using the same tools, an $\Omega(n)$ lower bound on the number of random bits necessary for the (information theoretic) private computation of the $k$-player, $n$-bit Disjointness function .