Multiparty Communication Complexity: Very Hard Functions

We consider the multiparty communication model defined in [DF89] using the formalism from [Hr97]. First, we correct an inaccuracy in the proof of the fundamental result of [DR95] providing a lower bound on the nondeterministic communication complexity of a function. Then we construct several very hard functions, i.e., functions such that those as well as their complements have the worst possible nondeterministic communication complexity. The problem to find a particular very haxd function was proposed in [Du99], where it has been shown that almost all functions are very hard. We also prove that combining two very hard functions by the boolean operation xor gives a very hard function.