Multiparty communication complexity and very hard functions

A boolean function f(x₁,...,x_n) with x_i ∈ {0,1}^m for each i is hard if its nondeterministic multiparty communication complexity (introduced in [in: Proceedings of the 30th IEEE FOCS, 1989, p. 428-433]), C(f), is at least nm. Note that C(f) ≤ nm for each f(x₁,...,x_n) with x_i ∈ {0, 1}^m for each i. A boolean function is very hard if it is hard and its complementary function is also hard. In this paper, we show that randomly chosen boolean function f(x₁,...,x_n) with x_i ∈ {0,1}^m for each i is very hard with very high probability (for n ≥ 3 and m large enough). In [in: Proceedings of the 12th Symposium on Theoretical Aspects of Computer Science, LNCS 900, 1995, p. 350-360], it has been shown that if f(x₁,...,x_k,...,x_n) = f₁(x₁,..., x_k) ċ f₂(x_k+1,...,x_n), where C(f₁) > 0 and C(f₂) > 0, then C(f) = C(f₁) + C(f₂). We prove here an analogical result: If f(x₁,..., x_k,..., x_n) = f₁(x₁,...,x_k) ⊕ f₂(x_k+1,..., x_n) then DC(f) = DC(f₁) + DC(f₂), where DC(g) denotes the deterministic multiparty communication complexity of the function g and "⊕" denotes the parity function.