
A boolean function f(x_1, . . . , 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) n*m for each f(x_1, . . . , 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(x_1, . . . , 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(x_1, . . . , x_k , . . . , x_n) = f_1 (x_1, . . . , x_k ) · f_2(x_{k+1}, . . . , x_n), where C(f_1) > 0 and C(f_2) > 0, then C(f) = C(f1) + C(f2).We prove here an analogical result: If f(x_1, . . . , x_k , . . . , x_n) = f_1(x_1, . . . , x_k ) ⊕ f_2(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. 