  Generalizing a boolean function from Cleve and Buhrman \cite{cb:sqec}, we
consider the class of {\it accumulative boolean functions} of the form
$f_B(X_1,X_2,..., X_m)=\bigoplus_{i=1}^n t_B(x_i^1x_i^2... x_i^m)$, where
$X_j=(x^j_1,x^j_2,..., x^j_n), 1\leq j\leq m$ and $t_B(x_i^1x_i^2... x_i^m)=1$
for input $m$-tuples $x_i^1x_i^2...x_i^m\in B\subseteq A\subseteq \{0,1\}^n$,
and 0, if $x_i^1x_i^2...x_i^m\in A\setminus B$. Here the set $A$ is the input
{\it promise} set for function $f_B$. The input vectors $X_j, 1\leq j\leq m$
are given to the $m\geq 3$ parties respectively, who communicate cbits in a
distributed environment so that one of them (say Alice) comes up with the value
of the function. We algebraically characterize entanglement assisted LOCC
protocols requiring only $m-1$ cbits of communication for such multipartite
boolean functions $f_B$, for certain sets $B\subseteq \{0,1\}^n$, for $m\geq 3$
parties under appropriate uniform parity promise restrictions on input
$m$-tuples $x_i^1x_i^2...x_i^m, 1\leq i\leq n$. We also show that these
functions can be computed using $2m-3$ cbits in a purely classical
deterministic setup. In contrast, for certain $m$-party accumulative boolean
functions ($m\geq 2$), we characterize promise sets of mixed parity for input
$m$-tuples so that $m-1$ cbits of communication suffice in computing the
functions in the absence of any a priori quantum entanglement. We compactly
represent all these protocols and the corresponding input promise restrictions
using uniform group theoretic and hamming distance characterizations.
