  In this paper we consider an application of the recently proposed quantum
hashing technique for computing Boolean functions in the quantum communication
model. The combination of binary functions on non-binary quantum hash function
is done via polynomial presentation, which we have called a characteristic of a
Boolean function. Based on the characteristic polynomial presentation of
Boolean functions and quantum hashing technique we present a method for
computing Boolean functions in the quantum one-way communication model, where
one of the parties performs his computations and sends a message to the other
party, who must output the result after his part of computations. Some of the
results are also true in a more restricted Simultaneous Message Passing model
with no shared resources, in which communicating parties can interact only via
the referee. We give several examples of Boolean functions whose polynomial
presentations have specific properties allowing for construction of quantum
communication protocols that are provably exponentially better than classical
ones in the simultaneous message passing setting.
