  Polynomial functions on the group of units Q_n of the ring Z_{2^n} are
considered. A finite set of reduced polynomials RP_n in Z[x] that induces the
polynomial functions on Q_n is determined. Each polynomial function on Q_n is
induced by a unique reduced polynomial - the reduction being made using a
suitable ideal in Z[x]. The set of reduced polynomials forms a multiplicative
2-group. The obtained results are used to efficiently construct families of
exponential cardinality of, so called, huge k-ary quasigroups, which are useful
in the design of various types of cryptographic primitives. Along the way we
provide a new (and simpler) proof of a result of Rivest characterizing the
permutational polynomials on Z_{2^n}.
