Polynomial functions on the group of units Qn of the ring Z2n are considered. A nite set of reduced polynomials RPn in Z[x] that induces the polynomial functions on Qn is determined. Each polynomial function on Qn 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 eciently 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 Z2n.