Expansions of the group $Z_8$ (Part I)

We investigate clones in the interval between the group polynomials and the ring polynomials of ${\mathbb{Z}}_8$. This is the simplest open case of the problem, as the answer is known for ${\mathbb{Z}}_{p^2}$ (with $p$ prime) and, in general, ${\mathbb{Z}}_n$ reduces to the case when $n$ is a prime power. The investigated structure proves to be very complicated, so we provide only a partial description. We restrict our attention to polynomials whose nonlinear monomials have even coefficients.