For a given elliptic curve E over a finite local ring, we denote by E∞ its subgroup at infinity. Every point P ∈ E∞ can be described solely in terms of its x-coordinate Px, which can be therefore used to parameterize all its multiples nP. We refer to the coefficient of (Px)i in the parameterization of (nP)x as the i-th multiplication polynomial. We show that this coefficient is a degree-i rational polynomial without a constant term in n. We also prove that no primes greater than i may appear in the denominators of its terms. As a consequence, for every finite field and any , we prescribe the group structure of a generic elliptic curve defined over .