For a given elliptic curve $E$ over a finite local ring, we denote by $E^{\infty}$ its subgroup at infinity. Every point $P \in E^{\infty}$ can be described solely in terms of its $x$-coordinate $P_x$, which can be therefore used to parameterize all its multiples $nP$. We refer to the coefficient of $(P_x)^i$ 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 $\mathbb{F}_q$ and any $k\in\mathbb{N}^*$, we prescribe the group structure of a generic elliptic curve defined over $\mathbb{F}_q[X]/(X^k)$, and we show that their ECDLP on $E^{\infty}$ may be efficiently solved.

翻译：对于给定有限局部环上的椭圆曲线 $E$，我们记 $E^{\infty}$ 为其无穷远子群。每个点 $P \in E^{\infty}$ 可以仅由其 $x$ 坐标 $P_x$ 描述，因此可用 $P_x$ 参数化其所有倍数 $nP$。我们将 $(nP)_x$ 的参数化中 $(P_x)^i$ 的系数称为第 $i$ 个乘法多项式。我们证明该系数是关于 $n$ 的 $i$ 次有理多项式且不含常数项，并进一步证明其项分母中不会出现大于 $i$ 的素数。作为推论，对任意有限域 $\mathbb{F}_q$ 和任意 $k\in\mathbb{N}^*$，我们确定了定义在 $\mathbb{F}_q[X]/(X^k)$ 上的通用椭圆曲线的群结构，并证明 $E^{\infty}$ 上的 ECDLP 可被高效求解。