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：设给定有限局部环上的椭圆曲线$E$，记$E^{\infty}$为其无穷远子群。$E^{\infty}$中的每个点$P$可仅由其$x$坐标$P_x$描述，因此可用$P_x$参数化其所有倍数$nP$。我们将$(nP)_x$参数化中$(P_x)^i$的系数定义为第$i$个乘法多项式。我们证明该系数是$n$的无常数项$i$次有理多项式，并进一步证明其分母中不出现大于$i$的素数因子。作为推论，对每个有限域$\mathbb{F}_q$及任意$k\in\mathbb{N}^*$，我们刻画了定义在$\mathbb{F}_q[X]/(X^k)$上的通用椭圆曲线的群结构，并证明其$E^{\infty}$上的ECDLP可高效求解。