Multiplication polynomials for elliptic curves over finite local rings. (arXiv:2302.03650v1 [math.NT])

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 $kinmathbb{N}^*$, 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.