# Point-halving/solving quartic equations over the elliptic curve E(Z_N)/ring Z_N where N = pq

I am wondering whether there are any results/whether there is any knowledge about the following problem:

Given a univariate polynomial (say, a quartic) equation defined over $$\mathbb{Z}_N$$, is it possible to find its roots/factorisation in polynomial time? I know that there exist randomized/average-case polynomial algorithms for doing so over finite fields $$\mathbb{Z}_p$$, and given $$p$$, $$q$$, where $$N = pq$$, one can reduce the problem to finding roots in $$\mathbb{Z}_p$$ and $$\mathbb{Z}_q$$ with the Chinese Remainder Theorem, but I am interested in the situation where the factorisation of $$N$$ is not known for cryptographic applications where $$N$$ is a RSA modulus.

Considering the case of the quartic equation, the closed solution requires computing square roots modulo $$N$$, which is as difficult as factoring $$N$$, but I am wondering if there may be a different way to do so in this case that bypasses this need (and is polynomial).

Indeed, halving a point over an elliptic curve reduces to solving such a quartic equation, which motivates my question. Perhaps there is a way to halve points over $$\mathbb{Z}_N$$ without needing to do this? I am not sure. This is my main concern, because it would seem like solving a general quartic reduces directly to the quadratic residuosity problem, but the group law gives you a quite specific quartic equation.