Let $n$ be composite with unknown factorization.

Can one efficiently find elliptic curve $E$ over $\mathbb{Z}/n\mathbb{Z}$ and a point $P$ on $E$ of known order $m > 5$?

$P$ should be nontorsion when lifted to $\mathbb{Q}$, so to avoid such cases one may consider $m$ larger.

$E$ might be singular (though better not be).

What I know

Singular cases like $E : y^2 = x^3$ are easy.

Singular cases when there is an isomorphism to a multiplicative subgroup are easy if one knows an element of known multiplicative order (say for Mersenne numbers), but this doesn't work for general $n$.

Tried division polynomials, but this leads to solving nonlinear equation over the ring so didn't work for me.

Subquestions

Are there $n$ of special form (except the described) for which it is possible?

The problem is not harder than factoring.

Is the non-singular case as hard as factoring?