One can note that, given an elliptic curve mod $p$, that the set of points together with the usual addition law gives a finite Abelian group.
Now by the fundamental theorem of finite abelian groups, $$E\cong\mathbb{Z}/p_1^{a_1}\mathbb{Z}\otimes\dots\otimes\mathbb{Z}/p_k^{a_k}\mathbb{Z},$$ where $$E=\{(x,y)\in\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}:y^2\equiv x^3+ax+b\}\cup\{O\}$$ and $n=p_1^{a_1}\dots p_k^{a_k}$.
Are there any cryptographic attacks on elliptic curve cryptography that utilize this decomposition? $n$ can be obtained easily enough by Schoof's algorithm.