Hyperelliptic curves and integer factorization

Let $C$ be a hyperelliptic curve over $\mathbb Z_n$ with the ramified or split model, where $n$ is a composite, square-free odd integer. If one assumes that there exists an oracle which for $D$ in the Picard group ${\rm Pic}^0_{\mathbb Z_n}(C)$ returns a non-zero multiple of the order ${\rm ord}(D)$, then we show that outputs of the oracle can be used to efficiently factorize $n$. To show this we transfer to $\mathbb Z_n$ some methods of representation and addition of divisor classes in the Picard group of a hyperelliptic curve with the ramified or split model over a field.