Hyperelliptic curves and integer factorization
Volume 179 / 2025
Colloquium Mathematicum 179 (2025), 123-146
MSC: Primary 11Y05; Secondary 14H45, 11Y16, 14Q05, 14H52, 11G20
DOI: 10.4064/cm9490-8-2025
Published online: 22 October 2025
Abstract
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.
