• Corpus ID: 119278392

Factoring integer using elliptic curves over rational number field $\mathbb{Q}$

@article{Li2012FactoringIU,
  title={Factoring integer using elliptic curves over rational number field \$\mathbb\{Q\}\$},
  author={Xiumei Li and Jinxiang Zeng},
  journal={arXiv: Number Theory},
  year={2012}
}
For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity conjecture, that the elliptic curve has rank one and $v_p(x([k]Q))\not=v_q(x([k]Q))$ for odd $k$ and a generator $Q$ of the free part of $E_{2rD}(\mathbb Q)$. Thus we can recover $p$ and $q$ from the data $D$ and $ x([k]Q))$. Furthermore, under the… 

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