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Paper: Factoring Polynomials for Constructing Pairing-friendly Elliptic Curves

Authors:
Zhitu su
Hui Li
Jianfeng Ma
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URL: http://eprint.iacr.org/2008/008
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Abstract: In this paper we present a new method to construct a polynomial $u(x) \in \mathbb{Z}[x]$ which will make $\mathrm{\Phi}_{k}(u(x))$ reducible. We construct a finite separable extension of $\mathbb{Q}(\zeta_{k})$, denoted as $\mathbb{E}$. By primitive element theorem, there exists a primitive element $\theta \in \mathbb{E}$ such that $\mathbb{E}=\mathbb{Q}(\theta)$. We represent the primitive $k$-th root of unity $\zeta_{k}$ by $\theta$ and get a polynomial $u(x) \in \mathbb{Q}[x]$ from the representation. The resulting $u(x)$ will make $\mathrm{\Phi}_{k}(u(x))$ factorable.
BibTeX
@misc{eprint-2008-17685,
  title={Factoring Polynomials for Constructing Pairing-friendly Elliptic Curves},
  booktitle={IACR Eprint archive},
  keywords={public-key cryptography /  pairing-friendly curves, polynomial factoring, primitive element theorem},
  url={http://eprint.iacr.org/2008/008},
  note={ ztsu@mail.xidian.edu.cn 14012 received 6 Jan 2008, last revised 12 May 2008},
  author={Zhitu su and Hui Li and Jianfeng Ma},
  year=2008
}