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**2008/008**

**Title:**

Factoring Polynomials for Constructing Pairing-friendly Elliptic Curves

**Authors:**
Zhitu su, Hui Li, Jianfeng Ma

**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.

**ePrint:**
https://eprint.iacr.org/2008/008

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