Cryptology ePrint Archive: Report 2008/008

Factoring Polynomials for Constructing Pairing-friendly Elliptic Curves

Zhitu su, Hui Li and 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.

Category / Keywords: public-key cryptography / pairing-friendly curves, polynomial factoring, primitive element theorem

Date: received 6 Jan 2008, last revised 12 May 2008

Contact author: ztsu at mail xidian edu cn

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Version: 20080513:030253 (All versions of this report)

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