Cryptology ePrint Archive: Report 2007/110

Construction of Pairing-Friendly Elliptic Curves

Woo Sug Kang

Abstract: We explain a method of finding the polynomials representing $\sqrt{-D}$ and $\zeta_k$ over the field containing $\sqrt{-D}$ and $\zeta_k$ and how to construct a pairing friendly elliptic curves over the cyclotomic fields containing ${\mathbb Q} (\zeta_k, \sqrt{-D})$ for arbitrary $k$ and $D$ by CP method. By using the factorization of the cyclotomic polynomial combined some polynomial, we extend the construction over cyclotomic fields to the construction over some extensions of the cyclotomic fields containing ${\mathbb Q} (\zeta_k, \sqrt{-D})$. We explain the limitation of finding more families of pairing friendly elliptic curves with embedding degree 10. For all computation, we use the PARI-GP \cite{GP}.

Category / Keywords: implementation / elliptic curves, embedding degree, pairing based cryptography

Date: received 23 Mar 2007, last revised 18 May 2007

Contact author: wsgkang at korea ac kr

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Version: 20070518:120843 (All versions of this report)

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