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**2007/110**

**Title:**

Construction of Pairing-Friendly Elliptic Curves

**Authors:**
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}.

**ePrint:**
https://eprint.iacr.org/2007/110

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