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Paper: Construction of Pairing-Friendly Elliptic Curves

Authors:
Woo Sug Kang
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URL: http://eprint.iacr.org/2007/110
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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}.
BibTeX
@misc{eprint-2007-13392,
  title={Construction of Pairing-Friendly Elliptic Curves},
  booktitle={IACR Eprint archive},
  keywords={implementation / elliptic curves, embedding degree, pairing based cryptography},
  url={http://eprint.iacr.org/2007/110},
  note={ wsgkang@korea.ac.kr 13651 received 23 Mar 2007, last revised 18 May 2007},
  author={Woo Sug Kang},
  year=2007
}