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Designs, Codes and Cryptography

, Volume 37, Issue 1, pp 133–141 | Cite as

Elliptic Curves Suitable for Pairing Based Cryptography

  • Friederike BrezingEmail author
  • Annegret Weng
Article

Abstract

For pairing based cryptography we need elliptic curves defined over finite fields \(\mathbb{F}_{q}\) whose group order is divisible by some prime \(\ell\) with \(\ell | q^{k-1}\) where k is relatively small. In Barreto et al. and Dupont et al. [Proceedings of the Third Workshop on Security in Communication Networks (SCN 2002), LNCS, 2576, 2003; Building curves with arbitrary small Mov degree over finite fields, Preprint, 2002], algorithms for the construction of ordinary elliptic curves over prime fields \(\mathbb{F}_{p}\) with arbitrary embedding degree k are given. Unfortunately, p is of size \(O(\ell^{2})\).

We give a method to generate ordinary elliptic curves over prime fields with p significantly less than \(\ell^{2}\) which also works for arbitrary k. For a fixed embedding degree k, the new algorithm yields curves with \(p \approx \ell^{s}\) where \(s = 2 - 2/\varphi(k)\) or \(s = 2 - 1/\varphi(k)\) depending on k. For special values of k even better results are obtained.

We present several examples. In particular, we found some curves where \(\ell\) is a prime of small Hamming weight resp. with a small addition chain.

Keywords

elliptic curves pairing based cryptagraphy 

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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Fachbereich MathematikJohann Wolfgang Goethe-UniversitätFrankfurtGermany
  2. 2.Fachbereich MathematikJohannes Gutenberg UniversitätMainzGermany

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