Paper 2002/088

Constructing Elliptic Curves with Prescribed Embedding Degrees

Paulo S. L. M. Barreto, Ben Lynn, and Michael Scott


Pairing-based cryptosystems depend on the existence of groups where the Decision Diffie-Hellman problem is easy to solve, but the Computational Diffie-Hellman problem is hard. Such is the case of elliptic curve groups whose embedding degree is large enough to maintain a good security level, but small enough for arithmetic operations to be feasible. However, the embedding degree is usually enormous, and the scarce previously known suitable elliptic groups had embedding degree $k \leqslant 6$. In this note, we examine criteria for curves with larger $k$ that generalize prior work by Miyaji et al. based on the properties of cyclotomic polynomials, and propose efficient representations for the underlying algebraic structures.

Note: Fixed the last example in appendix B and updated the references.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Accepted for presentation at SCN'02 (to be published in LNCS)
elliptic curve cryptosystem
Contact author(s)
pbarreto @ larc usp br
2005-02-22: last of 3 revisions
2002-07-04: received
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Creative Commons Attribution


      author = {Paulo S.  L.  M.  Barreto and Ben Lynn and Michael Scott},
      title = {Constructing Elliptic Curves with Prescribed Embedding Degrees},
      howpublished = {Cryptology ePrint Archive, Paper 2002/088},
      year = {2002},
      note = {\url{}},
      url = {}
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