Paper 2006/372

A taxonomy of pairing-friendly elliptic curves

David Freeman, Michael Scott, and Edlyn Teske


Elliptic curves with small embedding degree and large prime-order subgroup are key ingredients for implementing pairing-based cryptographic systems. Such "pairing-friendly" curves are rare and thus require specific constructions. In this paper we give a single coherent framework that encompasses all of the constructions of pairing-friendly elliptic curves currently existing in the literature. We also include new constructions of pairing-friendly curves that improve on the previously known constructions for certain embedding degrees. Finally, for all embedding degrees up to 50, we provide recommendations as to which pairing-friendly curves to choose to best satisfy a variety of performance and security requirements.

Note: Bibliography update.

Available format(s)
Public-key cryptography
Publication info
Published elsewhere. Unknown where it was published
elliptic curvespairing-based cryptosystemsembedding degreeefficient implementation
Contact author(s)
dfreeman @ cs stanford edu
2009-11-20: last of 8 revisions
2006-11-03: received
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      author = {David Freeman and Michael Scott and Edlyn Teske},
      title = {A taxonomy of pairing-friendly elliptic curves},
      howpublished = {Cryptology ePrint Archive, Paper 2006/372},
      year = {2006},
      note = {\url{}},
      url = {}
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