A cycle of elliptic curves is a list of elliptic curves over finite fields such that the number of points on one curve is equal to the size of the field of definition of the next, in a cyclic way. We study cycles of elliptic curves in which every curve is pairing-friendly. These have recently found notable applications in pairing-based cryptography, for instance, in improving the scalability of distributed ledger technologies. We construct a new cycle of length 4 consisting of MNT curves, and characterize all the possibilities for cycles consisting of MNT curves. We rule out cycles of length 2 for particular choices of small embedding degrees. We show that long cycles cannot be constructed from families of curves with the same complex multiplication discriminant, and that cycles of composite order elliptic curves cannot exist. We show that there are no cycles consisting of curves from only the Freeman or Barreto--Naehrig families.


  1. elliptic curves
  2. Weil pairing
  3. cryptography

MSC codes

  1. 14H52
  2. 14G50
  3. 11T71

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Information & Authors


Published In

cover image SIAM Journal on Applied Algebra and Geometry
SIAM Journal on Applied Algebra and Geometry
Pages: 175 - 192
ISSN (online): 2470-6566


Submitted: 5 March 2018
Accepted: 29 January 2019
Published online: 2 April 2019


  1. elliptic curves
  2. Weil pairing
  3. cryptography

MSC codes

  1. 14H52
  2. 14G50
  3. 11T71



Funding Information

University of California Berkeley https://doi.org/10.13039/100006978

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