Paper 2008/013

Twisted Edwards Curves

Daniel J. Bernstein, Peter Birkner, Marc Joye, Tanja Lange, and Christiane Peters


This paper introduces ``twisted Edwards curves,'' a generalization of the recently introduced Edwards curves; shows that twisted Edwards curves include more curves over finite fields, and in particular every elliptic curve in Montgomery form; shows how to cover even more curves via isogenies; presents fast explicit formulas for twisted Edwards curves in projective and inverted coordinates; and shows that twisted Edwards curves save time for many curves that were already expressible as Edwards curves.

Note: Included material on isogenies.

Available format(s)
Publication info
Published elsewhere. Unknown where it was published
Elliptic curvesEdwards curvestwisted Edwards curvesMontgomery curvesisogenies.
Contact author(s)
tanja @ hyperelliptic org
2008-03-13: last of 2 revisions
2008-01-14: received
See all versions
Short URL
Creative Commons Attribution


      author = {Daniel J.  Bernstein and Peter Birkner and Marc Joye and Tanja Lange and Christiane Peters},
      title = {Twisted Edwards Curves},
      howpublished = {Cryptology ePrint Archive, Paper 2008/013},
      year = {2008},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.