Abstract
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.
Permanent ID of this document: c798703ae3ecfdc375112f19dd0787e4. Date of this document: 2008.03.12. This work has been supported in part by the European Commission through the IST Programme under Contract IST–2002–507932 ECRYPT, and in part by the National Science Foundation under grant ITR–0716498. Part of this work was carried out during a visit to INRIA Lorraine (LORIA).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bernstein, D.J.: Curve25519: New Diffie-Hellman Speed Records. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) PKC 2006. LNCS, vol. 3958, pp. 207–228. Springer, Heidelberg (2006), http://cr.yp.to/papers.html#curve25519
Bernstein, D.J., Birkner, P., Lange, T., Peters, C.: ECM using Edwards curves (2007) (Citations in this document: §1), http://eprint.iacr.org/2008/016
Bernstein, D.J., Lange, T.: Explicit-formulas database (2007) (Citations in this document: §5, §6), http://hyperelliptic.org/EFD
Bernstein, D.J., Lange, T.: Faster addition and doubling on elliptic curves. In: Asiacrypt 2007 [19], pp. 29–50 (2007) (Citations in this document: §1, §1, §2, §2, §2, §3, §3, §3, §3, §3, §3, §4, §6, §6, §6, §7, §7), http://cr.yp.to/papers.html#newelliptic
Bernstein, D.J., Lange, T.: Inverted Edwards coordinates. In: AAECC 2007 [8], pp. 20–27 (2007) (Citations in this document: §1, §6, §7), http://cr.yp.to/papers.html#inverted
Billet, O., Joye, M.: The Jacobi model of an elliptic curve and side-channel analysis. In: AAECC 2003 [14], pp. 34–42 (2003) MR 2005c:94045 (Citations in this document: §5), http://eprint.iacr.org/2002/125
Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic curves in cryptography. Cambridge University Press, Cambridge (2000) (Citations in this document: §5)
Boztaş, S., Lu, H.-F(F.) (eds.): AAECC 2007. LNCS, vol. 4851. Springer, Heidelberg (2007)
Brier, É., Joye, M.: Fast point multiplication on elliptic curves through isogenies. In: AAECC 2003 [14], pp. 43–50 (2003) (Citations in this document: §5)
Cohen, H., Frey, G. (eds.): Handbook of elliptic and hyperelliptic curve cryptography. CRC Press, Boca Raton (2005) See [11]
Doche, C., Lange, T.: Arithmetic of elliptic curves. In: [10] (2005), pp. 267– 302. MR 2162729 (Citations in this document: §3)
Duquesne, S.: Improving the arithmetic of elliptic curves in the Jacobi model. Information Processing Letters 104, 101–105 (2007) (Citations in this document: §5)
Edwards, H.M.: A normal form for elliptic curves. Bulletin of the American Mathematical Society 44, 393–422 (2007) (Citations in this document: §1), http://www.ams.org/bull/2007-44-03/S0273-0979-07-01153-6/home.html
Fossorier, M.P.C., Høholdt, T., Poli, A. (eds.): AAECC 2003. LNCS, vol. 2643. Springer, Heidelberg (2003) See [6], [9]
Galbraith, S.D., McKee, J.: The probability that the number of points on an elliptic curve over a finite field is prime. Journal of the London Mathematical Society 62, 671–684 (2000) (Citations in this document: §4), http://www.isg.rhul.ac.uk/~sdg/pubs.html
Hisil, H., Carter, G., Dawson, E.: New formulae for efficient elliptic curve arithmetic. In: INDOCRYPT 2007 [23] (2007) (Citations in this document: §5)
Hisil, H., Wong, K., Carter, G., Dawson, E.: Faster group operations on elliptic curves. 25 Feb 2008 version (2008) (Citations in this document: §5), http://eprint.iacr.org/2007/441
Imai, H., Zheng, Y. (eds.): PKC 2000. LNCS, vol. 1751. Springer, Heidelberg (2000) see [21]
Kurosawa, K. (ed.): ASIACRYPT 2007. LNCS, vol. 4833. Springer, Heidelberg (2007)
Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Mathematics of Computation 48, 243–264 (1987), (Citations in this document: §3, §7), http://links.jstor.org/sici?sici=0025-571819870148:177243:STPAEC2.0.CO;2-3
Okeya, K., Kurumatani, H., Sakurai, K.: Elliptic curves with the Montgomery-form and their cryptographic applications. In: PKC 2000, pp. 238–257 (2000) (Citations in this document: §3 §3)
Silverman, J.H.: The arithmetic of elliptic curves. Graduate Texts in Mathematics 106 (1986)
Srinathan, K., Rangan, C.P., Yung, M. (eds.): INDOCRYPT 2007. LNCS, vol. 4859. Springer, Heidelberg (2007) See [16]
Stein, W. (ed.): Sage Mathematics Software (Version 2.8.12), The Sage Group (2008) (Citations in this document: §3), http://www.sagemath.org
Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.): PKC 2006. LNCS, vol. 3958. Springer, Heidelberg (2006) See [1]
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bernstein, D.J., Birkner, P., Joye, M., Lange, T., Peters, C. (2008). Twisted Edwards Curves. In: Vaudenay, S. (eds) Progress in Cryptology – AFRICACRYPT 2008. AFRICACRYPT 2008. Lecture Notes in Computer Science, vol 5023. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68164-9_26
Download citation
DOI: https://doi.org/10.1007/978-3-540-68164-9_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68159-5
Online ISBN: 978-3-540-68164-9
eBook Packages: Computer ScienceComputer Science (R0)