Paper 2017/121
Twisted $\mu_4$normal form for elliptic curves
David Kohel
Abstract
We introduce the twisted $\mu_4$normal form for elliptic curves, deriving in particular addition algorithms with complexity $9M + 2S$ and doubling algorithms with complexity $2M + 5S + 2m$ over a binary field. Every ordinary elliptic curve over a finite field of characteristic 2 is isomorphic to one in this family. This improvement to the addition algorithm is comparable to the $7M + 2S$ achieved for the $\mu_4$normal form, and replaces the previously best known complexity of $13M + 3S$ on LópezDahab models applicable to these twisted curves. The derived doubling algorithm is essentially optimal, without any assumption of special cases. We show moreover that the Montgomery scalar multiplication with point recovery carries over to the twisted models, giving symmetric scalar multiplication adapted to protect against side channel attacks, with a cost of $4M + 4S + 1m_t + 2m_c$ per bit. In characteristic different from 2, we establish a linear isomorphism with the twisted Edwards model. This work, in complement to the introduction of $\mu_4$normal form, fills the lacuna in the body of work on efficient arithmetic on elliptic curves over binary fields, explained by this common framework for elliptic curves if $\mu_4$normal form in any characteristic. The improvements are analogous to those which the Edwards and twisted Edwards models achieved for elliptic curves over finite fields of odd characteristic and extend $\mu_4$normal form to cover the binary NIST curves.
Metadata
 Available format(s)
 Category
 Publickey cryptography
 Publication info
 Published by the IACR in EUROCRYPT 2017
 Keywords
 elliptic curve cryptographybinary curves
 Contact author(s)
 David Kohel @ univamu fr
 History
 20170216: received
 Short URL
 https://ia.cr/2017/121
 License

CC BY
BibTeX
@misc{cryptoeprint:2017/121, author = {David Kohel}, title = {Twisted $\mu_4$normal form for elliptic curves}, howpublished = {Cryptology ePrint Archive, Paper 2017/121}, year = {2017}, note = {\url{https://eprint.iacr.org/2017/121}}, url = {https://eprint.iacr.org/2017/121} }