# [Resource Topic] 2017/121: Twisted $\mu_4$-normal form for elliptic curves

Welcome to the resource topic for 2017/121

Title:
Twisted \mu_4-normal form for elliptic curves

Authors: 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 López-Dahab 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.

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