Paper 2010/390

Elliptic curves in Huff's model

Hongfeng Wu and Rongquan Feng

Abstract

This paper introduce generalizes the Huff curves $x(ay^2-1)=y(bx^2-1)$ which contains Huff's model $ax(y^2-1)=by(x^2-1)$ as a special case. It is shown that every elliptic curve over the finite field with three points of order $2$ is isomorphic to a general Huff curve. Some fast explicit formulae for general Huff curves in projective coordinates are presented. These explicit formulae for addition and doubling are almost as fast in the general case as they are for the Huff curves in \cite{Joye}. Finally, the number of isomorphism classes of general Huff curves defined over the finite field $\mathbb{F}_q$ is enumerated.