Paper 2010/608 Original: eprint.iacr.org/2010/608 ↗

A New Model of Binary Elliptic Curves with Fast Arithmetic

Abstract

This paper presents a new model of ordinary elliptic curves with fast arithmetic over field of characteristic two. In addition, we propose two isomorphism maps between new curves and Weierstrass curves. This paper proposes new explicit addition law for new binary curves and prove the addition law corresponds to the usual addition law on Weierstrass curves. This paper also presents fast unified addition formulae and doubling formulae for these curves. The unified addition formulae cost $12M+2D$, where $M$ is the cost of a field multiplication, and $D$ is the cost of multiplying by a curve parameter. These formulae are more efficient than other formulae in literature. Finally, this paper presents explicit formulae for $w$-coordinates differential addition. In a basic step of Montgomery ladder, the cost of a projective differential addition and doubling are $5M$ and $1M+1D$ respectively, and the cost of mixed $w$-coordinates differential addition is $4M$.