Complete and Unified Group Laws are not Enough for Elliptic Curve Cryptography

We analyze four recently proposed normal forms for elliptic curves. Though these forms are mathematically appealing and exhibit some cryptographically desirable properties, they nonetheless fall short of cryptographic viability, especially when compared to various types of Edwards Curves. In this paper, we present these forms and demonstrate why they fail to measure up to the standards set by Edwards Curves

A new model of binary elliptic curves with fast arithmetic

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