Binary Edwards Curves

Abstract

This paper presents a new shape for ordinary elliptic curves over fields of characteristic 2. Using the new shape, this paper presents the first complete addition formulas for binary elliptic curves, i.e., addition formulas that work for all pairs of input points, with no exceptional cases. If n ≥ 3 then the complete curves cover all isomorphism classes of ordinary elliptic curves over .

This paper also presents dedicated doubling formulas for these curves using 2M + 6S + 3D, where M is the cost of a field multiplication, S is the cost of a field squaring, and D is the cost of multiplying by a curve parameter. These doubling formulas are also the first complete doubling formulas in the literature, with no exceptions for the neutral element, points of order 2, etc.

Finally, this paper presents complete formulas for differential addition, i.e., addition of points with known difference. A differential addition and doubling, the basic step in a Montgomery ladder, uses 5M + 4S + 2D when the known difference is given in affine form.

Permanent ID of this document: 592248bfa170d87d90a8d543cb645788. Date of this document: 2008.05.16. This work has been supported in part by the European Commission through the IST Programme under Contract IST–2002–507932 ECRYPT, in part by the National Science Foundation under grant ITR–0716498, and in part by the Ministry of Science, Research and Technology of I. R. Iran under scholarship no. 800.147.