Differential addition on binary elliptic curves

This paper presents new and fast differential addition (i.e., the addition of two points with the known difference) and doubling formulas, as the core step in Montgomery scalar multiplication, for various forms of elliptic curves over binary fields. The formulas are provided for binary Edwards, binary Hessian and binary Huff elliptic curves with cost of 5M + 4S + 1D when the given difference point is in affine form. Here, M, S, D denote the costs of a field multiplication, a field squaring and a field multiplication by a constant, respectively. In this paper, we show that every ordinary elliptic curve E over binary field F2n has Lopez and Dahab differential addition and doubling formulas with suitable function. This paper also presents, new complete differential addition formulas for binary Edwards curves with cost of 5M + 4S + 2D. Also, new complete Montgomery ladder and scalar multiplication algorithms are presented using the complete differential addition formulas.