Parallel and Regular Algorithm of Elliptic Curve Scalar Multiplication over Binary Fields

Table 1

Complexity comparison for -bit in different single algorithms.

Method	Point operations	Field operations (I = 10M)
Montgomery-D		6mM + 10M + I	1418M	2474M
Montgomery-H		3mM + m (2M + I)	3495M	6135M
Algorithm 2-D (-projective)		10mM + 11M + I	2351M	4111M
Algorithm 2-D (twisted )		9mM + 11M + I	2118M	3702M
Algorithm 1-H		10mM + 11M + I	2351M	4111M

Montgomery-D = Montgomery double-and-add algorithm, Montgomery-H = Montgomery halve-and-add algorithm, Algorithm. 2-D (-Projective) = Algorithm 2 using the -projective coordinate system, Algorithm. 2-D (twisted ) = Algorithm 2 using the twisted coordinate system, Algorithm. 1-H = Algorithm 1 for halve-and-add.