Abstract
We introduce efficient algorithms for scalar multiplication on elliptic curves defined over \(\mathbb{F}_p\). The algorithms compute 2k P directly from P, where P is a random point on an elliptic curve, without computing the intermediate points, which is faster than k repeated doublings. Moreover, we apply the algorithms to scalar multiplication on elliptic curves, and analyze their computational complexity. As a result of their implementation with respect to affine (resp. weighted projective) coordinates, we achieved an increased performance factor of 1.45 (45%) (resp. 1.15 (15%)) in the scalar multiplication of the elliptic curve of size 160-bit.
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