Refined Computations for Points of the Form 2kP Based on Montgomery Trick

This paper focuses on algorithms for an efficient scalar multiplication. It proposes two algorithms for computing points of the form 2kP in affine coordinates. One works for k = 2, and the other works for an arbitrary natural number k. The efficiency of these algorithms is based on a trade-off between a field inversion and several field multiplications. Montgomery trick is used to implement this trade-off. Since a field inversion is usually more expensive than 10 field multiplications, the proposed algorithms are efficient in comparison with existing ones. key words: scalar multiplication, elliptic curve arithmetic, Montgomery trick, window method