Scalar Multiplication

A point-scalar multiplication for an elliptic curve is defined as follows:

$$kP = k*(x,y)$$

Here, $k$ is a scalar or base field element of an elliptic curve and $P$ or $(x,y)$ is a point on an elliptic curve.

Optimizing point-scalar multiplication for elliptic curve points is of significant interest. As point addition proves very cheap, we must reduce point multiplications into a series of point additions instead. The simplest and most naive method to achieve this is the Double-and-add method.

Perhaps the more significant problem to solve is the MSM problem where the question of how to most efficiently optimize scalar multiplication is generalized to how to most efficiently optimize the sum of multiple scalar multiplications.