[slooonz]

Yes. This is partly why this article is crap. k*G is never defined, and is the core operation in ECC (also: the article insist on using an elliptic curve in R, but you need to do it on a finite field, because on a smooth curve you can just use a smooth interpolation to solve the logarithm — and obviously once you go on a finite field the curve no longer looks nice, it’s just a seemingly random cloud of points).

Very roughly speaking, putting the complication of "point at infinity" problem under the rug, a characteristic feature of a EC is that a straight line passing through two points on the curve will pass through a third point on the curve (yes, unless you take a vertical line, point at infinity). So you can define an "addition of points on the curve" : take two points A and B, draw a straight line passing through them, name the third intersection point between the line and the curve C, declare A + B = C (actually there’s a symmetry around the x axis involved for the usual properties of addition to hold, another complication, let's sweep it under the rug too).

(for A = B, take the tangent of the curve at A ; in R you can see that it works because you can take the limit as B goes arbitrarily close to A : that gives you the tangent ; in a finite field that’s less obvious but the algebraic proof is the same)

So k*G is just G + G + ... + G, k times.

If you want more details, your favorite reasoning LLM can do a better job explaining what I’ve swept under the rug.