Given an elliptical curve e.g. from “Understanding Cryptography” by Parr & Pelzl §9.2 Example 9.5:
$y^2 = x^3 + 2x + 2~~~~ mod~17$
And given a primitive $P = (5, 1)$, the book indicates:
We compute now all the "powers" of P.
They then provide a table:
$2P = (5, 1) + (5, 1) = (6, 3)$
$3P = 2P + P = (10, 6)$
$18P = (5, 16)$
Unfortunately it is not apparent to me what action they are performing with the addition ($+$), or what they mean by "powers". What operation is being performed to go from $(5, 1) + (5, 1)$ to $(6, 3)$, and so on?
The obvious operation (i.e. what Wolfram alpha does) of $(5, 1) + (5, 1) = (5 + 5, 1 + 1)$ yields $(10, 2)$. There are a plethora of other possible combinations of operations that one could try, but I'd just be guessing.
Generally, given the secret, ordinarily labeled $d$, how ought one calculate $dP = P + P~ +~ ... ~+~ P$?
(This presumably computes the public key $X$ & $Y$ points, from $d$ and the $Gx$ and $Gy$ parameters)
While it is a rather core operation, and likely quite basic, I have found no illustrative examples and at-best convoluted implementations. I'll keep looking, but I thought that a good answer might be a useful for the next person searching.
This exact example also appears and is discussed at: https://math.stackexchange.com/questions/430836