I have come across this question here: Points on elliptic curve over finite field
Which describes how you can take every value $x$ from $0$ to $p-1$ and calculate the points of an elliptic curve.
Thing is... in the example, $p$ is only $17$. What if $p$ was a much larger number where it might be infeasible to calculate using every value of $-1 < x < p-1$?
Is there some formula or way to calculate possible points on an elliptic curve given $p$? I know there are theorems to calculate the number of points on a curve, but what about the points themselves?
Additionally, a comment under the answer says "Plus the point at infinity". What is the point at infinity for that example - $y2=x3+2x+2$ in $F17$? Is the point at infinity an abstract concept or an actual combination of $(x,y)$ that can be calculated?
The resource linked in the question says:
Suppose the points $P1 = (x1, y1)$ and $P2 = (x2, y2)$ belong to an elliptic curve $y2 = x3 + αx + β$ over a field $F$ which does not have characteristic 2. Then the point $S = P1 ∗ P2$ is determined by the following rules:
(i) if $x1 = x2$ and $y1 = −y2$ then $S = I$
Earlier $I$ was defined as the Point at Infinity.
So say, if you have points $(4,4$) and $(4,13)$ where $x1 = x2$ and $y1 = −y2\ (mod\ 17)$ then.... $P1 ∗ P2 = S$ which $= I$... the point at infinity?
... What is $*$ in $P1 ∗ P2$? Multiplication?
I'm sorry if both of these questions sound moronic but I'm really struggling to wrap my head around these concepts :(
Point at infinity is an abstract concept. Briefly: given a plane, the point at infinity is a point not in the plane where we imagine all lines meet. You can think of folding up the plane and attaching the edges all to one point, making a sphere. It's convenient to consider this point as part of the elliptic curve. I know I said this quickly and it may not make any sense to you -- perhaps try consulting an introduction to projective geometry.
The $*$ is a new binary operation on points on the elliptic curve. It's not an operation like addition or multiplication you've heard of before, which is why we have to define it. (There isn't any obvious way to add or multiply points on the elliptic curve such that we still get a point on the curve as a result.) The important point is that once we define it, it has all the properties of addition inasmuch as the points form an abelian group under this operation. To get used to abstract operations like this, an introduction to group theory might be helpful.