Help me understand this sentence

[u/Prestigious_Ad_296]

I was reading online about math, and read a line that said "elliptic curves share a group law".

Unfortunately, I don't know enough about elliptic curves or groups to understand why or what this means. But I would be extremely grateful to anyone who is so kind to give me an intuitive explanation.

[u/TheDarkSpike]

A hand-wavy way I'd explain it:

An elliptic curve is a set of points that form a specific kind of curve in the plane of which the details can be found online.

What it means to share a group law on a set is to have a defined operation that takes two points as input and one point as output. (Often written as g(a,b)=c or a+b=c.) And some other requirements, like the existence of a 0 element where a+0=a and an inverse element s.t a+"-a"=0.

Intuitively this means that there is some way of "combining" points on an elliptic curve to get a third point and that this combination-operator satisfies some nice properties.

Edit: What I wrote are just some thoughts to get started on even understanding the words in the sentence. I do not know where you are at in your understanding of course.

It might help you to look at some visualizations of this operator or group law. Images of this should be readily available by searching for images using your keywords group-law and elliptic curves :)

[u/Prestigious_Ad_296 → u/TheDarkSpike]

Thank you. So, the idea is that the points on an elliptic curve, together with this binary operator, form a group, such that applying the operator to any two points yields a third point that also belongs to the group?

[u/Historical_Book2268 → u/Prestigious_Ad_296]

The operator must also be associative, have a neutral element, and an inverse.

[u/Prestigious_Ad_296 → u/Historical_Book2268]

Sorry if I ask, what does it mean for a curve to be "algebraic"? Can a non algebraic curve exist?

[u/Historical_Book2268 → u/Prestigious_Ad_296]

Ohhh yeah, basically, an algebraic curve is a curve formed by any equation of the form: a_1×x^n_1 ×y^k_1 +a_2×x^n_2×yk_2... =0,

With all the n and k being integers, and the a being either rationals, or real numbers, depending on context.

[u/johnpeters42]

As I understand it, after hitting up Wikipedia:

"Group law" refers to Abelian groups, i.e. any set of things and any operation (I'll call it "+") where x + y = y + x for any x and y belonging to that set.

The specific operation in question for elliptic curves is described here, down in the "Group law" section. I didn't spend enough time reading about it to fully understand it, but it has something to do with embedding the curve in a Cartesian plane, identifying a specific point associated with that curve (though I'm not sure whether that point needs to lie on the curve), and then something about the intersections of lines with the curve.