Elliptic curve cryptography is based on finding intersections of lines and elliptic curves:
$$y^2 = x^3 + ax + b ~~\text{and}~~ y = ax + b$$
It make sense when you see it on the graph, but the algorithm itself is using modular arithmetic where those curves are just a messy set of points. Is there any theorem that proves that elliptic curves work the same way on the finite field and the real number field? As per my research it has something to do with modular forms (each EC has a modular form, although that theorem was proven after that cryptography algorithm was introduced)
The whole story is based on the fact that one can add two points $P$ and $Q$ on an elliptic curve and obtain a third point $P+Q$ on the curve. The points of the curve form an abelian group under this addition. There is a basic access to this result. For this, it is better to work in the projective plane ${\Bbb P}^2({\Bbb K})$ over the field ${\Bbb K}$. The kind of field does not really matter here. A basic fact is that the intersection multiplicity between the curve and a projective line is 0, 1 or 3. Moreover, the only point on the curve not in the affine plane is the base point (i.e., unit element of the corresponding group). All necessary results can be established rather easily. Maybe, the use of modular forms is less boring but I haven't looked into this yet.