Operations on elliptic curves
The first observation is that an elliptic curve is not an ellipse. The general mathematical form of an elliptic curve is as follows:
E: y^2 = x^3 + ax^2 + bx + c
Important Note
E: represents the form of the elliptic curve, and the parameters (a, b, and c) are coefficients of the curve.
Just to give evidence of what we are discussing, we'll try to plot the following curve:
E: y2 = x3 + 73
As we can see in the following figure, I have plotted this elliptic curve with WolframAlpha represented in its geometric form:
Figure 7.1 – Elliptic curve: E: y^2 = x^3 + 73
We can start to analyze geometrically and algebraically how these curves work and their prerogatives. Since they are not linear, they are easy to implement for cryptographic scopes, making them adaptable.
For example, let's take the curve plotted previously:
E : y^2 = x^3 + 73
When (y = 0), we can see that, geometrically, the curve...