Elliptic Curve

A set of coordinate pairs $(x,y)$ that satisfy the equation $y^2=x^3+ax+b$.

⚠️ Prerequisites: Finite fields.

An elliptic curve is defined as the set of all coordinate pairs $(x,y)$ such that:

$$y^2=x^3+ax+b$$

In elliptic curve cryptography, we pick $x$ and $y$ to be elements of a finite field. We refer to this field as the base field.

We then define a group operation which we call “point addition” and usually denote with the ‘$+$’ symbol. Repeated applications of the group operation can be counted like below:

$$A+A+A=3A$$

The number ‘$3$’ above is referred to as a scalar. If we are in a cyclic group, then the scalars also define a finite field: we call this field the scalar field.