What are elliptic curves?

An elliptic curve $E$ is a cubic curve. The elliptic curves can be of two types:

1. SingularSingular elliptical curves are the ones that do not have a well-defined tangent line at a specific point on the curve due to a cusp .
2. Non-singular

They can be classified into these categories based on the values of the discriminant $\Delta$ of their equations. If $\Delta = 0$, then the curve is singular else, it is non-singular. This answer will focus on the non-singular elliptical curves due to their use case in elliptic curve cryptography (ECC).

The figure below shows sketches of $y^2 = x^3$ where $\Delta = 0$ and $y^2 = x^3 -1x + 3$ where $\Delta \neq 0$. They represent a singular and a non-singular curve, respectively:

Weierstrass equation

An elliptic curve $E$ is a smooth non-singular cubic curve defined over the field $K$ with $char(K)\neq 2, 3$, and is represented by the Weierstrass general form given below:

Where $a_i ∈ K$ . However, the short form of the Weierstrass equation is usually preferred to represent the elliptic curve, which is defined as follows:

Where $A, B ∈ K$, and A and B are constants that satisfy the following discriminant $\Delta$ equation condition:

An elliptic curve E defined over field K is the set of solutions $(x, y) \in \mathbb{K}^{2}$ of equation 2 and an extra point $\mathcal{O}$ at infinity:

Geometric properties

The elliptic curves have two geometrical properties given as follows:

Horizontal symmetry

The elliptic curves are horizontally symmetrical to the x-axis. This means that if there exists a point P(x, y), then a point Q(x, -y) also exists. This property is useful in ECC while calculating the addition of two points, that is, $P_1 \oplus P_2$.

A straight line will intersect the elliptic curve at most three locations

This property is useful in the ECC, as it helps set the basis for the addition operation used in ECC. The line drawn between two points on the curve will give a third point that can be mirrored horizontally to obtain the result of the addition operation.

Application of elliptic curves

Elliptic curves are mainly used in ECC, a key-based encryption technique used to encrypt data. It consists of a public-private key similar to RSA, but it provides more security with a shorter key length, so it saves computational power while encrypting and decrypting. ECC has many applications, such as encryption, digital signatures, and pseudo-random generators.

Code example

The code below is used to draw an elliptic curve. Change the values of variables a and b to draw different elliptic curves: