Elliptic Curves

In this section, we only consider elliptic curves with $char(\mathbb{K}) \neq 2,3$, which are popular choices for cryptographic use. We define an elliptic curve as being a smooth Weierstrass curve in the short Weierstrass form.

Elliptic curve

An elliptic curve $E$ over a field $\mathbb{K}$ with $char(\mathbb{K}) \neq 2,3$ is the set of solutions $(x, y) \in \mathbb{K}^{2}$ to the equation

$$E: y^{2}=x^{3}+A x+B, \quad \quad (1)$$

together with an extra point $\mathcal{O}$ at infinity, i.e.,

$$E(\mathbb{K})=\left\{(x, y) \in \mathbb{K} \times \mathbb{K}: y^{2}=x^{3}+A x+B\right\} \cup\{\mathcal{O}\},$$

where $A$ and $B$ must satisfy the condition

$$4 A^{3}+27 B^{2} \neq 0.$$