Pedro Adiel Hernández Leal: Elliptic curves

Elliptic curves have been an interesting subject in cryptography. In particular, elliptic curves had an important role in the development of the Fermat's Last Theorem's proof.
\begin{equation}
The \ equation \ Diophantine \ X^n+Y^n=Z^n \ has \ no \ integer \ solutions \ for \ all \ n>2 \ and \ X,Y,Z\neq 0.
\end{equation}
Since Andrew Wiles proved it in 1995, the elliptic curves' theory was estimulated and mathematicians have implemented these ones to cryptography.

Some important aspects about elliptic curves
An elliptic curve defined over a field $K$ is a pair $E=(E,O)$ where $E$ is an projective algebraic curve non singular defined over $K$ of gender $g=1$ and $O\in E$ es a fixed point called distinguished point. We can denoted an elliptic curve as $E/K$.

For computationals purposes we work over fields $K$ such that $\mathrm{char}(K)\neq 2,3$ and for this fields $E$ can be written in its Weierstrass' form
\begin{equation}
E: \ y^2=x^3+Ax+B \ where \ A,B\in K.
\end{equation}
Its homogeneous equation is $Y^2Z=X^3+AXZ^2+BZ^3$ which has just one point to the infinity, this is $O:=[0:1:0]$. Therefore $E/K$ has $K$-rational points
\begin{equation}
E(K):=\{ (x,y)\in K\times K: y^2=x^3+Ax+B \}\cup \{O\}
\end{equation}
The next picture shows two elliptic curves over $K=\mathbb{R}$.

$Fig \ 1. \ Two \ elliptic \ curves \ in \ \mathbb{R}$.

We show also two elliptic curves over $\mathbb{F}_{11}$.

$Fig\ 2.\ \ E: \ y^2=x^3+2x+4$

$Fig\ 3.\ \ E: \ y^2=x^3+6x+9$

In general over every field $K$ we can define an operation of geometric origin. This can be seen better over $\mathbb{R}$. We define the sum $P+Q$ with $P,Q\in E(K)$ joining the line which cointains $P,Q$ obteining a third intersection point. Then we take its symmetric respect to the x-axis like in the next picture.