# Group Structure

Learn the group structure of elliptic curves and their special properties in this lesson.

## Overview

This section describes the group structure of elliptic curves. Hereinafter, $\mathbb{Z}_{n}$ denotes a cyclic group of order $n$.

### Theorem 1: group structure of an elliptic curve

Let E be an elliptic curve over a finite field $\mathbb{F}_{p}$. Then, $E\left(\mathbb{F}_{p}\right)$ is isomorphic to $\mathbb{Z}_{n_{1}} \oplus \mathbb{Z}_{n_{2}}$, where $n_{1}$ and $n_{2}$ are unique positive integers such that $n_{2} \mid n_{1}$ and $n_{2} \mid p-1$. Furthermore, they give the following statement:

It holds that $\# E\left(\mathbb{F}_{p}\right)=n_{1} n_{2}$. If $n_{2}=1$, then $E\left(\mathbb{F}_{p}\right)$ is a cyclic group. If $n_{2}>1$ is a small integer, $E\left(\mathbb{F}_{p}\right)$ is said to be almost cyclic.

#### Example

We consider the elliptic curve $E: y^{2}=x^{3}+1$ over $\mathbb{F}_{5}$ of Example 1

$\langle(0,1)\rangle:(0,1) \rightarrow(0,4) \rightarrow \mathcal{O}$

or

$\langle(2,2)\rangle:(2,2) \rightarrow(0,4) \rightarrow(4,0) \rightarrow(0,1) \rightarrow(2,3) \rightarrow \mathcal{O}.$

This example shows that the choice of the point that generates the cyclic subgroup is of great importance. For the intractability of ECC algorithms, we usually want subgroups with high order $n$, so in the most favorable case, $\# E\left(\mathbb{F}_{p}\right)$ is prime itself because then the entire group is a cyclic group by this corollary

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