# Group Law

Learn about group law and basic group operations in this lesson.

In this section, we present the basic group operations on elliptic curves, namely **point addition** and **point doubling**. We’ll show that the points on an elliptic curve $E$ have an algebraic structure, i.e., (

## Point addition

Let $E(\mathbb{K})$ be an elliptic curve

$y^{2}=x^{3}+A x+B$

over a field $\mathbb{K}$ with $char(\mathbb{K}) \neq 2,3$. Furthermore, let $P=\left(x_{1}, y_{1}\right)$ and $Q=\left(x_{2}, y_{2}\right)$ be two distinct points on the elliptic curve $E(\mathbb{K}).$ Let $\bar{P}$ denote the conjugate point of $P$, i.e., if $P=(x, y)$, then $\bar{P}=(x,-y)$ (i.e., the point $P$ is reflected across the $x$-axis by changing the sign of the $y$-coordinate). Then, the sum $R$ of the addition $P+Q$ is defined as follows (

We draw a line through the two points $P$ and $Q$ as depicted in the figure given below. This line intersects $E$ at a unique third point $R^{\prime}$, which we denote by $R^{\prime}=P * Q$. Then, we obtain the sum $R$ by the reflection of $R^{\prime}$ across the $x$-axis.

### Figure 1

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