Group Law

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., ( Bezout’s theoremconcept_4), we’ll show that adding two points $P$ and $Q$ in $E(\mathbb{K})$ gives a third point in $E(\mathbb{K})$, whereas the set of $E(\mathbb{K})$ together with the addition operation forms an abelian group with $\mathcal{O}$ as its identity. The best way to explain this operation is to consider the elliptic curves over the reals $\mathbb{R}$ and to explain the rule geometrically. As already mentioned, we only consider fields with characteristic $char(\mathbb{K})>3$.

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 (Darrel Hankerson et al. (2006)Darrel Hankerson, Alfred J. Menezes, and Scott Vanstone. Guide to Elliptic Curve Cryptography. Springer Professional Computing. New York, 2006. Springer., Lawrence C. Washington (2008)Lawrence C. Washington. Elliptic Curves: Number Theory and Cryptography, Second Edition. Discrete Mathematics and Its Applications. New York, 2008. CRC Press.):

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