Introduction to Cryptography with Coding Theory, 3rd Edition

21.1 The Addition Law

An elliptic curve $E$ is the graph of an equation

$$\begin{array}{cc}E:& y^2=x^3+a{x}^2+bx+c\text{,}\end{array}$$

where $a\text{,}b\text{,}c$ are in whatever is the appropriate set (rational numbers, real numbers, integers mod $p\text{,}$ etc.). In other words, let $K$ be the rational numbers, the real numbers, or the integers mod a prime $p$ (or, for those who know what this means, any field of characteristic not 2; but see Section 21.4). Then we assume $a\text{,}b\text{,}c\in K$ and take $E$ to be

$$\{(x\text{,}y)\mid x\text{,}y\in K\text{,}{y}^2=x^3+a{x}^2+bx+c\}\text{.}$$

As will be discussed below, it is also convenient to include a point $(\text{∞}\text{,}\text{∞})\text{,}$ which often will be denoted simply by $\text{∞}\text{.}$

Let’s consider the case of real numbers first, since this case allows us to work with pictures. The graph $E$ has two possible forms, depending on whether the cubic polynomial ...