## Elliptic Curves (Over real numbers) An **Elliptic Curve** is a plane curve of the form $$ y^2 = x^3 + ax + b $$ where $a$ and $b$ are real numbers. This type if equation is called a _Weierstrass's elliptic function_ and can be considered a popular choice for elliptic curves. There is an important property for a curve to be elliptic in nature; It should be non-singular in nature. This means, the curve should not have cusps, self-intersections or isolated points. To enforce this condition, the discriminant of the above equation should obey: $$ \begin{align} \Delta = & -16 (4a^3 + 27b^3) \\ \Delta \neq & 0 \\ \end{align} $$ For the case of $\Delta$ less than zero, a non-singular curve has one component or one continuous curve. eg:

or

List of contents ---------------- * [Group Theory for elliptic curves](group.html) * [Addition Theory](addition.html) * [Multiplication Theory](multiplication.html) * [Elliptic Curve Diffie Hellman](ecdh.html) * [Points on the curve](points.html) * [Quadratic Congruences](quadratic.html) * [Elliptic Curve Digital Signing Algorithm](signature.html)