# Elliptic Curves over ℝ

__Elliptic curves__ have applications in various mathematical fields, including __number theory__ and __cryptography__.

## Definition

An elliptic curve over ℝ is defined by a __cubic equation__ of the form:

*y^2=x^3+Ax+B*

where *A* and *B* are __real coefficients__.

When these coefficients satisfy the condition:

*4A^3 + 27B^2 ≠ 0*

Elliptic curves that do not satisfy the above conditions are called "singular elliptic curves".

**The ****Weierstrass Equation form**: in this form (*y^2=x^3+Ax+B*), an Elliptic Curve is referred to as being in the Weierstrass Equation form.

There are __more general equations that define Elliptic Curves__ over ℝ (__Washington____, L. C., 2008,____ pp35-42, Section 2.5__). However, Elliptic Curves can be __converted into the ____Weierstrass Equation form__, when defined across fields with __a characteristic other than 2 or 3__.

## Animation of ECs in the Weierstrass form

See below, an animation of different elliptic curves in the Weierstrass Form, for varying values of the coefficients *A* and *B*:

## Geometric Features

**Shape of the Curve**: Over ℝ, in the Weierstrass Equation form, elliptic curves typically look like a smooth, symmetric curve crossing the x-axis at one point, or like two disjoint loops, depending on __the discriminant__ *Δ = −16(4A^3+27B^2) *(__Washington____, L. C., 2008, pp9-11____, Section 2.1__).

*If Δ < 0, the curv*e is a single, connected loop - the cubic equation: x^3 + Ax + B = 0 has 1 solution*If Δ > 0, the curv*e consists of two disjoint loops - the cubic equation: x^3 + Ax + B has 3 solutions

*Please Note: the definition of these "loops" will become clearer once we *__define "the point at infinity"__*. For now, we simply accept that the far end of each curve connects "at infinity".*

**Symmetry**: Every elliptic curve, in the Weierstrass Equation form, is symmetric about the x-axis. If *(x, y)* is on the curve, then so is *(x, −y)*.

**Visualisation of the Discriminant**: Replay the video above, to see the curve transform through the two forms (*Δ<0* and *Δ>0 *), and change from intersecting the x-axis once, to three times. At one point (when *Δ=0 *) the shape created only touches the x-axis at 2 points.

## Group LaW

** Addition of Points**: Elliptic curves over ℝ form a group under the operation of point addition. If

*P*and

*Q*are points on the curve, their sum

*P*+

*Q*is defined geometrically by drawing a line through

*P*and

*Q*and finding its third intersection point with the curve, then reflecting that point over the

*x-axis*.

**NOTE**: Imagine *P* approaches *Q *: eventually, when:

*P = Q*

the line that passes through *P* and *Q* becomes a tangent line, to the curve *E*, at the point *P*, and we say that:

*P + P = 2P = -R*

the other intersection point with the curve.

** Identity Element**: Any vertical line intersects the curve at "

__the point at infinity__" ("

*O*").

__The point at infinity__acts as the identity element in the group defined for an elliptic curve. That is:

*P + O = P*.

This concept will be more clearly defined,* ** in the section "the point at infinity"*.

** Additive Inverse Elements**: The additive inverse of a point (

*x , y*), in the Weierstrass Equation form, under addition is (

*x*, −

*y*), reflecting the point over the x-axis.

#### Animation of Point Addition

## Special Cases

**When ***P = Q*

(i.e the points you are adding together are equal), as described above under "Group Law":

*P + Q = 2P*

so the equation for the gradient:

*m = ( y_Q - y_P ) / ( x_Q - x_P )*

no longer makes any sense - because:

*x_Q - x_P = 0*

and therefore, the denominator would be *0*.

So we need to find the slope of *E*:

*m = ( ( x_P )^2 + a ) / ( 2 y_P )*

The concept of finding the slope of an Elliptic Curve will be further explored in __the section on "Formal Derivatives"__. For now, we will simply say that we use the derivative for finding the slope at a point on the curve.

**When y = 0**

That is: the tangent to the curve *E* is a vertical line.

So:

*2P = the point at infinity*

It can also be shown, related to the definition of addition for elliptic curves, that any straight line that passes through 3 point on the curve (*P*, *Q* and *R *) represents the fact that:

*P + Q + R = 0*

For this, and many related reasons, the "point at infinity" is taken to be the "additive neutral element" for the Group defined for Elliptic Curves (__Washington, Section 2.3, pp18-20__).

*NOTE*: When defined in ℝ, it can easily be shown that, if the tangent line to the curve *E* is anything other than vertical, __it will intersect with ____E____ (in the Affine plane), at another point__.

## Applications

** Cryptography**: Elliptic curves over ℝ serve as simpler models for understanding more complex cryptographic schemes over finite fields and rings, including those used in Elliptic Curve Cryptography (ECC).

** Theoretical Mathematics**: They are studied for their properties and structures in algebraic geometry and number theory.

## Challenges / Considerations

** Visualisation**: Unlike elliptic curves over finite fields used (for cryptography), curves over ℝ can be more graphically represented in a more well-known way, which makes visual learning and understanding more straight forward.

** Complexity in Handling**: While the real-number setting is more intuitive due to familiar geometric concepts, it introduces complexities in ensuring mathematical rigour, especially concerning limits, continuity, and differentiability.

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