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Elliptic Curves over ℝ

Elliptic curves over ℝ, the real numbers, have applications in various mathematical fields, including number theory and cryptography.


Definition

An elliptic curve over  is defined by a cubic equation of the form:

y^2=x^3+ax+b

where a and b are real coefficients that satisfy the condition:

4a^3 + 27b^2 ≠ 0

to ensure the curve is non-singular (i.e., it has no cusps or self-intersections).


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.




Δ < 0

Geometric Features

Shape of the Curve: Over , 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 curve is a single, connected loop

  • If Δ > 0, the curve consists of two disjoint loops


Δ > 0

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).


Intersection with the x-axis: Over , in the Weierstrass Equation form, all elliptic curves intersect with the x-axis either 1 or 3 times.


  • If Δ < 0, intersects with the x-axis once

  • If Δ > 0, intersects with the x-axis three times


In the case where the discriminant of the curve is equal to 0, the curve is said to be "singular" and is not considered a proper Elliptic Curve.


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. But this shape is not considered an elliptic curve, as elliptic curves, by definition, cannot have cusps or self-intersections.


Group LaW

Addition of Points: Elliptic curves over  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: The point at infinity, often denoted as O, acts as the identity element in this group. That is:

P + O = P.


Inverse Elements: The 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 gradient of E:

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




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  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  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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