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 curve is a single, connected loop - the cubic equation: x^3 + Ax + B = 0 has 1 solution
If Δ > 0, the curve 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.
