Elliptic curve

The operation is defined e.g. at en.wikipedia.org/w/index.php?title=Elliptic_curve_point_multiplication&oldid=1168754060#Point_operations. For example, consider the most common case for two different points different. If the two points are given in coordinates:

P (x_{p}, y_{p}) + Q + (x_{q}, y_{q}) = R = (x_{r}, y_{r})

(1)

then the addition is defined in the general case as:

λ x_{r} y_{r} = \frac{y _{q} - y _{p}}{x _{q} - x _{p}} = λ^{2} - x_{p} - x_{q} = λ (x_{p} - x_{r}) - y_{p}

(2)

with some slightly different definitions for point doubling

P + P

and the identity point.

This definition relies only on operations that we know how to do on arbitrary fields:

addition $+$
multiplication $\times$

and it therefore works for elliptic curves defined over any field.

Just remember that:

x / y

(3)

means:

x \times y^{- 1}

(4)

and that

y^{- 1}

always exists because it is the inverse element, which is guaranteed to exist for multiplication due to the group axioms it obeys.

The group function is usually called elliptic curve point addition, and repeated addition as done for DHKE is called elliptic curve point multiplication.

Figure 1.
Visualisation of elliptic curve point addition
. Source.

Elliptic curve point multiplication

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Not every $x$ belongs to the elliptic curve over a non quadratically closed field

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One major difference between the elliptic curve over a finite field or the elliptic curve over the rational numbers the elliptic curve over the real numbers is that not every possible

x

generates a member of the curve.

This is because on the Equation "Definition of the elliptic curves" we see that given an

x

, we calculate

x^{3} + a x + b

, which always produces an element

y^{2}

But then we are not necessarily able to find an

y

for the

y^{2}

, because not all fields are not quadratically closed fields.

For example: with

a = 1

and

b = 1

, taking

x = 1

gives:

y^{2} = 1^{3} + 1 \times 1 + 1 = 3

(1)

and therefore there is no

y \in Q

that satisfies the equation. So

x = 1

is not on the curve if we consider this elliptic curve over the rational numbers.

That

x

would also not belong to Elliptic curve over the finite field

F_{4}

, because doing everything

m o d 4

we have:

0 * 0 1 * 1 2 * 2 3 * 3 = 0 = 1 = 4 = 9 = 0 = 1 m o d 4 m o d 4 m o d 4 m o d 4

(2)

Therefore, there is no element

y \in F_{4}

such that

y \times y = 2

y \times y = 3

, i.e.

2

and

3

don't have a multiplicative inverse.

For the real numbers, it would work however, because the real numbers are a quadratically closed field, and

3 \in R

For this reason, it is not necessarily trivial to determine the number of elements of an elliptic curve.

Number of elements of an elliptic curve

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Elliptic curve over the real numbers ( $E (F)$ )

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Elliptic curve over the rational numbers ( $E (Q)$ )

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Number of elements of an elliptic curve over the rational numbers

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Can be finite or infinite! TODO examples. But it is always a finitely generated group.

Mordell's theorem (1922)

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The elliptic curve group of all elliptic curve over the rational numbers is always a finitely generated group.

The number of points may be either finite or infinite. But when infinite, it is still a finitely generated group.

For this reason, the rank of an elliptic curve over the rational numbers is always defined.

TODO example.

Rank of an elliptic curve over the rational numbers ( $r$ )

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Mordell's theorem guarantees that the rank (number of elements in the generating set of the group) is always well defined for an elliptic curve over the rational numbers. But as of 2023 there is no known algorithm which calculates the rank of any curve!

It is not even known if there are elliptic curves of every rank or not: Largest known ranks of an elliptic curve over the rational numbers, and it has proven extremely hard to find new ones over time.

TODO list of known values and algorithms? The Birch and Swinnerton-Dyer conjecture would immediately provide a stupid algorithm for it.