Ciro Santilli @cirosantilli 34

 Incoming links: Elliptic curve group

Birch and Swinnerton-Dyer conjecture in two minutes by Ciro Santilli  Updated 2024-12-15  +Created 1970-01-01

 View more

Summary:

overview of the formula of the BSD conjecture
definition of elliptic curve
domain of an elliptic curve. Prerequisite: field
elliptic curve group. Prerequisite: group
Mordell's theorem lets us define the rank of an elliptic curve over the rational numbers, which is the $r$ . Prerequisite: generating set of a group
reduction of an elliptic curve from $E (Q)$ to $E (F_{p}) m o d p$ lets us define $N_{r}$ as the number of elements of the generated finite group

Video 1. Source.

 Read the full article

Elliptic-curve Diffie-Hellman  Updated 2024-12-15  +Created 1970-01-01

 View more

The algorithm is completely analogous to Diffie-Hellman key exchange in that you efficiently raise a number to a power

n

times and send the result over while keeping

n

as private key.

The only difference is that a different group is used: instead of using the cyclic group, we use the elliptic curve group of an elliptic curve over a finite field.

Video 1.

Elliptic curves by Computerphile (2018)

Source. youtu.be/NF1pwjL9-DE?t=143 shows the continuous group well, but then fails to explain the discrete part.

Variant of Diffie-Hellman key exchange based on elliptic curve cryptography.

 Read the full article

Elliptic curve group  Updated 2024-12-15  +Created 1970-01-01

 View more

The elliptic curve group of an elliptic curve is a group in which the elements of the group are points on an elliptic curve.

The group operation is called elliptic curve point addition.

Bibliography:

mathoverflow.net/questions/6870/why-is-an-elliptic-curve-a-group

 Read the full article

Elliptic curve point addition  Updated 2024-12-15  +Created 1970-01-01

 View more

Elliptic curve point addition is the group operation of an elliptic curve group, i.e. it is a function that takes two points of an elliptic curve as input, and returns a third point of the elliptic curve as its output, while obeying the group axioms.

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.

 Read the full article

Mordell's theorem  Updated 2024-12-15  +Created 1970-01-01

 View more

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.

 Read the full article