Endomorphism ring over an elliptic curve

In summary, an endomorphism ring over an elliptic curve refers to the set of all endomorphisms of the curve and is closely related to its group structure. The endomorphism ring can be finite or infinite, depending on the field over which the curve is defined. In cryptography, the endomorphism ring plays a crucial role in designing and analyzing secure schemes. There are still open problems related to the endomorphism ring over elliptic curves, such as finding efficient algorithms and studying its relationship to the arithmetic properties of curves.
  • #1
yavanna
12
0
I found that the endomorphism group over an elliptic curve is isomorphic to a complex quadratic order:

[itex]End(E)\simeq \mathbb{Z}[\delta]=\mathbb{Z}+\delta\mathbb{Z}[/itex],
where
[itex]\delta=\frac{\sqrt{\Delta}}{2}[/itex] if [itex]\Delta [/itex] is even
[itex]\delta=\frac{1+\sqrt{\Delta}}{2}[/itex] if [itex]\Delta [/itex] is odd

Does anyone know where I can find some info and a rigorous definition of a complex quadratic order, and the proof of that result?
Thanks
 
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FAQ: Endomorphism ring over an elliptic curve

1. What is an endomorphism ring over an elliptic curve?

An endomorphism ring over an elliptic curve refers to the set of all endomorphisms (homomorphisms from a mathematical object to itself) of an elliptic curve. It is a fundamental structure in the study of elliptic curves and plays a crucial role in understanding their algebraic properties.

2. How is the endomorphism ring related to the group structure of an elliptic curve?

The endomorphism ring of an elliptic curve is closely related to its group structure. In fact, the endomorphisms of an elliptic curve form a ring under composition, and this ring is isomorphic to the group of points on the curve. This connection allows us to use the tools of ring theory to study the group structure of elliptic curves.

3. Can the endomorphism ring of an elliptic curve be finite?

Yes, the endomorphism ring of an elliptic curve can be finite. In fact, for a curve defined over a finite field, the endomorphism ring is always finite. This is because the number of endomorphisms is limited by the size of the field. However, for curves defined over infinite fields, the endomorphism ring can be infinite.

4. What is the importance of the endomorphism ring in cryptography?

The endomorphism ring is crucial in the study of elliptic curve cryptography. It helps in designing secure cryptographic schemes and in analyzing their security. It also allows for efficient implementations of cryptographic algorithms, making elliptic curves a popular choice in modern cryptography.

5. Are there any open problems related to the endomorphism ring over elliptic curves?

Yes, there are several open problems related to the endomorphism ring over elliptic curves. Some of these include finding efficient algorithms for computing endomorphism rings, investigating the structure of endomorphism rings over certain classes of curves, and studying the relationship between endomorphism rings and the arithmetic properties of elliptic curves.

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