What Is an Elliptic Curve?

This section describes what is elliptic curve - A set of 2 dimensional points of (x, y) that satisfy the y*y = x*x*x + a*x + b equation with given values of a and b.



What Is an Elliptic Curve? - An elliptic curve is the set of 2 dimensional points of (x, y) that satisfy the following mathematical equation with given values of a and b:

y2 = x3 + ax + b

Here are two examples of elliptic curves (source: wikipedia.org):

Examples of Elliptic Curves
Examples of Elliptic Curves

Here are more examples of elliptic curves (source: wikipedia.org):

More Examples of Elliptic Curves
More Examples of Elliptic Curves


Here are some elliptic curves superposed like contour lines (source: cosec.bit.uni-bonn.de):

Elliptic Curves Superposed Like Contour Lines
Elliptic Curves Superposed Like Contour Lines

By the way, the earliest text book discussing elliptic curves is probably the "Arithmetica" by "the father of algebra", Diophantus of Alexandria in the third century.



 

Table of Contents

 About This Book

Geometric Introduction to Elliptic Curves

What Is an Elliptic Curve?

 Elliptic Curve Geometric Properties

 Addition Operation on an Elliptic Curve

 Prove of Elliptic Curve Addition Operation

 Same Point Addition on an Elliptic Curve

 Infinity Point on an Elliptic Curve

 Negation Operation on an Elliptic Curve

 Subtraction Operation on an Elliptic Curve

 Identity Element on an Elliptic Curve

 Commutativity of Elliptic Curve Operations

 Associativity of Elliptic Curve Operations

 Elliptic Curve Operation Summary

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 Terminology

 References

 Full Version in PDF/EPUB