**EC Cryptography Tutorials - Herong's Tutorial Examples** - Version 1.00, by Dr. Herong Yang

Niels Henrik Abel and Abelian Group

Abelian Groups are named after early 19th century mathematician Niels Henrik Abel.

Abelian Groups are named after early 19th century mathematician Niels Henrik Abel. Here is how Wikipedia describes him at en.wikipedia.org/wiki/Niels_Henrik_Abel:

*Niels Henrik Abel (/ˈɑːbəl/;
Norwegian: [ˈɑːbl̩]; 5 August 1802 - 6 April
1829) was a Norwegian mathematician who made pioneering contributions in
a variety of fields. His most famous single result is the first complete
proof demonstrating the impossibility of solving the general quintic
equation in radicals. This question was one of the outstanding open
problems of his day, and had been unresolved for over 350 years. He
was also an innovator in the field of elliptic functions, discoverer of
Abelian functions. Through the great works from Abel's hand he was known
to the world's mathematicians; he made his discoveries while living in
poverty and died at the age of 26 from tuberculosis.*

Here is what Niels Henrik Abel is known for:

Abel's binomial theorem Abelian category Abelian variety Abel equation Abel equation of the first kind Abelian extension Abel function Abelian group Abel's identity Abel's inequality Abel's irreducibility theorem Abel-Jacobi map Abel-Plana formula Abel-Ruffini theorem Abelian means Abel's summation formula Abelian and tauberian theorems Abel's test Abel's theorem Abel transform Abel transformation Abelian variety Abelian variety of CM-type Dual Abelian variety

Last update: 2019.

Table of Contents

Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

►Abelian Group and Elliptic Curves

►Niels Henrik Abel and Abelian Group

Multiplicative Notation of Abelian Group

Additive Notation of Abelian Group

Modular Addition of 10 - Abelian Group

Modular Multiplication of 10 - Not Abelian Group

Modular Multiplication of 11 - Abelian Group

Abelian Group on Elliptic Curve

Discrete Logarithm Problem (DLP)

Generators and Cyclic Subgroups

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)