**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.02, by Dr. Herong Yang

Elliptic Curve Subgroups

This chapter provides notes on subgroup generation from reduced elliptic curve groups, Ep(a,b). Python programs are provided to perform point addition, scalar multiplication, and subgroup generation.

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Subgroups of Reduced Elliptic Curve Groups

Python Program for Point Addition

Point Addition Extended with Identity

Python Program for Scalar Multiplication

Python Program for Generating Subgroup

Subgroup Example - G(15,13) of E17(0,7)

Takeaways:

- Subgroup groups can be generated with a base point from a reduced elliptic curve group, since it is a finite Abelian group.
- It is not so difficult to write Python programs to to perform point addition, scalar multiplication, and subgroup generation..

Table of Contents

Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

Abelian Group and Elliptic Curves

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)