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

Element Generated Subgroup Is Cyclic

This section describes the fact that all subgroups generated from elements in finite Abelian groups are cyclic groups.

If we take any element P from an finite Abelian group and generate a subgroup S, we know that S is a cyclic group. Because the way S is generated matches perfectly the definition of cyclic group.

**Cyclic Subgroup Example 1** -
Here is a cyclic subgroup of integers: 0, 3, 6, 9, 12, 15, 18, 21,
generated from integer 3 in the parent group of
0, 1, 2, 3, ..., 23 using the addition operation with modular reduction of 24.

P = 3 2P = 6 3P = 9 4P = 12 5P = 15 6P = 18 7P = 21 8P = 0

**Cyclic Subgroup Example 2** -
Here is a cyclic subgroup of polynomials: 0, x+1, 2x+2,
generated from polynomial x+1 in the parent group of
0, 1, 2, x, x+1, x+2, 2x, 2x+1, 2x+2
using the algebraic addition operation with modular reduction of 3 on coefficients.

P = x+1 2P = 2x+2 3P = 0

**Cyclic Subgroup Example 3** -
Here is a cyclic group of integers: 1, 3, 4, 5, 9,
generated from integer 3 in the parent group of
1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
using the multiplication operation with modular reduction of 11.

P = 3 P^{2}= 9 P^{3}= 27 = 5, (mod 11) P^{4}= 81 = 4, (mod 11) P^{5}= 243 = 1, (mod 11)

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

What Is Subgroup in Abelian Group

What Is Subgroup Generator in Abelian Group

Every Element Is Subgroup Generator

Order of Subgroup and Lagrange Theorem

►Element Generated Subgroup Is Cyclic

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)