**EC Cryptography Tutorials - Herong's Tutorial Examples** - Version 1.00, 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)

Last update: 2019.

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)