**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.03, by Herong Yang

What Is Cyclic Group

This section describes Cyclic Group, which is a finite Abelian group that can be generated by a single element using the scalar multiplication operation in additive notation (or exponentiation operation in multiplicative notation).

**What Is Cyclic Group?**
A Cyclic Subgroup is a finite Abelian group that can be
generated by a single element using the scalar multiplication operation
in additive notation (or exponentiation operation in multiplicative notation).

A Cyclic Group can be expressed in additive notation as:

A finite Abelian Group is a Cyclic Group, if it's elements can be expressed as: {P, P+P, P+P+P, ..., P+P+...+P} or {P, 2P, 3P, ..., nP}, where n is the order of the group nP = 0 is the identity element

A Cyclic Group can be expressed in multiplicative notation as:

A finite Abelian Group is a Cyclic Group, if it's elements can be expressed as: {P, P*P, P*P*P, ..., P*P*...*P} or {P, P^{2}, P^{3}, ..., P^{n}}, where n is the order of the group P^{n}= 1 is the identity element

**Cyclic Group Example 1** -
Here is a Cyclic group of integers: 0, 3, 6, 9, 12, 15, 18, 21
and the addition operation with modular reduction of 24.
Integer 3 is a group generator:

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

**Cyclic Group Example 2** -
Here is a Cyclic group of polynomials: 0, x+1, 2x+2,
and the algebraic addition operation with modular reduction of 3 on coefficients.
Polynomial x+1 is a group generator:

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

**Cyclic Group Example 3** -
Here is a Cyclic group of integers: 1, 3, 4, 5, 9,
and the multiplication operation with modular reduction of 11.
Integer 3 is a group generator:

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

**Cyclic Group Example 4** -
Here is a Cyclic group of integers: 1, 2, 3, 4, 5, 6,
and the multiplication operation with modular reduction of 7.
Integer 3 is a group generator:

P = 3 P^{2}= 9 = 2, (mod 7) P^{3}= 27 = 6, (mod 7) P^{4}= 81 = 4, (mod 7) P^{5}= 243 = 5, (mod 7) P^{6}= 729 = 1, (mod 7)

If express P^{6} as P^{0} in the last example and
replace P with Z, we can use the following graph to represent this Cyclic Group:

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)