EC Cryptography Tutorials - Herong's Tutorial Examples - v1.03, by 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 P2 = 9 P3 = 27 = 5, (mod 11) P4 = 81 = 4, (mod 11) P5 = 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)