**EC Cryptography Tutorials - Herong's Tutorial Examples** - Version 1.00, by Dr. Herong Yang

Modular Multiplication of 10 - Not Abelian Group

This section demonstrates that the modular arithmetic multiplication of 10 (integer multiplication operation followed by a modular reduction of 10) can not define an Abelian Group.

In the previous tutorial, we demonstrated that the first 10 non-negative integer can be defined as an Abelian Group with modular arithmetic addition of 10.

Now we want to see if we can defined as an Abelian Group with modular arithmetic multiplication of 10, G(10,*).

The binary operation is the integer multiplication operation followed by a modular reduction of 10.

The "identity" condition is easy to satisfy. Integer 1 should be the identity element. Because I * 1 mode 10 = I is true for any integer.

Now let's check the "symmetry" condition:

If I = 1, then -I = 1, because 1 * 1 mod 10 = 1. If I = 2, then there is no -I that 2 * (-I) mod 10 = 1. If I = 3, then -I = 7, because 3 * 7 mod 10 = 1. If I = 4, then there is no -I that 4 * (-I) mod 10 = 1 If I = 5, then there is no -I that 5 * (-I) mod 10 = 1 If I = 6, then there is no -I that 7 * (-I) mod 10 = 1 If I = 7, then -I = 3, because 7 * 3 mod 10 = 1. If I = 8, then there is no -I that 8 * (-I) mod 10 = 1 If I = 9, then -I = 9, because 9 * 9 mod 10 = 1.

As you can see, there are a number of elements that have no symmetric elements.

So G(10,*) is not an Abelian Group.

Last update: 2019.

Table of Contents

Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

►Abelian Group and Elliptic Curves

Niels Henrik Abel and Abelian Group

Multiplicative Notation of Abelian Group

Additive Notation of Abelian Group

Modular Addition of 10 - Abelian Group

►Modular Multiplication of 10 - Not Abelian Group

Modular Multiplication of 11 - Abelian Group

Abelian Group on Elliptic Curve

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)