Modular Multiplication of 11 - Abelian Group

This section provides an Abelian Group using the modular arithmetic multiplication of 11 (integer multiplication operation followed by a modular reduction of 11).

In the last tutorial, we demonstrated that the modular arithmetic multiplication of 10 can not be used to define an Abelian Group.

But if we change the modular base from 10 to 11, then we can use the modular arithmetic multiplication of 11 to define an Abelian Group.

You can verify that all 5 Abelian Group conditions are satisfied. For example:

(6 * 7) mod 11 = 9                The "closure" condition
(6 * 7) mod 11 = (7 * 6) mod 11   The "commutativity" condition
((6 * 7) mod 11) * 8) mod 11 = (6 * (7 * 8) mod 11) mod 11  
                                  The "associativity" condition
(3 * 1) mod 11 = 3                The "identity" condition

(1 * 1) mod 11 = 1                The "symmetry" condition
(2 * 6) mod 11 = 1                The "symmetry" condition
(3 * 4) mod 11 = 1                The "symmetry" condition
(4 * 3) mod 11 = 1                The "symmetry" condition
(5 * 9) mod 11 = 1                The "symmetry" condition
(6 * 2) mod 11 = 1                The "symmetry" condition
(7 * 8) mod 11 = 1                The "symmetry" condition
(8 * 7) mod 11 = 1                The "symmetry" condition
(9 * 5) mod 11 = 1                The "symmetry" condition

In fact the above example can be generalized to any prime integer p to define an Abelian Group with p-1 integers:

We can call the above as Integer Multiplicative Group of Order p, and denote it as G(p,*).

Last update: 2019.

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

Abelian Group and Elliptic Curves

 What Is Abelian Group

 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)

 Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 Terminology

 References

 Full Version in PDF/EPUB