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

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.

- The set of elements is the set of integers from 1, 2, ..., to 10.
- The binary operation is the integer multiplication operation followed by a modular reduction of 11 (also called modular arithmetic multiplication of 11).
- The identity element is the integer 1.

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:

- The set of elements is the set of integers from 1, 2, ..., to p-1.
- The binary operation is the integer multiplication operation followed by a modular reduction of p (also called modular arithmetic multiplication of p).
- The identity element is the integer 1.

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

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)