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

Multiplicative Notation of Abelian Group

This section describes the multiplicative notation of an Abelian Group. The multiplication sign, *, is used as the operator. Number 1 is used as the identity element.

An Abelian Group can be expressed in Multiplicative Notation using the multiplication sign, *, as the operator and number one, 1, as the identity element:

P * Q for F(P,Q) P * Q = Q * P for F(P, Q) = F(Q, P) (P * Q) * S = P * (Q * S) for F(F(P, Q), S) = F(P, F(Q, S)) P * I = P for F(P, I) = P P * Q = 1 for F(P, Q) = I

The Multiplicative Notation matches well with certain examples of Abelian Groups.

Let's consider the following Abelian Group:

- The set of elements is the set of all rational numbers without 0.
- The binary operation is the numeric multiplication.
- The identity element is the number 1.

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

2 * 3 = 6 The "closure" condition 1/6 * 1/6 = 1/3 The "closure" condition 2 * 3 = 3 * 2 The "associativity" condition (2 * 3) * 4 = 2 * (3 * 4) The "commutativity condition 3 * 1 = 3 The "identity" condition 3 * 1/3 = 1 The "symmetry" condition

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)