Additive Notation of Abelian Group

This section describes the Additive notation of an Abelian Group. The addition sign, +, is used as the operator. Number 0 is used as the identity element.

An Abelian Group can also be expressed in Additive Notation using the addition sign, +, as the operator and number zero, 0, 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:

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

2 + 3 = 6                   The "closure" condition
2 + 3 = 3 + 2               The "commutativity" condition
(2 + 3) + 4 = 2 + (3 + 4)   The "associativity" condition
3 + 0 = 3                   The "identity" condition
3 + (-3) = 0                The "symmetry" condition

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