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

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:

- The set of elements is the set of integers.
- The binary operation is the numeric addition.
- The identity element is the integer 0.

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

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)