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

Modular Addition of 10 - Abelian Group

This section provides an Abelian Group using the modular arithmetic addition of 10 (integer addition operation followed by a modular reduction of 10).

An Abelian Group can also be defined on at finite number of integers with the help of modular operation.

For example, here is an Abelian Group of 10 integers:

- The set of elements is the set of integers from 0, 1, 2, ..., to 9.
- The binary operation is the integer addition operation followed by a modular reduction of 10 (also called modular arithmetic addition of 10).
- The identity element is the integer 0.

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

(6 + 7) mod 10 = 3 The "closure" condition (6 + 7) mod 10 = (7 + 6) mod 10 The "commutativity" condition ((6 + 7) mod 10) + 8) mod 10 = (6 + (7 + 8) mod 10) mod 10 The "associativity" condition (3 + 0) mod 10 = 3 The "identity" condition (3 + 7) mod 10 = 0 The "symmetry" condition

In fact the above example can be generalized to any positive integer i to define an Abelian Group with i integers:

- The set of elements is the set of integers from 0, 1, 2, ..., to i-1.
- The binary operation is the integer addition operation followed by a modular reduction of i (also called modular arithmetic addition of i).
- The identity element is the integer 0.

We can call the above as Integer Additive Group of Order i, and denote it as G(i,+).

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)