**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.03, by 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 a 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,+).

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)