Modular Addition of 10 - Abelian Group

EC Cryptography Tutorials - Herong's Tutorial Examples

∟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.