Modular Multiplication of 10 - Not Abelian Group

EC Cryptography Tutorials - Herong's Tutorial Examples

∟Modular Multiplication of 10 - Not Abelian Group

This section demonstrates that the modular arithmetic multiplication of 10 (integer multiplication operation followed by a modular reduction of 10) can not define an Abelian Group.

In the previous tutorial, we demonstrated that the first 10 non-negative integer can be defined an Abelian Group with modular arithmetic addition of 10.

Now we want to see if we can defined as an Abelian Group with modular arithmetic multiplication of 10, G(10,*).

The binary operation is the integer multiplication operation followed by a modular reduction of 10.

The "identity" condition is easy to satisfy. Integer 1 should be the identity element. Because I * 1 mode 10 = I is true for any integer.

Now let's check the "symmetry" condition:

If I = 1, then -I = 1, because 1 * 1 mod 10 = 1.
If I = 2, then there is no -I that 2 * (-I) mod 10 = 1.
If I = 3, then -I = 7, because 3 * 7 mod 10 = 1.
If I = 4, then there is no -I that 4 * (-I) mod 10 = 1
If I = 5, then there is no -I that 5 * (-I) mod 10 = 1
If I = 6, then there is no -I that 7 * (-I) mod 10 = 1
If I = 7, then -I = 3, because 7 * 3 mod 10 = 1.
If I = 8, then there is no -I that 8 * (-I) mod 10 = 1
If I = 9, then -I = 9, because 9 * 9 mod 10 = 1.