For The Love of Cycling — ℤₙ* The Multiplicative group for ℤₙ modulo n

Okay … let me ask you. For the numbers up to 10, which numbers do not share any factors with 10? Remember that 10 has factors of 2 and 5, so you can’t use any number which have any of these factors.

Let’s try … 1 … that’s okay.
Let’s try … 2 … nope! It is 2 ｘ1. Reject!
Let’s try … 3 … that’s okay.
Let’s try … 4 .. nope. It is 2 ｘ2. Reject!
Let’s try … 5 … nope. It is 5 ｘ 1. Reject!
Let’s try … 6 … It is 3 ｘ2. Reject!
Let’s try … 7… that’s okay.
Let’s try … 8… It is 2 ｘ2ｘ2. Reject!
Let’s try … 9… that’s okay.

And so we get 1, 3, 7 and 9 [here]. This has the name of the multiplicative group for ℤₙ modulo n.

Sometimes the symbols used on cryptography are a little difficult to understand, but often they are not too difficult, as the operations themselves are quite simple. If you can get to grips with the modulus operator, the rest is fairly easy.

In number theory, ℤₙ is the set of non-negative integers less than n ({0,1,2,3…n-1}). ℤₙ* is then a subnet of this which is the multiplicative group for ℤₙ modulo n. The set ℤₙ* is the set of integers between…