Group of integers modulo n

From CRYPTUTOR

Additive group modulo $n$

The set $\mathbb{Z}_n=\{0,1,\dots,n-1\}$ forms an abelian group under addition modulo $n$ , with 0 representing the identity element. It is also a cyclic group since 1 generates the group. In fact, any number $g\in\mathbb{Z}_n$ which satisfies ${\rm gcd}(g,n)=1\,$ is a generator of the group.

When $n$ is a product of two primes $p$ and $q$ , $\mathbb{Z}_n$ is isomorphic to the product group $\mathbb{Z}_p \times \mathbb{Z}_q$ .

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Multiplicative group modulo $n$

The set $\mathbb{Z}^*_n=\{x \,|\, {\rm gcd}(x,n) = 1\}$ forms an abelian group under multiplication modulo $n$ , with 1 representing the identity element.