# What $\mathbb{Z}_n^*$ stands for in number theory?

I think that is a very simple/silly question but i can't find the meaning of the asterisk in $$\mathbb{Z}_n^*$$.

I'm trying to understand some basic concepts in number theory, while studying cryptography (for example yesterday i came across these lectures) and I've been seeing this symbolism.

I know that $$\mathbb{Z}_n = {0, 1, ..., n-1}$$, but what the extra $$'*'$$ stands for?

Until today, i knew that the asterisk above a set of numbers "means" the same set with $$'0'$$ excluded, but i don't think it's the same in this case.

Also i found that it may mean the non-negative integers of the set, but (at least in my lectures), the $$\mathbb{Z}_n$$ already doesn't have negative integers.

In every cryptography - number theory lectures/course i've came across they just write it down without exlain it, but it really confuses me.

Any help?

• It likely stands for the multiplicative group of units of $\Bbb Z/n\Bbb Z$. That is, the group consisting of elements of $\Bbb Z/n\Bbb Z$ that are coprime to $n$, with multiplication as its operation. Commented Dec 27, 2018 at 11:19

It stands for the set of elements of $$\mathbb{Z}_n$$ which are invertible modulo $$n$$. In particular, if $$n$$ is prime, then (and only then) $$\mathbb{Z}_n^*=\mathbb{Z}_n\setminus\{0\}$$.