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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?

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    $\begingroup$ 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. $\endgroup$ – ÍgjøgnumMeg Dec 27 '18 at 11:19
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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) $\mathbb{Z}_n^*=\mathbb{Z}_n\setminus\{0\}$.

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  • $\begingroup$ thanks for the answer! $\endgroup$ – Don S Dec 27 '18 at 11:33
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Basically it refers to the invertible elements in the group, which in this case are those who are relatively prime to n.

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