1
$\begingroup$

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?

$\endgroup$
1
  • 1
    $\begingroup$ 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. $\endgroup$ Commented Dec 27, 2018 at 11:19

2 Answers 2

0
$\begingroup$

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) $\mathbb{Z}_n^*=\mathbb{Z}_n\setminus\{0\}$.

$\endgroup$
1
  • $\begingroup$ thanks for the answer! $\endgroup$
    – Don S
    Commented Dec 27, 2018 at 11:33
0
$\begingroup$

Basically it refers to the invertible elements in the group, which in this case are those who are relatively prime to n.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .