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Can you give me an example of generator of multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{*}=\{1, 2, \ldots, p-1\}.$

Thanks.

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    You should make the title of your post a question or similar statement; as it is, your title is very uninformative.2011-11-06

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I am putting together the question with the comments to figure out the real question. And, this is the answer.

As Henning said, the group is always cyclic. This is a special case of the fact that the multiplicative group of units from a finite field is always cyclic. In a cyclic group of order $n$, there is always an element of order $d$ for any $d$ that divides $n$.

The group $\mathbb{Z} / p \mathbb{Z}$ is order $p-1$. So, if $p = 4k+1$, as you mentioned in a comment, then this group is order $4k$. In that case, 4 divides the order of the group so there is guaranteed to be an element of order 4.

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    @pki, why would it? The cryptographic applications of discrete logarithms are usually for moduli where a generator is known and public already.2011-11-06