ElGamal in $Z^*_{p^n}$ If $p$ is an odd prime and $n$ natural,it is known that the group $Z^*_{p^n}$ is cyclic.Explain why the selection-choice of the group $Z^*_{{3^{1000}}}$ for the construction of a cryptosystem ElGamal it's not good.

Can anyone explain me why this happens?
The order of $(\mathbf{Z}/3^{1000}\mathbf{Z})^*$ is $\varphi(3^{1000}) = 2\times 3^{999}$, which is a highly composite number, and hence the discrete logarithm in this group is highly vulnerable to the Pohlig-Hellman algorithm.

If you are not familiar with the Pohlig-Hellman algorithm, you can peruse for example Section 2.9 of the book by Hoffstein, Pipher and Silverman. Sadly, the Wikipedia article about it is of quite low quality.