ElGamal in $Z^*_{p^n}$
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.