Is MOV attack against ECDLP fundamentally impossible?

Question

The main idea of the MOV attack is to map EC additive group of order $n$ to multiplicative group in the finite field extension $p^k$. For this, the groups must have the same order, what fully relies on property that multiplicative group over $p^k$ has order $p^k-1$, but it hasn't. The fundamental mistake is in the fact that such multiplicative group has order $p^{k-1}(p-1)$ which wouldn't be divisible by $n$ if $n≠p$, so the main idea of such attack is (as i think) completely broken.

And there is my question: is there any example of usage of MOV attack against any curve except those with $p=n$? I don't believe that since 90s there wasn't any practical usage of this method for, even, weak curves, so the inapplicability of such attack would be visible. Or am I just mistaken and don't know something important, and the MOV attack really works?

Answer 1

Actually, the multiplicative group of the extension field $GF(p^k)$ does have order $p^k-1$. In particular, for any $m$ that does not have $p$ as a factor, there will exist a $k$ such that $GF(p^k)$ will have a subgroup of order $m$.

You appear to be confusing that with the multiplicative group $\mathbb{Z}_{p^k}^*$. That does have order $p^{k-1}(p-1)$, however that's not the group that MOV maps the point into.