Let $\mathbb F_p$ be the finite field of a prime order $p$, $f(x)\in \mathbb F_p[x]$ an irreducible polynomial, $E = \mathbb F_p[x]/\langle f(x)\rangle$ a finite extension of $\mathbb F_p$, $\lambda\in E$ is a zero of $f$. Is there an efficient algorithm for computing the order of $\lambda$ in $E^\ast$ or, more generally, the order of an arbitrary $\alpha\in E^\ast$?

Edit: Thanks a lot for the comments. My impression is that this problem can be computationally hard (not harder than factorization of the numbers $p^n-1$, of course).