Applications of the Hermite's criterion?

Question

I found this statement on permutation polynomials and I was wondering in which domain we can find applications and what is its aim.

Here is the criterion : «If $q=p^n$ with $p$ a prime number then $f\in \mathbb{F}_q[X]$ is a permutation polynomial of $\mathbb{F}_q$ iff the two following conditions work : $f$ has exactly one root in $\mathbb{F}_q$ and for all $t\in [\vert 1,q-2\vert ]$, $p \not\mid t$ we have $\deg(f^t \pmod{(X^q-X)})\le q-2.$ »

I heard that we can encounter these polynomials in cryptography.

Thanks in advance !