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I wondered about the classes of invertible functions over $n$-dimensional Galois fields, $f: GF(q)^n \to GF(q)^n$ s.t. $f^{-1}$ exists.

I know that all linear mappings, e.g. of the form $f(x) = Gx$ with some matrix $G\in GF(q)^{n\times n}$, are invertible if the respective matrix is invertible. But are there any other, non-linear, invertible mappings?

Currently, I'm looking for examples of existence or arguments of non-existence. Ultimately I'd like to characterize and construct all invertible mappings $f: GF(q)^n \to GF(q)^n$.

Uroc327
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    It’s just the symmetric group on $q^n$ elements, no? – Aphelli Jul 29 '21 at 09:53
  • @Mindlack could you expand on this a bit? I'm not too experienced in group theory. My background on Galois fields stems mostly from algebraic coding theory. – Uroc327 Jul 29 '21 at 10:03
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    Take a look at the smallest case $q=n=2$. Then $f$ is just any bijection from the set $S={00, 01,10,11}$ to itself. That is, any function that permutes the elements of $S$. There are $4!$ of those but only six of them are invertible linear transformations (namely those that keep $00$ fixed, but this is a unique quirk of the $q=n=2$ case). With larger parameters it gets worse. No other feasible characterization exists. You can represent all such a functions as vector values polynomials in $n$ variables, but then we lose the ability to test invertibility by non brute-force methods. – Jyrki Lahtonen Aug 04 '21 at 08:58

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