Invertible function that randomizes order

I am looking for an invertible discrete function $f:\{0,1,2,\dots,n-1\} \to \{0,1,2,\dots,n-1\}$ for some given integer $n$. I want $f(0),f(1),\dots,f(n-1)$ to return all the integers in range $[0..n)$ exactly once, but in a "messy", random-seeming arrangement. I anticipate that $n$ will be not bigger than $2^{30}$.

I thought about finding a generator for the group ``, but I'm not sure if it would work for any given $n$ (would it?). Any other ideas?

You are looking for a pseudorandom permutation on the set $\{0,1,2,\dots,n-1\}$. In cryptography, this has been studied under the (counter-intuitive) name "format-preserving encryption". There are a number of constructions you could use for your purposes.

There's a bunch of research literature on the problem, with different schemes that are optimized for different values of $n$. You can also find some summaries on Cryptography.SE.

I recommend you start by reading the question and the answers at Lazily computing a random permutation of the positive integers and Encrypting a 180-bit plaintext into a 180 bit ciphertext with a 128-bit block cipher and What are the examples of the easily computable "wild" permutations?.

StackExchange Computer Science Q#54227, answer score: 4