How would one go about selecting an appropriate bijective function for introducing permutations into a cipher or hash?
For example, $f(x) = x+1 \space mod \space n$ is a bijective function, but isn't particularly good as a permutation, in that the outputs do not vary sufficiently when an input bit is changed.
The same applies to $f(X_i) = X_{\sigma(i)}$, where $\sigma(i) \equiv i+1\space mod\space |X|$, i.e. a circular shift.
If I understand it correctly, modular exponentiation can be bijective if you select appropriate parameters, but it's rather slow.