Consider a single round of a cryptographic permutation, what is so special about it that lacking some bits of information that either get mixed into it as a subkey or omitted at the end (sponge construction) makes you unable to revert it efficiently once it gets looped in sufficient number?
The best analogy I could come up with is a chaotic dynamical system, where any tiny change eventually gets amplified out of control (where the evolution of the two systems - changed and unchanged - no longer encodes any information about the change). And even though it is in a continuous domain where gradients can be analytically computed, after a certain point they either explode to infinity or vanish to 0.
There, there is the jargon of stretching and folding - the fractal dimension, how points get stretched out into the fractal dimension and then get folded back (this achieves a mixing of sort and eventually any two points no matter how close get further and further apart until eventually they completely dissociate as they each get sent into different parts of the attractor).
This seems similar to confusion and diffusion.
Is what's being done in cryptographic permutations the same phenomenon but in a discrete domain?