Are there full cycle cryptographic/one-way hash primitives?
I'm sorry, but I don't believe that there are any known 'one way permutations' for small inputs (that stay one way if you know the 'key').

When we create a cryptographic permutation (or bijection; that's what we call a 'full cycle' function), we have two basic ways to do it:


We can take a series of easy to invert permutations ("round functions"), and glue them together to form a function that we hope is hard to invert.


This is the general design principle behind most block ciphers; we believe that if the key is secret, this can work very well.  However, if you give someone the key, it becomes easy to invert (by just inverting each individual round function).


We can take advantage of some mathematical relation which is one-to-one (that is, it avoids collisions), but it doesn't do it for any naively obvious reasons.


We often find these sorts of things within public key cryptography; one such is $f(x) = g^x \bmod p$ (for appropriate $g, p$); this function can map the inputs $(1, p-1)$ into the range $(1, p-1)$ such that every output is possible.

One problem with this is that these types of functions are large; in the specific case I mentioned, we need $p$ to be perhaps 2048 bits to be really secure, and that's hard to describe as a small range.