Nifty functions

About a month ago, I worked out how many functions had the following properties:

Since then, I've figured out how to write out pretty equations in Google Docs. (Can you tell?) The number of such functions, expressed prettily, is:

This type of function probably has a proper name, but I don't know what it is. For the purpose of this blog post, I'll call these "nifty functions". A month ago, I thought I had figured out how to generate the truth table for the nth nifty function, but it turns out I was wrong. Painfully wrong.

A couple weeks ago, I wrote some code that will take a truth table and solve for the simplest modular polynomial that would generate the truth table. I thought it would be neat to see what the nifty functions looked like as polynomials.

That's where I realized I was painfully wrong about how easy it would be to generate the truth table for the nth nifty function. While it appears I was right about the number of these functions, generating the truth table turned out to be far more complicated than I thought. However, I have emerged tired but victorious. I can now list out polynomials that represent the nifty functions for a given (mod P).

Naturally, quite a bit of processing is required relative to P, so I am only exploring small primes at the moment. I'm interested in two questions:

1. If you have a polynomial expression that is nifty in (mod P_a), is it also nifty in (mod P_b)?
2. Are there nifty functions where f() can be processed in polynomial time, but its inverse g() cannot be?

Given the tools I have developed so far, I should be able to test these questions experimentally for small P. In the mean time, I leave you with one of the most complex nifty polynomials for (mod 5).