Learn English – Is “noncompatible” a legitimate synonym of “incompatible”

It's definitely not a standard English term, and as such will likely sound wrong to most people, even if it is shown somehow to be a legitimate term.

I understand your interest in using it to contrast against "complexity" but I think you would find that "simple" would work just as well, as simple is quite commonly used as the opposite of complex. And you'd get far fewer people wondering if it were a real word.

If I were trying to write the title, I'd probably go with, "A Simplicity-based Introduction to Complexity-based Cryptography." Snappy, and you have full part-of-speech agreement between your contrasting phrases--which makes it crystal clear that you're going for a fun contrast.

A string of letters doesn’t have to be in a dictionary to be a word, but, as it happens, there is an entry for legit in the Oxford English Dictionary, where it is given as both an adjective and a noun and defined as being a colloquial abbreviation of legitimate. The earliest citation is from 1897.

Whether and how you use it is up to you.