yes, deriving all of the math cryptography depends on independently would not be easy. Fortunately, that's not really how anybody learns.

Along those lines, you do not need to understand the proof of Euler's totient theorem to understand cryptography. It is a distraction. All you need (at most) is to know that the result is true, and even then it's only fundamentally important for RSA, which you likely shouldn't bother learning about these days. RSA simultaneously

1. looks very simple (though the simple version is horrendously insecure), and 2. does not have particularly good performance, and 3. does not have particularly good security (either post or pre quantum), and 4. has been in the process of being phased out for quite some time now.

this is not a good combination of properties. The fact that cryptography textbooks cover it is mostly due to historical tradition. I would personally argue it is time to omit it from instruction materials.