> hashes (galois fields)
Huh? What do hashes like Sha256 have to do with Galois fields?
Huh? What do hashes like Sha256 have to do with Galois fields?
You have a set field size as you go over it, it loops and so output fields are never unique, you have a 1:infinite relationship. That property to determinism, there are problems computers cannot solve but we can. A computer requires determinism to do work, which in effect is that 1:1 state->next state mapping on the state graph. There's a lot of computer science theory and literature on this if you want to dig deeper. MIT has an OCW course in signals and systems.
Anywhere you see modulus math happening, its working with finite fields.
Just because there exists a finite field of size 2^n doesn't mean that any length-n-bit-string that is the output of a hash function lives in a finite field of size 2^n.
Adding and multiplying hashes doesn't make any sense. If it does then you have a very weird hash function. Not one of the standard cryptographic hashes. So saying that a hash function maps onto a finite field sounds pretty confused to me.