[than3]

Math, the hash function use magic numbers which in turn initialize and eventually map to a specific Galois Field, they are also called finite 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.

[deadbeef57]

That's flat out wrong. Working modulo 4 is not working in a finite field, because 22 = 0 when you work mod 4. When you work modulo a prime, then you are working in a finite field. Working in the finite field of 4 elements is not* the same as working mod 4. Finite fields of size p^n are not the same as (not isomorphic to!) the ring Z/(p^nZ) whenever n > 1.

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.

[deadbeef57]

2 times 2 = 0 mod 4. Sorry for the markdown mess-up.