hash-no-gal-fld

I don't think you intended "propositionally" equal in your final sentence. Equality is data in HoTT. If you take the propositional truncation then you usually throw away too much.

Just because finite fields are finite doesn't mean that every finite "thing" is a finite field.

Hashes don't really map onto finite fields. They purposefully try to not preserve any algebra.

Cryptography is not all based on primes.

You are making a lot of over-general assumptions in your posts.

> hashes (galois fields)

Huh? What do hashes like Sha256 have to do with Galois fields?