there is a lot of mystification around the subject of how this stuff works

for the most part "finite field" literally means the number sequence of a hash chain, as in Galois Field

The hash function is really centrtal to the security of it all, and then second to that is how hard it is to trivially inject a substantial fraction of the seed of a cipher such that you can break the rest, this is a problem both with feedback and counter mode ciphers, but a greater problem with feedback and code book style, and i think the purpose of a GCM style

my inclination is to just crank it all up to 256 bits, make 32 byte nonces, to go with the 32 byte shared secret, and use a 32 byte galois field that is derived from the hash of the nonce and secret, i don't see how you can manipulate that, it's way too big numbers, and trivial for most modern hardware to do it