Are there any (asymmetric) cryptographic primitives not relying on arithmetic over prime fields and/or finite fields? Trying to figure out if any (asymmetric) cryptographic primitives exists, which do not rely on arithmetic over a prime field and/or arithmetic over a finite field, some people might get lost in research. Therefore, here’s the related question (inspired by this comment):

Does any asymmetric cryptographic primitive exist, that doesn’t rely on arithmetic over prime fields and/or finite fields?


If, which one(s) would that be?
If not, it would be nice if you would also explain why not?
Braid cryptography?  Knapsack cryptosystems, like Nasako–Murikami?

Lattice-based cryptography tends to work in polynomial rings or modules with coefficients in finite fields, but whose higher-level structure is not a field.

Also: don't forget RSA!  RSA works in a ring, not a field.