The ABC of Symmetric Primitives over Integer Rings: Milk Before Meat

Designing a secure symmetric-key cipher over a vector space over a field $\mathbb F_{p^n}^t$ is well known and understood by the cryptographic community. Even if the attacks are continuously improving, our current understanding regarding the design and security of the majority of the symmetric-key primitives has not fundamentally changed in the last 20 years.

How does this picture change when we move to an integer ring $\mathbb Z_{p^n}^t$? Although the question is easy to state, it turns out to