My setting is the following: $n$ is a product of two big primes (RSA-like), and I am given a $R = \mathbb{Z}/n^s\mathbb{Z}$ as a space to work with. I would like to represent elements of $R$ as polynomials or vectors, to avoid the operations in the $R$ directly (real parameters make $n^s$ really big - thousands of bits - which is cumbersome). I am free to choose the particular $s$ and $n$ (the conditions on $n$ are much more restrictive due to the security reasons, but I am considering it as well).
As far as I understand, there is no obvious and efficiently computable isomorphism between $R$ and $\mathbb{Z}_n[X]/F(X)$, even if I am free to choose $F$ and its degree $s$, but correct me if I am wrong. Another thing is that with high probability the elements of $R \setminus R^{\times}$ will not be a part of arithmetic operations over $R$. So my homomorphism can be less strict on these elements - I only need the homomorphic property to apply in case when both operands and the result of the binary operation are in $R^{\times}$. Also, I do not need the whole $R$ - if some particular value of $s$ or $F$ allows some polynomial ring to be embedded into $R$, this would be a good solution too.
I ran out of ideas, could it be that I am missing the important impossibility result? Any help would be appreciated.