I'm implementing a public key cryptosystem in a proprietary System On Chip that only allows modular reductions by $n$.
However, that cryptosystem requires ($n^2$) modular reductions. Perhaps someone knows a strategy for performing those reductions relying on a modular reducer by $n$? The Montgomery reduction requires a value $R > n$ that makes ill-suited my approach.
I can perform any modular arithmetic operation mod $n$ but nothing $n^2$.