I'm working on implementation of some algorithm where the input is an arbitrarily long message m that I need to reduces to its residue modulo $p^n$ . I'm working on binary fields (characteristic-two finite fields) so p=2 and all the reduction algorithm that I have seen required the input to be of degree at most 2n −2, which makes sense, since you only need to do a reduction after squaring or multiplication and this is the maximum value that you could get. However, is it possible to do a reduction when the input of degree greater than 2n −2?
I read about linear folding in CRC and GCM but couldn't find any good resources that explain it.
Thank you for any input or thoughts!