I have a ring $R=\Bbb Z[x]/(x^m+1)$ with $m$ some power of two and a polynomial $g \in R$, which has relatively small coefficients and some other properties that I believe to be irrelevant for this question. Lastly I have an $a \in \Bbb Z$.
I would like to reduce $a$ modulo $(g)$ to create a small polynomial $\hat a = a\mod g$ in $R$. So far I am simply assuming that $\hat a = a$, because that seems to be trivially correct, but unfortunately I am running into some trouble with this assumption later on in the construction that I am trying to implement. Thus now I am wondering if there is a different way of computing a polynomial in $R$ which is congruent to $a\mod g$. Any help is greatly appreciated!
Background: These computations are necessary when implementing the Multilinear Jigsaw Puzzles that are specified in this paper on page 31+.