Efficient polynomial ring multiplication with coefficients in GF(2^s)

Question

Is there something similar (or an other efficient approach) to the NTT for the multiplication of polynomials in polynomial rings modulo $X^{2^n} + 1$ with coefficients in $\operatorname{GF}(p)$ with $p \equiv 1 \pmod{2^{n+1}}$, but instead for polynomial rings with coefficients in $\operatorname{GF}(2^s)$?