Polynomial coefficients from GF(2^k) to GF(2)

Question

I'm working with polynomials over $GF(2^k)^n$, so with polynomials with coefficients over $GF(2^k)$, the base field. Is there a way to transform these polynomials so that I can work (e.g., multiply them) with polynomials with coefficients over $GF(2)$?

Thank you!

Answer 1

Since $GF(2^k)\cong\mathbb Z_p[y]/(f(y))$ where $f(y)$ is an irreducible polynomial mod $p$ and of degree $k$, you could say that $F_{p^k}[x]\cong \mathbb Z_p[x,y]/(f(y))$.

So you could compute all your products in $\mathbb Z_2[x,y]$ and then reduce mod $f(y)$.