Multiplicative inverse of a polynomial in $GF(8)$

Question

I am trying to find the inverse of $x ^3+x +1$ in $GF(8)$ defined on the quotient ring $GF(2)[x]/\langle x^3+x+1\rangle$. I have done the Euclidean algorithm but I am stuck in the forward process to get the inverse. Please explain how to do it from reverse.

Answer 1

Hint: The field $GF(8)$ is isomorphic to $GF(2)[x]/\langle x^3+x+1\rangle$ or to $GF(2)[x]/\langle x^3+x^2+1\rangle$. The polynomials $x^3+x+1$ and $x^3+x^2+1$ are the only irreducible polynomials over $GF(2)$ (and they are conjugate). The elements of $GF(8)$ are the residue classes of the polynomials $ax^2+bx+c$, where $a,b,c\in GF(2)$. So maybe you should rethink about your question.