Algorithm 2 Itoh-Tsujii Inversion for ${\mathbb{F}}_{2^m}$ Where $m=2^r+1$.

Input:$A\left(x\right)\in {\mathbb{F}}_{2^m}$ with $m=2^r+1$, $f\left(x\right)$ the irreducible polynomial of ${\mathbb{F}}_{2^m}$ Output: $C\left(x\right)=A^{-1}\left(x\right)\mathrm{mod}f\left(x\right)$  $C\left(x\right)\leftarrow A\left(x\right)$  for $i=0$ to $r-1$ do   $B\left(x\right)\leftarrow C\left(x\right)$   for $j=0$ to $2^i$ do    $B\left(x\right)\leftarrow B^2\left(x\right)\mathrm{mod}f\left(x\right)$    end for   $C\left(x\right)\leftarrow C\left(x\right)B\left(x\right)\mathrm{mod}f\left(x\right)$  end for  $C\left(x\right)\leftarrow C^2\left(x\right)\mathrm{mod}f\left(x\right)$  return $C\left(x\right)$