Algorithm 5 Digit Multiplication in ${\mathbb{F}}_{2^m}$ Where d is the Digit Size [55].

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