Comment on Nizam Chew, L.C.; Ismail, E.S. S-box Construction Based on Linear Fractional Transformation and Permutation Function. Symmetry 2020, 12, 826

The aim of this comment paper is to identify a technical error in the published article of Nizam Chew et al. [1]. A new algorithm for substitution box (S-Box) construction has been proposed by Nizam Chew et al. [1]. The proposed algorithm uses linear fractional transformation and permutation function on Galois field$GF\left(2^8\right)$. The linear fractional transformation used in the construction of the S-Box is defined as

$$Z\left(t\right)=\frac{35t+15}{9z+5}$$

where, $35, 15, 9, 5 \in GF\left(2^8\right)$. The algebraic structure of $GF\left(2^8\right)$used in this work is defined as

$$GF\left(2^8\right)=\raisebox{1ex}{$Z_2\left[t\right]$}\!\left/ \!\raisebox{-1ex}{$\left(P\left(t\right)\right)$}\right.$$

where, $P\left(t\right)=t^8+t^4+t^3+t^2+t+1$ is an irreducible polynomial over $GF\left(2\right)$.

However, it is not so as claimed by the authors. We found that the polynomial $P\left(t\right)$ is not an irreducible polynomial over $GF\left(2\right)$. The polynomial $P\left(t\right)$ can be factored as$P\left(t\right)={\left(t+1\right)}^3(t^2+t+1)\left(t^3+t+1\right)$. The finite field theorem [2] states that

Theorem 1.

Let$F$ be a field, and $f\left(t\right)$ is a polynomial in$F\left[t\right]$.$E=\raisebox{1ex}{$F\left[t\right]$}\!\left/ \!\raisebox{-1ex}{$f\left(t\right)$}\right.$ is a field if and only if $f\left(t\right)$ is irreducible.

The algebraic structure of $GF\left(2^8\right)$with $P\left(t\right)=t^8+t^4+t^3+t^2+t+1$ defined by the authors is not a finite field. This led to errors in the calculation of the S-Box. Therefore, it is suggested that

Table 1. List of Irreducible Polynomials over $GF\left(2\right)$.

Irreducible Polynomials over GF(2)
$“P_1\left(t\right)=t^8+t^6+t^3+t^2+1”$
$“P_2\left(t\right)=t^8+t^6+t^4+t^3+t^2+t+1”$
$“P_3\left(t\right)=t^8+t^4+t^3+t+1”$
$“P_4\left(t\right)=t^8+t^4+t^3+t^2+1”$
$“P_5\left(t\right)=t^8+t^5+t^3+t+1”$
$“P_6\left(t\right)=t^8+t^5+t^3+t^2+1”$
$“P_7\left(t\right)=t^8+ t^5+t^4+t^3+1”$
$“P_8\left(t\right)=t^8+t^5+t^4+t^3+t^2+t+1”$
$“P_9\left(t\right)=t^8+t^6+t^5+t+1”$
$“P_{10}\left(t\right)=t^8+t^6+t^5+t^2+1”$
$“P_{11}\left(t\right)=t^8+t^6+t^5+t^3+1”$
$“P_{12}\left(t\right)=t^8+t^6+t^5+t^4+1”$
$“P_{13}\left(t\right)=t^8+t^6+t^5+t^4+t^2+t+1”$
$“P_{14}\left(t\right)=t^8+ t^6+t^5+t^4+t^3+t+1”$
$“P_{15}\left(t\right)=t^8+t^7+t^2+t+1”$
$“P_{16}\left(t\right)=t^8+ t^7+t^3+t+1”$
$“P_{17}\left(t\right)=t^8+t^7+t^3+t^2+1”$
$“P_{18}\left(t\right)=t^8+t^7+t^4+t^3+t^2+t+1”$
$“P_{19}\left(t\right)=t^8+ t^7+t^5+t+1”$
$“P_{20}\left(t\right)=t^8+t^7+t^5+t^3+1”$
$“P_{21}\left(t\right)=t^8+t^7+t^5+t^4+1”$
$“P_{22}\left(t\right)=t^8+t^7+t^5+t^4+t^3+t^2+1”$
$“P_{23}\left(t\right)=t^8+t^7+t^6+t+1”$
$“P_{24}\left(t\right)=t^8+t^7+t^6+t^3+t^2+t+1”$
$“P_{25}\left(t\right)=t^8+t^7+t^6+t^4+t^2+t+1”$
$“P_{26}\left(t\right)=t^8+t^7+t^6+t^4+t^3+t^2+1”$
$“P_{27}\left(t\right)=t^8+t^7+t^6+t^5+t^2+t+1”$
$“P_{28}\left(t\right)=t^8+t^7+t^6+t^5+t^4+t+1”$
$“P_{29}\left(t\right)=t^8+t^7+t^6+t^5+t^4+t^2+1”$
$“P_{30}\left(t\right)=t^8+ t^7+t^6+t^5+t^4+t^3+1”$

[3] must be considered while choosing an irreducible polynomial over a finite field. Table 1 may act as a reference for any future work undertaken in this direction.