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

Razi Arshad; Mudassar Jalil

doi:10.3390/sym15051005

and

¹

Department of Computing, School of Electrical Engineering and Computer Sciences, National University of Sciences and Technology, Islamabad 44000, Pakistan

²

Department of Mathematics, Comsats University Islamabad, Islamabad 44000, Pakistan

^*

Author to whom correspondence should be addressed.

Symmetry2023, 15(5), 1005;https://doi.org/10.3390/sym15051005

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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

G F (2^{8})

. The linear fractional transformation used in the construction of the S-Box is defined as

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

where,

35, 15, 9, 5 \in G F (2^{8})

. The algebraic structure of

G F (2^{8})

used in this work is defined as

G F (2^{8}) = \frac{Z_{2} [t]}{(P (t))}

where,

P (t) = t^{8} + t^{4} + t^{3} + t^{2} + t + 1

is an irreducible polynomial over

G F (2)

.

However, it is not so as claimed by the authors. We found that the polynomial

P (t)

is not an irreducible polynomial over

G F (2)

. The polynomial

P (t)

can be factored as

P (t) = {(t + 1)}^{3} (t^{2} + t + 1) (t^{3} + t + 1)

. The finite field theorem [2] states that

Theorem 1.

Let

F

be a field, and

f (t)

is a polynomial in

F [t]

.

E = \frac{F [t]}{f (t)}

is a field if and only if

f (t)

is irreducible.

The algebraic structure of

G F (2^{8})

with

P (t) = 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 [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.

Table 1. List of Irreducible Polynomials over

G F (2)

.

Author Contributions

R.A. and M.J. have equally contributed to this research. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data sharing is not applicable to this article.

Conflicts of Interest

The authors declare no conflict of interest.

References

Nizam Chew, L.C.; Ismail, E.S. S-box Construction Based on Linear Fractional Transformation and Permutation Function. Symmetry 2020, 12, 826. [Google Scholar] [CrossRef]
Lidl, R.; Niederreiter, H. Finite Fields. In Encyclopedia of Mathematics and Its Applications, 2nd ed.; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar] [CrossRef]
Pardo, J.L.G. Introduction to Cryptography with Maple; Springer: Berlin/Heidelberg, Germany, 2013; ISBN 978-3-642-32165-8. [Google Scholar] [CrossRef]

Table 1. List of Irreducible Polynomials over

G F (2)

.

Table 1. List of Irreducible Polynomials over

G F (2)

.

Irreducible Polynomials over GF(2)
$“ P_{1} (t) = t^{8} + t^{6} + t^{3} + t^{2} + 1 ”$
$“ P_{2} (t) = t^{8} + t^{6} + t^{4} + t^{3} + t^{2} + t + 1 ”$
$“ P_{3} (t) = t^{8} + t^{4} + t^{3} + t + 1 ”$
$“ P_{4} (t) = t^{8} + t^{4} + t^{3} + t^{2} + 1 ”$
$“ P_{5} (t) = t^{8} + t^{5} + t^{3} + t + 1 ”$
$“ P_{6} (t) = t^{8} + t^{5} + t^{3} + t^{2} + 1 ”$
$“ P_{7} (t) = t^{8} + t^{5} + t^{4} + t^{3} + 1 ”$
$“ P_{8} (t) = t^{8} + t^{5} + t^{4} + t^{3} + t^{2} + t + 1 ”$
$“ P_{9} (t) = t^{8} + t^{6} + t^{5} + t + 1 ”$
$“ P_{10} (t) = t^{8} + t^{6} + t^{5} + t^{2} + 1 ”$
$“ P_{11} (t) = t^{8} + t^{6} + t^{5} + t^{3} + 1 ”$
$“ P_{12} (t) = t^{8} + t^{6} + t^{5} + t^{4} + 1 ”$
$“ P_{13} (t) = t^{8} + t^{6} + t^{5} + t^{4} + t^{2} + t + 1 ”$
$“ P_{14} (t) = t^{8} + t^{6} + t^{5} + t^{4} + t^{3} + t + 1 ”$
$“ P_{15} (t) = t^{8} + t^{7} + t^{2} + t + 1 ”$
$“ P_{16} (t) = t^{8} + t^{7} + t^{3} + t + 1 ”$
$“ P_{17} (t) = t^{8} + t^{7} + t^{3} + t^{2} + 1 ”$
$“ P_{18} (t) = t^{8} + t^{7} + t^{4} + t^{3} + t^{2} + t + 1 ”$
$“ P_{19} (t) = t^{8} + t^{7} + t^{5} + t + 1 ”$
$“ P_{20} (t) = t^{8} + t^{7} + t^{5} + t^{3} + 1 ”$
$“ P_{21} (t) = t^{8} + t^{7} + t^{5} + t^{4} + 1 ”$
$“ P_{22} (t) = t^{8} + t^{7} + t^{5} + t^{4} + t^{3} + t^{2} + 1 ”$
$“ P_{23} (t) = t^{8} + t^{7} + t^{6} + t + 1 ”$
$“ P_{24} (t) = t^{8} + t^{7} + t^{6} + t^{3} + t^{2} + t + 1 ”$
$“ P_{25} (t) = t^{8} + t^{7} + t^{6} + t^{4} + t^{2} + t + 1 ”$
$“ P_{26} (t) = t^{8} + t^{7} + t^{6} + t^{4} + t^{3} + t^{2} + 1 ”$
$“ P_{27} (t) = t^{8} + t^{7} + t^{6} + t^{5} + t^{2} + t + 1 ”$
$“ P_{28} (t) = t^{8} + t^{7} + t^{6} + t^{5} + t^{4} + t + 1 ”$
$“ P_{29} (t) = t^{8} + t^{7} + t^{6} + t^{5} + t^{4} + t^{2} + 1 ”$
$“ P_{30} (t) = t^{8} + t^{7} + t^{6} + t^{5} + t^{4} + t^{3} + 1 ”$

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

Author Contributions

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