# A Novel Method to Generate Key-Dependent S-Boxes with Identical Algebraic Properties. (arXiv:1908.09168v2 [cs.CR] UPDATED)

May 4, 2021

The s-box plays the vital role of creating confusion between the ciphertext
and secret key in any cryptosystem, and is the only nonlinear component in many
block ciphers. Dynamic s-boxes, as compared to static, improve entropy of the
system, hence leading to better resistance against linear and differential
attacks. It was shown in [2] that while incorporating dynamic s-boxes in
cryptosystems is sufficiently secure, they do not keep non-linearity invariant.
This work provides an algorithmic scheme to generate key-dependent dynamic
\$ntimes n\$ clone s-boxes having the same algebraic properties namely
bijection, nonlinearity, the strict avalanche criterion (SAC), the output bits
independence criterion (BIC) as of the initial seed s-box. The method is based
on group action of symmetric group \$S_n\$ and a subgroup \$S_{2^n}\$ respectively
on columns and rows of Boolean functions (\$GF(2^n)to GF(2)\$) of s-box.
Invariance of the bijection, nonlinearity, SAC, and BIC for the generated clone
copies is proved. As illustration, examples are provided for \$n=8\$ and \$n=4\$
along with comparison of the algebraic properties of the clone and initial seed
s-box. The proposed method is an extension of [3,4,5,6] which involved group
action of \$S_8\$ only on columns of Boolean functions (\$GF(2^8)to GF(2)\$ ) of
s-box. For \$n=4\$, we have used an initial \$4times 4\$ s-box constructed by
Carlisle Adams and Stafford Tavares [7] to generated \$(4!)^2\$ clone copies. For
\$n=8\$, it can be seen [3,4,5,6] that the number of clone copies that can be
constructed by permuting the columns is \$8!\$. For each column permutation, the
proposed method enables to generate \$8!\$ clone copies by permuting the rows.