May 9, 2021

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

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.