A Novel Method to Generate Key-Dependent S-Boxes with Identical Algebraic Properties

TL;DR

Addressing the need for secure, dynamic s-boxes, the paper proposes a method to generate key-dependent clone s-boxes that preserve bijection, nonlinearity, SAC, and BIC. The approach leverages two permutations on inputs and outputs, realized via permutation matrices $P_1$, $P_2$, and the induced row permutation $Q_1$, to construct clone copies $Q_1 Y P_2$ with invariant properties. A main theorem proves invariance under these transformations, and the algorithm provides steps plus a fixed-point handling mechanism and key-extraction via the factorial number system. Empirical demonstrations for $n=4$ and $n=8$ show substantial clone families ($(4!)^2$ clones) and AES s-box cloning with preserved properties, illustrating practical applicability.

Abstract

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 $n\times n$ 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 $4\times 4$ 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.

Figures (1)

• Figure 1: Flowchart of constructing cloned key-dependent s-box

Theorems & Definitions (12)

• Proposition 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• Theorem 2.5
• Definition 3.1
• Lemma 3.2
• Theorem 3.3
• Remark 3.4
• Example 4.1