A novel method to generate key-dependent s-boxes with identical algebraic properties

• DOI: 10.1016/j.jisa.2021.103065
• Publication Date: 2021-12-01T23:44:32Z
• Authors: Ahmad Y. Al-Dweik; Iqtadar Hussain; Moutaz Saleh; M.T. Mustafa
• Keywords: FOS: Computer and information sciences; Invariant; Computer Science - Cryptography and Security; Permutation; 005; Group Theory (math.GR); 02 engineering and technology; Strict avalanche criterion; 01 natural sciences; Key-dependent s-boxes; Bits independence criterion; 0103 physical sciences; Cryptography; FOS: Mathematics; 0202 electrical engineering, electronic engineering, information engineering; Nonlinearity; Mathematics - Group Theory; Cryptography and Security (cs.CR)

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.