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

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 Easttom (2018) 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 x 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-2n respectively on columns and rows of Boolean functions (GF(2(n)) -> 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 Hussain et al. (2012); Hussain et al. (2012); Hussain et al. (2018); Anees and Chen (2020) which involved group action of S-8 only on columns of Boolean functions (GF(2(8)) -> GF(2) ) of s-box. For n = 4, we have used an initial 4 x 4 s-box constructed by Carlisle Adams and Stafford Tavares (Adams and Tavares, 1990) to generated (4!)(2) clone copies. For n = 8, it can be seen (Hussain et al. (2012); Hussain et al. (2012); Hussain et al. (2018); Anees and Chen (2020)) 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.