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 × 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 ( 𝐺 ⁡ 𝐹 ⁡ ( 2 𝑛 ) → 𝐺 ⁡ 𝐹 ⁡ ( 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 ( 𝐺 ⁡ 𝐹 ⁡ ( 2 8 ) → 𝐺 ⁡ 𝐹 ⁡ ( 2 ) ) of s-box. For 𝑛 = 4 , we have used an initial 4 × 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.

Abstract We have proposed a robust, secure and efficient image encryption algorithm based on chaotic maps and algebraic structure. Nowadays, the chaotic cryptosystems gained more attention due to their efficiency, the assurance of robustness and high sensitivity corresponding to initial conditions. In literature, there are many encryption algorithms that can simply guarantees security while the schemes based on chaotic systems only promises the uncertainty, both of them can not encounter the needs of current scenario. To tackle this issue, this article proposed an image encryption algorithm based on Lorenz chaotic system and primitive irreducible polynomial substitution box. First, we have proposed 16 different S-boxes based on projective general linear group and 16 primitive irreducible polynomials of Galois field of order 256, and then utilized these S-boxes with combination of chaotic map in image encryption scheme. Three chaotic sequences can be produced by the disturbed of Lorenz chaotic system corresponding to variables x , y and z . We have constructed a new pseudo random chaotic sequence k i based on x , y and z . The plain image is encrypted by the use of chaotic sequence k i and XOR operation to get a ciphered image. To show the strength of presented image encryption, some renowned analyses are performed.

The fractional order model of a glucose-insulin regulatory system is derived and presented. It has been extensively proved in the literature that fractional order analysis of complex systems can reveal interesting and unexplored features of the system. In our investigations we have revealed that the glucose-insulin regulatory system shows multistability and antimonotonicity in its fractional order form. To show the effectiveness of fractional order analysis, all numerical investigations like stability of the equilibrium points, Lyapunov exponents, and bifurcation plots are derived. Various biological disorders caused by an unregulated glucose-insulin system are studied in detail. This may help better understand the regulatory system.

The protection of copyrights of digital media uploaded to the Internet is a growing problem. In this paper, first, we present a unified framework for embedding and detecting watermark in digital data. Second, a new robust watermarking scheme is proposed considering this concern. The proposed work incorporates three chaotic maps which specify the location for embedding the watermark. Third, a new chaotic map, the Extended Logistic map, is proposed in this work. The proposed map has a bigger range than logistic and cubic maps. It has shown good results in a bifurcation, sensitivity to initial conditions, and randomness tests. Furthermore, with the detailed analysis of initial parameters, it is justified that Extended Logistic map can be used in secure communication, particularly watermarking. Fourth, to check the robustness of proposed watermarking scheme, we have done a series of analyses and standard attacks. The results confirm that the proposed watermarking scheme is robust against visual and statistical analysis and can resist the standard attacks.