Abstract
This paper proposes to present a novel method of generating cryptographic dynamic substitution-boxes, which makes use of the combined effect of discrete hyperchaos mapping and algebraic group theory. Firstly, an improved 2D hyperchaotic map is proposed, which consists of better dynamical behaviour in terms of large Lyapunov exponents, excellent bifurcation, phase attractor, high entropy, and unpredictability. Secondly, a hyperchaotic key-dependent substitution-box generation process is designed, which is based on the bijectivity-preserving effect of multiplication with permutation matrix to obtain satisfactory configuration of substitution-box matrix over the enormously large problem space of 256!. Lastly, the security strength of obtained S-box is further elevated through the action of proposed algebraic group structure. The standard set of performance parameters such as nonlinearity, strict avalanche criterion, bits independent criterion, differential uniformity, and linear approximation probability is quantified to assess the security and robustness of proposed S-box. The simulation and comparison results demonstrate the effectiveness of proposed method for the construction of cryptographically sound S-boxes.
Highlights
E substitution layer in a symmetric-key cryptosystem is meant to perform the process of data substitution with the help of S-boxes
The scenario when n m corresponds to n × n bijective S-boxes in which there exists a one-to-one mapping from input domain to output domain; that is, each input of n-bit long uniquely maps to an n-bit long output. ere exist a vast number of bijective S-boxes for n 8 whose count is 256!, which is more than 10506 [9]. us, it is challenging for the cryptographers to construct and find cryptographically strong S-boxes configurations out of this vast state space. e cryptographic strength of S-box decides the security of block ciphers
Gao et al presented a new way of constructing S-boxes by using the algebraic action of modular groups PSL (2, Z) on a particular projective line in [19]. e obtained S-box met all the performance criteria well and was deemed suitable for encryption applications
Summary
0.0726 − 0.0121 0.01187 − 0.0049 bifurcation analysis of the proposed map (1) is investigated for k, r ∈ (0, 10]. e bifurcation plots of different 2D discrete chaotic maps are shown in Figure 1 (right column). E obtained ApEn scores justify that the proposed map possesses better complexity and unpredictability compared to existing 2D logistic chaotic map, 2D Henon chaotic map, and 2D SLMM map as evident from Table 1. Phase 1: Require: initial conditions of hyperchaos map (2) as x(0), y(0), a, b, c, k, r, itr n0, and sbox [] Returns: initial S-box A, last x and y chaotic variables x-series.
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