S-box generation through 2D Schaffer–Gompertz chaotic map and CHDNA-SNEA optimization algorithm

Saba Naz, Zohaib Zahid, Rashad Ali and Muhammad Kamran Jamil

Chaos, Solitons & Fractals, 2026, vol. 210, issue P2

Abstract: The security strength of cryptographic algorithms is directly determined by substitution boxes (S-boxes), which are essential nonlinear components in contemporary block cipher systems. This work introduces an effective S-box generating method based on a 2-Dimensional Schaffer–Gompertz (2D SG) chaotic map and the DNA-Stochastic Nonlinearity Enhancement Algorithm (CHDNA-SNEA). With a well-defined range of control parameters and high sensitivity to initial conditions, the proposed 2D SG chaotic map, which is formed from the Schaffer F6 function combined with Gompertz terms and hyperbolic tangent coupling, exhibits excellent chaotic properties. By applying Deoxyribonucleic Acid (DNA) encoding procedures to chaotic sequences, the CHDNA-SNEA optimization method generates initial S-boxes and systematically enhances their nonlinearity (NL) levels through the analysis of Walsh spectrum distributions and the deliberate strengthening of weak linear combinations. The generated S-boxes achieve superior cryptographic properties, such as high NL (NLmin=112), optimal strict avalanche criterion (SAC ≈0.5), low differential uniformity, and enhanced bit independence criterion (BIC), according to a thorough performance analysis. The efficacy and efficiency of the proposed method are validated by comparison with current state-of-the-art techniques, demonstrating its promise for reliable cryptographic applications.

Keywords: S-box construction; 2D SG chaotic map; DNA encoding; Nonlinearity optimization; Cryptography; Block cipher; Schaffer function; Walsh spectrum (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:210:y:2026:i:p2:s0960077926008192

DOI: 10.1016/j.chaos.2026.118678

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