Framework for Secure Substitution Box Construction and Its Application in Image Encryption
Umar Hayat1,2,*, Ikram Ullah2, Muhammad Bilal2
1 Department of Computer Science, University of Warwick, Coventry, CV4 7AL, UK
2 Department of Mathematics, Quaid-i-Azam University, Islamabad, 44000, Pakistan
* Corresponding Author: Umar Hayat. Email: 
Computers, Materials & Continua https://doi.org/10.32604/cmc.2025.073097
Received 10 September 2025; Accepted 03 December 2025; Published online 25 December 2025
Abstract
Elliptic curve (EC) based cryptosystems gained more attention due to enhanced security than the existing public key cryptosystems. A substitution box (S-box) plays a vital role in securing modern symmetric key cryptosystems. However, the recently developed EC based algorithms usually trade off between computational efficiency and security, necessitating the design of a new algorithm with the desired cryptographic strength. To address these shortcomings, this paper proposes a new scheme based on Mordell elliptic curve (MEC) over the complex field for generating distinct, dynamic, and highly uncorrelated S-boxes. Furthermore, we count the exact number of the obtained S-boxes, and demonstrate that the permuted version of the presented S-box is statistically optimal. The nonsingularity of the presented algorithm and the injectivity of the resultant output are explored. Rigorous theoretical analysis and experimental results demonstrate that the proposed method is highly effective in generating a large number of dynamic S-boxes with adequate cryptographic properties, surpassing current state-of-the-art S-box generation algorithms in terms of security. Apart from this, the generated S-box is benchmarked using side-channel attacks, and its performance is compared with highly nonlinear S-boxes, demonstrating comparable results. In addition, we present an application of our proposed S-box generator by incorporating it into an image encryption technique. The encrypted and decrypted images are tested by employing extensive standard security metrics, including the Number of Pixel Change Rate, the Unified Average Changing Intensity, information entropy, correlation coefficient, and histogram analysis. Moreover, the analysis is extended beyond conventional metrics to validate the new method using advanced tests, such as the NIST statistical test suite, robustness analysis, and noise and cropping attacks. Experimental outcomes show that the presented algorithm strengthens the existing encryption scheme against various well-known cryptographic attacks.
Keywords
Substitution box; Mordell elliptic curve; Möbius transformation; nonlinearity; image encryption; cryptanalysis; data security
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