A key research challenge in modern cryptography is the construction of robust nonlinear components that simultaneously achieve high nonlinearity, resistance to linear and differential cryptanalysis, and efficient implementation for real-time systems. Existing S-box construction methods—whether chaos-based or algebraic—struggle to balance these requirements, especially in RGB image encryption where large data size, high redundancy, and inter-pixel correlations complicate security. To address these challenges, this paper presents a novel technique for constructing block cipher nonlinear elements and developing an RGB image encryption scheme by integrating Gaussian integer rings with Mordell elliptic curves. The proposed method generates strong \(\:n\times\:n\) S-boxes by first establishing a Mordell elliptic curve on a finite field, selecting unique \(\:y-\) values, and applying Gaussian integer transformations to construct nonlinear substitution boxes. The resulting S-boxes exhibit high nonlinearity and effective resistance to linear and differential attacks. Building on this, a new RGB encryption system is designed using a Substitution–Permutation Network (SPN) framework that applies input preparation, S-box substitution, a secondary permutation, and XOR operations to ensure confusion and diffusion across color channels. Experimental evaluation shows that the scheme achieves high entropy, low pixel correlation, and strong robustness against statistical, brute-force, and differential attacks. The integration of Gaussian integer rings with Mordell elliptic curves advances cryptographic security by providing an efficient algebraic framework for image encryption. The proposed methodology is well-suited for real-time applications, including cloud security, IoT data protection, and secure medical image handling. The main originality of this study lies in combining Gaussian integers with elliptic curves to construct secure nonlinear components, offering a new algebraic paradigm for modern encryption systems.
