A four-dimensional discrete second-order memristive chaotic map with GOA-based keystream optimization for image encryption

Although discrete chaotic maps are commonly employed in image encryption, certain low-dimensional variants are limited by narrow chaotic parameter ranges, inadequate orbit complexity, and randomness loss in finite-precision implementations. To overcome these drawbacks, this work develops a four-dimensional Discrete Second-order Memristor Map (4D-DSM) by coupling a discrete second-order memristor with a two-dimensional generalized square map (2D-GSM). A modulo- 2 π folding operation is embedded into the update equation of the internal state variable q n of the second-order memristor, which confines the evolution of q n to a bounded interval and introduces a repeated folding mechanism. Through its coupling with the 2D-GSM, this folding mechanism contributes to more intricate orbital evolution of the overall 4D map. Dynamical analysis reveals that the 4D-DSM exhibits rich nonlinear behaviors, including attractor crises, transitions among periodic, chaotic, and hyperchaotic states, and complex attractor evolution. Sequence complexity analysis and NIST tests demonstrate the good statistical properties of the generated chaotic sequences, while digital signal processor implementation verifies the feasibility of digital realization. Based on the 4D-DSM, an image encryption algorithm incorporating the Gannet Optimization Algorithm (GOA) is designed. GOA is employed to search for the optimal control vector X best , which is then used to modulate chaotic sequences for generating the global scrambling sequence and diffusion key streams. A global permutation strategy and a chaos-selected dual-rule cascaded diffusion mechanism are combined to disturb pixel positions and gray values. Experimental results demonstrate that the proposed algorithm achieves low correlation between adjacent pixels, strong plaintext and key sensitivity, and good resistance to differential attacks. In addition, cropping and noise tests indicate a certain degree of robustness against transmission disturbances.