A 3D-Cascading Crossing Coupling Framework for Hyperchaotic Map Construction and Its Application to Color Image Encryption

Abstract

This paper focuses on hyperchaotic-map construction and proposes a 3D-Cascading Crossing Coupling framework (3D-CCC), which cascades, crosses, and couples three one-dimensional chaotic maps to form a three-dimensional hyperchaotic system. The framework avoids modulo-1 operations and introduces bounded-state and denominator safeguards for stable digital implementation. A general 3D-CCC formulation is established, and its derivative/Jacobian structure is analyzed to characterize multidirectional expansion. By instantiating ICMIC, Logistic, and Sine maps, a concrete system (3D-ILS) is derived. Phase portraits, bifurcation behavior, sensitivity tests, and Lyapunov-exponent analysis indicate pronounced ergodicity and hyperchaotic dynamics. As an application of the constructed map, a one-round RGB image-encryption scheme is developed using cross-channel bit mixing with joint permutation-diffusion. Under the reported settings, the cipher reaches near-ideal entropy (average 7.9993), NPCR of 96.61\%, UACI of 33.46\%, and an effective key space of about $2^{309}$. These results support the effectiveness of 3D-CCC as a practical framework for hyperchaotic-system design, with image encryption as one representative application.