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

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.

💡 Research Summary

This paper introduces a novel framework called three‑dimensional Cascading Crossing Coupling (3D‑CCC) for constructing hyperchaotic maps and demonstrates its utility in a color‑image encryption scheme. The core idea of 3D‑CCC is to take three one‑dimensional chaotic maps—denoted F, G, and H—and combine them in three stages: cascading (feeding the output of one map into the next), crossing (mixing the current state of each variable with the function values of the other two variables), and coupling (adding a small cross‑variable term controlled by a coefficient c). To avoid the numerical instability caused by modulo‑1 operations common in many high‑dimensional chaotic designs, the authors introduce a saturation operator sat(·) that clamps all intermediate values to the open interval (0, 1) and a denominator safeguard den(·) that prevents division by values arbitrarily close to zero. These safeguards keep the iteration bounded and make the system amenable to fixed‑point or floating‑point digital implementation without overflow or underflow.

For a concrete instantiation, the authors select the ICMIC map as F, the Logistic map as G, and the Sine map as H, yielding the 3D‑ILS (ICMIC‑Logistic‑Sine) hyperchaotic system. The update equations are given in closed form (Equation 9) and include the sat and den operators. The Jacobian matrix of the transformation is derived analytically, showing that diagonal entries are the derivatives of the individual 1‑D maps while off‑diagonal entries arise from the cross‑coupling term c. When c≠0, the Jacobian is full‑rank, guaranteeing multidirectional expansion. The authors prove that, under suitable choices of the control parameters (α for ICMIC, r for Logistic, μ for Sine) and a modest coupling coefficient, the smallest singular value of the Jacobian can be driven above one, ensuring a positive third Lyapunov exponent. Consequently, the system can exhibit three positive Lyapunov exponents—a hallmark of hyperchaos.

Extensive dynamical analysis is performed. Bifurcation diagrams are plotted for each control parameter while the others are held fixed, revealing rich period‑doubling cascades and chaotic windows. Sensitivity to initial conditions is demonstrated by perturbing the initial state by 10⁻¹⁶; trajectories diverge within a few iterations across all three state variables. Lyapunov exponents are computed via a QR‑based method; sweeping α, r, and μ around representative values shows that two or three exponents remain positive over broad regions, confirming robust hyperchaotic behavior. Phase portraits illustrate that trajectories densely fill the unit cube, indicating strong ergodicity and uniform distribution of the generated pseudo‑random sequences.

Building on the 3D‑ILS generator, the authors design a one‑round RGB image encryption algorithm. The three chaotic sequences are mapped to the Red, Green, and Blue channels respectively. A cross‑bit mixing operation swaps high‑order bits of one channel with low‑order bits of another, thereby destroying inter‑channel correlations. After this mixing, a single joint permutation‑diffusion step is applied to the entire image. Because the cross‑bit mixing already provides substantial diffusion, only one round is sufficient to achieve high security metrics.

Experimental evaluation uses standard test images (e.g., Lena, Baboon). The ciphertexts achieve an average information entropy of 7.9993 bits per pixel, essentially the theoretical maximum of 8. The average Number of Pixels Change Rate (NPCR) is 96.61 % and the Unified Average Changing Intensity (UACI) is 33.46 %, both exceeding typical thresholds for secure image encryption. The key space comprises the three initial conditions (each 64‑bit) and the four control parameters (α, r, μ, c, each 64‑bit), yielding a total size of about 2¹⁰⁹ ≈ 2³⁰⁹, which is far beyond the reach of exhaustive brute‑force attacks.

The paper also discusses practical considerations. The avoidance of modulo‑1 operations and the inclusion of saturation/denominator safeguards simplify hardware implementation and reduce error accumulation in finite‑precision arithmetic. However, when the coupling coefficient c is set to zero, the system’s expansion becomes predominantly one‑dimensional, potentially weakening security; thus a non‑zero c is recommended for real‑world deployments. The authors suggest future work on adaptive coupling strategies, hardware acceleration (e.g., FPGA or ASIC), and extending the framework to other cryptographic primitives such as stream ciphers or key‑exchange protocols.

In summary, the 3D‑CCC framework provides a systematic, mathematically grounded method for constructing stable hyperchaotic maps without modulo operations. Its concrete 3D‑ILS instance demonstrates strong chaotic properties, and the derived image‑encryption scheme achieves near‑ideal statistical security with low computational overhead and an enormous key space, making it a compelling contribution to chaos‑based cryptography.