Gain-tunable cross-coupled n-dimensional sinusoidal discrete hyperchaotic system and its application to image encryption

This paper proposes a gain-tunable fully cross-coupled n-dimensional sinusoidal discrete hyperchaotic system, whose core dynamics is given by xk(i+1)=sin(axk(i)∑ℓ≠kbkℓxℓ(i)+bk),k=1,…,n,and develops a dual-chaotic dynamic-parameter-modulation image encryption scheme together with an FPGA (Field Programmable Gate Array)-oriented implementation of the chaotic core. By injecting multivariable fully cross-coupled sinusoidal terms into discrete maps, the proposed system exhibits rich nonlinear dynamics and can achieve maximum-dimensional hyperchaos with n positive Lyapunov exponents in broad parameter regions. A Jacobian factorization yields a verifiable spectral-shifting rule: if the normalized residuals admit a uniform lower bound m∗ over a gain interval of interest, then LEj(a)≥ln(a)+m∗, and any a>a∗=exp(−m∗) guarantees a fully positive Lyapunov spectrum. The encryption scheme uses the outputs of a continuous hyperchaotic system to dynamically modulate several parameters of the discrete hyperchaotic map, forming a time-varying encryption framework with an enlarged key space and improved resistance to common cryptanalytic attacks; representative tests yield a ciphertext information entropy of 7.999330 and strong differential-attack resistance with NPCR=99.6068% and UACI=33.4661%. For hardware realization, a pipelined CORDIC (Coordinate Rotation Digital Computer)-based sine architecture is adopted to enable efficient fixed-point computation with high numerical fidelity and low resource cost. Theoretical analysis and numerical experiments validate the hyperchaotic characteristics and encryption effectiveness, while hardware results further confirm the FPGA feasibility and numerical consistency of the chaotic core for image-protection-oriented applications.