An n-dimensional modulo chaotic system with expected Lyapunov exponents and its application in image encryption

Abstract

One-dimensional (1D) chaotic maps have many shortcomings, including simple structures, uncomplicated chaotic behaviors, and narrow chaotic ranges. Multi-dimensional (MD) chaotic maps certainly have more complex constructions and more abundant chaotic phenomena. However, it is still challenging to artificially customize its Lyapunov exponents (LEs), which are characterization indicators of the chaotic state. This paper proposed a novel n-dimensional (nD) modulo chaotic system (nD-MCS) based on preceding works that can generate nD chaotic maps by using existing 1D chaotic maps as seed chaotic maps. The proposed nD-MCS can customize its LEs by setting system parameters. Three 2D and three 3D chaotic maps are presented as examples. Performance evaluations demonstrated that they both exhibit hyperchaotic behaviors, broad chaotic ranges, and large complexities. We present an encryption scheme to confirm the application of the developed chaotic maps to image encryption. Finally, a hardware experiment utilizing the field-programmable gate array (FPGA) is adapted to verify the proposed cryptosystem.