A class of 2n+1 dimensional simplest Hamiltonian conservative chaotic systems and fast image encryption schemes

Abstract

Compared with dissipative systems, conservative systems do not have attractors, thus they can better cope with the security risks brought by phase space reconstruction. In this paper, a paradigm for constructing 2n+1 dimensional simplest Hamiltonian conservative chaotic systems (HCCSs) is proposed, and their Hamiltonian energy conservation, Casimir energy conservation, volume conservation, and boundedness are proved. Based on this construction paradigm, we propose a five-dimensional HCCS (FHCCS) that contains the fewest terms but has complex dynamic behaviors, including energy-controlled hyperchaotic multi-scroll flow, hyperchaos under wide parameter range, energy-induced extremely multistability, initial value enhancement of hyperchaos. To solve the problem of high-dimensional continuous chaotic systems seriously slowing down the speed of chaotic image encryption, a new fractal scrambling and batch diffusion algorithm is proposed. They only need a small number of chaotic sequences to achieve excellent encryption effects, such as strong plaintext sensitivity, strong key sensitivity, high information entropy, low correlation, and huge key space. After sufficient testing, on the Matlab platform, for high-resolution images with a resolution of 2048×2048, only 0.186886s is needed to generate all pseudorandom numbers required for encryption. In addition, they can also make full use of the SIMD technology provided by the CPU to maximize the encryption speed.