Generating digital chaotic systems of the simplest structure via a strongly connected graph inverse approach

This paper designs the simplest m-dimensional (m=2,3,4,…) digital chaotic system using a strongly connected graph inverse approach. Compared to previous systems, this approach significantly simplifies the system structure while enhancing statistical performance. First, in the m-dimensional digital iterative system, we construct 2m-state transition graph with bidirectional direct paths between any two states. Under the condition that the bitwise XOR result between any two states equals the combination of the current m unilateral infinite sequence outputs, we derive the corresponding simplest uncoupled m-dimensional iterative functions based on the inverse approach. Second, based on the simplest uncoupled m-dimensional iterative functions, we develop a cascaded closed-loop coupling approach to obtain the corresponding simplest fully coupled m-dimensional iterative functions, theoretically proving that they satisfy Devaney’s chaos definition. Compared to previous systems, this closed-loop coupling method not only simplifies the system structure, making it the simplest form among all fully coupled m-dimensional iterative functions, but also significantly improves statistical performance, as evidenced by passing both NIST and TestU01 tests. Finally, we validate the effectiveness and superiority of the simplest m-dimensional digital chaotic system through circuit design and FPGA simulation experiments.