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

Abstract

This paper designs the simplest $m$-dimensional ($m=2,3,4,\dots$) 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 $2^m$-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.