Chaotic Nature of Integer Sequences From Primitive Linear Feedback Shift Registers

In this brief, we investigate the chaotic characteristics of the integer sequences generated by primitive linear feedback shift registers (LFSRs) by interpreting the internal states as integers. We prove that the discrete Lyapunov exponent (dLE) of the permutations induced by these sequences from an L-stage primitive LFSR approches to the range between $\ln (\sqrt {3})$ and $\ln (2)$ as L increases indefinitely and hence the dynamic systems satisfy the definition of discrete chaos. Furthermore, the 0–1 test of the sequences yields statistics close to 1, supporting the conclusion that these sequences exhibit chaotic dynamics under both theoretical and empirical evaluations.