Permutation XOR cellular automata: partial-shift connections and half-primitive polynomials

@misc{suzuki2026permutation,
  title = {{Permutation XOR cellular automata: partial-shift connections and half-primitive polynomials}},
  author = {Yosuke Suzuki and Naoto Okuyama and Hiroki Takahashi and Toshimichi Saito},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34542},
  eprint = {2609.34542},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.34542},
}

TY  - UNPB
TI  - Permutation XOR cellular automata: partial-shift connections and half-primitive polynomials
AU  - Yosuke Suzuki
AU  - Naoto Okuyama
AU  - Hiroki Takahashi
AU  - Toshimichi Saito
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.34542
UR  - https://arxiv.org/abs/2609.34542
AB  - This paper presents a permutation XOR cellular automaton, a simple digital dynamical system composed of an elementary cellular automaton of the XOR rule and a partial-shift permutation connection characterized by a shift-part parameter. The proposed system is suitable for precise analysis and simple hardware implementation. Depending on the partial-shift connection and system dimension, the automaton can generate long binary periodic orbits (LBPOs) with strong stability. As a principal result, we identify the partial-shift parameters and dimensions that enable the generation of LBPOs. In our analysis, these LBPOs are characterize using half-primitive polynomials over GF(2). Furthermore, we present a simple FPGA-based hardware implementation. The hardware transforms typical LBPOs into electrical signals for potential engineering applications.
ER  -