Developing NSRTD framework for 5-neighborhood cellular automata under null boundary condition

Cellular automata (CAs) have gained significant attention from researchers across various fields due to their applications in diverse areas such as cryptography, VLSI design, test pattern generation and random number generation. While most of the research works on CAs have focused on 3-neighborhood CAs, 5-neighborhood CAs have shown distinct advantages in applications like cryptography and random number generation, particularly due to enhanced randomness and diffusion properties associated with the larger neighborhood size. However, most of the existing cellular automaton (CA) characterization tools — such as reachability tree (RT), next-state RMT transition diagram (NSRTD), rule vector graph (RVG) and explicit rule vector graph (ERVG) — are primarily tailored to 3-neighborhood CAs. This underscores the need for specialized tools to support 5-neighborhood CAs. In light of this, this work presents the design and development of NSRTD framework for 2-state 5-neighborhood null-boundary CA (5N NBCA). An algorithm has been developed that formalizes the NSRTD framework for 5N NBCA. Further, another algorithm has been presented for identifying a CA rule that forms only a single fixed-point (termed single-length cycle single attractor CA, or SACA) and multiple fixed-points (termed single-length cycle multiattractor CA, or MACA), along with the number of attractors and attractor states, for any arbitrary CA length n. A subset of CA rules (0 to 255) that form SACA and MACA has been identified and reported. The number of fixed-point attractors for all the identified MACA rules with varying CA length n has also been reported.