Integrable cellular automata on finite fields of order $2^n$
Aoi Araoka, Tetsuji Tokihiro
TL;DR
This work develops integrable cellular automata over finite fields $\mathbb{F}_{2^n}$ by constructing set-theoretic Yang–Baxter maps via a single function $f$, deriving the YBE conditions, and exploiting the simplifications in characteristic two. An exhaustive search yields $16$, $736$, and $269{,}056$ bijective $f$ for field orders $4$, $8$, and $16$, respectively, enabling explicit CA constructions under helical boundary conditions. A striking finding is that, for bijective $f$, the CA period matches the field order $q$, with analytical proofs for $q=4$ and $q=8$ and numerical support suggesting the phenomenon extends to general $q=2^n$. Non-bijective $f$ break this periodicity, underscoring the central role of bijectivity in the integrable CA dynamics. Together, these results illuminate a tight link between the algebraic structure of YBE maps over finite fields and the global periodic behavior of discrete integrable systems, with potential implications for discrete soliton-like dynamics and finite-field cryptographic constructions.
Abstract
This paper explores cellular automata (CA) constructed from Yang-Baxter maps over finite fields $F_{2^n}$. We define $R$-matrices using a map $f$ on $F_{2^n}$ and establish necessary and sufficient conditions for $f$ to satisfy the Yang-Baxter equation. We show that these conditions become remarkably streamlined in characteristic two. An exhaustive search for bijective solutions in fields of order 4, 8, and 16 yields 16, 736, and 269,056 maps, respectively. Analysis of the resulting CA under helical boundary conditions reveals a consistent alignment between the temporal period and the field order. We propose the conjecture that this periodic identity holds generally for $F_{2^n}$, supported by analytical proofs for $n=2$ and $n=3$. Our results further indicate that bijectivity is a fundamental requirement for this periodic behavior.