Algebraic Characterization of Reversible First Degree Cellular Automata over $\mathbb{Z}_d$

There exists algorithms to detect reversibility of cellular automaton (CA) for both finite and infinite lattices taking quadratic time. But, can we identify a $d$-state CA rule in constant time that is always reversible for every lattice size $n\in \mathbb{N}$? To address this issue, this paper explores the reversibility properties of a subset of one-dimensional, $3$-neighborhood, $d$-state finite cellular automata (CAs), known as the first degree cellular automata (FDCAs) for any number of cells $(n\in \mathbb{N})$ under the null boundary condition. {In a first degree cellular automaton (FDCA), the local rule is defined using eight parameters. To ensure that the global transition function of $d$-state FDCA is reversible for any number of cells $(n\in \mathbb{N})$, it is necessary and sufficient to verify only three algebraic conditions among the parameter values. Based on these conditions, for any given $d$, one can synthesize all reversible FDCAs rules. Similarly, given a FDCA rule, one can check these conditions to decide its reversibility in constant time.