We prove that important properties describing complex behaviours as ergodicity, chaos, topological transitivity, and topological mixing, are decidable for one-dimensional linear cellular automata (LCA) over ${(\mathbb{Z}/m\mathbb{Z})}^n$ (Theorem 6 and Corollary 7), a large and important class of cellular automata (CA) which are able to exhibit the complex behaviours of general CA and are used in applications. In particular, we provide a decidable characterization of ergodicity, which is known to be equivalent to all the above mentioned properties, in terms of the characteristic polynomial of the matrix associated with LCA. We stress that the setting of LCA over ${(\mathbb{Z}/m\mathbb{Z})}^n$ with $n>1$ is more expressive, gives rise to much more complex dynamics, and is more difficult to deal with than the already investigated case $n=1$. The proof techniques from [23], [25] used when $n=1$ for obtaining decidable characterizations of dynamical and ergodic properties can no longer be exploited when $n>1$ for achieving the same goal. Indeed, in order to get the decision algorithm (Algorithm 1) we need to prove a non trivial result of abstract algebra (Theorem 5) which is also of interest in its own.

We also illustrate the impact of our results in real-world applications concerning the important and growing domain of cryptosystems which are often based on one-dimensional LCA over ${(\mathbb{Z}/m\mathbb{Z})}^n$ with $n>1$. As a matter of facts, since cryptosystems have to satisfy the so-called confusion and diffusion properties (ensured by ergodicity and chaos, respectively, of the involved LCA) Algorithm *1 turns out to be an important tool for building chaotic/ergodic one-dimensional linear CA over ${(\mathbb{Z}/m\mathbb{Z})}^n$ and, hence, for improving the existing methods based on them.

我们证明了描述复杂行为（如遍历，混沌，拓扑可传递性和拓扑混合）的重要属性对于一维线性元胞自动机（LCA）可以决定 ${（\mathbb{ž}/米\mathbb{ž}）}^{ñ}$（定理6和推论7），是一类重要的细胞自动机（CA），能够展现一般CA的复杂行为，并在应用中使用。尤其是，我们提供了可确定的遍历性表征，据称与LCA相关的矩阵的特征多项式等同于所有上述属性。我们强调，将LCA设置为${（\mathbb{ž}/米\mathbb{ž}）}^{ñ}$ 与 $ñ>\mathrm{1个}$ 比已经调查的案例更具表现力，产生更复杂的动态，并且更难处理 $ñ=\mathrm{1个}$。[23]，[25]中的证明技术用于$ñ=\mathrm{1个}$ 用于获得动态和遍历属性的可确定特征的描述 $ñ>\mathrm{1个}$为了实现相同的目标。确实，为了获得决策算法（算法1），我们需要证明抽象代数（定理5）的非平凡结果，该结果本身也很有趣。

我们还说明了结果在现实应用中的影响，该应用涉及密码系统的重要且不断增长的领域，这些领域通常基于一维LCA， ${（\mathbb{ž}/米\mathbb{ž}）}^{ñ}$ 与 $ñ>\mathrm{1个}$。事实上，由于密码系统必须满足所谓的混淆和扩散属性（分别由所涉及的LCA的遍历和混沌来确保），算法* 1成为构建混沌/遍历的重要工具，尺寸线性CA${（\mathbb{ž}/米\mathbb{ž}）}^{ñ}$ 因此，用于改进基于它们的现有方法。