Orthogonal Cellular Automata (OCA) have been recently investigated in the literature as a new approach to construct orthogonal Latin squares for cryptographic applications such as secret sharing schemes. In this paper, we consider OCA for a different cryptographic task, namely the generation of pseudorandom sequences. The idea is to iterate a dynamical system where the output of an OCA pair is fed back as a new set of coordinates on the superposed squares. The main advantage is that OCA ensure a certain amount of diffusion in the generated sequences, a property which is usually missing from traditional CA-based pseudorandom number generators. We study the problem of finding OCA pairs with maximal period by first performing an exhaustive search up to local rules of diameter $d=5$, and then focusing on the subclass of linear bipermutive rules. In this case, we characterize the periods of the sequences in terms of the order of the subgroup generated by an invertible Sylvester matrix. We finally devise an algorithm based on Lagrange's theorem to efficiently enumerate all linear OCA pairs of maximal period up to diameter $d=11$.

中文翻译：

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Hip to Be (Latin) Square：来自正交元胞自动机的最大周期序列

正交元胞自动机 (OCA) 最近在文献中被研究为一种为密码应用（例如秘密共享方案）构建正交拉丁方的新方法。在本文中，我们将 OCA 用于不同的密码任务，即伪随机序列的生成。这个想法是迭代一个动态系统，其中 OCA 对的输出作为叠加正方形上的一组新坐标反馈。主要优点是 OCA 确保生成序列中的一定量的扩散，这是传统的基于 CA 的伪随机数生成器通常缺少的属性。我们通过首先对直径 $d=5$ 的局部规则进行穷举搜索，然后关注线性双置换规则的子类来研究寻找具有最大周期的 OCA 对的问题。在这种情况下，我们根据可逆 Sylvester 矩阵生成的子群的阶数来表征序列的周期。我们最终设计了一种基于拉格朗日定理的算法，以有效地枚举最大周期的所有线性 OCA 对，最大周期为直径 $d=11$。