Abstract: The periodicity theorems of modular series of Fibonacci series were firstly presented, and then the periodicity property of Fibonacci _Q transform matrix was generalized. The relationship between the period of the Fibonacci _Q matrix and that of cat map was discussed, and the inherent relationship between the period of cat map and that of series generated from Fibonacci series was disclosed. The best transform period was secondly defined, and the best periodicity theorem for cat map was given. Finally, the relevant theory was verified correct by several simulation experiments of cat map in image scrambling, which provided mathematical foundation for image scrambling.

Key words: Fibonacci mode series, Fibonacci matrix, cat map, best period, image scrambling

摘要： 介绍了Fibonacci数列的模数列周期性定理，并推广至Fibonacci_Q变换矩阵的模周期。通过探讨Fibonacci_Q矩阵的周期与猫映射的周期的关系，揭示了猫映射的模周期与Fibonacci数列的模周期的内在联系；定义了矩阵变换的最佳周期概念，给出了猫映射的最佳周期性定理。通过图像置乱的仿真实验，验证了相关理论的正确性，从而为图像置乱提供了数学理论依据。

关键词: Fibonacci模数列, Fibonacci矩阵, 猫映射, 最佳周期, 图像置乱