三维随机矩阵置乱变换的周期及其应用

The Period of 3-D Random Matrix Scrambling Transformation and Its Applications

为了适合数字多媒体特性,实施多媒体加密与信息隐藏,生成充分大的密钥空间,使用了数论、近世代数、算法分析等工具,对高维随机矩阵置乱变换的精确周期进行了研究。给出三维随机整数矩阵A决定的置乱变换在任意模N下,其周期T(A,N)的精确表达式及上界估计,构造了求周期的快速算法,仅耗费O(log2N)2次模N乘法便可得到T(A,N)。大量的算例和应用范例与理论结果相吻合。结论可用于建立数字多媒体的新型密码体制,实施高效率的加密。

For implementing the encryption/decryption and information hiding for digital multimedia, and aiming to generate enough large cipher key space, the accurate period of high dimension random matrix scrambling transformation is studied with the help of number theory and algebraic theory. An accurate expression and an upper bound estimation for the period T （A, N） of a 3 - D random integer matrix scrambling transformation under any modular N is presented. The efficient algorithm for computing the period is constructed. It is proved that the algorithm needs only O（ log2N）^ 2 times multiplications modulo N for determining the period T （A, N）. Many practical demonstration examples verified the results. This approach can be used to construct new efficient cryptosystems for digital multimedia encryption/decryption.