一类高维随机矩阵置乱变换的周期

为了适合多媒体信息量庞大、存在数据冗余的特点,实施可证明安全、高效率的加密解密,使用了数论、近世代数、矩阵变换、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。将实数域上线性代数的若干结果,推广到模素数有限域上,得到一类整数矩阵及其相关同余方程组之解的若干新性质;在此基础上将用于置乱的矩阵由2维扩展到任意高维,给出广泛一类高维随机整数矩阵A决定的置乱变换,在任意素数幂N=pr模数下,其周期T(A,N)的精确表达式,给出求精确周期算法的时间复杂度。结论可用于建立新型数字多媒体密码体制和信息隐藏体制,扩大其密钥空间,增加其安全性。

On Periods of Higher Dimensional Random Matrix Scrambling Permutations

For efficiently implementing the encryption/decryption for digital multimedia, which is often with huge amount of data and much redundancy, the accurate period of high dimension random matrix scrambling permutation is studied with the help of number theory and algebraic theory. Some new properties of a class of integer matrices and the solutions of its correlative congruent equations are obtained by generalizing some results for linear algebra in real fields to the finite fields over modulo prime numbers. Based on these properties, random scrambling permutation can be extended to any high dimension matrix A and the period T(A,N) with an arbitrary prime power modulo N=pr can be accurately expressed. The complexity of the computation of T(A,N) is presented. The results can be used to construct new cryptosystems for digital multimedia and information hiding systems with bigger key spaces to improve their security levels.