模N具有给定周期的变换方阵的构造

Construction of Transformation Matrix with a Given Period Modulo N

置乱技术在数字图像信息隐藏和图像加密中都具有重要的作用;而变换方阵因其理论简洁、实现简单而得到了广泛的关注和研究,但已有工作主要集中在对给定方阵周期的研究方面.分析了变换方阵模素数幂周期的上确界,得到了变换方阵模素数幂的周期达到上确界的充要条件.在此基础上给出了模素数幂具有最大周期的变换方阵的构造方法及两个改进方案.进而分析了变换方阵模一般整数N周期的上界,并使用中国剩余定理给出了两种算法,可构造模一般整数N具有给定周期的变换方阵.

The scrambling transformation technology plays a key role in digital image information hiding and digital image encryption to obtain security. Transformation matrix which has a big period is one of the basic tools of scrambling and thus is very important in practice. Many works in some literature have been done on investigating transformation matrix. However, up to our knowledge, all known works focused on determining the periods of certain transformation matrices modulo positive integers. Different from any known ideas, the general methods to generate transformation matrices for a given period and a given modulus are studied. The supremum of the period of transformation matrix modulo a power of a given prime is analyzed. The necessary and sufficient conditions when the supremum is reached are also presented. A new algorithm and two improvements on constructing transformation matrix which has the maximum period modulo a power of a prime are proposed. Furthermore, the upper bound of the period of transformation matrix modulo a general integer N is investigated, based on which two algorithms on constructing transformation matrix modulo the given N using Chinese Remainder Theorem are designed. The result of the former one has maximum period, while those of the latter one have foreseeable periods.