Construction of Transformation Matrix with a Given Period Modulo N

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.