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.