Abstract: Based on an arithmetic progression with an input secret key, a method is proposed to construct n-dimensional generalized Arnold transformation matrixes.Direct calculation algorithms are also presented for the transformation matrix and the inverse transformation matrix.The algorithms are only relevant to the secret key and their time complexity is equal to n(n+1)/2 times multiplication operation.Using the n-dimensional generalized Arnold transformation matrix as a transform matrix, and adopting double product-like scrambling in the image position space and the hue space, the image scrambling method has long period and is public, and can prevent many attacks and thus greatly enhances the system’s security.Moreover, when the inverse transformation matrix is applied to restore the scrambled image, the period of the transformation matrix is not needed to calculate.Simulation experiments show that the proposed method is effective and very secure