中文摘要：

为了改善传统二维Arnold变换用于图像置乱加密的效果，提出了离散Arnold变换的改进方法，并将其用于图像置乱加密测试研究。该方法利用现有离散标准映射的构造思想，将传统离散二维Arnold变换表达式中第一个变换表达式所对应变换结果非线性融入第二个变换表达式，实现经典离散二维Arnold 变换的非线性去拟仿射化修改，以便快速改善图像置乱加密效果。数学证明改进方法不再保持现有离散二维Arnold变换所具有的拟仿射不变性，但是改进变换仍是一种具有周期性的可逆映射，将其用于图像置乱加密时，利用其周期性或逆变换能恢复置乱前原图像。大量实验结果表明，本文所建议的改进方法是有效的，相比现有的离散Arnold变换更具有实用价值意义。

英文摘要：

To improve the image scrambling and encryption effect in traditional two-dimensional discrete Arnold transform, a new nonlinear transform for image scrambling is proposed which improves the classical discrete Arnold transform with quasi-affine properties, and can be applied in image scrambling and encryption researching. This method first makes good use of the construction thought in classical discrete standard map, and embeds the nonlinear expressions of output results of one congruence equation for classical two-dimensional discrete Arnold transform into the input item of the other congruence equation for two-dimensional discrete Arnold transform. Then a new transform with good nonlinear characteristics is constructed on the basis of classical two-dimensional discrete Arnold transform in order to quickly improve the scrambling effect of the gray image. In the end, through mathematical proof it is shown that the proposed transform no longer has the quasi-affine invariance properties in the existing two-dimensional discrete Arnold transform, but it is still a reversible mapping with periodic properties; and when it is applied in image scrambling encryption, the original image can be restored from the scrambling and encryption in gray image for its periodic properties or inverse transform. Some experimental results show that the proposed nonlinear transform is effective, and can obtain better scrambling and encryption quality than the existing discrete two-dimensional Arnold transform, meanwhile it is more practical than the standard Arnold transform in view of security.