目的 作为一门以数学为基础的学科,数学领域的新进展经常能够促进加密技术及密码学的研究与发展。半张量积理论是一种新的数学工具,是传统矩阵乘法理论的推广,它对于实现不同阶的高维矩阵数字信号处理提供了一个非常好的途径。为研究半张量积理论在数字图像处理中的应用,提出一种基于半张量积的图像加密算法。方法 算法以明文内容为密钥参数,以张量积运算构建阶数远小于输入图像尺寸的可逆密钥矩阵,将低阶密钥矩阵与高阶输入图像进行半张量积运算实现图像的加密与解密。结果 实验采用8×8密钥矩阵对多个不同尺寸的输入图像进行了验证,比较验证表明,从算法安全性、运行效率来说本文算法较现有的一些算法更有优势,能降低约10%~30%的计算时间。结论 提出了一种新的加密算法,只要满足密钥矩阵阶数与输入图像尺寸之间的整数倍关系,一个固定阶数的密钥就能实现对不同尺寸图像的加密及解密,有效降低了数据运算量。实验结果表明,该算法具有较高的安全性和运行效率,可在实时数据加密中得到应用。

Objective The development of new disciplines in the field of mathematics lies in promoting research on encryption technology and cryptography. The semi-tensor product, a new mathematical tool, is a generalization of the traditional matrix multiplication; it provides a new approach by which high-dimensional matrix digital signal processing can achieve a different dimension. In this paper, a novel image-encryption algorithm based on the semi-tensor product is proposed. Method The content of the plaintext is used as a key parameter. Then, a small reversible key matrix is constructed by using the Kronecker product; the key matrixis then used to change the values of pixels in the original image by applying the semi-tensor product. As a result, the dimensions of the original image are much larger than those of the key matrix. Experiments are performed by using an 8×8 key matrix featuring images of various sizes. Result After comparing the experimental results with those of previous methods, the proposed method demonstrated a high level of security with a suitable processing performance. Conclusion A small encryption matrix is proposedto encrypt and decrypt images, wherein the dimensions of the original image are larger thanthose of the key matrix. The computations of the data are effectively reduced, and the operational efficiency of the encryption process is enhanced. The experimental results also demonstrate that the proposedalgorithm offers secure information protection and satisfies the processing time required by standard applications.