Abstract: The large-scale MMC (matrix multiplication computation), MIC (matrix inversion computation) and MDC (matrix determinant computation) algorithm in secure outsourcing computing is widely researched and applied, while its existing problems are also increasingly highlighted. The problem of hiding the number of zero elements in the original matrix is one of them. However, there are few researches on it, and the existing researches can only protect the position of zero elements. There is no encryption method aiming at protecting the number of zero elements and at the same time achieving high efficiency, which is very unsafe in the cloud computing. To solve this problem, from the perspective of algorithm, the encryption method of permutation matrix is improved, and a new algorithm of secure outsourcing MMC, MIC and MDC is designed. The algorithm extends the algebraic structure to the finite domain. Firstly, the initial matrix is permuted randomly, then transformed by the unimodular matrix, and the encrypted matrix is sent to the cloud server. The cloud computes it and returns the results to the client. The client decrypts and verifies it. Through theoretical proof, the algorithm not only protects the number of zero elements of the original matrix, but also achieves the purposes of correctness, privacy and verifiability. Finally, experiments show that the encryption method based on unimodular matrix is efficient.

Key words: cloud computing, outsourcing computing, matrix determinant, matrix multiplication, inversion of matrix

摘要： 关于大规模矩阵相乘（MMC）、矩阵求逆（MIC）和矩阵行列式（MDC）的算法在安全外包计算中得到广泛研究与运用，其存在的问题也日益凸显，隐藏原始矩阵中零元素的数目问题就是其中之一。然而，目前学术界关于保护零元素数目的研究较少，现有的研究也仅能保护零元素的位置，没有针对保护零元素的数目同时又能达到高效性的加密方法，这在大规模云计算环境中是很不安全的。针对这个问题，从算法的角度出发，改进了原有的置换矩阵的加密方法，并设计了一种新的安全外包MMC、MIC和MDC的算法。该算法将代数结构扩展到有限域中，首先对初始矩阵进行随机置换，然后进行幺模矩阵变换，并将加密后的矩阵发送给云服务端；云经过计算之后把结果返回给客户端，随后客户端进行解密和验证。通过理论证明，设计的三个协议不仅保护了原始矩阵零元素的数目，而且实现了正确性、隐私性和可验证性的目标。最后，通过实验证明了基于幺模矩阵的加密方法是高效的。

关键词: 云计算, 外包计算, 矩阵行列式, 矩阵相乘, 矩阵的逆