A Secure and Efficient Framework for Outsourcing Large-scale Matrix Determinant and Linear Equations

Large-scale matrix determinants and linear equations are two basic computational tools in science and engineering fields. However, it is difficult for a resource-constrained client to solve large-scale computational tasks. Cloud computing service provides additional computing resources for resource-constrained clients. To solve the problem of large-scale computation, in this article, a secure and efficient framework is proposed to outsource large-scale matrix determinants and linear equations to a cloud. Specifically, the proposed framework contains two protocols, which solve large-scale matrix determinant and linear equations, respectively. In the outsourcing protocols of large-scale matrix determinants and linear equations, the task matrix is encrypted and sent to the cloud by the client. The encrypted task matrix is directly computed by using LU factorization in the cloud. The computed result is returned and verified by the cloud and the client, respectively. The computed result is decrypted if it passes the verification. Otherwise, it is returned to the cloud for recalculation. The framework can protect the input privacy and output privacy of the client. The framework also can guarantee the correctness of the result and reduce the local computational complexity. Furthermore, the experimental results show that the framework can save more than 70% of computing resources after outsourcing computing. Thus, this article provides a secure and efficient alternative for solving large-scale computational tasks.