New Algorithms for Secure Outsourcing of Large-Scale Systems of Linear Equations With the rapid development in availability of cloud services, the techniques for securely outsourcing the prohibitively expensive computations to untrusted servers are getting more and more attentions in the scientific community. In this paper, we investigate secure outsourcing for large-scale systems of linear equations, which are the most popular problems in various engineering disciplines. For the first time, we utilize the sparse matrix to propose a new secure outsourcing algorithm of large-scale linear equations in the fully malicious model. Compared with the state-of-the-art algorithm, the proposed algorithm only requires (optimal) one round communication (while the algorithm requires $L$ rounds of interactions between the client and cloud server, where $L$ denotes the number of iteration in iterative methods). Furthermore, the client in our algorithm can detect the misbehavior of cloud server with the (optimal) probability 1. Therefore, our proposed algorithm is superior in both efficiency and checkability. We also provide the experimental evaluation that demonstrates the efficiency and effectiveness of our algorithm.