Research On Secure Outsourcing Calculation For Solving Systems Of Linear Equations

Solving a system of linear equations is not only a fundamental problem in numerical computation,but also an important problem in engineering applications.With the rapid development of science and technology and the proliferation of data volume,more and more users outsource their computational problems to resource-rich cloud servers due to their own limited computing resources,and this new computing model is the outsourced computing in cloud computing.However,while users use this model to improve efficiency,it is often accompanied by leakage of private data,and even the correctness of calculation results cannot be guaranteed.To address the above problems,this paper investigates a secure outsourcing computational protocol for solving systems of linear equations based on elementary matrices,data partitioning techniques,singular value decomposition and optimization theory,respectively,and the main research results are as follows:(1)An outsourcing computation protocol based on elementary matrices.This computational protocol investigates two operations,matrix inversion and matrix multiplication,used in the inverse matrix method to solve systems of linear equations,making it possible to ensure the safety and efficiency of these two operations in solving systems of linear equations when using the inverse matrix method.In the computational protocol,not only the matrix elements are rearranged and the size of non-zero elements are changed by randomly displacing the matrix,but also the Sherman-Morrison formula is introduced in order to hide the zero elements.By sending the blinded original matrix to the cloud,users can achieve fast computation while ensuring data privacy and verifying the correctness of the results.(2)Data partitioning-based outsourcing computation protocol.The protocol first partitions the coefficient matrix A of the linear equation system Ax(28)b into the original equation system to generate a homogeneous equation system A’x(28)b’.Afterwards,it performs a primary row transformation and then partitions the vector b’.Thus,the original equation system is blinded into two new linear equation systems.The user sends the blinded system of equations to two non-complicit cloud servers,and then verifies the returned results separately,ensuring the design goal of safe and efficient verification of outsourced computing.In addition,the protocol does not increase the computational complexity for the user compared to previous research results.(3)An outsourced computational protocol for solving a superdeterministic system of linear equations based on singular value decomposition and optimization theory.The computational protocol transforms the problem of solving the superdeterministic linear equation system into an optimization theory problem,blinding the coefficient matrix A of the superdeterministic linear equation system by real numbers greater than 0 and sparse triangular matrix,then verifying the singular value decomposition of the blinded matrix returned from the cloud,and finally solving the least squares solution of the superdeterministic equation system and unblinding it,which ensures the security,efficiency and verifiability of the outsourcing computation and achieves the outsourcing computation.The design goal of outsourcing computing is achieved.