Research On Secure Outsourcing Algorithms For Large-scale Matrix Operations

With the recent growth and commercialization of cloud computing,outsourcing computation has become one of the most important cloud services,which allows the re-source-constrained clients to outsource large-scale computation to the cloud.Meanwhile,outsourcing large scale computing problems and computationally intensive applications to the cloud has become prevalent in the science and engineering computing community.Large-scale matrix multiplication computation(MMC),matrix inversion computation(MIC),matrix determinant computation(MDC),and matrix eigen decomposition(MED)as important fundamental operations,its algorithm design and analysis of outsourcing in the background of a cloud environment,have been widely studied and frequently used of academic and industry community.However,how to efficiently implement outsourcing data privacy protection problem is becoming more and more attention.In many applica-tions,the number of zero elements often contain sensitive information,but the existing scheme to protect the number of zero elements study is less,only consider the location of the hidden zero element,therefore,the design is to ensure the safety of the zero element information and can achieve high efficiency outsourcing algorithm has important theoret-ical significance and application value.To solve this problem,this article has mainly done the following work:(1)A simple and novel matrix encryption method is proposed,and based on this method,the outsourcing algorithms of MMC,MIC and MDC on the security outsourcing finite field are designed respectively.Firstly,the position information of input matrix el-ements is hidden by random permutation,then the value information of input matrix ele-ments is hidden by unimodular matrix transformation,and the encrypted matrix is sent to the cloud server.The cloud performs the corresponding operation and returns the result to the client,which then decrypts and verifies the result.Through the strict theoretical analy-sis,the three algorithms not only protect the number of zero elements in the original ma-trix,but also achieve the purpose of correctness and verifiability.Finally,the effectiveness of the scheme is verified by extensive experimental simulation.(2)Aiming at the problem of insufficient confusion of input matrix information in scheme(1),the proposed MMC,MIC and MDC outsourcing algorithms over finite fields were further improved.The main idea is to use continuous sparse unimodular matrix transformation to achieve dense matrix transformation encryption effect.Compared with the previous schemes,this kind of transformation confuses the input matrix elements suf-ficiently,improves the security and reliability of the scheme,and guarantees the efficien-cy of the scheme with the combinability of matrix multiplication.In addition,this tech-nique has a wide range of applications and has potential applications to other matrix op-erations.Our theoretical analysis indicates that the proposed algorithms reduce the time overhead on the client side from O(7)n ~3(8)to O(7)n ~2(8).Finally,the extensive experimental evaluations demonstrate the actual performance of the algorithm.(3)Aiming at the problem that the zero-element information of the input matrix is easy to be disclosed in the MED-outsourcing scheme over the real field proposed by Zhou et al.,we improved the proposed MED algorithms.Due to the product of the continuous sparse unimodular matrix of scheme(2)generates "explosive" growth in the real domain,the improved scheme adopted a single unimodular matrix transformation technology similar to scheme(1)to protect the number of input matrix zero elements,and theoretically strict prove the validity of the improved scheme,the input/output privacy,verifiability and efficiency.