Secure and Efficient Outsourcing of Large-Scale Overdetermined Systems of Linear Equations

  • Shiran Pan
  • Wen-Tao Zhu
  • Qiongxiao Wang
  • Bing Chang
Conference paper
Part of the Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering book series (LNICST, volume 254)


We address overdetermined systems of linear equations, where the number of unknowns is smaller than the number of equations so that only approximate solutions exist instead of exact solutions. Such systems are prevalent in many areas of science and engineering, and finding the optimal solutions is mathematically known as the linear least squares (LLS) problem. Real-world overdetermined systems are often large-scale and computationally expensive to solve. Consequently, we are interested in connecting the LLS problem with cloud computing, where a resource-constrained client outsources the problem to a powerful but untrusted cloud. Among several security considerations is that the input of and solution to the LLS problem usually contain the client’s private information, which necessitates privacy-preserving outsourcing. In this paper, we present a construction called Sells, which employs a mathematical method called QR decomposition to solve the above problem, in a masked yet verifiable manner. One advantage of adopting QR decomposition is that in certain circumstances, solving a batch of LLS problems only requires fully executing Sells once, where certain intermediate result can be reused and the overall efficiency is greatly improved. Theoretical analysis shows that our proposal is verifiable, recoverable, and privacy-preserving. Experiments demonstrate that a client can benefit from the scheme not only reduced computation cost but also accelerated problem solving.


Linear equations Overdetermined system Linear least squares Cloud computing Verifiable outsourcing Privacy preserving 



The authors would like to thank the anonymous reviewers for their valuable comments. This work was supported by the National Basic Research Program of China (973 Program) under Grant 2014CB340603.


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Copyright information

© ICST Institute for Computer Sciences, Social Informatics and Telecommunications Engineering 2018

Authors and Affiliations

  • Shiran Pan
    • 1
    • 2
    • 3
  • Wen-Tao Zhu
    • 2
  • Qiongxiao Wang
    • 1
    • 2
  • Bing Chang
    • 4
  1. 1.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina
  2. 2.Data Assurance and Communication Security Research CenterChinese Academy of SciencesBeijingChina
  3. 3.School of Cyber SecurityUniversity of Chinese Academy of SciencesBeijingChina
  4. 4.School of Information SystemsSingapore Management UniversitySingaporeSingapore

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