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

  • Shiran Pan
  • Wen-Tao ZhuEmail author
  • 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)

Abstract

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.

Keywords

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

Notes

Acknowledgment

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.

References

  1. 1.
    Baldwin, M.P., Stephenson, D., Thompson, D.W., Dunkerton, T.J., Charlton, A.J., O’Neill, A.: Stratospheric memory and skill of extended-range weather forecasts. Science 301, 636–640 (2003)CrossRefGoogle Scholar
  2. 2.
    So, H.C., Lin, L.: Linear least squares approach for accurate received signal strength based source localization. IEEE Trans. Signal Process. 59, 4035–4040 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Lobos, T., Kozina, T., Koglin, H.J.: Power system harmonics estimation using linear least squares method and SVD. IEE Proc. Gener. Transm. Distrib. 148, 567–572 (1999)CrossRefGoogle Scholar
  4. 4.
    Bjorck, A.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)CrossRefGoogle Scholar
  5. 5.
    Edelman, A.: Large dense numerical linear algebra in 1993: the parallel computing influence. Int. J. High Perform. Comput. Appl. 7, 113–128 (1993)Google Scholar
  6. 6.
    Ren, K., Wang, C., Wang, Q.: Security challenges for the public cloud. IEEE Internet Comput. 16, 69–73 (2012)CrossRefGoogle Scholar
  7. 7.
    Chen, X., Huang, X., Li, J., Ma, J., Lou, W., Wong, D.S.: New algorithms for secure outsourcing of large-scale systems of linear equations. IEEE Trans. Inf. Forensics Secur. 10, 69–78 (2015)CrossRefGoogle Scholar
  8. 8.
    Pan, S., Zheng, F., Zhu, W.-T., Wang, Q.: Harnessing the cloud for secure and efficient outsourcing of non-negative matrix factorization. In: the 6th IEEE Conference on Communications and Network Security (CNS 2018). IEEE (2018)Google Scholar
  9. 9.
    Salinas, S., Luo, C., Chen, X., Li, P.: Efficient secure outsourcing of large-scale linear systems of equations. In: 2015 IEEE Conference on Computer Communications (INFOCOM 2015), pp. 1035–1043. IEEE (2015)Google Scholar
  10. 10.
    Yu, Y., Luo, Y., Wang, D., Fu, S., Xu, M.: Efficient, secure and non-iterative outsourcing of large-scale systems of linear equations. In: 2016 IEEE International Conference on Communications (ICC 2016), p. 6. IEEE (2016)Google Scholar
  11. 11.
    Lawson, C.L., Hanson, R.J.: Solving Least Squares Problems. SIAM, Philadelphia (1974)zbMATHGoogle Scholar
  12. 12.
    Demmel, J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)CrossRefGoogle Scholar
  13. 13.
    Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. CM, vol. 89. Springer, Heidelberg (2001).  https://doi.org/10.1007/978-3-642-61896-3CrossRefzbMATHGoogle Scholar
  14. 14.
    Wang, C., Ren, K., Wang, J., Wang, Q.: Harnessing the cloud for securely outsourcing large-scale systems of linear equations. IEEE Trans. Parallel Distrib. Syst. 24, 1172–1181 (2013)CrossRefGoogle Scholar
  15. 15.
    Wang, C., Ren, K., Wang, K., Urs, K.M.R.: Harnessing the cloud for securely solving large-scale systems of linear equations. In: 31st IEEE International Conference on Distributed Computing Systems (ICDCS 2011), pp. 549–558. IEEE (2011)Google Scholar
  16. 16.
    Atallah, M.J., Pantazopoulos, K.N., Rice, J.R., Spafford, E.E.: Secure outsourcing of scientific computations. Adv. Comput. 54, 215–272 (2002)CrossRefGoogle Scholar
  17. 17.
    Lei, X., Liao, X., Huang, T., Li, H., Hu, C.: Outsourcing large matrix inversion computation to a public cloud. IEEE Trans. Cloud Comput. 1, 78–87 (2013)Google Scholar
  18. 18.
    Zhou, L., Li, C.: Outsourcing eigen-decomposition and singular value decomposition of large matrix to a public cloud. IEEE ACCESS 4, 869–879 (2016)CrossRefGoogle Scholar
  19. 19.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  20. 20.
    Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409–436 (1952)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Plemmons, R.J.: FFT-based RLS in signal processing. In: 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 1993), pp. 571–574 (1993)Google Scholar
  22. 22.
    Solomon, C., Breckon, T.: Fundamentals of Digital Image Processing: A Practical Approach with Examples in MATLAB. Wiley-Blackwell, Hoboken (2011)Google Scholar
  23. 23.
    Smith, C.F., Peterson, A.F., Mittra, R.: A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields. IEEE Trans. Antennas Propag. 37, 1490–1493 (1989)CrossRefGoogle Scholar
  24. 24.
    Lei, X., Liao, X., Huang, T., Li, H.: Cloud computing service: the case of large matrix determinant computation. IEEE Trans. Serv. Comput. 8, 688–700 (2015)CrossRefGoogle Scholar
  25. 25.
    Salinas, S., Luo, C., Liao, W., Li, P.: Efficient secure outsourcing of large-scale quadratic programs. In: 11th ACM on Asia Conference on Computer and Communications Security (ASIA CCS 2016), pp. 281–292. ACM (2016)Google Scholar

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
    Email author
  • 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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