Subscribe

Issue No.06 - June (2013 vol.24)

pp: 1172-1181

Cong Wang , Illinois Institute of Technology, Chicago

Kui Ren , Illinois Institute of Technology, Chicago and The State University of New York at Buffalo

Jia Wang , Illinois Institute of Technology, Chicago

Qian Wang , Wuhan University, Wuhan and Illinois Institute of Technology, Chicago

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TPDS.2012.206

ABSTRACT

Cloud computing economically enables customers with limited computational resources to outsource large-scale computations to the cloud. However, how to protect customers' confidential data involved in the computations then becomes a major security concern. In this paper, we present a secure outsourcing mechanism for solving large-scale systems of linear equations (LE) in cloud. Because applying traditional approaches like Gaussian elimination or LU decomposition (aka. direct method) to such large-scale LEs would be prohibitively expensive, we build the secure LE outsourcing mechanism via a completely different approach—iterative method, which is much easier to implement in practice and only demands relatively simpler matrix-vector operations. Specifically, our mechanism enables a customer to securely harness the cloud for iteratively finding successive approximations to the LE solution, while keeping both the sensitive input and output of the computation private. For robust cheating detection, we further explore the algebraic property of matrix-vector operations and propose an efficient result verification mechanism, which allows the customer to verify all answers received from previous iterative approximations in one batch with high probability. Thorough security analysis and prototype experiments on Amazon EC2 demonstrate the validity and practicality of our proposed design.

INDEX TERMS

Outsourcing, Iterative methods, Equations, Cryptography, Vectors, Servers, Mathematical model, cloud computing, Confidential data, computation outsourcing, system of linear equations

CITATION

Cong Wang, Kui Ren, Jia Wang, Qian Wang, "Harnessing the Cloud for Securely Outsourcing Large-Scale Systems of Linear Equations",

*IEEE Transactions on Parallel & Distributed Systems*, vol.24, no. 6, pp. 1172-1181, June 2013, doi:10.1109/TPDS.2012.206REFERENCES

- [1] C. Wang, K. Ren, J. Wang, and K. Mahendra Raje Urs, "Harnessing the Cloud for Securely Solving Large-Scale Systems of Linear Equations,"
Proc. 31st Int'l Conf. Distributed Computing Systems (ICDCS), pp. 549-558, 2011.- [2] M. Armbrust et al., "A View of Cloud Computing,"
Comm. ACM, vol. 53, no. 4, pp. 50-58, Apr. 2010.- [3] Cloud Security Alliance, "Security Guidance for Critical Areas of Focus in Cloud Computing," http:/www.cloudsecurityalliance. org, 2009.
- [4] C. Gentry, "Computing Arbitrary Functions of Encrypted Data,"
Comm. ACM, vol. 53, no. 3, pp. 97-105, 2010.- [5] K. Forsman, W. Gropp, L. Kettunen, D. Levine, and J. Salonen, "Solution of Dense Systems of Linear Equations Arising from Integral-Equation Formulations,"
IEEE Antennas and Propagation Magazine, vol. 37, no. 6, pp. 96-100, Dec. 1995.- [6] A. Edelman, "Large Dense Numerical Linear Algebra in 1993: The Parallel Computing Influence,"
Int'l J. High Performance Computing Applications, vol. 7, no. 2, pp. 113-128, 1993.- [7] V. Prakash, S. Kwon, and R. Mittra, "An Efficient Solution of a Dense System of Linear Equations Arising in the Method-of-Moments Formulation,"
Microwave and Optical Technology Letters, vol. 33, no. 3, pp. 196-200, 2002.- [8] B. Carpentieri, "Sparse Preconditioners for Dense Linear Systems from Electromagnetic Applications," PhD dissertation, CERFACS, Toulouse, France, 2002.
- [9] R. Cramer and I. Damgård, "Secure Distributed Linear Algebra in a Constant Number of Rounds,"
CRYPTO: Proc. Ann. Int'l Cryptology Conf. Advances in Cryptology, 2001.- [10] K. Nissim and E. Weinreb, "Communication Efficient Secure Linear Algebra,"
Proc. Third Conf. Theory of Cryptography (TCC), pp. 522-541, 2006.- [11] E. Kiltz, P. Mohassel, E. Weinreb, and M.K. Franklin, "Secure Linear Algebra Using Linearly Recurrent Sequences,"
Proc. Fourth Conf. Theory of Cryptography (TCC), pp. 291-310, 2007.- [12] P. Mohassel and E. Weinreb, "Efficient Secure Linear Algebra in the Presence of Covert or Computationally Unbounded Adversaries,"
CRYPTO: Proc. 28th Ann. Int'l Cryptology Conf., pp. 481-496, 2008.- [13] J.R. Troncoso-Pastoriza, P. Comesaña, and F. Pérez-González, "Secure Direct and Iterative Protocols for Solving Systems of Linear Equations,"
Proc. First Int'l Workshop Signal Processing in the EncryptEd Domain (SPEED), pp. 122-141, 2009.- [14] W. Du and M.J. Atallah, "Privacy-Preserving Cooperative Scientific Computations,"
Proc. IEEE 14th Computer Security Foundations Workshop (CSFW), pp. 273-294, 2001.- [15] Y. Saad,
Iterative Methods for Sparse Linear Systems, second ed. Soc. for Industrial and Applied Math., 2003.- [16] P. Paillier, "Public-Key Cryptosystems Based on Composite Degree Residuosity Classes,"
EUROCRYPT: Proc. 17th Int'l Conf. Theory and Application of Cryptographic Techniques, pp. 223-238, 1999.- [17] Amazon.com, "Amazon Elastic Compute Cloud," http://aws. amazon.comec2/, 2009.
- [18] R. Gennaro, C. Gentry, and B. Parno, "Non-Interactive Verifiable Computing: Outsourcing Computation to Untrusted Workers,"
CRYPTO: Proc. 30th Ann. Conf. Advances in Cryptology , pp. 465-482, 2010.- [19] C. Wang, N. Cao, J. Li, K. Ren, and W. Lou, "Secure Ranked Keyword Search over Encrypted Cloud Data,"
Proc. IEEE 30th Int'l Conf. Distributed Computing Systems (ICDCS), pp. 253-262, 2010.- [20] S. Yu, C. Wang, K. Ren, and W. Lou, "Achieving Secure, Scalable, and Fine-Grained Access Control in Cloud Computing,"
Proc. IEEE INFOCOM, pp. 534-542, 2010.- [21] C. Wang, K. Ren, S. Yu, and K. Mahendra Raje Urs, "Achieving Usable and Privacy-Assured Similarity Search Over Outsourced Cloud Data,"
Proc. IEEE INFOCOM, pp. 451-459, 2012.- [22] T.H. Cormen, C.E. Leiserson, R.L. Rivest, and C. Stein,
Introduction to Algorithms, second ed. MIT press, 2008.- [23] D. Benjamin and M.J. Atallah, "Private and Cheating-Free Outsourcing of Algebraic Computations,"
Proc. Sixth Conf. Privacy, Security, and Trust (PST), pp. 240-245, 2008.- [24] M. Atallah and K. Frikken, "Securely Outsourcing Linear Algebra Computations,"
Proc. Fifth ACM Symp. Information, Computer and Comm. Security (ASIACCS), pp. 48-59, 2010.- [25] G. Dahlquist and A. Bjorck,
Numerical Methods. Dover Publications, 2003.- [26] M. Bellare, J. Garay, and T. Rabin, "Fast Batch Verification for Modular Exponentiation and Digital Signatures,"
Eurocrypt: Proc. Int'l Conf. the Theory and Application of Cryptographic Techniques, pp. 236-250, 1998.- [27] J. Camenisch, S. Hohenberger, and M. Pedersen, "Batch Verification of Short Signatures,"
EUROCRYPT: Proc. 26th Ann. Int'l Conf. Advances in Cryptology, pp. 243-263, 2007.- [28] J. Bethencourt, D.X. Song, and B. Waters, "New Techniques for Private Stream Searching,"
ACM Trans. Information Systems Security, vol. 12, no. 3,article 16, 2009.- [29] S. Han, W.K. Ng, L. Wan, and V.C. Lee, "Privacy-Preserving Gradient-Descent Methods,"
IEEE Trans. Knowledge and Data Eng., vol. 22, no. 6, pp. 884-899, June 2010.- [30] A.C.-C. Yao, "Protocols for Secure Computations (Extended Abstract),"
Proc. IEEE 23rd Symp. Foundations of Computer Science (FOCS), pp. 160-164, 1982.- [31] C. Gentry, "Fully Homomorphic Encryption Using Ideal Lattices,"
Proc. 41st Ann. ACM Symp. Theory of computing (STOC), pp. 169-178, 2009.- [32] M.J. Atallah, K.N. Pantazopoulos, J.R. Rice, and E.H. Spafford, "Secure Outsourcing of Scientific Computations,"
Advances in Computers, vol. 54, pp. 216-272, 2001.- [33] C. Wang, K. Ren, and J. Wang, "Secure and Practical Outsourcing of Linear Programming in Cloud Computing,"
Proc. IEEE INFOCOM, pp. 820-828, 2011.- [34] M. Blanton, Y. Zhang, and K.B. Frikken, "Secure and Verifiable Outsourcing of Large-Scale Biometric Computations,"
Proc. IEEE Third Int'l Conf. Privacy, Security, Risk, and Trust (PASSAT), pp. 1185-1191, 2011.- [35] C. Orlandi, A. Piva, and M. Barni, "Oblivious Neural Network Computing via Homomorphic Encryption,"
EURASIP J. Information Security, vol. 2007, pp. 1-10, 2007.- [36] S. Han and W.K. Ng, "Privacy-Preserving Linear Fisher Discriminant Analysis,"
Proc. 12th Pacific-Asia Conf. Advances in Knowledge Discovery and Data Mining, pp. 136-147, 2008.- [37] S. Goldwasser, Y.T. Kalai, and G.N. Rothblum, "Delegating Computation: Interactive Proofs for Muggles,"
Proc. ACM Symp. Theory of Computing (STOC), pp. 113-122, 2008.- [38] P. Golle and I. Mironov, "Uncheatable Distributed Computations,"
CT-RSA: Proc. Conf. Topics in Cryptology: The Cryptographer's Track at RSA , pp. 425-440, 2001.- [39] D. Szajda, B.G. Lawson, and J. Owen, "Hardening Functions for Large Scale Distributed Computations,"
Proc. IEEE Symp. Security and Privacy, pp. 216-224, 2003.- [40] W. Du, J. Jia, M. Mangal, and M. Murugesan, "Uncheatable Grid Computing,"
Proc. 24th Int'l Conf. Distributed Computing Systems (ICDCS), pp. 4-11, 2004. |