Abstract
In this paper, we study the system of linear equation problems in the two-party computation setting. Consider that P 1 holds an m × m matrix M 1 and an m-dimensional column vector B 1. Similarly, P 2 holds M 2 and B 2. Via executing a secure linear system computation, P 1 gets the output x (or ⊥) conditioned on (M 1 + M 2)x = (B 1 + B 2), and the rank of matrix M 1 + M 2, while P 2 gets nothing. This also can be used to settle other cooperative linear system problems. We firstly design an efficient protocol to solve this problem in the presence of malicious adversaries, then propose a simple way to modify our protocol for having a precise functionality, in which the rank of matrix M 1+M 2 is not necessary. We note that our protocol is more practical than these existing malicious secure protocols. We also give comparisons with other protocols and extensions to similar functions.
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References
Yao A C. How to generate and exchange secrets. Foundations of Computer Science. Canada: IEEE, 1986. 162–167
Goldreich O, Micali S, Wigderson A. How to play any mental game. In: STOC 1987. 218–229
Ben-Or M, Goldwasser S, Wigderson A. Completeness theorems for non-cryptographic fault-tolerent distributed computation. In: STOC 1988. 1–10
Chaum D, Crepeau C, Damgård I. Multiparty unconditionally secure protocols. In: STOC, Santa Barbara, 1988. 11–19
Noar M, Pinkas B. Efficient oblivious transfer protocols. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, 2001. 448–457
Aumann Y, Lindell Y. Security against covert adversaries: Efficient protocols for realistic adversaries. In: Theory of Cryptography Conference, the Netherlands. Berlin/Heidelberg: Springer, 2007. 137–156
Malkhi D, Nisan N, Pinkas B, et al. Fairplay—a secure two-party computation system. In: USENIX Security Symposium, San Diego, 2004. 287–302
Lindell Y, Pinkas B. Secure two-party computation via cut-and-choose oblivious transfer. J Crypt, 2012, 25: 680–722. Full version in Cryptology ePrint Archive, report 2010/284
Shelat A, Shen C. Two-outputs secure computation with malicious adversaries. In: EUROCRYPT 2011, UK. Berlin/Heidelberg: Springer, 2011. 386–405
Cramer R, Damgård I. Secure distributed linear algebra in a constant number of rounds. In: CRYPTO 2001, USA. Berlin/Heidelberg: Springer, 2001. 119–136
Cramer R, Kiltz E, Padro C. A note on secure computation of the moore-penrose pseudoinverse and its application to secure linear algebra. In: CRYPTO 2007, USA. Berlin/Heidelberg: Springer, 2007. 613–630
Nissim K, Weinreb E. Communication efficient secure linear algerbra. In: Theory of Cryptography, USA. Berlin/Heidelberg: Springer, 2006. 522–541
Kiltz E, Mohassel P, Weinreb E, et al. Secure linear algebra using linearly recurrent sequences. In: Theory of Cryptography, USA. Berlin/Heidelberg: Springer, 2007. 291–310
Du W, Atallah M J. Privacy-preserving cooperative scientific computations. Computer Security Foundations Workshop, IEEE, Canada, 2001. 0273–0273
Mohassel P, Weinreb E. Efficient secure linear algebra in the presence of covert or computationally unbounded adversaries. In: CRYPTO 2008, USA. Berlin/Heidelberg: Springer, 2008. 481–496
Jarecki S, Shmatikov V. Efficient two-party secure computation on committed inputs. In: EUROCRYPT 2007, Spain. Berlin/Heidelberg: Springer, 2007. 97–114
Coppersmith D, Winograd S. Matrix multiplication via arithmetic progressions. J Symb Comp, 1990, 9: 251–280
Ishai Y, Prabhakaran M, Sahai A. Secure arithmetic computation with no honest majority. In: Theory of Cryptography, USA. Berlin/Heidelberg: Springer, 2009. 294–314
Peikert C, Vaikuntanathan V, Waters B. A framework for efficient and composable oblivious transfer. In: CRYPTO 2008, USA. Berlin/Heidelberg: Springer, 2008. 554–571
Goldreich O. Basic application. Foundations of Cryptography, vol.2. Cambridge: Cambridge University Press, 2004
Canetti R. Security and composition of multiparty cryptographic protocols. J Crypt, 2000, 13: 143–202
Pedersen T P. Non-interactive and information-theoretic secure verifiable secret sharing. In: CRYPTO 1992, USA. Berlin/Heidelberg: Springer, 1992. 129–140
Chaum D, Pedersen T P. Wallet databases with observers. In: CRYPTO 1993, USA. Berlin/Heidelberg, 1993. 89–105
Jacobson N. Basic Algebra II, 2nd ed. New York: W.H. Freeman and Company, 1985
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Zhang, B., Zhang, F. Secure linear system computation in the presence of malicious adversaries. Sci. China Inf. Sci. 57, 1–10 (2014). https://doi.org/10.1007/s11432-014-5160-2
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DOI: https://doi.org/10.1007/s11432-014-5160-2