Abstract
Solving linear programming (LP) problems can be used to solve many different types of problems. Immediate examples include certain types of auctions as well as benchmarking. However, the input data may originate from different, mistrusting sources, which implies the need for a privacy preserving solution.
We present a protocol solving this problem using black-box access to secure modulo arithmetic. The solution can be instantiated in various settings: Adversaries may be both active and adaptive, but passive and/or static ones can be employed, e.g. for efficiency reasons. Perfect security can be obtained in the information theoretic setting (up to 1/3 corruptions), while corruption-of-all-but-one is possible in the cryptographic setting. The latter allows a two-party protocol.
The solution is based on the well known simplex method. Letting n denote the number of initial variables and m the number of constraints, each pivot requires only \(\mathcal{O}({\rm loglog}(m))\) rounds in which \(\mathcal{O}(m(m+ n))\) multiplication protocols and \(\mathcal{O}(m+n)\) comparison protocols are invoked; this is equivalent to the base-algorithm. A constant-rounds variation is also possible, this increases the number of comparisons to \(\mathcal{O}(m^2+n)\).
Work partially performed at Aarhus University. Supported by the research program Sentinels (http://www.sentinels.nl). Sentinels is being financed by Technology Foundation STW, the Netherlands Organization for Scientific Research (NWO), and the Dutch Ministry of Economic Affairs.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bar-Ilan, J., Beaver, D.: Non-cryptographic fault-tolerant computing in a constant number of rounds of interaction. In: Rudnicki, P. (ed.) Proceedings of the eighth annual ACM Symposium on Principles of distributed computing, pp. 201–209. ACM Press, New York (1989)
Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for noncryptographic fault-tolerant distributed computations. In: 20th Annual ACM Symposium on Theory of Computing, pp. 1–10. ACM Press, New York (1988)
Bogetoft, P., Nielsen, K.: Dea based auctions. European Journal of Operational Research 184(2), 685–700 (2008)
Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols. In: 20th Annual ACM Symposium on Theory of Computing, pp. 11–19. ACM Press, New York (1988)
Cramer, R., Damgård, I., Nielsen, J.: Multiparty computation from threshold homomorphic encryption. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 280–300. Springer, Heidelberg (2001)
Chvátal, V.: Linear Programming. W.H. Freeman, New York (1983)
Damgård, I., Fitzi, M., Kiltz, E., Nielsen, J., Toft, T.: Unconditionally secure constant-rounds multi-party computation for equality, comparison, bits and exponentiation. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 285–304. Springer, Heidelberg (2006)
Damgård, I., Jurik, M.: A generalization, a simplification and some applications of Paillier’s probabilistic public-key system. In: Kim, K.-c. (ed.) PKC 2001. LNCS, vol. 1992, pp. 110–136. Springer, Heidelberg (2001)
Damgård, I., Nielsen, J.: Universally composable efficient multiparty computation from threshold homomorphic encryption. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 247–264. Springer, Heidelberg (2003)
Feigenbaum, J., Ishai, Y., Malkin, T., Nissim, K., Strauss, M., Wright, R.: Secure multiparty computation of approximations. ACM Transactions on Algorithms 2(3), 435–472 (2006)
Fouque, P., Stern, J., Wackers, G.: CryptoComputing with rationals. In: Financial Cryptography 2002. LNCS. Springer, Berlin (2002)
Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game. In: STOC 1987: Proceedings of the nineteenth annual ACM conference on Theory of computing, pp. 218–229. ACM Press, New York (1987)
Goemans, M.: Linear programming. Course notes (October 1994), http://www-math.mit.edu/~goemans/notes-lp.ps
Jájá, J.: An Introduction to Parallel Algorithms. Addison-Wesley, Reading (1992)
Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)
Khachiyan, L.: A polynomial algorithm in linear programming. Soviet Mathematics Doklady 20 (1979)
Li, J., Atallah, M.: Secure and private collaborative linear programming. In: Collaborative Computing: Networking, Applications and Worksharing, 2006. CollaborateCom (2006)
Nishide, T., Ohta, K.: Multiparty computation for interval, equality, and comparison without bit-decomposition protocol. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 343–360. Springer, Heidelberg (2007)
Nielsen, K., Toft, T.: Secure relative performance scheme. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 396–403. Springer, Heidelberg (2007)
Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999)
Rosenberg, G.: Enumeration of all extreme equlibria of bimatrix games with integer pivoting and improved degeneracy check, CDAM Research Report LSE-CDAM-2005-18 (2005), http://www.cdam.lse.ac.uk/Reports/Abstracts/cdam-2005-18.html
Reistad, T., Toft, T.: Secret sharing comparison by transformation and rotation. In: Proceedings of the International Conference on Information Theoretic Security (ICITS) 2007. LNCS. Springer, Heidelberg (2007) (to appear)
Silaghi, M., Faltings, B., Petcu, A.: Secure combinatorial optimization simulating dfs tree-based variable elimination. In: AI and Math 2006 Proceedings (2006), http://anytime.cs.umass.edu/aimath06/proceedings.html
Shamir, A.: How to share a secret. Communications of the ACM 22(11), 612–613 (1979)
Silaghi, M.: A suite of secure multi-party computation algorithms for solving distributed constraint satisfaction and optimization problems. Technical Report CS-2004-04, Florida Institute of Technology (2004)
Yao, A.: How to generate and exchange secrets. In: Proceedings of the 27th IEEE Symposium on Foundations of Computer Science, pp. 162–167 (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Toft, T. (2009). Solving Linear Programs Using Multiparty Computation. In: Dingledine, R., Golle, P. (eds) Financial Cryptography and Data Security. FC 2009. Lecture Notes in Computer Science, vol 5628. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03549-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-03549-4_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03548-7
Online ISBN: 978-3-642-03549-4
eBook Packages: Computer ScienceComputer Science (R0)