Abstract
In this work, we design two-party and multiparty protocols for evaluating multivariate polynomials at participants’ inputs with security against a malicious adversary who may corrupt all but one of the parties. Our protocols are round and communication efficient, and use the underlying cryptographic primitives in a black-box way. Our construction achieves optimal communication complexity for degree 2 and 3 polynomials.
Our constructions can be used to securely and efficiently realize a wide range of functionalities. For instance, we demonstrate how our techniques lead to efficient protocols for secure linear algebra with security against malicious adversaries. Other applications include secure evaluation of DNF/CNF formulas, and conditional secret reconstruction (or conditional oblivious transfer) for a large family of condition functions.
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References
Aumann, Y., Lindell, Y.: Security against covert adversaries: Efficient protocols for realistic adversaries. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 137–156. Springer, Heidelberg (2007)
Benaloh, J.: Verifiable secret-ballot elections. Yale University, New Haven (1987)
Boneh, D., Goh, E., Nissim, K.: Evaluating 2-DNF formulas on ciphertexts. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 325–341. Springer, Heidelberg (2005)
Chen, H., Cramer, R., Goldwasser, S., de Haan, R., Vaikuntanathan, V.: Secure computation from random error correcting codes. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 291–310. Springer, Heidelberg (2007)
Dachman-Soled, D., Malkin, T., Raykova, M., Yung, M.: Efficient Robust Private Set Intersection. In: Abdalla, M., Pointcheval, D., Fouque, P.-A., Vergnaud, D. (eds.) ACNS 2009. LNCS, vol. 5536, p. 142. Springer, Heidelberg (2009)
Franklin, M.K., Yung, M.: Communication complexity of secure computation. In: Proc. of the 24th ACM Symp. on the Theory of Computing, pp. 699–710 (1992)
El Gamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 10–18. Springer, Heidelberg (1985)
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC, pp. 169–178 (2009)
Goldreich, O.: Foundations of Cryptography. Basic Applications, vol. II. Cambridge University Press, Cambridge (2004)
Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game. In: Proc. of the 19th ACM Symp. on the Theory of Computing, pp. 218–229 (1987)
Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity or all languages in NP have zero-knowledge proof systems. JACM, 691–729 (1991)
Goldwasser, S., Micali, S.: Probabilistic encryption & how to play mental poker keeping secret all partial information. In: STOC ’82, pp. 365–377 (1982)
Goyal, V., Mohassel, P., Smith, A.: Efficient two party and multi party computation against covert adversaries. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 289–306. Springer, Heidelberg (2008)
Guruswami, V., Rudra, A.: Error Correction Up to the Information-Theoretic Limit. Communications of the ACM 52(3), 87–95 (2009)
Ishai, Y., Kushilevitz, E.: Randomizing polynomials: A new representation with applications to round-efficient secure computation. In: Proc. of the 41st IEEE Symp. on Foundations of Computer Science, pp. 294–304 (2000)
Ishai, Y., Kushilevitz, E.: Perfect constant-round secure computation via perfect randomizing polynomials. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 244–256. Springer, Heidelberg (2002)
Ishai, Y., Kushilevitz, E., Lindell, Y., Petrank, E.: Black-box constructions for secure computation. In: STOC, pp. 99–108 (2006)
Ishai, Y., Prabhakaran, M., Sahai, A.: Founding cryptography on oblivious transfer - efficiently. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 572–591. Springer, Heidelberg (2008)
Ishai, Y., Prabhakaran, M., Sahai, A.: Secure arithmetic computation with no honest majority. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 294–314. Springer, Heidelberg (2009)
Kilian, J.: A note on efficient zero-knowledge proofs and arguments (extended abstract). In: Proceedings of STOCS, pp. 723–732 (1992)
Kiltz, E., Mohassel, P., Weinreb, E., Franklin, M.: Secure linear algebra using linearly recurrent sequences. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, p. 291. Springer, Heidelberg (2007)
Lindell, Y., Pinkas, B.: An Efficient Protocol for Secure Two-Party Computation in the Presence of Malicious Adversaries. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 52–78. Springer, Heidelberg (2007)
Mohassel, P., Franklin, M.K.: Efficiency tradeoffs for malicious two-party computation. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 458–473. Springer, Heidelberg (2006)
Mohassel, P., Weinreb, E.: Efficient secure linear algebra in the presence of covert or computationally unbounded adversaries. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 481–496. Springer, Heidelberg (2008)
Naor, M., Nissim, K.: Communication preserving protocols for secure function evaluation. In: Proc. of the 33rd ACM Symp. on the Theory of Computing (2001)
Nissim, K., Weinreb, E.: Communication efficient secure linear algebra. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 522–541. Springer, Heidelberg (2006)
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)
Welch, L.R., Berlekamp, E.R.: Error correction for algebraic block codes. US Patent 4,633,470, December 30 (1986)
Woodruff, D.P.: Revisiting the efficiency of malicious two-party computation. In: Naor, M. (ed.) EUROCRYPT 2007. LNCS, vol. 4515, pp. 79–96. Springer, Heidelberg (2007)
Yao, A.C.: How to generate and exchange secrets. In: Proc. of the 27th IEEE Symp. on Foundations of Computer Science, pp. 162–167 (1986)
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Franklin, M., Mohassel, P. (2010). Efficient and Secure Evaluation of Multivariate Polynomials and Applications. In: Zhou, J., Yung, M. (eds) Applied Cryptography and Network Security. ACNS 2010. Lecture Notes in Computer Science, vol 6123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13708-2_15
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DOI: https://doi.org/10.1007/978-3-642-13708-2_15
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