Abstract
Secure computation is one of the most fundamental cryptographic tasks. It is known that all functions can be computed securely in the information theoretic setting, given access to a black box for some complete function such as AND. However, without such a black box, not all functions can be securely computed. This gives rise to two types of functions, those that can be computed without a black box (“easy”) and those that cannot (“hard”). However, no further distinction among the hard functions is made.
In this paper, we take a quantitative approach, associating with each function f the minimal number of calls to the black box that are required for securely computing f. Such an approach was taken before, mostly in an ad-hoc manner, for specific functions f of interest. We propose a systematic study, towards a general characterization of the hierarchy according to the number of black-box calls. This approach leads to a better understanding of the inherent complexity for securely computing a given function f. Furthermore, minimizing the number of calls to the black box can lead to more efficient protocols when the calls to the black box are replaced by a secure protocol.
We take a first step in this study, by considering the two-party, honest-but-curious, information-theoretic case. For this setting, we provide a complete characterization for deterministic protocols. We explore the hierarchy for randomized protocols as well, giving upper and lower bounds, and comparing it to the deterministic hierarchy. We show that for every Boolean function the largest gap between randomized and deterministic protocols is at most exponential, and there are functions which exhibit such a gap.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Beaver, D.: Perfect privacy for two-party protocols. Technical Report TR-11-89, Computer Science, Harvard University (1989)
Beaver, D.: Correlated pseudorandomness and the complexity of private computations. In: The 28th Symp. on the Theory of Computing, pp. 479–488 (1996)
Beimel, A., Malkin, T.G., Micali, S.: The all-or-nothing nature of two-party secure computation. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 80–97. Springer, Heidelberg (1999)
Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for noncryptographic fault-tolerant distributed computations. In: The 20th Symp. on the Theory of Computing, pp. 1–10 (1988)
Brassard, G., Crépeau, C.: Oblivious transfers and privacy amplification. In: Fumy, W. (ed.) EUROCRYPT 1997. LNCS, vol. 1233, pp. 334–347. Springer, Heidelberg (1997)
Brassard, G., Crépeau, C., Robert, J.-M.: Information theoretic reductions among disclosure problems. In: The 27th Symp. on Foundations of Computer Science, pp. 168–173 (1986)
Brassard, G., Crépeau, C., Sántha, M.: Oblivious transfers and intersecting codes. IEEE Trans. on Information Theory 42(6), 1769–1780 (1996)
Canetti, R.: Security and composition of multiparty cryptographic protocols. J. of Cryptology 13(1), 143–202 (2000)
Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols. In: The 20th Symp. on the Theory of Computing, pp. 11–19 (1988)
Chor, B., Kushilevitz, E.: A zero-one law for Boolean privacy. SIAM J. on Discrete Mathematics 4(1), 36–47 (1991)
Crépeau, C.: Equivalence between two flavours of oblivious transfers. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 350–354. Springer, Heidelberg (1988)
Crépeau, C., Kilian, J.: Achieving oblivious transfer using weakened security assumptions. In: 29th Symp. on Found. of Computer Science, pp. 42–52 (1988)
Damgård, I.B., Kilian, J., Salvail, L.: On the (im)possibility of basing oblivious transfer and bit commitment on weakened security assumptions. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 56–73. Springer, Heidelberg (1999)
Dodis, Y., Micali, S.: Lower bounds for oblivious transfer reductions. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 42–55. Springer, Heidelberg (1999)
Forster, J.: A linear lower bound on the unbounded error probabilistic communication complexity. In: 16th Conf. on Comput. Complexity, pp. 100–106 (2001)
Goldreich, O., Vainish, R.: How to solve any protocol problem—an efficiency improvement. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 73–86. Springer, Heidelberg (1988)
Ishai, Y., Kilian, J., Nissim, K., Petrank, E.: Extending oblivious transfers efficiently. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 145–161. Springer, Heidelberg (2003)
Kilian, J.: Basing cryptography on oblivious transfer. In: Proc. of the 20th Symp. on the Theory of Computing, pp. 20–31 (1988)
Kilian, J.: A general completeness theorem for two-party games. In: Proc. of the 23rd Symp. on the Theory of Computing, pp. 553–560 (1991)
Kilian, J.: More general completeness theorems for two-party games. In: Proc. of the 32nd Symp. on the Theory of Computing, pp. 316–324 (2000)
Kilian, J., Kushilevitz, E., Micali, S., Ostrovsky, R.: Reducibility and completeness in private computations. SIAM J. on Computing 28(4), 1189–1208 (2000)
Kushilevitz, E.: Privacy and communication complexity. SIAM J. on Discrete Mathematics 5(2), 273–284 (1992)
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)
Maurer, U.: Information-theoretic cryptography. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 47–64. Springer, Heidelberg (1999)
Naor, M., Nissim, K.: Communication preserving protocols for secure function evaluation. In: Proc. of the 33rd Symp. on the Theory of Computing (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Beimel, A., Malkin, T. (2004). A Quantitative Approach to Reductions in Secure Computation. In: Naor, M. (eds) Theory of Cryptography. TCC 2004. Lecture Notes in Computer Science, vol 2951. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24638-1_14
Download citation
DOI: https://doi.org/10.1007/978-3-540-24638-1_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21000-9
Online ISBN: 978-3-540-24638-1
eBook Packages: Springer Book Archive