Abstract
We consider a generalized adaptive and active adversary model for unconditionally secure Multi-Party Computation (MPC) in the zero error case.
Cramer et al. proposed a generic approach to build a multiplicative Monotone Span Programs (MSP) – the special property of a Linear Secret Sharing Schemes (LSSS) that is needed to perform a multiplication of shared values. They give an efficient generic construction to build verifiability into every LSSS and to obtain from any LSSS a multiplicative LSSS for the same access structure. But the multiplicative property guarantees security against passive adversary only. For an active adversary a strong multiplicative property is required. Unfortunately there is no known efficient construction to obtain a strongly multiplicative LSSS yet.
Recently Nikov et al. have expanded the construction of Cramer et al. using a different approach. Multiplying two different MSP M 1 and M 2 computing the access structures Γ1 and Γ2 a new MSP M called “resulting” is obtained. M computes a new access structure Γ ⊂ Γ1 (or Γ2). The goal of this construction is to enable the investigation of how the properties that Γ should fulfil are linked to the initial access structures Γ1 and Γ2. It is proved that Γ2 should be a dual access structure of Γ1 in order to have a multiplicative resulting MSP. But there are still not known requirements for initial access structures in order to obtain strongly multiplicative resulting MSP. Nikov et al. proved that to have unconditionally secure MPC the following minimal conditions for the resulting access structure should be satisfied \((\Gamma_{A}\uplus \Gamma_{A})^{\bot}\subseteq \Gamma\).
In this paper we assume that the resulting MSP could be constructed such that the corresponding access structure Γ satisfies the required properties. Our goal is to study the requirements that Γ should fulfil in order to have an MPC unconditionally secure against adaptive and active adversary in the zero error case. First, we prove that Γ could satisfy weaker conditions than those in Nikov et al., namely \(\Gamma^{\bot}_{A}\subseteq \Gamma\). Second, we propose a commitment “degree reduction” protocol which allows the players to “reduce” one access structure, e.g. Γ, to another access structure Γ3. This reduction protocol appears to be a generalization of the reduction protocol of Cramer et al. in the sense that we can choose to reduce Γ to the initial access structures Γ1 or Γ2, or to a new one Γ3. This protocol is also more efficient, since it requires less Verifiable Secret Sharing Schemes to be used.
Chapter PDF
Similar content being viewed by others
Keywords
References
Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness Theorems for Non- Cryptographic Fault-Tolerant Distributed Computation. In: Proc. ACM STOC 1988, pp. 1–10 (1988)
Chaum, D., Crepeau, C., Damgard, I.: Multi-Party Unconditionally Secure Protocols. In: Proc. ACM STOC 1988, pp. 11–19 (1988)
Cramer, R.: Introduction to Secure Computation. In: Damgård, I.B. (ed.) EEF School 1998. LNCS, vol. 1561, pp. 16–62. Springer, Heidelberg (1999)
Cramer, R., Damgard, I., Maurer, U.: General Secure Multi-Party Computation from any Linear Secret Sharing Scheme. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 316–334. Springer, Heidelberg (2000)
Cramer, R., Fehr, S.: Optimal Black-Box Secret Sharing over Arbitrary Abelian Groups. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 272–287. Springer, Heidelberg (2002)
Chor, B., et al.: Verifiable Secret Sharing and Achieving Simultaneity in the Presence of Faults. In: Proc. of the IEEE 26th Annual Symp. on Foundations of Computer Science, pp. 383–395 (1985)
Damgard, I.: An Error in the Mixed Adversary Protocol by Fitzi, Hirt and Maurer, Bricks Report, RS-99-2 (1999)
Fehr, S., Maurer, U.: Linear VSS and distributed commitments based on secret sharing and pairwise checks. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 565–580. Springer, Heidelberg (2002)
Fitzi, M., Hirt, M., Maurer, U.: Trading Correctness for Privacy in Unconditional Multi-Party Computation. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 121–136. Springer, Heidelberg (1998)
Gennaro, R., Rabin, M., Rabin, T.: Simplified VSS and Fast-Track Multi-party Computations with Applications to Threshold Cryptography. In: Proc. ACM PODC 1998 (1998)
Goldreich, O., Micali, S., Wigderson, A.: How to Play Any Mental Game or a Completeness Theorem for Protocols with Honest Majority. In: Proc. ACM STOC, 1987, pp. 218–229 (1987)
Hirt, M., Maurer, U.: Player Simulation and General Adversary Structures in Perfect Multi-party Computation. J. of Cryptology 13, 31–60 (2000)
Karchmer, M., Wigderson, A.: On Span Programs. In: Proc. of 8-th Annual Structure in Complexity Theory Conference, San Diego, California, May 18-21, pp. 102–111. IEEE Computer Society Press, Los Alamitos (1993)
Maurer, U.: Secure Multi-Party Computation Made Simple. In: Cimato, S., Galdi, C., Persiano, G. (eds.) SCN 2002. LNCS, vol. 2576, pp. 14–28. Springer, Heidelberg (2003)
Nikov, V., Nikova, S., Preneel, B., Vandewalle, J.: Applying General Access Structure to Proactive Secret Sharing Schemes. In: Proc. of the 23rd Symposium on Information Theory in the Benelux, May 29-31, pp. 197–206. Universite Catolique de Lovain (UCL), Lovain-la-Neuve, Belgium (2002)
Nikov, V., et al.: On Distributed Key Distribution Centers and Unconditionally Secure Proactive Verifiable Secret Sharing Schemes based on General Access Structure. In: Menezes, A., Sarkar, P. (eds.) INDOCRYPT 2002. LNCS, vol. 2551, pp. 422–437. Springer, Heidelberg (2002)
Nikov, V., Nikova, S., Preneel, B.: Multi-Party Computation from any Linear Secret Sharing Scheme Secure against Adaptive Adversary: The Zero-Error Case, Cryptology ePrint Archive: Report 2003/006
Shamir, A.: How to share a secret. Commun. ACM 22, 612–613 (1979)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nikov, V., Nikova, S., Preneel, B. (2003). Multi-party Computation from Any Linear Secret Sharing Scheme Unconditionally Secure against Adaptive Adversary: The Zero-Error Case. In: Zhou, J., Yung, M., Han, Y. (eds) Applied Cryptography and Network Security. ACNS 2003. Lecture Notes in Computer Science, vol 2846. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45203-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-540-45203-4_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20208-0
Online ISBN: 978-3-540-45203-4
eBook Packages: Springer Book Archive