Abstract
We show that verifiable secret sharing (VSS) and secure multi-party computation (MPC) among a set of n players can efficiently be based on any linear secret sharing scheme (LSSS) for the players, provided that the access structure of the LSSS allows MPC or VSS at all. Because an LSSS neither guarantees reconstructability when some shares are false, nor verifiability of a shared value, nor allows for the multiplication of shared values, an LSSS is an apparently much weaker primitive than VSS or MPC.
Our approach to secure MPC is generic and applies to both the information-theoretic and the cryptographic setting. The construction is based on 1) a formalization of the special multiplicative property of an LSSS that is needed to perform a multiplication on shared values, 2) an efficient generic construction to obtain from any LSSS a multiplicative LSSS for the same access structure, and 3) an efficient generic construction to build verifiability into every LSSS (always assuming that the adversary structure allows for MPC or VSS at all).
The protocols are efficient. In contrast to all previous information-theoretically secure protocols, the field size is not restricted (e.g, to be greater than n). Moreover, we exhibit adversary structures for which our protocols are polynomial in n while all previous approaches to MPC for non-threshold adversaries provably have super-polynomial complexity.
Supported by the Swiss SNF, grant no. 5003-045293.
Supported by the Swiss SNF.
(Basic Reseach in Computer Science, center of the Danish National Research Foundation), work done while employed at ETH Zürich.
Chapter PDF
Similar content being viewed by others
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
D. Beaver, Foundations of secure interactive computing, Proc. CRYPTO’ 91, Springer Verlag LNCS, vol. 576, pp. 377–391.
D. Beaver and A. Wool, Quorum-based multi-party computations, Proc. EUROCRYPT’ 98, Springer Verlag LNCS, vol. 1403, pp. 375–390.
A. Beimel, Secure schemes for secret sharing and key distribution, Ph.D.-thesis, Technion, Haifa, June 1996.
J. Benaloh, J. Leichter, Generalized secret sharing and monotone functions, Proc. CRYPTO’ 88, Springer Verlag LNCS, vol. 403, pp. 25–35.
M. Ben-Or, S. Goldwasser, A. Wigderson, Completeness theorems for non-cryptographic fault-tolerant distributed computation, Proc. ACM STOC’ 88, pp. 1–10.
M. Bertilsson, I. Ingemarsson, A construction of practical secret sharing schemes using linear block codes, Proc. AUSCRYPT’ 92, Springer Verlag LNCS, vol. 718, pp. 67–79.
E. F. Brickell, Some ideal secret sharing schemes, J. Combin. Maths. & Combin. Comp. 9 (1989), pp. 105–113.
R. Canetti, Studies in secure multi-party computation and applications, Ph. D. thesis, Weizmann Institute of Science, Rehovot, 1995.
R. Canetti, U. Feige, O. Goldreich, M. Naor, Adaptively secure multi-party computation, Proc. ACM STOC’ 96, pp. 639–648.
D. Chaum, C. Crépeau, I. Damgård, Multi-party unconditionally secure protocols, Proc. ACM STOC’ 88, pp. 11–19.
B. Chor, S. Goldwasser, S. Micali, B. Awerbuch, Verifiable secret sharing and achieving simultaneity in the presence of faults, Proc. FOCS’ 85, pp. 383–395.
R. Cramer, I. Damgård, Zero Knowledge for Finite Field Arithmetic or: Can Zero Knowledge be for Free?, Proc. CRYPTO’98, Springer Verlag LNCS, vol. 1462, pp. 424–441.
R. Cramer, I. Damgård, S. Dziembowski, On the complexity of verifiable secret sharing and multi-party computation, Proceedings of the 32nd ACM Symposium on Theory of Computing (STOC’ 00), Portland, Oregon, May 2000.
R. Cramer, I. Damgård, S. Dziembowski, M. Hirt and T. Rabin, Efficient multiparty computations secure against an adaptive adversary, Proc. EUROCRYPT’ 99, Springer Verlag LNCS, vol. 1592, pp. 311–326.
C. Crépeau, J. van de Graaf and A. Tapp, Committed oblivious transfer and private multi-party computation, proc. CRYPTO’ 95, Springer Verlag LNCS, vol. 963, pp. 110–123.
M. van Dijk, Secret key sharing and secret key generation, Ph.D. Thesis, Eindhoven University of Technology, 1997.
S. Fehr, Efficient construction of dual MSP, manuscript 1999.
M. Fitzi, U. Maurer, Efficient Byzantine agreement secure against general adversaries, Proc. 12th Int. Symp. on Distributed Computing (DISC’ 98), Springer Verlag LNCS, vol. 1499, pp. 134–148.
A. Gál, A characterization of span program size and improved lower bounds for monotone span programs, Proceedings of the 30th ACM Symposium on the Theory of Computing, 1998, pp. 429–437.
A. Gál, Combinatorial methods in Boolean function complexity, Ph.D.-thesis, University of Chicago, 1995.
Z. Galil, S. Haber and M. Yung, Cryptographic computation: Secure fault-tolerant protocols and the public-key model, Proc. CRYPTO’87, Springer Verlag LNCS, vol. 293, pp. 135–155.
R. Gennaro, Theory and practice of veri_able secret sharing, Ph.D. thesis, MIT, 1996.
R. Gennaro, M. Rabin, T. Rabin, Simplified VSS and fast-track multi-party computations with applications to threshold cryptography, Proc. ACM PODC’98.
O. Goldreich, S. Micali and A. Wigderson, How to play any mental game or a completeness theorem for protocols with honest majority, Proc. ACM STOC’ 87, pp. 218–229.
M. Hirt, U. Maurer, Player simulation and general adversary structures in perfect multi-party computation, Journal of Cryptology, vol. 13, no. 1, pp. 31–60, 2000. (Preliminary version in Proc. ACM PODC’97, pp. 25–34.)
M. Ito, A. Saito and T. Nishizeki, Secret sharing schemes realizing general access structures, Proc. IEEE GlobeCom’ 87 Tokyo, pp. 99–102.
M. Karchmer, A. Wigderson, On span programs, Proc. of Structure in Complexity’ 93, pp. 102–111.
S. Micali and P. Rogaway, Secure computation, Manuscript, Preliminary version in Proc. CRYPTO’ 91, Springer Verlag LNCS, vol. 576, pp. 392–404
T. Rabin, M. Ben-Or, Verifiable secret sharing and multi-party protocols with honest majority, Proc. ACM STOC’ 89, pp. 73–85.
T. Rabin, Robust sharing of secrets when the dealer is honest or cheating, J. ACM, 41(6):1089–1109, November 1994.
A. Shamir, How to share a secret, Communications of the ACM 22 (1979) 612–613.
Technical report, full version of this paper. Will be posted on the Web and is available from the authors. Obsolete are the earlier versions: Span programs and general secure multi-party computation, BRICS Report RS-97-28, Nov. 1997, and Enforcing the multiplication property on MSPs, with only constant overhead, Jan. 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cramer, R., Damgård, I., Maurer, U. (2000). General Secure Multi-party Computation from any Linear Secret-Sharing Scheme. In: Preneel, B. (eds) Advances in Cryptology — EUROCRYPT 2000. EUROCRYPT 2000. Lecture Notes in Computer Science, vol 1807. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45539-6_22
Download citation
DOI: https://doi.org/10.1007/3-540-45539-6_22
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67517-4
Online ISBN: 978-3-540-45539-4
eBook Packages: Springer Book Archive