Abstract
Secret-sharing schemes are an important tool in cryptography that is used in the construction of many secure protocols. However, the shares’ size in the best known secret-sharing schemes realizing general access structures is exponential in the number of parties in the access structure, making them impractical. On the other hand, the best lower bound known for sharing of an ℓ-bit secret with respect to an access structure with n parties is Ω(ℓn/logn) (Csirmaz, EUROCRYPT 94). No major progress on closing this gap has been obtained in the last decade.
Faced by our lack of understanding of the share complexity of secret sharing schemes, we investigate a weaker notion of privacy in secrets sharing schemes where each unauthorized set can never rule out any secret (rather than not learn any “probabilistic” information on the secret). Such schemes were used previously to prove lower bounds on the shares’ size of perfect secret-sharing schemes. Our main results is somewhat surprising upper-bounds on the shares’ size in weakly-private schemes.
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For every access structure, we construct a scheme for sharing an ℓ-bit secret with (ℓ + c)-bit shares, where c is a constant depending on the access structure (alas, c can be exponential in n). Thus, our schemes become more efficient as ℓ – the secret size – grows. For example, for the above mentioned access structure of Csirmaz, we construct a scheme with shares’ size ℓ + nlogn.
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We construct efficient weakly-private schemes for threshold access structures for sharing a one bit secret. Most impressively, for the 2-out-of-n threshold access structure, we construct a scheme with 2-bit shares (compared to Ω(logn) in any perfect secret sharing scheme).
The work of the first author was done while on sabbatical at the University of California, Davis, partially supported by the Packard Foundation. The second author is partially supported by the NSF and the Packard Foundation.
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References
Applebaum, B., Ishai, Y., Kushilevitz, E.: Cryptography in NC0. In: Proc. of the 45th Symp. on Foundations of Computer Science, pp. 166–175 (2004)
Beguin, P., Cresti, A.: General short computational secret sharing schemes. In: Guillou, L.C., Quisquater, J.-J. (eds.) EUROCRYPT 1995. LNCS, vol. 921, pp. 194–208. Springer, Heidelberg (1995)
Beimel, A.: Secure Schemes for Secret Sharing and Key Distribution. PhD thesis, Technion – Israel Institute of Technology (1996)
Beimel, A.: On private computation in incomplete networks. In: Distributed Computing (2006)
Beimel, A., Chor, B.: Communication in key distribution schemes. IEEE Trans. on Information Theory 42(1), 19–28 (1996)
Beimel, A., Ishai, Y.: On the power of nonlinear secret-sharing. SIAM J. on Discrete Mathematics 19(1), 258–280 (2005)
Beimel, A., Livne, N.: On matroids and non-ideal secret sharing. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 482–501. Springer, Heidelberg (2006)
Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for noncryptographic fault-tolerant distributed computations. In: Proc. of the 20th STOC, pp. 1–10 (1988)
Benaloh, J., Leichter, J.: Generalized secret sharing and monotone functions. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 27–35. Springer, Heidelberg (1990)
Blakley, G.R.: Safeguarding cryptographic keys. In: Proc. of the 1979 AFIPS National Computer Conference, pp. 313–317 (1979)
Blundo, C., De Santis, A., de Simone, R., Vaccaro, U.: Tight bounds on the information rate of secret sharing schemes. Designs, Codes and Cryptography 11(2), 107–122 (1997)
Blundo, C., De Santis, A., Giorgio Gaggia, A., Vaccaro, U.: New bounds on the information rate of secret sharing schemes. IEEE Trans. on Information Theory 41(2), 549–553 (1995)
Blundo, C., De Santis, A., Herzberg, A., Kutten, S., Vaccaro, U., Yung, M.: Perfectly secure key distribution for dynamic conferences. Info. and Comput. 146(1), 1–23 (1998)
Blundo, C., Stinson, D.R.: Anonymous secret sharing schemes. Discrete Applied Math. and Combin. Operations Research and Comp. Sci. 77, 13–28 (1997)
Brickell, E.F.: Some ideal secret sharing schemes. Journal of Combin. Math. and Combin. Comput. 6, 105–113 (1989)
Brickell, E.F., Davenport, D.M.: On the classification of ideal secret sharing schemes. J. of Cryptology 4(73), 123–134 (1991)
Capocelli, R.M., De Santis, A., Gargano, L., Vaccaro, U.: On the size of shares for secret sharing schemes. J. of Cryptology 6(3), 157–168 (1993)
Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols. In: Proc. of the 20th STOC, pp. 11–19 (1988)
Cramer, R., Damgård, 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)
Csirmaz, L.: The size of a share must be large. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 223–231. Springer, Heidelberg (1995), Also in: J. of Cryptology, 10(4), 223–231, 1997
Csirmaz, L.: The dealer’s random bits in perfect secret sharing schemes. Studia Sci. Math. Hungar. 32(3–4), 429–437 (1996)
Damgård, I., Thorbek, R.: Linear integer secret sharing and distributed exponentiation. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T.G. (eds.) PKC 2006. LNCS, vol. 3958, pp. 75–90. Springer, Heidelberg (2006)
Desmedt, Y., Frankel, Y.: Shared generation of authenticators and signatures. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 457–469. Springer, Heidelberg (1992)
van Dijk, M.: On the information rate of perfect secret sharing schemes. Designs, Codes and Cryptography 6, 143–169 (1995)
van Dijk, M.: A linear construction of secret sharing schemes. Designs, Codes and Cryptography 12(2), 161–201 (1997)
Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute based encryption for fine-grained access control of encrypted data. In: CCS 2006 (2006)
Ishai, Y.: Personal communication (2006)
Ito, M., Saito, A., Nishizeki, T.: Secret sharing schemes realizing general access structure. In: Proc. of Globecom 87, pp. 99–102 (1987)
Jackson, W.-A, Martin, K.M.: Combinatorial models for perfect secret sharing schemes. J. of Comb. Mathematics and Comb. Computing 28, 249–265 (1998)
Karchmer, M., Wigderson, A.: On span programs. In: Proc. of the 8th Structure in Complexity Theory, pp. 102–111 (1993)
Karnin, E.D., Greene, J.W., Hellman, M.E.: On secret sharing systems. IEEE Trans. on Information Theory 29(1), 35–41 (1983)
Kilian, J., Nisan, N.: Private communication (1990)
Kishimoto, W., Okada, K., Kurosawa, K., Ogata, W.: On the bound for anonymous secret sharing schemes. Discrete Appl. Math. 121(1-3), 193–202 (2002)
Krawczyk, H.: Secret sharing made short. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 136–146. Springer, Heidelberg (1994)
Kurosawa, K., Okada, K.: Combinatorial lower bounds for secret sharing schemes. Inform. Process. Lett. 60(6), 301–304 (1996)
Kushilevitz, E.: Privacy and communication complexity. SIAM J. on Discrete Mathematics 5(2), 273–284 (1992)
Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. of the ACM 41(5), 960–981 (1994)
Martí-Farré, J., Padró, C.: On secret sharing schemes, matroids and polymatroids. Technical Report 2006/077, Cryptology ePrint Archive (2006)
Miao, Y.: A combinatorial characterization of regular anonymous perfect threshold schemes. Inform. Process. Lett. 85(3), 131–135 (2003)
Naor, M., Wool, A.: Access control and signatures via quorum secret sharing. IEEE Transactions on Parallel and Distributed Systems 9(1), 909–922 (1998)
Padró, C., Sáez, G.: Secret sharing schemes with bipartite access structure. IEEE Trans. on Information Theory 46, 2596–2605 (2000)
Rabin, M.O.: Randomized Byzantine generals. In: Proc. of the 24th IEEE Symp. on Foundations of Computer Science, pp. 403–409. IEEE Computer Society Press, Los Alamitos (1983)
Seymour, P.D.: On secret-sharing matroids. J. of Combinatorial Theory, Series B 56, 69–73 (1992)
Shamir, A.: How to share a secret. Communications of the ACM 22, 612–613 (1979)
Simmons, G.J., Jackson, W., Martin, K.M.: The geometry of shared secret schemes. Bulletin of the ICA 1, 71–88 (1991)
Stinson, D.R.: Decomposition construction for secret sharing schemes. IEEE Trans. on Information Theory 40(1), 118–125 (1994)
Stinson, D.R., Vanstone, S.A.: A combinatorial approach to threshold schemes. SIAM J. on Discrete Mathematics 1(2), 230–236 (1988)
Vinod, V., Narayanan, A., Srinathan, K., Pandu Rangan, C., Kim, K.: On the power of computational secret sharing. In: Johansson, T., Maitra, S. (eds.) INDOCRYPT 2003. LNCS, vol. 2904, pp. 162–176. Springer, Heidelberg (2003)
Yao, A.C.: Unpublished manuscript, Presented at Oberwolfach and DIMACS workshops (1989)
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Beimel, A., Franklin, M. (2007). Weakly-Private Secret Sharing Schemes . In: Vadhan, S.P. (eds) Theory of Cryptography. TCC 2007. Lecture Notes in Computer Science, vol 4392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70936-7_14
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