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Annual International Conference on the Theory and Applications of Cryptographic Techniques

EUROCRYPT 2019: Advances in Cryptology – EUROCRYPT 2019 pp 441–471Cite as

Secret-Sharing Schemes for General and Uniform Access Structures

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11478))

Abstract

A secret-sharing scheme allows some authorized sets of parties to reconstruct a secret; the collection of authorized sets is called the access structure. For over 30 years, it was known that any (monotone) collection of authorized sets can be realized by a secret-sharing scheme whose shares are of size \(2^{n-o(n)}\) and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to \(O(2^{0.994n})\). Our first contribution is improving the exponent of secret sharing down to 0.892. For the special case of linear secret-sharing schemes, we get an exponent of 0.942 (compared to 0.999 of Liu and Vaikuntanathan).

Motivated by the construction of Liu and Vaikuntanathan, we study secret-sharing schemes for uniform access structures. An access structure is k-uniform if all sets of size larger than k are authorized, all sets of size smaller than k are unauthorized, and each set of size k can be either authorized or unauthorized. The construction of Liu and Vaikuntanathan starts from protocols for conditional disclosure of secrets, constructs secret-sharing schemes for uniform access structures from them, and combines these schemes in order to obtain secret-sharing schemes for general access structures. Our second contribution in this paper is constructions of secret-sharing schemes for uniform access structures. We achieve the following results:

  • A secret-sharing scheme for k-uniform access structures for large secrets in which the share size is \(O(k^2)\) times the size of the secret.

  • A linear secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is \(\tilde{O}(2^{h(k/n)n/2})\) (where h is the binary entropy function). By counting arguments, this construction is optimal (up to polynomial factors).

  • A secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is \(2^{\tilde{O}(\sqrt{k \log n})}\).

Our third contribution is a construction of ad-hoc PSM protocols, i.e., PSM protocols in which only a subset of the parties will compute a function on their inputs. This result is based on ideas we used in the construction of secret-sharing schemes for k-uniform access structures for a binary secret.

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Notes

  1. 1.

    Formally, such a statement implicitly refers to an infinite sequence of (collections of) access structures that is parameterized by the number of participants n.

  2. 2.

    The notation \(\mathbf {M}\) stands for “middle”.

  3. 3.

    The notation \(\mathbf {X}\) stands for eXternal slices.

References

  1. Aiello, B., Ishai, Y., Reingold, O.: Priced oblivious transfer: how to sell digital goods. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 119–135. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44987-6_8

    Chapter  Google Scholar 

  2. Applebaum, B., Arkis, B.: On the power of amortization in secret sharing: d-uniform secret sharing and CDS with constant information rate. In: Beimel, A., Dziembowski, S. (eds.) TCC 2018. LNCS, vol. 11239, pp. 317–344. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03807-6_12

    Chapter  MATH  Google Scholar 

  3. Applebaum, B., Arkis, B., Raykov, P., Vasudevan, P.N.: Conditional disclosure of secrets: amplification, closure, amortization, lower-bounds, and separations. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 727–757. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_24

    Chapter  Google Scholar 

  4. Applebaum, B., Beimel, A., Farràs, O., Nir, O., Peter, N.: Secret-sharing schemes for general and uniform access structures. Technical report 2019/231, IACR Cryptology ePrint Archive (2019)

    Google Scholar 

  5. Applebaum, B., Holenstein, T., Mishra, M., Shayevitz, O.: The communication complexity of private simultaneous messages, revisited. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10821, pp. 261–286. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78375-8_9

    Chapter  Google Scholar 

  6. Applebaum, B., Vasudevan, P.: Placing conditional disclosure of secrets in the communication complexity universe. In: 10th Innovations in Theoretical Computer Science Conference, ITCS. LIPIcs, vol. 124, pp. 4:1–4:14 (2019)

    Google Scholar 

  7. Attrapadung, N.: Dual system encryption via doubly selective security: framework, fully secure functional encryption for regular languages, and more. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 557–577. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_31

    Chapter  Google Scholar 

  8. Beimel, A., Chor, B.: Universally ideal secret-sharing schemes. IEEE Trans. Inf. Theory 40(3), 786–794 (1994)

    Article  MathSciNet  Google Scholar 

  9. Beimel, A., Farràs, O., Mintz, Y., Peter, N.: Linear secret-sharing schemes for forbidden graph access structures. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10678, pp. 394–423. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70503-3_13

    Chapter  Google Scholar 

  10. Beimel, A., Farràs, O., Peter, N.: Secret sharing schemes for dense forbidden graphs. In: Zikas, V., De Prisco, R. (eds.) SCN 2016. LNCS, vol. 9841, pp. 509–528. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44618-9_27

    Chapter  MATH  Google Scholar 

  11. Beimel, A., Ishai, Y., Kumaresan, R., Kushilevitz, E.: On the cryptographic complexity of the worst functions. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 317–342. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54242-8_14

    Chapter  MATH  Google Scholar 

  12. Beimel, A., Kushilevitz, E., Nissim, P.: The complexity of multiparty PSM protocols and related models. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10821, pp. 287–318. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78375-8_10

    Chapter  Google Scholar 

  13. Beimel, A., Peter, N.: Optimal linear multiparty conditional disclosure of secrets protocols. In: Peyrin, T., Galbraith, S. (eds.) ASIACRYPT 2018. LNCS, vol. 11274, pp. 332–362. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03332-3_13

    Chapter  Google Scholar 

  14. Beimel, A., Ishai, Y., Kushilevitz, E.: Ad hoc PSM protocols: secure computation without coordination. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10212, pp. 580–608. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56617-7_20

    Chapter  Google Scholar 

  15. Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for noncryptographic fault-tolerant distributed computations. In: Proceedings of the 20th ACM Symposium on the Theory of Computing, pp. 1–10 (1988)

    Google Scholar 

  16. Benaloh, J., Leichter, J.: Generalized secret sharing and monotone functions. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 27–35. Springer, New York (1990). https://doi.org/10.1007/0-387-34799-2_3

    Chapter  Google Scholar 

  17. Bertilsson, M., Ingemarsson, I.: A construction of practical secret sharing schemes using linear block codes. In: Seberry, J., Zheng, Y. (eds.) AUSCRYPT 1992. LNCS, vol. 718, pp. 67–79. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-57220-1_53

    Chapter  Google Scholar 

  18. Blakley, G.R.: Safeguarding cryptographic keys. In: Proceedings of the 1979 AFIPS National Computer Conference, vol. 48, pp. 313–317 (1979)

    Google Scholar 

  19. Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols. In: Proceedings of the 20th ACM Symposium on the Theory of Computing, pp. 11–19 (1988)

    Google Scholar 

  20. Chor, B., Kushilevitz, E.: Secret sharing over infinite domains. J. Cryptol. 6(2), 87–96 (1993)

    Article  MathSciNet  Google Scholar 

  21. 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). https://doi.org/10.1007/3-540-45539-6_22

    Chapter  Google Scholar 

  22. Csirmaz, L.: The size of a share must be large. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 13–22. Springer, Heidelberg (1995). https://doi.org/10.1007/BFb0053420. Journal Version in: J. Cryptol. 10(4), 223–231 (1997)

    Chapter  Google Scholar 

  23. Csirmaz, L.: The dealer’s random bits in perfect secret sharing schemes. Studia Sci. Math. Hungar. 32(3–4), 429–437 (1996)

    MathSciNet  MATH  Google Scholar 

  24. 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). https://doi.org/10.1007/3-540-46766-1_37

    Chapter  Google Scholar 

  25. Erdös, P., Spencer, J.: Probabilistic Methods in Combinatorics. Academic Press, Cambridge (1974)

    MATH  Google Scholar 

  26. Feige, U., Kilian, J., Naor, M.: A minimal model for secure computation. In: 26th STOC 1994, pp. 554–563 (1994)

    Google Scholar 

  27. Gay, R., Kerenidis, I., Wee, H.: Communication complexity of conditional disclosure of secrets and attribute-based encryption. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 485–502. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48000-7_24

    Chapter  Google Scholar 

  28. Gertner, Y., Ishai, Y., Kushilevitz, E., Malkin, T.: Protecting data privacy in private information retrieval schemes. J. Comput. Syst. Sci. 60(3), 592–629 (2000)

    Article  MathSciNet  Google Scholar 

  29. Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: Proceedings of the 13th ACM Conference on Computer and Communications Security, pp. 89–98 (2006)

    Google Scholar 

  30. Ishai, Y., Kushilevitz, E.: Private simultaneous messages protocols with applications. In: 5th Israel Symposium on Theory of Computing and Systems, pp. 174–183 (1997)

    Google Scholar 

  31. Ishai, Y., Wee, H.: Partial garbling schemes and their applications. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 650–662. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43948-7_54

    Chapter  MATH  Google Scholar 

  32. Ito, M., Saito, A., Nishizeki, T.: Secret sharing schemes realizing general access structure. In: Globecom 1987, pp. 99–102 (1987). Journal Version: Multiple assignment scheme for sharing secret. J. Cryptol. 6(1), 15–20 (1993)

    Google Scholar 

  33. Karchmer, M., Wigderson, A.: On span programs. In: 8th Structure in Complexity Theory, pp. 102–111 (1993)

    Google Scholar 

  34. Liu, T., Vaikuntanathan, V.: Breaking the circuit-size barrier in secret sharing. In: 50th STOC 2018, pp. 699–708 (2018)

    Google Scholar 

  35. Liu, T., Vaikuntanathan, V., Wee, H.: Conditional disclosure of secrets via non-linear reconstruction. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 758–790. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_25

    Chapter  Google Scholar 

  36. Liu, T., Vaikuntanathan, V., Wee, H.: Towards breaking the exponential barrier for general secret sharing. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10820, pp. 567–596. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_21

    Chapter  Google Scholar 

  37. Naor, M., Wool, A.: Access control and signatures via quorum secret sharing. In: 3rd ACM Conference on Computer and Communications Security, pp. 157–167 (1996)

    Google Scholar 

  38. Shamir, A.: How to share a secret. Commun. ACM 22, 612–613 (1979)

    Article  MathSciNet  Google Scholar 

  39. Shankar, B., Srinathan, K., Rangan, C.P.: Alternative protocols for generalized oblivious transfer. In: Rao, S., Chatterjee, M., Jayanti, P., Murthy, C.S.R., Saha, S.K. (eds.) ICDCN 2008. LNCS, vol. 4904, pp. 304–309. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-77444-0_31

    Chapter  Google Scholar 

  40. Stinson, D.R.: Decomposition construction for secret sharing schemes. IEEE Trans. Inf. Theory 40(1), 118–125 (1994)

    Article  MathSciNet  Google Scholar 

  41. Sun, H., Shieh, S.: Secret sharing in graph-based prohibited structures. In: INFOCOM 1997, pp. 718–724 (1997)

    Google Scholar 

  42. Tassa, T.: Generalized oblivious transfer by secret sharing. Des. Codes Crypt. 58(1), 11–21 (2011)

    Article  MathSciNet  Google Scholar 

  43. Waters, B.: Ciphertext-policy attribute-based encryption: an expressive, efficient, and provably secure realization. In: Catalano, D., Fazio, N., Gennaro, R., Nicolosi, A. (eds.) PKC 2011. LNCS, vol. 6571, pp. 53–70. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19379-8_4

    Chapter  Google Scholar 

  44. Wee, H.: Dual system encryption via predicate encodings. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 616–637. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54242-8_26

    Chapter  Google Scholar 

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Acknowledgement

The first and fourth authors are supported by the European Union’s Horizon 2020 Programme (ERC-StG-2014-2020) under grant agreement no. 639813 ERC-CLC, and the Check Point Institute for Information Security. Part of this work was done while the second author was visiting Georgetown university, supported by NSF grant no. 1565387, TWC: Large: Collaborative: Computing Over Distributed Sensitive Data. The second author is also supported by ISF grant 152/17 and by a grant from the Cyber Security Research Center at Ben-Gurion University of the Negev. The third author is supported by the Spanish Government through TIN2014-57364-C2-1-R and by the Government of Catalonia through Grant 2017 SGR 705. The fifth author is supported by ISF grant 152/17, by a grant from the Cyber Security Research Center at Ben-Gurion University of the Negev, and by the Frankel center for computer science.

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Authors and Affiliations

  1. Tel Aviv University, Tel Aviv, Israel

    Benny Applebaum & Oded Nir

  2. Ben-Gurion University of the Negev, Be’er-Sheva, Israel

    Amos Beimel & Naty Peter

  3. Universitat Rovira i Virgili, Tarragona, Catalonia, Spain

    Oriol Farràs

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  1. Benny Applebaum
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  2. Amos Beimel
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Correspondence to Benny Applebaum .

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Editors and Affiliations

  1. Technion, Haifa, Israel

    Yuval Ishai

  2. COSIC Group, KU Leuven, Heverlee, Belgium

    Vincent Rijmen

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Applebaum, B., Beimel, A., Farràs, O., Nir, O., Peter, N. (2019). Secret-Sharing Schemes for General and Uniform Access Structures. In: Ishai, Y., Rijmen, V. (eds) Advances in Cryptology – EUROCRYPT 2019. EUROCRYPT 2019. Lecture Notes in Computer Science(), vol 11478. Springer, Cham. https://doi.org/10.1007/978-3-030-17659-4_15

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