Performing Computations on Hierarchically Shared Secrets

  • Giulia Traverso
  • Denise Demirel
  • Johannes Buchmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10831)


Hierarchical secret sharing schemes distribute a message to a set of shareholders with different reconstruction capabilities. In distributed storage systems, this is an important property because it allows to grant more reconstruction capability to better performing storage servers and vice versa. In particular, Tassa’s conjunctive and disjunctive hierarchical secret sharing schemes are based on Birkhoff interpolation and perform equally well as Shamir’s threshold secret sharing scheme. Thus, they are promising candidates for distributed storage systems. A key requirement is the possibility to perform function evaluations over shared data. However, practical algorithms supporting this have not been provided yet with respect to hierarchical secret sharing schemes. Aiming at closing this gap, in this work, we show how additions and multiplications of shares can be practically computed using Tassa’s conjunctive and disjunctive hierarchical secret sharing schemes. Furthermore, we provide auditing procedures for operations on messages shared hierarchically, which allow to verify that functions on the shares have been performed correctly. We close this work with an evaluation of the correctness, security, and efficiency of the protocols we propose.


Hierarchical secret sharing Birkhoff interpolation Verifiable secret sharing Auditing Multi-party computation Distributed storage systems Cloud computing 



The authors thank Lucas Schabüser and Denis Butin for useful discussions. This work was in part funded by the European Commission through grant agreement no. 644962 (PRISMACLOUD). Furthermore, it received funding from the DFG as part of project S6 within the CRC 1119 CROSSING.


  1. 1.
    Beaver, D.: Efficient multiparty protocols using circuit randomization. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 420–432. Springer, Heidelberg (1992). Scholar
  2. 2.
    Beimel, A.: Secret-sharing schemes: a survey. In: Chee, Y.M., Guo, Z., Ling, S., Shao, F., Tang, Y., Wang, H., Xing, C. (eds.) IWCC 2011. LNCS, vol. 6639, pp. 11–46. Springer, Heidelberg (2011). Scholar
  3. 3.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: STOC 1988 (1988)Google Scholar
  4. 4.
    Blakley, G.R., et al.: Safeguarding cryptographic keys. In: Proceedings of the National Computer Conference (1979)Google Scholar
  5. 5.
    Blundo, C., Cresti, A., De Santis, A., Vaccaro, U.: Fully dynamic secret sharing schemes. In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 110–125. Springer, Heidelberg (1994). Scholar
  6. 6.
    Boneh, D., Franklin, M.: Identity-based encryption from the Weil pairing. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 213–229. Springer, Heidelberg (2001). Scholar
  7. 7.
    Brickell, E.F.: Some ideal secret sharing schemes. In: Quisquater, J.-J., Vandewalle, J. (eds.) EUROCRYPT 1989. LNCS, vol. 434, pp. 468–475. Springer, Heidelberg (1990). Scholar
  8. 8.
    Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols. In: STOC 1988 (1988)Google Scholar
  9. 9.
    Chor, B., Goldwasser, S., Micali, S., Awerbuch, B.: Verifiable secret sharing and achieving simultaneity in the presence of faults (extended abstract). In: FOCS (1985)Google Scholar
  10. 10.
    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). Scholar
  11. 11.
    Damgård, I., Nielsen, J.B.: Scalable and unconditionally secure multiparty computation. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 572–590. Springer, Heidelberg (2007). Scholar
  12. 12.
    Desmedt, Y., Jajodia, S.: Redistributing secret shares to new access structures and its applications. Technical report ISSE TR-97-01, George Mason University (1997)Google Scholar
  13. 13.
    Doganay, M.C., Pedersen, T.B., Saygin, Y., Savas, E., Levi, A.: Distributed privacy preserving k-means clustering with additive secret sharing. In: PAIS (2008)Google Scholar
  14. 14.
    Farràs, O., Padró, C.: Ideal hierarchical secret sharing schemes. In: TCC (2010)Google Scholar
  15. 15.
    Feldman, P.: A practical scheme for non-interactive verifiable secret sharing. In: 28th Annual Symposium on Foundations of Computer Science (1987)Google Scholar
  16. 16.
    Gennaro, R., Rabin, M.O., Rabin, T.: Simplified VSS and fact-track multiparty computations with applications to threshold cryptography. In: PODC 1998 (1998)Google Scholar
  17. 17.
    Ghodosi, H., Pieprzyk, J., Safavi-Naini, R.: Secret sharing in multilevel and compartmented groups. In: Boyd, C., Dawson, E. (eds.) ACISP 1998. LNCS, vol. 1438, pp. 367–378. Springer, Heidelberg (1998). Scholar
  18. 18.
    Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or a completeness theorem for protocols with honest majority. In: STOC 1990 (1990)Google Scholar
  19. 19.
    Gupta, V., Gopinath, K.: \({\rm G}_{\rm its}^{2}\) VSR: an information theoretical secure verifiable secret redistribution protocol for long-term archival storage. In: SISW 2007 (2007)Google Scholar
  20. 20.
    Heather, J., Lundin, D.: The append-only web bulletin board. In: Degano, P., Guttman, J., Martinelli, F. (eds.) FAST 2008. LNCS, vol. 5491, pp. 242–256. Springer, Heidelberg (2009). Scholar
  21. 21.
    Herzberg, A., Jarecki, S., Krawczyk, H., Yung, M.: Proactive secret sharing or: how to cope with perpetual leakage. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 339–352. Springer, Heidelberg (1995). Scholar
  22. 22.
    Käsper, E., Nikov, V., Nikova, S.: Strongly multiplicative hierarchical threshold secret sharing. In: Desmedt, Y. (ed.) ICITS 2007. LNCS, vol. 4883, pp. 148–168. Springer, Heidelberg (2009). Scholar
  23. 23.
    Loruenser, T., Happe, A., Slamanig, D.: ARCHISTAR: towards secure and robust cloud based data sharing. In: CloudCom 2015 (2015)Google Scholar
  24. 24.
    Nojoumian, M., Stinson, D.R.: Social secret sharing in cloud computing using a new trust function. In: PST 2012 (2012)Google Scholar
  25. 25.
    Nojoumian, M., Stinson, D.R., Grainger, M.: Unconditionally secure social secret sharing scheme. Inf. Secur. IET 4, 202–211 (2010)CrossRefGoogle Scholar
  26. 26.
    Pakniat, N., Eslami, Z., Nojoumian, M.: Ideal social secret sharing using Birkhoff interpolation method. IACR 2014 (2014)Google Scholar
  27. 27.
    Pedersen, T.P.: Non-interactive and information-theoretic secure verifiable secret sharing. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 129–140. Springer, Heidelberg (1992). Scholar
  28. 28.
    Schabhüser, L., Demirel, D., Buchmann, J.A.: An unconditionally hiding auditing procedure for computations over distributed data. In: CNS 2016 (2016)Google Scholar
  29. 29.
    Shamir, A.: How to share a secret. Commun. ACM 22, 612–613 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Simmons, G.J.: How to (really) share a secret. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 390–448. Springer, New York (1990). Scholar
  31. 31.
    Tassa, T.: Hierarchical threshold secret sharing. J. Cryptology 20, 237–264 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Traverso, G., Demirel, D., Buchmann, J.: Dynamic and verifiable hierarchical secret sharing. In: Nascimento, A.C.A., Barreto, P. (eds.) ICITS 2016. LNCS, vol. 10015, pp. 24–43. Springer, Cham (2016). Scholar
  33. 33.
    Traverso, G., Demirel, D., Habib, S.M., Buchmann, J.A.: As\({}^{\text{3}}\): adaptive social secret sharing for distributed storage systems. In: PST 2016 (2016)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Giulia Traverso
    • 1
  • Denise Demirel
    • 1
  • Johannes Buchmann
    • 1
  1. 1.Technische Universität DarmstadtDarmstadtGermany

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