Advertisement

Conditional Disclosure of Secrets via Non-linear Reconstruction

  • Tianren LiuEmail author
  • Vinod Vaikuntanathan
  • Hoeteck Wee
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10401)

Abstract

We present new protocols for conditional disclosure of secrets (CDS), where two parties want to disclose a secret to a third party if and only if their respective inputs satisfy some predicate.

  • For general predicates \(\mathsf {P}: [N] \times [N] \rightarrow \{0,1\}\), we present two protocols that achieve \(o(N^{1/2})\) communication: the first achieves \(O(N^{1/3})\) communication and the second achieves sub-polynomial \(2^{O(\sqrt{\log N \log \log N})} = N^{o(1)}\) communication.

  • As a corollary, we obtain improved share complexity for forbidden graph access structures. Namely, for every graph on N vertices, there is a secret-sharing scheme for N parties in which each pair of parties can reconstruct the secret if and only if the corresponding vertices in G are connected, and where each party gets a share of size \(2^{O(\sqrt{\log N \log \log N})} = N^{o(1)}\).

Prior to this work, the best protocols for both primitives required communication complexity \(\tilde{O}(N^{1/2})\). Indeed, this is essentially the best that all prior techniques could hope to achieve as they were limited to so-called “linear reconstruction”. This is the first work to break this \(O(N^{1/2})\) “linear reconstruction” barrier in settings related to secret sharing. To obtain these results, we draw upon techniques for non-linear reconstruction developed in the context of information-theoretic private information retrieval.

We further extend our results to the setting of private simultaneous messages (PSM), and provide applications such as an improved attribute-based encryption (ABE) for quadratic polynomials.

Supplementary material

References

  1. [AARV17]
    Applebaum, B., Arkis, B., Raykov, P., Vasudevan, P.N.: Conditional disclosure of secrets: amplification, closure, amortization, lower-bounds, and separations. IACR Cryptology ePrint Archive 2017:164 (2017)Google Scholar
  2. [ACC+14]
    Ada, A., Chattopadhyay, A., Cook, S.A., Fontes, L., Koucký, M., Pitassi, T.: The hardness of being private. TOCT 6(1), 1:1–1:24 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Att14]
    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). doi: 10.1007/978-3-642-55220-5_31 CrossRefGoogle Scholar
  4. [Bei11]
    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). doi: 10.1007/978-3-642-20901-7_2 CrossRefGoogle Scholar
  5. [BFG06]
    Beigel, R., Fortnow, L., Gasarch, W.I.: A tight lower bound for restricted pir protocols. Comput. Complex. 15(1), 82–91 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BGP95]
    Beimel, A., Gál, A., Paterson, M.: Lower bounds for monotone span programs. In: FOCS, pp. 674–681 (1995)Google Scholar
  7. [BI01]
    Beimel, A., Ishai, Y.: On the power of nonlinear secret-sharing. In: Proceedings of the 16th Annual IEEE Conference on Computational Complexity, Chicago, Illinois, USA, 18–21 June 2001, pp. 188–202. IEEE Computer Society (2001)Google Scholar
  8. [BIKK14]
    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). doi: 10.1007/978-3-642-54242-8_14 CrossRefGoogle Scholar
  9. [BIKO12]
    Beimel, A., Ishai, Y., Kushilevitz, E., Orlov, I.: Share conversion and private information retrieval. In: Proceedings of the 27th Conference on Computational Complexity, CCC 2012, Porto, Portugal, 26–29 June 2012, pp. 258–268. IEEE Computer Society (2012)Google Scholar
  10. [BSGV96]
    Blundo, C., De Santis, A., Gargano, L., Vaccaro, U.: On the information rate of secret sharing schemes. Theor. Comput. Sci. 154(2), 283–306 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [BSSV97]
    Blundo, C., De Santis, A., De Simone, R., Vaccaro, U.: Tight bounds on the information rate of secret sharing schemes. Des. Codes Crypt. 11(2), 107–122 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Bub86]
    Bublitz, S.: Decomposition of graphs and monotone formula size of homogeneous functions. Acta Inf. 23(6), 689–696 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [CGW15]
    Chen, J., Gay, R., Wee, H.: Improved dual system ABE in prime-order groups via predicate encodings. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 595–624. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46803-6_20 Google Scholar
  14. [CK91]
    Chor, B., Kushilevitz, E.: A zero-one law for boolean privacy. SIAM J. Discrete Math. 4(1), 36–47 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [CKGS98]
    Chor, B., Kushilevitz, E., Goldreich, O., Sudan, M.: Private information retrieval. J. ACM 45(6), 965–981 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Csi97]
    Csirmaz, L.: The size of a share must be large. J. Cryptol. 10(4), 223–231 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [Csi05]
    Csirmaz, L.: Secret sharing schemes on graphs. IACR Cryptology ePrint Archive 2005:59 (2005)Google Scholar
  18. [DG15]
    Dvir, Z., Gopi, S.: 2-server PIR with sub-polynomial communication. In: STOC, pp. 577–584 (2015)Google Scholar
  19. [DGY11]
    Dvir, Z., Gopalan, P., Yekhanin, S.: Matching vector codes. SIAM J. Comput. 40(4), 1154–1178 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [DPP14]
    Data, D., Prabhakaran, M.M., Prabhakaran, V.M.: On the communication complexity of secure computation. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8617, pp. 199–216. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-44381-1_12 CrossRefGoogle Scholar
  21. [Efr09]
    Efremenko, K.: 3-query locally decodable codes of subexponential length. In: STOC, pp. 39–44 (2009)Google Scholar
  22. [EP97]
    Erdös, P., Pyber, L.: Covering a graph by complete bipartite graphs. Discrete Math. 170(1–3), 249–251 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [FKN94]
    Feige, U., Kilian, J., Naor, M.: A minimal model for secure computation (extended abstract). In: Leighton, F.T., Goodrich, M.T. (eds.) Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, 23–25 May 1994, Montréal, Québec, Canada, pp. 554–563. ACM (1994)Google Scholar
  24. [GIKM00]
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [GKW15]
    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). doi: 10.1007/978-3-662-48000-7_24 CrossRefGoogle Scholar
  26. [GPSW06]
    Goyal, V., Pandey, O., Sahai, A., Waters, B.: Attribute-based encryption for fine-grained access control of encrypted data. In: ACM Conference on Computer and Communications Security, pp. 89–98 (2006)Google Scholar
  27. [Gro00]
    Grolmusz, V.: Superpolynomial size set-systems with restricted intersections mod 6 and explicit ramsey graphs. Combinatorica 20(1), 71–86 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [IK97]
    Ishai, Y., Kushilevitz, E.: Private simultaneous messages protocols with applications. In: ISTCS, pp. 174–184 (1997)Google Scholar
  29. [IK00]
    Ishai, Y., Kushilevitz, E.: Randomizing polynomials: a new representation with applications to round-efficient secure computation. In: 41st Annual Symposium on Foundations of Computer Science, FOCS 2000, 12–14 November 2000, Redondo Beach, California, USA, pp. 294–304. IEEE Computer Society (2000)Google Scholar
  30. [IK02]
    Ishai, Y., Kushilevitz, E.: Perfect constant-round secure computation via perfect randomizing polynomials. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 244–256. Springer, Heidelberg (2002). doi: 10.1007/3-540-45465-9_22 CrossRefGoogle Scholar
  31. [ISN89]
    Ito, M., Saito, A., Nishizeki, T.: Secret sharing scheme realizing general access structure. Electron. Commun. Jpn (Part III: Fundam. Electron. Sci.) 72(9), 56–64 (1989)MathSciNetCrossRefGoogle Scholar
  32. [KNR99]
    Kremer, I., Nisan, N., Ron, D.: On randomized one-round communication complexity. Comput. Complex. 8(1), 21–49 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  33. [Lew12]
    Lewko, A.: Tools for simulating features of composite order bilinear groups in the prime order setting. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 318–335. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-29011-4_20 CrossRefGoogle Scholar
  34. [LOS+10]
    Lewko, A.B., Okamoto, T., Sahai, A., Takashima, K., Waters, B.: Fully secure functional encryption: attribute-based encryption and (hierarchical) inner product encryption. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 62–91. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-13190-5_4 CrossRefGoogle Scholar
  35. [LW10]
    Lewko, A.B., Waters, B.: New techniques for dual system encryption and fully secure HIBE with short ciphertexts. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 455–479. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-11799-2_27 CrossRefGoogle Scholar
  36. [LW11]
    Lewko, A.B., Waters, B.: Decentralizing attribute-based encryption. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 568–588. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-20465-4_31 CrossRefGoogle Scholar
  37. [Nay99]
    Nayak, A.: Optimal lower bounds for quantum automata and random access codes. In: FOCS, pp. 369–377 (1999)Google Scholar
  38. [OS08]
    Ostrovsky, R., Skeith III, W.E.: Communication complexity in algebraic two-party protocols. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 379–396. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-85174-5_21 CrossRefGoogle Scholar
  39. [OT10]
    Okamoto, T., Takashima, K.: Fully secure functional encryption with general relations from the decisional linear assumption. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 191–208. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14623-7_11 CrossRefGoogle Scholar
  40. [OT12]
    Okamoto, T., Takashima, K.: Adaptively attribute-hiding (hierarchical) inner product encryption. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 591–608. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-29011-4_35. Also, Cryptology ePrint Archive, Report 2011/543CrossRefGoogle Scholar
  41. [RPRC16]
    Robere, R., Pitassi, T., Rossman, B., Cook, S.A.: Exponential lower bounds for monotone span programs. In: FOCS, pp. 406–415 (2016)Google Scholar
  42. [Sha79]
    Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  43. [SS97]
    Sun, H.-M., Shieh, S.-P.: Secret sharing in graph-based prohibited structures. In: INFOCOM, pp. 718–724 (1997)Google Scholar
  44. [SW05]
    Sahai, A., Waters, B.: Fuzzy identity-based encryption. In: Cramer, R. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 457–473. Springer, Heidelberg (2005). doi: 10.1007/11426639_27 CrossRefGoogle Scholar
  45. [vD95]
    van Dijk, M.: On the information rate of perfect secret sharing schemes. Des. Codes Crypt. 6(2), 143–169 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  46. [VV15]
    Vaikuntanathan, V., Vasudevan, P.N.: Secret sharing and statistical zero knowledge. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9452, pp. 656–680. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48797-6_27 CrossRefGoogle Scholar
  47. [Wat09]
    Waters, B.: Dual system encryption: realizing fully secure IBE and HIBE under simple assumptions. In: Halevi, S. (ed.) CRYPTO 2009. LNCS, vol. 5677, pp. 619–636. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-03356-8_36 CrossRefGoogle Scholar
  48. [Wee14]
    Wee, H.: Dual system encryption via predicate encodings. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 616–637. Springer, Heidelberg (2014). doi: 10.1007/978-3-642-54242-8_26 CrossRefGoogle Scholar
  49. [WY05]
    Woodruff, D.P., Yekhanin, S.: A geometric approach to information-theoretic private information retrieval. In: CCC, pp. 275–284 (2005)Google Scholar
  50. [Yek08]
    Yekhanin, S.: Towards 3-query locally decodable codes of subexponential length. J. ACM 55(1), 1:1–1:16 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2017

Authors and Affiliations

  • Tianren Liu
    • 1
    Email author
  • Vinod Vaikuntanathan
    • 1
  • Hoeteck Wee
    • 2
  1. 1.MITCambridgeUSA
  2. 2.CNRS and ENSParisFrance

Personalised recommendations