Skip to main content
SpringerLink
Log in

Homomorphic Secret Sharing from Paillier Encryption

  • Conference paper
  • First Online:
Provable Security (ProvSec 2017)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10592))

Included in the following conference series:

Abstract

A recent breakthrough by Boyle et al. [7] demonstrated secure function evaluation protocols for branching programs, where the communication complexity is sublinear in the size of the circuit (indeed just linear in the size of the inputs, and polynomial in the security parameter). Their result is based on the Decisional Diffie-Hellman assumption (DDH), using (variants of) the ElGamal cryptosystem. In this work, we extend their result to show a construction based on the circular security of the Paillier encryption scheme. We also offer a few optimizations to the scheme, including an alternative to the “Las Vegas”-style share conversion protocols of [7, 9] which directly checks the correctness of the computation. This allows us to reduce the number of required repetitions to achieve a desired overall error bound by a constant fraction for typical cases, and for large programs, reduces the total computation cost.

R. Gennaro—supported by NSF Grant 1565403.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    There is no contradiction here with the hardness of discrete log, since this works only for small values of b.

  2. 2.

    For example in our DCRA-based construction, this would be equivalent to decryption.

  3. 3.

    Differently than in [7] we do not use \(\langle y \rangle \) in the multiplication step – The additive sharing of y however needs to be stored so that we can compute the output at the end.

  4. 4.

    \(\ell \le t\) since additive shares start of size \(\ell \) and then they can grow as the result of addition operations.

  5. 5.

    At least the test is efficient if the factorization of the order of the group is known, as is the case if \(n\) was a product of safe primes.

  6. 6.

    We note that naive polynomial evaluation could also be made reasonable by raising to \(\alpha ^i \bmod n\), since in any abelian group, if \(\prod h_i\in H<G\) with \(|H| = n\), then for any \(k\in \mathbb {Z}\), \((\prod h_i)^k = (\prod h_i)^{k\bmod n} = \prod (h_i^{k\bmod n})\).

References

  1. Acar, T., Belenkiy, M., Bellare, M., Cash, D.: Cryptographic agility and its relation to circular encryption. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 403–422. Springer, Heidelberg (2010). doi:10.1007/978-3-642-13190-5_21

    Chapter  Google Scholar 

  2. Alamati, N., Peikert, C.: Three’s compromised too: circular insecurity for any cycle length from (Ring-)LWE. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 659–680. Springer, Heidelberg (2016). doi:10.1007/978-3-662-53008-5_23

    Chapter  Google Scholar 

  3. Applebaum, B., Ishai, Y., Kushilevitz, E.: Computationally private randomizing polynomials and their applications. Comput. Complex. 15(2), 115–162 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: STOC, pp. 1–10 (1988)

    Google Scholar 

  5. Bishop, A., Hohenberger, S., Waters, B.: New circular security counterexamples from decision linear and learning with errors. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015. LNCS, vol. 9453, pp. 776–800. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48800-3_32

    Chapter  Google Scholar 

  6. Boyle, E., Gilboa, N., Ishai, Y.: Function secret sharing. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9057, pp. 337–367. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46803-6_12

    Google Scholar 

  7. Boyle, E., Gilboa, N., Ishai, Y.: Breaking the circuit size barrier for secure computation under DDH. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9814, pp. 509–539. Springer, Heidelberg (2016). doi:10.1007/978-3-662-53018-4_19

    Chapter  Google Scholar 

  8. Boyle, E., Gilboa, N., Ishai, Y.: Function secret sharing: improvements and extensions. In: Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pp. 1292–1303. ACM (2016)

    Google Scholar 

  9. Boyle, E., Gilboa, N., Ishai, Y.: Group-based secure computation: optimizing rounds, communication, and computation. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017. LNCS, vol. 10211, pp. 163–193. Springer, Cham (2017). doi:10.1007/978-3-319-56614-6_6

    Chapter  Google Scholar 

  10. Cash, D., Green, M., Hohenberger, S.: New definitions and separations for circular security. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 540–557. Springer, Heidelberg (2012). doi:10.1007/978-3-642-30057-8_32

    Chapter  Google Scholar 

  11. Chaum, D., Crépeau, C., Damgard, I.: Multiparty unconditionally secure protocols. In: Proceedings of the Twentieth Annual ACM symposium on Theory of Computing, pp. 11–19. ACM (1988)

    Google Scholar 

  12. Chor, B., Gilboa, N.: Computationally private information retrieval. In: Proceedings of the Twenty-Ninth Annual ACM symposium on Theory of Computing, pp. 304–313. ACM (1997)

    Google Scholar 

  13. Chor, B., Kushilevitz, E., Goldreich, O., Sudan, M.: Private information retrieval. J. ACM 45(6), 965–981 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cohen, J.D., Fischer, M.J.: A robust and verifiable cryptographically secure election scheme (extended abstract). In: 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, USA, pp. 372–382, 21–23 October 1985

    Google Scholar 

  15. Damgård, I., Jurik, M.: A length-flexible threshold cryptosystem with applications. In: Safavi-Naini, R., Seberry, J. (eds.) ACISP 2003. LNCS, vol. 2727, pp. 350–364. Springer, Heidelberg (2003). doi:10.1007/3-540-45067-X_30

    Chapter  Google Scholar 

  16. Damgård, I., Jurik, M.: A generalisation, a simplification and some applications of Paillier’s probabilistic public-key system. In: Kim, K. (ed.) PKC 2001. LNCS, vol. 1992, pp. 119–136. Springer, Heidelberg (2001). doi:10.1007/3-540-44586-2_9

    Chapter  Google Scholar 

  17. Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pp. 169–178. ACM, New York (2009)

    Google Scholar 

  18. Gentry, C., Halevi, S.: Implementing gentry’s fully-homomorphic encryption scheme. Cryptology ePrint Archive, Report 2010/520 (2010)

    Google Scholar 

  19. Gentry, C., Halevi, S., Smart, N.P.: Fully homomorphic encryption with polylog overhead. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 465–482. Springer, Heidelberg (2012). doi:10.1007/978-3-642-29011-4_28

    Chapter  Google Scholar 

  20. Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or a completeness theorem for protocols with honest majority. In: STOC, pp. 218–229 (1987)

    Google Scholar 

  21. Goldwasser, S., Micali, S.: Probabilistic encryption. JCSS 28(2), 270–299 (1984)

    MathSciNet  MATH  Google Scholar 

  22. Goyal, R., Koppula, V., Waters, B.: Separating IND-CPA and circular security for unbounded length key cycles. In: Fehr, S. (ed.) PKC 2017. LNCS, vol. 10174, pp. 232–246. Springer, Heidelberg (2017). doi:10.1007/978-3-662-54365-8_10

    Chapter  Google Scholar 

  23. Ishai, Y., Kushilevitz, E.: Randomizing polynomials: a new representation with applications to round-efficient secure computation. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 294–304. IEEE (2000)

    Google Scholar 

  24. Jurik, M.J.: Extensions to the Paillier cryptosystem with applications to cryptological protocols. In: BRICS (2003)

    Google Scholar 

  25. Koppula, V., Waters, B.: Circular security separations for arbitrary length cycles from LWE. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9815, pp. 681–700. Springer, Heidelberg (2016). doi:10.1007/978-3-662-53008-5_24

    Chapter  Google Scholar 

  26. Kushilevitz, E., Ostrovsky, R.: Replication is not needed: single database, computationally-private information retrieval. In: FOCS, pp. 364–373 (1997)

    Google Scholar 

  27. Marcedone, A., Orlandi, C.: Obfuscation \(\Rightarrow \) (IND-CPA security \(\nRightarrow \) circular security). In: Abdalla, M., De Prisco, R. (eds.) SCN 2014. LNCS, vol. 8642, pp. 77–90. Springer, Cham (2014). doi:10.1007/978-3-319-10879-7_5

    Google Scholar 

  28. Naccache, D., Stern, J.: A new public key cryptosystem based on higher residues. In: Proceedings of the 5th ACM Conference on Computer and Communications Security, pp. 59–66. ACM (1998)

    Google Scholar 

  29. Naehrig, M., Lauter, K., Vaikuntanathan, V.: Can homomorphic encryption be practical? In: Proceedings of the 3rd ACM Workshop on Cloud Computing Security Workshop, pp. 113–124. ACM (2011)

    Google Scholar 

  30. Okamoto, T., Uchiyama, S.: A new public-key cryptosystem as secure as factoring. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 308–318. Springer, Heidelberg (1998). doi:10.1007/BFb0054135

    Google Scholar 

  31. Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999). doi:10.1007/3-540-48910-X_16

    Google Scholar 

  32. Rothblum, R.D.: On the circular security of bit-encryption. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 579–598. Springer, Heidelberg (2013). doi:10.1007/978-3-642-36594-2_32

    Chapter  Google Scholar 

  33. Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wichs, D., Zirdelis, G.: Obfuscating compute-and-compare programs under LWE. Technical report, Cryptology ePrint Archive, Report 2017/276 (2017). http://eprint.iacr.org/2017/276

  35. Yao, A.C.C.: Protocols for secure computations (extended abstract). In: FOCS, pp. 160–164 (1982)

    Google Scholar 

  36. Yao, A.C.C.: How to generate and exchange secrets. In: 27th Annual Symposium on Foundations of Computer Science, pp. 162–167. IEEE (1986)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate Center, The City College, CUNY, New York, NY, USA

    Nelly Fazio, Rosario Gennaro & William E. Skeith III

  2. The Graduate Center, CUNY, New York, NY, USA

    Tahereh Jafarikhah

Authors
  1. Nelly Fazio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rosario Gennaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tahereh Jafarikhah
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. William E. Skeith III
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to William E. Skeith III .

Editor information

Editors and Affiliations

  1. NTT Laboratories, Tokyo, Japan

    Tatsuaki Okamoto

  2. Shaanxi Normal University, Xi’an, China

    Yong Yu

  3. Hong Kong Polytechnic University, Hong Kong, China

    Man Ho Au

  4. University of Wollongong, Wollongong, New South Wales, Australia

    Yannan Li

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Fazio, N., Gennaro, R., Jafarikhah, T., Skeith, W.E. (2017). Homomorphic Secret Sharing from Paillier Encryption. In: Okamoto, T., Yu, Y., Au, M., Li, Y. (eds) Provable Security. ProvSec 2017. Lecture Notes in Computer Science(), vol 10592. Springer, Cham. https://doi.org/10.1007/978-3-319-68637-0_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-68637-0_23

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-68636-3

  • Online ISBN: 978-3-319-68637-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation