Advertisement

Leveled Hierarchical Identity-Based Fully Homomorphic Encryption from Learning with Rounding

  • Fucai Luo
  • Kunpeng Wang
  • Changlu Lin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11125)

Abstract

Hierarchical identity-based fully homomorphic encryption (HIBFHE) aggregates the advantages of both fully homomorphic encryption (FHE) and hierarchical identity-based encryption (HIBE) that permits data encrypted by HIBE to be processed homomorphically. This paper mainly constructs a new leveled HIBFHE scheme based on Learning with Rounding (\(\textsf {LWR}\)) problem, which removes Gaussian noise sampling in encryption process. In more detail, we use the lattice basis delegation method proposed by Agrawal, Boneh and Boyen at CRYPTO 2010 to generate delegated basis, while cleverly exploit a scaled rounding function of LWR problem to hide plaintext rather than adding an auxiliary Gaussian noise matrix. Besides, Gentry, Sahai and Waters constructed the first leveled LWE-based HIBFHE schemes from identity-based encryption scheme at CRYPTO 2013, in this work, however, we also focus on improving their leveled HIBFHE scheme, using Alperin-Sheriff and Peikert’s technically simpler method. We prove that our schemes are adaptively secure under classic lattice hardness assumptions.

Keywords

FHE Hierarchical identity-based encryption Learning with Rounding 

Notes

Acknowledgments

The authors would like to thank the anonymous reviewers for their detailed reviews and helpful comments. This research is supported in part by the National Nature Science Foundation of China (Nos. 61672030, 61272040 and U1705264; Nos. 61572132 and U1705264).

References

  1. 1.
    Agrawal, S., Boneh, D., Boyen, X.: Efficient lattice (H)IBE in the standard model. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 553–572. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13190-5_28CrossRefzbMATHGoogle Scholar
  2. 2.
    Agrawal, S., Boneh, D., Boyen, X.: Lattice basis delegation in fixed dimension and shorter-ciphertext hierarchical IBE. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 98–115. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-14623-7_6CrossRefzbMATHGoogle Scholar
  3. 3.
    Alperin-Sheriff, J., Peikert, C.: Faster bootstrapping with polynomial error. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8616, pp. 297–314. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-44371-2_17CrossRefGoogle Scholar
  4. 4.
    Alwen, J., Krenn, S., Pietrzak, K., Wichs, D.: Learning with rounding, revisited. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 57–74. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40041-4_4CrossRefGoogle Scholar
  5. 5.
    Banerjee, A., Peikert, C., Rosen, A.: Pseudorandom functions and lattices. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 719–737. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-29011-4_42CrossRefGoogle Scholar
  6. 6.
    Bogdanov, A., Guo, S., Masny, D., Richelson, S., Rosen, A.: On the hardness of learning with rounding over small modulus. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016. LNCS, vol. 9562, pp. 209–224. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49096-9_9CrossRefzbMATHGoogle Scholar
  7. 7.
    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).  https://doi.org/10.1007/3-540-44647-8_13CrossRefGoogle Scholar
  8. 8.
    Boneh, D., Gentry, C., Hamburg, M.: Space-efficient identity based encryption without pairings. In: 48th Annual IEEE Symposium on Foundations of Computer Science 2007. FOCS 2007, pp. 647–657. IEEE (2007)Google Scholar
  9. 9.
    Brakerski, Z.: Fully homomorphic encryption without modulus switching from classical GapSVP. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 868–886. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32009-5_50CrossRefGoogle Scholar
  10. 10.
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Innovations in Theoretical Computer Science 2012, Cambridge, MA, USA, 8–10 January 2012, pp. 309–325 (2012)Google Scholar
  11. 11.
    Brakerski, Z., Langlois, A., Peikert, C., Regev, O., Stehlé, D.: Classical hardness of learning with errors. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp. 575–584. ACM (2013)Google Scholar
  12. 12.
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: IEEE 52nd Annual Symposium on Foundations of Computer Science. FOCS 2011, Palm Springs, CA, USA, 22–25 October 2011, pp. 97–106 (2011)Google Scholar
  13. 13.
    Groot Bruinderink, L., Hülsing, A., Lange, T., Yarom, Y.: Flush, gauss, and reload – a cache attack on the BLISS lattice-based signature scheme. In: Gierlichs, B., Poschmann, A.Y. (eds.) CHES 2016. LNCS, vol. 9813, pp. 323–345. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53140-2_16CrossRefGoogle Scholar
  14. 14.
    Cash, D., Hofheinz, D., Kiltz, E., Peikert, C.: Bonsai trees, or how to delegate a lattice basis. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 523–552. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13190-5_27CrossRefGoogle Scholar
  15. 15.
    Clear, M., Hughes, A., Tewari, H.: Homomorphic encryption with access policies: characterization and new constructions. In: Youssef, A., Nitaj, A., Hassanien, A.E. (eds.) AFRICACRYPT 2013. LNCS, vol. 7918, pp. 61–87. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38553-7_4CrossRefGoogle Scholar
  16. 16.
    Clear, M., McGoldrick, C.: Bootstrappable identity-based fully homomorphic encryption. In: Gritzalis, D., Kiayias, A., Askoxylakis, I. (eds.) CANS 2014. LNCS, vol. 8813, pp. 1–19. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-12280-9_1CrossRefGoogle Scholar
  17. 17.
    Cocks, C.: An Identity based encryption scheme based on quadratic residues. In: Honary, B. (ed.) Cryptography and Coding 2001. LNCS, vol. 2260, pp. 360–363. Springer, Heidelberg (2001).  https://doi.org/10.1007/3-540-45325-3_32CrossRefGoogle Scholar
  18. 18.
    Fang, F., Li, B., Lu, X., Liu, Y., Jia, D., Xue, H.: (Deterministic) hierarchical identity-based encryption from learning with rounding over small modulus. In: Proceedings of the 11th ACM on Asia Conference on Computer and Communications Security, pp. 907–912. ACM (2016)Google Scholar
  19. 19.
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing. STOC 2009, Bethesda, MD, USA, 31 May–2 June 2009, pp. 169–178 (2009)Google Scholar
  20. 20.
    Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pp. 197–206. ACM (2008)Google Scholar
  21. 21.
    Gentry, C., Sahai, A., Waters, B.: Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40041-4_5CrossRefGoogle Scholar
  22. 22.
    Gentry, C., Silverberg, A.: Hierarchical ID-based cryptography. In: Zheng, Y. (ed.) ASIACRYPT 2002. LNCS, vol. 2501, pp. 548–566. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-36178-2_34CrossRefGoogle Scholar
  23. 23.
    Halevi, S., Shoup, V.: Bootstrapping for HElib. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 641–670. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46800-5_25CrossRefGoogle Scholar
  24. 24.
    Micciancio, D., Peikert, C.: Trapdoors for lattices: simpler, tighter, faster, smaller. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 700–718. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-29011-4_41CrossRefGoogle Scholar
  25. 25.
    Peikert, C.: Public-key cryptosystems from the worst-case shortest vector problem. In: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, pp. 333–342. ACM (2009)Google Scholar
  26. 26.
    Pessl, P.: Analyzing the shuffling side-channel countermeasure for lattice-based signatures. In: Dunkelman, O., Sanadhya, S.K. (eds.) INDOCRYPT 2016. LNCS, vol. 10095, pp. 153–170. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-49890-4_9CrossRefGoogle Scholar
  27. 27.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. J. ACM (JACM) 56(6), 34 (2009)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Shamir, A.: Identity-based cryptosystems and signature schemes. In: Blakley, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 47–53. Springer, Heidelberg (1985).  https://doi.org/10.1007/3-540-39568-7_5CrossRefGoogle Scholar
  29. 29.
    Sun, X., Yu, J., Wang, T., Sun, Z., Zhang, P.: Efficient identity-based leveled fully homomorphic encryption from RLWE. Secur. Commun. Netw. 9(18), 5155–5165 (2016)CrossRefGoogle Scholar
  30. 30.
    Wang, F., Wang, K., Li, B.: An efficient leveled identity-based FHE. Network and System Security. LNCS, vol. 9408, pp. 303–315. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-25645-0_20CrossRefGoogle Scholar
  31. 31.
    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).  https://doi.org/10.1007/978-3-642-03356-8_36CrossRefGoogle Scholar
  32. 32.
    Xie, X., Xue, R., Zhang, R.: Deterministic public key encryption and identity-based encryption from lattices in the auxiliary-input setting. In: Visconti, I., De Prisco, R. (eds.) SCN 2012. LNCS, vol. 7485, pp. 1–18. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-32928-9_1CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Cyber SecurityUniversity of Chinese Academy of SciencesBeijingChina
  2. 2.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina
  3. 3.College of Mathematic and InformaticsFujian Normal UniversityFuzhouChina

Personalised recommendations