Compact (Targeted Homomorphic) Inner Product Encryption from LWE

  • Jie Li
  • Daode ZhangEmail author
  • Xianhui Lu
  • Kunpeng Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10631)


Inner product encryption (IPE) is a public-key encryption mechanism that supports fine-grained access control. Agrawal et al. (ASIACRYPT 2011) proposed the first IPE scheme from the Learning With Errors (LWE) problem. In their scheme, the public parameter size and ciphertext size are \(O(un^2\log ^3n)\) and \(O(un\log ^3n)\), respectively. Then, Xagawa (PKC 2013) proposed the improved scheme with public parameter of size \(O(un^2\log ^2n)\) and ciphertext of size \(O(un\log ^2n)\).

In this paper, we construct a more compact IPE scheme under the LWE assumption, which has public parameter of size \(O(un^2\log n)\) and ciphertext of size \(O(un\log n)\). Thus our scheme improves the size of Xagawa’s IPE scheme by a factor of \(\log n\).

Inspired by the idea of Brakerski et al. (TCC 2016), we propose a targeted homomorphic IPE (THIPE) scheme based on our IPE scheme. Compared with Brakerski et al.’s scheme, our THIPE scheme has more compact public parameters and ciphertexts. However, our scheme can only apply to the inner product case, while in their scheme the predicate f can be any efficiently computable polynomial.


Inner product encryption Homomorphic encryption Learning with errors 



We thank the anonymous ICICS’2017 reviewers for their helpful comments. This work is supported by the National Basic Research Program of China (973 project, No. 2014CB340603) and the National Nature Science Foundation of China (No. 61672030).


  1. 1.
    Apon, D., Fan, X., Liu, F.: Compact identity based encryption from LWE.
  2. 2.
    Agrawal, S., Freeman, D.M., Vaikuntanathan, V.: Functional encryption for inner product predicates from learning with errors. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 21–40. Springer, Heidelberg (2011). Scholar
  3. 3.
    Ajtai, M.: Generating hard instances of the short basis problem. In: Wiedermann, J., van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 1–9. Springer, Heidelberg (1999). Scholar
  4. 4.
    Attrapadung, N., Libert, B.: Functional encryption for inner product: achieving constant-size ciphertexts with adaptive security or support for negation. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 384–402. Springer, Heidelberg (2010). Scholar
  5. 5.
    Alwen, J., Peikert, C.: Generating shorter bases for hard random lattices. Theory Comput. Syst. 48, 535–553 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brakerski, Z., Cash, D., Tsabary, R., Wee, H.: Targeted homomorphic attribute-based encryption. In: Hirt, M., Smith, A. (eds.) TCC 2016. LNCS, vol. 9986, pp. 330–360. Springer, Heidelberg (2016). Scholar
  7. 7.
    Boneh, D., Gentry, C., Gorbunov, S., Halevi, S., Nikolaenko, V., Segev, G., Vaikuntanathan, V., Vinayagamurthy, D.: Fully key-homomorphic encryption, arithmetic circuit ABE and compact Garbled circuits. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 533–556. Springer, Heidelberg (2014). Scholar
  8. 8.
    Boneh, D., Waters, B.: Conjunctive, subset, and range queries on encrypted data. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 535–554. Springer, Heidelberg (2007). Scholar
  9. 9.
    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). Scholar
  10. 10.
    Katz, J., Sahai, A., Waters, B.: Predicate encryption supporting disjunctions, polynomial equations, and inner products. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 146–162. Springer, Heidelberg (2008). Scholar
  11. 11.
    Lewko, A., 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). Scholar
  12. 12.
    Okamoto, T., Takashima, K.: Hierarchical predicate encryption for inner-products. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 214–231. Springer, Heidelberg (2009). Scholar
  13. 13.
    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). Scholar
  14. 14.
    Okamoto, T., Takashima, K.: Achieving short ciphertexts or short secret-keys for adaptively secure general inner-product encryption. In: Lin, D., Tsudik, G., Wang, X. (eds.) CANS 2011. LNCS, vol. 7092, pp. 138–159. Springer, Heidelberg (2011). Scholar
  15. 15.
    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). Scholar
  16. 16.
    Park, J.-H.: Inner-product encryption under standard assumptions. Des. Codes Crypt. 58, 235–257 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Xagawa, K.: Improved (Hierarchical) Inner-Product Encryption from Lattices. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 235–252. Springer, Heidelberg (2013). Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Jie Li
    • 1
    • 2
    • 3
  • Daode Zhang
    • 1
    • 2
    • 3
    Email author
  • Xianhui Lu
    • 1
    • 2
    • 3
  • Kunpeng Wang
    • 1
    • 2
    • 3
  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.Science and Technology on Communication Security LaboratoryBeijingChina

Personalised recommendations