Leveled FHE with Matrix Message Space

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10726)

Abstract

Up to now, almost all fully homomorphic encryption (FHE) schemes can only encrypt bit or vector. In PKC 2015, Hiromasa et al. [12] constructed the only leveled FHE scheme that encrypts matrices and supports homomorphic matrix addition and multiplication. But the ciphertext size of their scheme is somewhat large and the security of their scheme depends on some special kind of circular security assumption.

We propose a leveled FHE scheme that encrypts matrices and supports homomorphic matrix addition, multiplication and Hadamard product. It can be viewed as matrix-packed FHE, and has much smaller ciphertext size. Its security is only based on LWE assumption. In particular, the advantages of our scheme are:
  1. 1.

    Supporting homomorphic matrix Hadamard product. All entries in plaintext matrices can be viewed as plaintext slots. While the scheme in [12] doesn’t support this homomorphic operation and only the diagonal entries of plaintext matrix can be viewed as plaintext slots.

     
  2. 2.

    Small ciphertext size. For a plaintext matrix \(\varvec{M} \in \{0,1\}^{r\times r}\), the size of ciphertext matrix is \(r\times (n+r)\), in contrast to \((n+r)\times (n+r)\lceil \log q\rceil \) in [12].

     
  3. 3.

    Standard assumption. The security is based on LWE assumption merely, while the security of scheme in [12] depends additionally on some special kind of circular security assumption.

     

As Brakerski’s work [3] in CRYPTO 2012, our scheme can be improved in efficiency by using ring-LWE (RLWE).

Keywords

Fully homomorphic encryption LWE Matrix Packing 

Notes

Acknowledgment

This work is supported by National Natural Science Foundation of China (No. 61402471, 61472414, 61602061, 61772514).

References

  1. 1.
    Brakerski, Z., Gentry, C., Halevi, S.: Packed ciphertexts in LWE-based homomorphic encryption. In: Kurosawa, K., Hanaoka, G. (eds.) PKC 2013. LNCS, vol. 7778, pp. 1–13. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36362-7_1 CrossRefGoogle Scholar
  2. 2.
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: ITCS, pp. 309–325 (2012), Full Version, http://people.csail.mit.edu/vinodv/6892-Fall2013/BGV.pdf
  3. 3.
    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_50 CrossRefGoogle Scholar
  4. 4.
    Brakerski, Z., Vaikuntanathan, V.: Fully homomorphic encryption from ring-LWE and security for key dependent messages. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 505–524. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22792-9_29 CrossRefGoogle Scholar
  5. 5.
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: FOCS, pp. 97–106 (2011)Google Scholar
  6. 6.
    van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully homomorphic encryption over the integers. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 24–43. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13190-5_2 CrossRefGoogle Scholar
  7. 7.
    Gentry, C.: A Fully Homomorphic Encryption Scheme. PhD thesis. Stanford University (2009). http://crypto.stanford.edu/craig
  8. 8.
    Gentry, C.: Fully homomorphic encryption using ideal lattices. In: STOC, pp. 169–178 (2009)Google Scholar
  9. 9.
    Gentry, C., Halevi, S., Smart, N.P.: Better bootstrapping in fully homomorphic encryption. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 1–16. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-30057-8_1 CrossRefGoogle Scholar
  10. 10.
    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).  https://doi.org/10.1007/978-3-642-29011-4_28 CrossRefGoogle Scholar
  11. 11.
    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_5 CrossRefGoogle Scholar
  12. 12.
    Hiromasa, R., Abe, M., Okamoto, T.: Packing messages and optimizing bootstrapping in GSW-FHE. In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 699–715. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46447-2_31 Google Scholar
  13. 13.
    Micciancio, D., Voulgaris, P.: A deterministic single exponential time algorithm for most lattice problems based on voronoi cell computations. In: Schulman, L.J. (ed.) STOC, pp. 351–358. ACM (2010)Google Scholar
  14. 14.
    Peikert, C.: Public-key cryptosystems from the worst-case shortest vector problem. In: STOC, pp. 333–342. ACM (2009)Google Scholar
  15. 15.
    Peikert, C., Vaikuntanathan, V., Waters, B.: A framework for efficient and composable oblivious transfer. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 554–571. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-85174-5_31 CrossRefGoogle Scholar
  16. 16.
    Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) STOC, pp. 84–93. ACM, New York (2005)Google Scholar
  17. 17.
    Rivest, R., Adleman, L., Dertouzos, M.: On data banks and privacy homomorphisms. In: Foundations of Secure Computation, pp. 169–180 (1978)Google Scholar
  18. 18.
    Rothblum, R.: Homomorphic encryption: from private-key to public-key. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 219–234. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-19571-6_14 CrossRefGoogle Scholar
  19. 19.
    Smart, N.P., Vercauteren, F.: Fully homomorphic encryption with relatively small key and ciphertext sizes. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 420–443. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-13013-7_25 CrossRefGoogle Scholar
  20. 20.
    Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. Des. Codes Crypt. 71(1), 57–81 (2014)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina
  2. 2.School of Cyber SecurityUniversity of Chinese Academy of SciencesBeijingChina

Personalised recommendations