Accelerating Homomorphic Computations on Rational Numbers

  • Angela JäschkeEmail author
  • Frederik Armknecht
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9696)


Fully Homomorphic Encryption (FHE) schemes are conceptually very powerful tools for outsourcing computations on confidential data. However, experience shows that FHE-based solutions are not sufficiently efficient for practical applications yet. Hence, there is a huge interest in improving the performance of applying FHE to concrete use cases. What has been mainly overlooked so far is that not only the FHE schemes themselves contribute to the slowdown, but also the choice of data encoding. While FHE schemes usually allow for homomorphic executions of algebraic operations over finite fields (often \(\mathbb {Z}_2\)), many applications call for different algebraic structures like signed rational numbers. Thus, before an FHE scheme can be used at all, the data needs to be mapped into the structure supported by the FHE scheme.

We show that the choice of the encoding can already incur a significant slowdown of the overall process, which is independent of the efficiency of the employed FHE scheme. We compare different methods for representing signed rational numbers and investigate their impact on the effort needed for processing encrypted values. In addition to forming a new encoding technique which is superior under some circumstances, we also present further techniques to speed up computations on encrypted data under certain conditions, each of independent interest. We confirm our results by experiments.


Confidential machine learning Fully homomorphic encryption Encoding Implementation 


  1. 1.
  2. 2.
  3. 3.
    Armknecht, F., Boyd, C., Carr, C., Gjøsteen, K., Jäschke, A., Reuter, C.A., Strand, M.: A guide to fully homomorphic encryption. IACR Cryptology ePrint Archive (2015/1192)Google Scholar
  4. 4.
    Armknecht, F., Katzenbeisser, S., Peter, A.: Group homomorphic encryption: characterizations, impossibility results, and applications. DCC 67(2), 209–232 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Armknecht, F., Strufe, T.: An efficient distributed privacy-preserving recommendation system. In: Med-Hoc-Net (2011)Google Scholar
  6. 6.
    Aslett, L.J.M., Esperança, P.M., Holmes, C.C.: Encrypted statistical machine learning: new privacy preserving methods. CoRR abs/1508.06845 (2015)Google Scholar
  7. 7.
    Bos, J.W., Lauter, K.E., Naehrig, M.: Private predictive analysis on encrypted medical data. J. Biomed. Inform. 50, 234–243 (2014)CrossRefGoogle Scholar
  8. 8.
    Bost, R., Popa, R.A., Tu, S., Goldwasser, S.: Machine learning classification over encrypted data. In: NDSS (2015)Google Scholar
  9. 9.
    Brakerski, Z., Gentry, C., Vaikuntanathan, V.: Fully homomorphic encryption without bootstrapping. ECCC 18, 111 (2011)Google Scholar
  10. 10.
    Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: FOCS (2011)Google Scholar
  11. 11.
    Cheon, J.H., Kim, M., Lauter, K.: Homomorphic computation of edit distance. In: Brenner, M., Christin, N., Johnson, B., Rohloff, K. (eds.) FC 2015 Workshops. LNCS, vol. 8976, pp. 194–212. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  12. 12.
    Chung, H., Kim, M.: Encoding rational numbers for fhe-based applications. IACR Cryptology ePrint Archive (2016/344)Google Scholar
  13. 13.
    Coron, J.-S., Lepoint, T., Tibouchi, M.: Scale-invariant fully homomorphic encryption over the integers. In: Krawczyk, H. (ed.) PKC 2014. LNCS, vol. 8383, pp. 311–328. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  14. 14.
    Coron, J.-S., Naccache, D., Tibouchi, M.: Public key compression and modulus switching for fully homomorphic encryption over the integers. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 446–464. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Costache, A., Smart, N.P., Vivek, S., Waller, A.: Fixed point arithmetic in SHE scheme. IACR Cryptology ePrint Archive (2016/250)Google Scholar
  16. 16.
    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)CrossRefGoogle Scholar
  17. 17.
    Dowlin, N., Gilad-Bachrach, R., Laine, K., Lauter, K., Naehrig, M., Wernsing, J.: Manual for using homomorphic encryption for bioinformatics. Technical report MSR-TR-2015-87, Microsoft Research (2015)Google Scholar
  18. 18.
    Ducas, L., Micciancio, D.: FHEW: bootstrapping homomorphic encryption in less than a second. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 617–640. Springer, Heidelberg (2015)Google Scholar
  19. 19.
    Gentry, C.: A fully homomorphic encryption scheme. Ph.D. thesis, Stanford University (2009)Google Scholar
  20. 20.
    Gentry, C., Halevi, S., Smart, N.P.: Homomorphic evaluation of the AES circuit. In: Canetti, R., Safavi-Naini, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 850–867. Springer, Heidelberg (2012)CrossRefGoogle 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, Part I. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  22. 22.
    Graepel, T., Lauter, K., Naehrig, M.: ML confidential: machine learning on encrypted data. In: Kwon, T., Lee, M.-K., Kwon, D. (eds.) ICISC 2012. LNCS, vol. 7839, pp. 1–21. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  23. 23.
    Henecka, W., Kögl, S., Sadeghi, A., Schneider, T., Wehrenberg, I.: TASTY: tool for automating secure two-party computations. In: CCS (2010)Google Scholar
  24. 24.
    Naehrig, M., Lauter, K.E., Vaikuntanathan, V.: Can homomorphic encryption be practical? In: CCSW (2011)Google Scholar
  25. 25.
    Sadeghi, A.-R., Schneider, T.: Generalized universal circuits for secure evaluation of private functions with application to data classification. In: Lee, P.J., Cheon, J.H. (eds.) ICISC 2008. LNCS, vol. 5461, pp. 336–353. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  26. 26.
    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)CrossRefGoogle Scholar
  27. 27.
    Songhori, E.M., Hussain, S.U., Sadeghi, A., Schneider, T., Koushanfar, F.: Tinygarble: highly compressed and scalable sequential garbled circuits. In: SP (2015)Google Scholar
  28. 28.
    Wu, D.J., Feng, T., Naehrig, M., Lauter, K.E.: Privately evaluating decision trees and random forests. IACR Cryptology ePrint Archive (2015/386)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of MannheimMannheimGermany

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