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(Finite) Field Work: Choosing the Best Encoding of Numbers for FHE Computation

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11261)

Abstract

Fully Homomorphic Encryption (FHE) schemes operate over finite fields while many use cases call for real numbers, requiring appropriate encoding of the data into the scheme’s plaintext space. However, the choice of encoding can tremendously impact the computational effort on the encrypted data. In this work, we investigate this question for applications that operate over integers and rational numbers using p-adic encoding and the extensions p’s Complement and Sign-Magnitude, based on three natural metrics: the number of finite field additions, multiplications, and multiplicative depth. Our results are partly constructive and partly negative: For the first two metrics, an optimal choice exists and we state it explicitly. However, for multiplicative depth the optimum does not exist globally, but we do show how to choose this best encoding depending on the use-case.

Keywords

Fully Homomorphic Encryption Encoding Efficiency 

References

  1. 1.
    Arita, S., Nakasato, S.: Fully homomorphic encryption for point numbers. IACR Cryptology ePrint Archive 2016/402 (2016)Google Scholar
  2. 2.
    Bonte, C., Bootland, C., Bos, J.W., Castryck, W., Iliashenko, I., Vercauteren, F.: Faster homomorphic function evaluation using non-integral base encoding. IACR Cryptology ePrint Archive 2017/333 (2017)Google Scholar
  3. 3.
    Chen, Y., Gong, G.: Integer arithmetic over ciphertext and homomorphic data aggregation. In: CNS (2015)Google Scholar
  4. 4.
    Cheon, J.H., Kim, A., Kim, M., Song, Y.: Homomorphic encryption for arithmetic of approximate numbers. IACR Cryptology ePrint Archive 2016/421 (2016)Google Scholar
  5. 5.
    Chung, H., Kim, M.: Encoding rational numbers for FHE-based applications. IACR Cryptology ePrint Archive 2016/344(2016)Google Scholar
  6. 6.
    Costache, A., Smart, N.P., Vivek, S., Waller, A.: Fixed point arithmetic in SHE scheme. IACR Cryptology ePrint Archive 2016/250 (2016)Google Scholar
  7. 7.
    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_2CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Jäschke, A., Armknecht, F.: Accelerating homomorphic computations on rational numbers. In: Manulis, M., Sadeghi, A.-R., Schneider, S. (eds.) ACNS 2016. LNCS, vol. 9696, pp. 405–423. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-39555-5_22CrossRefGoogle Scholar
  10. 10.
    Jäschke, A., Armknecht, F.: (Finite) field work: choosing the best encoding of numbers for FHE Computation. IACR Cryptology ePrint Archive 2017/582 (2017)Google Scholar
  11. 11.
    Kim, E., Tibouchi, M.: FHE over the integers and modular arithmetic circuits. In: CANS, pp. 435–450 (2016)CrossRefGoogle Scholar
  12. 12.
    Nuida, K., Kurosawa, K.: (Batch) fully homomorphic encryption over integers for non-binary message spaces. In: Oswald, E., Fischlin, M. (eds.) EUROCRYPT 2015. LNCS, vol. 9056, pp. 537–555. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46800-5_21CrossRefGoogle Scholar
  13. 13.
    Xu, C., Chen, J., Wu, W., Feng, Y.: Homomorphically encrypted arithmetic operations over the integer ring. In: Bao, F., Chen, L., Deng, R.H., Wang, G. (eds.) ISPEC 2016. LNCS, vol. 10060, pp. 167–181. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-49151-6_12CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of MannheimMannheimGermany

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