Skip to main content
Springer Nature Link
Log in

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

  • Conference paper
  • First Online:
Cryptology and Network Security (CANS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11261))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    The term \(p^k\)-adic encoding denotes the natural extension of p-adic encoding to the field \(GF(p^k)\) for \(k\ge 1\) and is explained in Sect. 6.

References

  1. Arita, S., Nakasato, S.: Fully homomorphic encryption for point numbers. IACR Cryptology ePrint Archive 2016/402 (2016)

    Google Scholar 

  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. Chen, Y., Gong, G.: Integer arithmetic over ciphertext and homomorphic data aggregation. In: CNS (2015)

    Google Scholar 

  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. Chung, H., Kim, M.: Encoding rational numbers for FHE-based applications. IACR Cryptology ePrint Archive 2016/344(2016)

    Google Scholar 

  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. 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

    Chapter  Google Scholar 

  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. 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_22

    Chapter  Google Scholar 

  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. Kim, E., Tibouchi, M.: FHE over the integers and modular arithmetic circuits. In: CANS, pp. 435–450 (2016)

    Chapter  Google Scholar 

  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_21

    Chapter  Google Scholar 

  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_12

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Mannheim, Mannheim, Germany

    Angela Jäschke & Frederik Armknecht

Authors
  1. Angela Jäschke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frederik Armknecht
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Angela Jäschke .

Editor information

Editors and Affiliations

  1. ETH Zürich, Zürich, Switzerland

    Srdjan Capkun

  2. Chinese University of Hong Kong, Shatin, Hong Kong

    Sherman S. M. Chow

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Jäschke, A., Armknecht, F. (2018). (Finite) Field Work: Choosing the Best Encoding of Numbers for FHE Computation. In: Capkun, S., Chow, S. (eds) Cryptology and Network Security. CANS 2017. Lecture Notes in Computer Science(), vol 11261. Springer, Cham. https://doi.org/10.1007/978-3-030-02641-7_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-02641-7_23

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-02640-0

  • Online ISBN: 978-3-030-02641-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics