Skip to main content

Hardware Optimization on FPGA for the Modular Multiplication in the AMNS Representation

  • Conference paper
  • First Online:
Risks and Security of Internet and Systems (CRiSIS 2019)

Abstract

This paper describes our results of the AMNS modular multiplication algorithm for efficient implementations of ECC over \(\mathbb {F}_p\) on the Hardware/Software (HW/SW) implementation in FPGA. We provide both arithmetic operators and computation architectures optimized for high speed. We also compare our results with the implementation of the CIOS method for modular multiplication.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abid, M.: System-level hardware synthesis of dataflow programs with HEVC as study use case. Ph.D. thesis, Bretagne Loire University, France (2016)

    Google Scholar 

  2. Bajard, J.-C., Imbert, L., Plantard, T.: Modular number systems: beyond the mersenne family. In: Handschuh, H., Hasan, M.A. (eds.) SAC 2004. LNCS, vol. 3357, pp. 159–169. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30564-4_11

    Chapter  Google Scholar 

  3. Bajard, J.C., Imbert, L., Plantard, T.: Arithmetic operations in the polynomial modular number system. In: 17th IEEE Symposium on Computer Arithmetic (ARITH-17) 2005, Cape Cod, MA, USA, pp. 206–213 (2005). https://hal-lirmm.ccsd.cnrs.fr/lirmm-00109201/document

  4. Barrett, P.: Implementing the Rivest Shamir and Adleman public key encryption algorithm on a standard digital signal processor. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 311–323. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-47721-7_24

    Chapter  Google Scholar 

  5. Blakely, G.R.: A computer algorithm for calculating the product AB modulo M. IEEE Trans. Comput. C–32(5), 497–500 (1983). https://doi.org/10.1109/TC.1983.1676262

    Article  Google Scholar 

  6. Didier, L.S., Dosso, F.Y., Véron, P.: Efficient modular operations using the adapted modular number system (2019). https://arxiv.org/abs/1901.11485

  7. Dussé, S.R., Kaliski, B.S.: A cryptographic library for the Motorola DSP56000. In: Damgård, I.B. (ed.) EUROCRYPT 1990. LNCS, vol. 473, pp. 230–244. Springer, Heidelberg (1991). https://doi.org/10.1007/3-540-46877-3_21

    Chapter  Google Scholar 

  8. El Mrabet, N., Gama, N.: Efficient multiplication over extension fields. In: Özbudak, F., Rodríguez-Henríquez, F. (eds.) WAIFI 2012. LNCS, vol. 7369, pp. 136–151. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31662-3_10. Kindly check and confirm the updated details are correct in Ref. [9].

    Chapter  MATH  Google Scholar 

  9. Alrimeih, H., Rakhmatov, D.: Fast and flexible hardware support for ECC over multiple standard prime fields. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 22, 2661–2674 (2014)

    Google Scholar 

  10. Hrica, J.: Floating-point design with Vivado HLS. https://www.xilinx.com/support/documentation/application_notes/xapp599-floating-point-vivado-hls.pdf

  11. Koc, C.K., Acar, T., Kaliski, B.S.: Analyzing and comparing montgomery multiplication algorithms. IEEE Micro 16(3), 26–33 (1996)

    Article  Google Scholar 

  12. Montgomery, P.L.: Modular multiplication without trial division. Math. Comput. 44(170), 519–521 (1985)

    Article  MathSciNet  Google Scholar 

  13. Negre, C., Plantard, T.: Efficient modular arithmetic in adapted modular number system using lagrange representation. In: Mu, Y., Susilo, W., Seberry, J. (eds.) ACISP 2008. LNCS, vol. 5107, pp. 463–477. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70500-0_34

    Chapter  Google Scholar 

  14. Plantard, T.: Arithmétique modulaire pour la cryptographie. Ph.D. thesis, Montpellier 2 University, France (2005)

    Google Scholar 

  15. Takagi, N.: A radix-4 modular multiplication hardware algorithm for modular exponentiation. IEEE Trans. Comput. 41(8), 949–956 (1992). https://doi.org/10.1109/12.156537

    Article  Google Scholar 

  16. Tambara, L.A., et al.: Analyzing reliability and performance trade-offs of HLS-based designs in SRAM-based FPGAS under soft errors. IEEE Trans. Nucl. Sci. 64(2), 874–881 (2017)

    Article  Google Scholar 

  17. Taylor, F.: Large moduli multipliers for signal processing. IEEE Trans. Circuits Syst. 28(7), 731–736 (1981)

    Article  MathSciNet  Google Scholar 

  18. Xilinx: Zynq evaluation and development (hardware user’s guide) (2014). http://zedboard.org/sites/default/files/documentations/ZedBoard_HW_UG_v2_2.pdf

  19. Xilinx: Zynq-7000 SoC data sheet: overview (2018). https://www.xilinx.com/support/documentation/data_sheets/ds190-Zynq-7000-Overview.pdf

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Asma Chaouch or Nadia El Mrabet .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Chaouch, A., Dosso, Y.F., Didier, LS., El Mrabet, N., Ouni, B., Bouallegue, B. (2020). Hardware Optimization on FPGA for the Modular Multiplication in the AMNS Representation. In: Kallel, S., Cuppens, F., Cuppens-Boulahia, N., Hadj Kacem, A. (eds) Risks and Security of Internet and Systems. CRiSIS 2019. Lecture Notes in Computer Science(), vol 12026. Springer, Cham. https://doi.org/10.1007/978-3-030-41568-6_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-41568-6_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-41567-9

  • Online ISBN: 978-3-030-41568-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics