Faster ECC over \(\mathbb {F}_{2^{521}-1}\) (feat. NEON)

  • Hwajeong Seo
  • Zhe Liu
  • Yasuyuki Nogami
  • Taehwan Park
  • Jongseok Choi
  • Lu Zhou
  • Howon KimEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9558)


In this paper, we present high speed parallel multiplication and squaring algorithms for the Mersenne prime \(2^{521}-1\). We exploit 1-level Karatsuba method in order to provide asymptotically faster integer multiplication and fast reduction algorithms. With these optimization techniques, ECDH on NIST’s (and SECG’s) curve P-521 requires 8.1/4 M cycles on an ARM Cortex-A9/A15, respectively. As a comparison, on the same architecture, the latest OpenSSL 1.0.2d’s ECDH speed test for curve P-521 requires 23.8/18.7 M cycles for ARM Cortex-A9/A15, respectively.


Elliptic Curve Cryptography P-521 Karatsuba SIMD NEON 


  1. 1.
    Bernstein, D.J., Chuengsatiansup, C., Lange, T.: Curve41417: karatsuba revisited. In: Batina, L., Robshaw, M. (eds.) CHES 2014. LNCS, vol. 8731, pp. 316–334. Springer, Heidelberg (2014)Google Scholar
  2. 2.
    Bos, J.W., Kaihara, M.E.: Montgomery multiplication on the cell. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds.) PPAM 2009, Part I. LNCS, vol. 6067, pp. 477–485. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Standard for Efficient Cryptography Group: Recommended elliptic curve domain parameters (2000)Google Scholar
  4. 4.
    Granger, R., Scott, M.: Faster ECC over \(\mathbb{F}_{2^{521}-1}\). In: Katz, J. (ed.) PKC 2015. LNCS, vol. 9020, pp. 539–553. Springer, Heidelberg (2015)Google Scholar
  5. 5.
    Hamburg, M.: Ed448-goldilocks, a new elliptic curveGoogle Scholar
  6. 6.
    Intel Corporation.: Using streaming SIMD extensions (SSE2) to perform big multiplications, Application note AP-941 (2000).
  7. 7.
    U.D. of Commerce/N.I.S.T. Federal information processing standards publication 186–2 fipps 186–2 digital signature standardGoogle Scholar
  8. 8.
    Pabbuleti, K.C., Mane, D.H., Desai, A., Albert, C., Schaumont, P.: SIMD acceleration of modular arithmetic on contemporary embedded platforms. In: High Performance Extreme Computing Conference (HPEC), pp. 1–6. IEEE (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Hwajeong Seo
    • 1
  • Zhe Liu
    • 2
  • Yasuyuki Nogami
    • 3
  • Taehwan Park
    • 1
  • Jongseok Choi
    • 1
  • Lu Zhou
    • 4
  • Howon Kim
    • 1
    Email author
  1. 1.School of Computer Science and EngineeringPusan National UniversityBusanRepublic of Korea
  2. 2.Laboratory of Algorithmics, Cryptology and Security (LACS)University of LuxembourgLuxembourg-KirchbergLuxembourg
  3. 3.Graduate School of Natural Science and TechnologyOkayama UniversityOkayamaJapan
  4. 4.School of Computer Science and TechnologyShandong UniversityJinanChina

Personalised recommendations