Efficient Scalar Multiplications on Elliptic Curves without Repeated Doublings and Their Practical Performance

  • Yasuyuki Sakai
  • Kouichi Sakurai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1841)

Abstract

We introduce efficient algorithms for scalar multiplication on elliptic curves defined over \(\mathbb{F}_p\). The algorithms compute 2 k P directly from P, where P is a random point on an elliptic curve, without computing the intermediate points, which is faster than k repeated doublings. Moreover, we apply the algorithms to scalar multiplication on elliptic curves, and analyze their computational complexity. As a result of their implementation with respect to affine (resp. weighted projective) coordinates, we achieved an increased performance factor of 1.45 (45%) (resp. 1.15 (15%)) in the scalar multiplication of the elliptic curve of size 160-bit.

Keywords

Elliptic Curve Cryptosystems Scalar Multiplication Window Method Coordinate System Implementation 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. CC86.
    Chudnovsky, D.V., Chudnovsky, G.V.: Sequences of numbers generated by addition in formal groups and new primality and factorization tests. Advances in Applied Math. 7, 385–434 (1986)MATHMathSciNetCrossRefGoogle Scholar
  2. CMO98.
    Cohen, H., Miyaji, A., Ono, T.: Efficient Elliptic Curve Exponentiation Using Mixed Coordinates. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 51–65. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. Go98.
    Gordon, D.M.: A survey of fast exponentiation methods. Journal of Algorithms 27, 129–146 (1998)MATHMathSciNetCrossRefGoogle Scholar
  4. GP97.
    Guajardo, J., Paar, C.: Efficient Algorithms for Elliptic Curve Cryptosystems. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 342–356. Springer, Heidelberg (1997)Google Scholar
  5. HT99.
    Han, Y., Tan, P.C.: Direct Computation for Elliptic Curve Cryptosystems. In: Pre-proceedings of CHES 1999, pp. 328–340. Springer, Heidelberg (1999)Google Scholar
  6. IEEE.
    IEEE P1363/D11 (Draft Version 11) (1999), http://grouper.ieee.org/groups/1363/
  7. ITTTK98.
    Itoh, K., Takenaka, M., Torii, N., Temma, S., Kurihara, Y.: Fast Implementation of Public-key Cryptography on a DSP TMS320C6201. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 61–72. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  8. Ko87.
    Koblitz, N.: Elliptic curve cryptosystems. Mathematics of Computation 48, 203–209 (1987)MATHMathSciNetCrossRefGoogle Scholar
  9. KT92.
    Koyama, K., Tsuruoka, Y.: Speeding up Elliptic Cryptosystems by Using a Signed Binary Window Method. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 345–357. Springer, Heidelberg (1993)Google Scholar
  10. Le38.
    Lehmer, D.H.: Euclid’s Algorithm for Large Numbers. American Mathematical Monthly 45, 227–233 (1938)MathSciNetCrossRefGoogle Scholar
  11. Mi85.
    Miller, V.: Uses of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)Google Scholar
  12. MOC97a.
    Miyaji, A., Ono, T., Cohen, H.: Efficient Elliptic Curve Exponentiation (I). Technical Report of IEICE, ISEC97-16 (1997) Google Scholar
  13. MOC97b.
    Miyaji, A., Ono, T., Cohen, H.: Efficient Elliptic Curve Exponentiation. In: Han, Y., Quing, S. (eds.) ICICS 1997. LNCS, vol. 1334, pp. 282–290. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  14. Mu97.
    Müller, V.: Efficient Algorithms for Multiplication on Elliptic Curves, Proceedings of GI-Arbeitskonferenz Chipkarten 1998, TU München (1998) Google Scholar
  15. NIST.
    Nist, Recommended Elliptic Curves for Federal Government Use (1999), http://csrc.nist.gov/encryption/
  16. So97.
    Solinas, J.A.: An Improved Algorithm for Arithmetic on a Family of Elliptic Curves. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 357–371. Springer, Heidelberg (1997)Google Scholar
  17. WMPW98.
    De Win, E., Mister, S., Preneel, B., Wiener, M.: On the Performance of Signature Schemes Based on Elliptic Curves. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 252–266. Springer, Heidelberg (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Yasuyuki Sakai
    • 1
  • Kouichi Sakurai
    • 2
  1. 1.Mitsubishi Electric CorporationKanagawaJapan
  2. 2.Kyushu UniversityFukuokaJapan

Personalised recommendations