Fast Implementation of Elliptic Curve Arithmetic in GF(pn)

  • Chae Hoon Lim
  • Hyo Sun Hwang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1751)

Abstract

Elliptic curve cryptosystems have attracted much attention in recent years and one of major interests in ECC is to develop fast algorithms for field/elliptic curve arithmetic. In this paper we present various improvement techniques for field arithmetic in GF(pn)(p a prime), in particular, fast field multiplication and inversion algorithms, and provide our implementation results on Pentium II and Alpha 21164 microprocessors.

References

  1. 1.
    Agnew, G.B., Mullin, R.C., Vanstone, S.A.: An implementation of elliptic curve cryptosystems over F\(_{2^155}\). IEEE J. Selected Areas in Commum. 11(5), 804–813 (1993)CrossRefGoogle Scholar
  2. 2.
    Bailey, D.V., Paar, C.: Optimal extension field for fast arithmetic in public key algorithms. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 472–485. Springer, Heidelberg (1998)Google Scholar
  3. 3.
    Bailey, D.V., Paar, C.: Elliptic curve cryptosystems over large characteristic extension fields (1999) (preprint) Google Scholar
  4. 4.
    Cheon, J.H., Park, S.M., Park, S.W., Kim, D.H.: Two efficient algorithms for arithmetic of elliptic curves using Frobenius map. In: Imai, H., Zheng, Y. (eds.) PKC 1998. LNCS, vol. 1431, pp. 195–202. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. 5.
    Cohen, H.: A course in computational number theory. Graduate Texts in Math., vol. 138. Springer, Heidelberg (1993) (Third corrected printing (1996))Google Scholar
  6. 6.
    Cohen, H., Miyaji, A., Ono, T.: Efficient elliptic curve exponentiation. In: Han, Y., Quing, S. (eds.) ICICS 1997. LNCS, vol. 1334, pp. 282–290. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  7. 7.
    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. 50–65. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Kobayashi, T., Morita, H., Kobayashi, K., Hoshino, F.: Fast elliptic curve algorithm combining frobenius map and table reference to adapt to higher characteristic. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 176–189. Springer, Heidelberg (1999)Google Scholar
  10. 10.
    Koblitz, N.: Elliptic curve cryptosystems. Math. Comp. 48, 203–209 (1987)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Koblitz, N.: CM curves with good cryptographic properties. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 279–287. Springer, Heidelberg (1992)Google Scholar
  12. 12.
    Koyama, K., Tsuruoka, Y.: Speeding up elliptic cryptosystems using a signed binary method. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 345–357. Springer, Heidelberg (1993)Google Scholar
  13. 13.
    Knuth, D.E.: The art of Computer Programming: Seminumerical Algorithms, 3rd edn. Addison Wesley, Reading (1998)MATHGoogle Scholar
  14. 14.
    Lim, C.H., Lee, P.J.: A key recovery attack on discrete log-based schemes using a prime order subgroup. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 249–263. Springer, Heidelberg (1997)Google Scholar
  15. 15.
    Lim, C.H., Hwang, H.S.: Fast elliptic scalar multiplication with precomputation (1999) (preprint)Google Scholar
  16. 16.
    Lopez, J., Dahab, R.: Improved algorithms for elliptic curve arithmetic in GF(2n). In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 201–212. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  17. 17.
    Lopez, J., Dahab, R.: Fast multiplication on elliptic curves over GF(2m) without precomputation. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, p. 316. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  18. 18.
    Meier, W., Staffelbach, O.: Efficient multiplication on certain non-supersingular elliptic curves. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 333–344. Springer, Heidelberg (1993)Google Scholar
  19. 19.
    Miller, V.S.: Use of elliptic curves in cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)Google Scholar
  20. 20.
    Montgomery, P.L.: Speeding the Pollard and elliptic curve methods of factorization. Math. of Computation 48(177), 243–264 (1987)MATHCrossRefGoogle Scholar
  21. 21.
    Muller, V.: Fast multiplication on elliptic curves over small fields of characteristic two. J. of Cryptology 11(4), 219–234 (1998)CrossRefGoogle Scholar
  22. 22.
    Schroeppel, A., Orman, H., O’Malley, S., Spatschek, O.: Fast key exchange with elliptic curve systems. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 43–56. Springer, Heidelberg (1995)Google Scholar
  23. 23.
    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
  24. 24.
    de Win, E., Bosselaers, A., Vandenberghe, S.: A fast software implementation for arithmetic operations in GF(2n). In: Kim, K.-c., Matsumoto, T. (eds.) ASIACRYPT 1996. LNCS, vol. 1163, pp. 65–76. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  25. 25.
    Wiener, M.J., Zuccherato, R.J.: Faster attacks on elliptic curve cryptosystems. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 190–200. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  26. 26.
    IEEE P1363: Standard Specifications for Public Key Cryptography, Working Draft (October 1998)Google Scholar
  27. 27.
    ANSI X9.62: The elliptic curve digital signature algorithm, Working Draft (October 1998) Google Scholar
  28. 28.
    ANSI X9.63: Elliptic curve key agreement and key transport protocols, Working Draft (October 1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Chae Hoon Lim
    • 1
  • Hyo Sun Hwang
    • 1
  1. 1.Future Systems, Inc.Information and Communications Research CenterSeoulKorea

Personalised recommendations