New Architecture for Multiplication in GF(2m) and Comparisons with Normal and Polynomial Basis Multipliers for Elliptic Curve Cryptography

  • Soonhak Kwon
  • Taekyoung Kwon
  • Young-Ho Park
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3935)


We propose a new linear multiplier which is comparable to linear polynomial basis multipliers in terms of the area and time complexity. Also we give a very detailed comparison of our multiplier with the normal and polynomial basis multipliers for the five binary fields GF(2 m ), m=163,233,283,409,571, recommended by NIST for elliptic curve digital signature algorithm.


linear multiplier NIST recommended binary fields elliptic curve cryptography 


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  1. 1.
    Massy, J.L., Omura, J.K.: Computational method and apparatus for finite field arithmetic, US Patent No. 4587627 (1986)Google Scholar
  2. 2.
    Agnew, G.B., Mullin, R.C., Onyszchuk, I., Vanstone, S.A.: An implementation for a fast public key cryptosystem. J. Cryptology 3, 63–79 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Wu, H., Hasan, M.A., Blake, I.F.: New low complexity bit-parallel finite field multipliers using weakly dual bases. IEEE Trans. Computers 47, 1223–1234 (1998)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Reyhani-Masoleh, A., Hasan, M.A.: Low complexity sequential normal basis multipliers over GF(2m). In: 16th IEEE Symposium on Computer Arithmetic, vol.16, pp. 188–195 (2003)Google Scholar
  5. 5.
    Kwon, S., Gaj, K., Kim, C., Hong, C.: Efficient linear array for multiplication in GF(2m) using a normal basis for elliptic curve cryptography. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 76–91. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Menezes, A.J., Blake, I.F., Gao, S., Mullin, R.C., Vanstone, S.A., Yaghoobian, T.: Applications of Finite Fields. Kluwer Academic Publishers, Dordrecht (1993)zbMATHGoogle Scholar
  7. 7.
    Berlekamp, E.R.: Bit-serial Reed-Solomon encoders. IEEE Trans. Inform. Theory 28, 869–874 (1982)CrossRefzbMATHGoogle Scholar
  8. 8.
    Wang, M., Blake, I.F.: Bit serial multiplication in finite fields. SIAM J. Disc. Math. 3, 140–148 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Morii, M., Kasahara, M., Whiting, D.L.: Efficient bit-serial multiplication and the discrete-time Wiener-Hopf equation over finite fields. IEEE Trans. Inform. Theory 35, 1177–1183 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fenn, S.T.J., Benaissa, M., Taylor, D.: GF(2m) multiplication and division over the dual basis. IEEE Trans. Computers 45, 319–327 (1996)CrossRefzbMATHGoogle Scholar
  11. 11.
    Stinson, D.R.: On bit-serial multiplication and dual bases in GF(2m). IEEE Trans. Inform. Theory 37, 1733–1736 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    NIST, Digital Signature Standard. FIPS Publication, 186-2 (February 2000)Google Scholar
  13. 13.
    Wu, H., Hasan, M.A., Blake, I.F., Gao, S.: Finite field multiplier using redundant representation. IEEE Trans. Computers 51, 1306–1316 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Feisel, S., von zur Gathen, J., Shokrollahi, M.: Normal bases via general Gauss periods. Math. Comp. 68, 271–290 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sunar, B., Koç, Ç.K.: An efficient optimal normal basis type II multiplier. IEEE Trans. Computers 50, 83–87 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handboook of Applied Cryptography. CRC Press, Boca Raton (1996)CrossRefGoogle Scholar
  17. 17.
    Zivkovic, M.: Table of primitive binary polynomials II. Math. Comp. 63, 301–306 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hankerson, D., Menezes, A.J., Vanstone, S.A.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  19. 19.
    Hankerson, D., Hernandez, J.L., Menezes, A.J.: Software implementation of elliptic curve cryptography over binary fields. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 1–24. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  20. 20.
    Song, L., Parhi, K.K.: Efficient finite field serial/parallel multiplication. In: International Conference on Application Specific Systems, Architectures and Processors, pp. 19–21 (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Soonhak Kwon
    • 1
  • Taekyoung Kwon
    • 2
  • Young-Ho Park
    • 3
  1. 1.Inst. of Basic Science and Dept. of MathematicsSungkyunkwan UniversitySuwonKorea
  2. 2.School of Computer EngineeringSejong UniversitySeoulKorea
  3. 3.Dept. of Information SecuritySejong Cyber UniversitySeoulKorea

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