Fast Finite Field Multiplication

  • Serdar Süer Erdem
  • Tuğrul Yanik
  • Çetin Kaya Koç


  1. 1.
    D. V. Bailey and C. Paar. Efficient arithmetic in finite field extensions with application in elliptic curve cryptography. Journal of Cryptology, 2000.Google Scholar
  2. 2.
    P. Barrett. Implementing the Rivest Shamir and Adleman public-key encryption algorithm on a standard digital signal processor. In A. M. Odlyzko, editor, Advances in Cryptology–-CRYPTO 86, Proceedings, Lecture Notes in Computer Science, vol. 263, pp. 311–323. Springer, Berlin, Germany, 1986.Google Scholar
  3. 3.
    E. Berlekamp. Algebraic Coding Theory. McGraw-Hill, New York, NY, 1968.MATHGoogle Scholar
  4. 4.
    R. Blahut. Theory and Practice of Error Control Codes. Addison-Wesley, Reading, MA, 1983.MATHGoogle Scholar
  5. 5.
    A. Bosselaers, R. Govaerts, and J. Vandewalle. Comparison of three modular reduction functions. In Crypto ’93, Lecture Notes in Computer Science, vol. 773, pp. 175–186, 1994.Google Scholar
  6. 6.
    M. Brown, D. Hankerson, J. López, and A. Menezes. Software implementation of the NIST elliptic curves over prime fields. Topics in Cryptology – CT-RSA 2001, Lecture Notes in Computer Science, vol. 2020, pp. 250–265, Springer, Berlin, Germany, 2001Google Scholar
  7. 7.
    J. F. Dhem. Efficient modular reduction algorithm in \(\mathcal{F}_q[x]\) and its application to “left to right” modular multiplication in \(\mathcal{F}_2[x]\). In C. D. Walter, editor, Cryptographic Hardware and Embedded Systems – CHES 2003, Lecture Notes in Computer Science, vol. 2779, pp. 203–213. Springer, Berlin, Germany, 2003.Google Scholar
  8. 8.
    W. Diffie and M. E. Hellman. New directions in cryptography. IEEE Transactions on Information Theory, 22:644–654, November 1976.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    IEEE P1363. Standard specifications for public-key cryptography.Google Scholar
  10. 10.
    D. E. Knuth. The Art of Computer Programming, Volume 2, Seminumerical Algorithms. Addison-Wesley, Reading, MA, Third edition, 1998.Google Scholar
  11. 11.
    N. Koblitz. Elliptic curve cryptosystems. Mathematics of Computation, 48(177):203–209, January 1987.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ç. K. Koç and T. Acar. Montgomery multiplication in GF\((2^k)\). Designs, Codes and Cryptography, 14(1):57–69, April 1998.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ç. K. Koç, T. Acar, and B. S. Kaliski Jr. Analyzing and comparing Montgomery multiplication algorithms. IEEE Micro, 16(3):26–33, June 1996.CrossRefGoogle Scholar
  14. 14.
    R. J. McEliece. Finite Fields for Computer Scientists and Engineers. Kluwer Academic Publishers, Boston, MA, Second edition, 1989.Google Scholar
  15. 15.
    A. Menezes, P. Van Oorschot, and S. Vanstone. Handbook of Applied Cryptography. CRC Press, Boca Raton, FL, 1997.MATHGoogle Scholar
  16. 16.
    A. J. Menezes, I. F. Blake, X. Gao, R. C. Mullen, S. A. Vanstone, and T. Yaghoobian. Applications of Finite Fields. Kluwer Academic Publishers, Boston, MA, 1993.MATHGoogle Scholar
  17. 17.
    V. Miller. Uses of elliptic curves in cryptography. In H. C. Williams, editor, Advances in Cryptology–-CRYPTO 85, Proceedings, Lecture Notes in Computer Science, No. 218, pp. 417–426. Springer, Berlin, Germany, 1985.Google Scholar
  18. 18.
    P. L. Montgomery. Modular multiplication without trial division. Mathematics of Computation, 44(170):519–521, April 1985.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    W. W. Peterson and E. J. Weldon Jr. Error-Correcting Codes. MIT Press, Cambridge, MA, 1972.MATHGoogle Scholar
  20. 20.
    J. Solinas. Generalized Mersenne numbers. Technical Report CORR 99-39, Dept. of C&O, University of Waterloo, 1999.Google Scholar
  21. 21.
    S. B. Wicker and V. K. Bhargava, editors. Reed-Solomon Codes and Their Applications. IEEE Press, New York, NY, 1994.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Serdar Süer Erdem
    • 1
  • Tuğrul Yanik
    • 2
  • Çetin Kaya Koç
    • 3
  1. 1.Gebze Institute of TechnologyGebze
  2. 2.Fatih UniversityIstanbul
  3. 3.City University of Istanbul & University of California Santa BarbaraSanta Barbara

Personalised recommendations