Acta Applicandae Mathematica

, Volume 93, Issue 1–3, pp 33–55 | Cite as

Polynomial Basis Multiplication over GF(2m)

  • Serdar S. Erdem
  • Tuğrul Yanık
  • Çetin K. Koç
Article

Abstract

In this paper, we describe, analyze and compare various \(GF(2^m)\) multipliers. Particularly, we investigate the standard modular multiplication, the Montgomery multiplication, and the matrix–vector multiplication techniques.

Mathematics Subject Classifications (2000)

12-XX 68-XX 94-XX 

Key words

finite fields binary fields computer arithmetic modular multiplication modular reduction 

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References

  1. 1.
    Halbutoğulları, A., Koç, Ç.K.: Mastrovito multipliers for general irreducible polynomials. IEEE Trans. Comput. 49(5), 503–518 (May 2000)CrossRefGoogle Scholar
  2. 2.
    Hasan, M.A., Wang, M.Z., Bhargava, V.K.: Modular construction of low complexity parallel multipliers for a class of finite fields GF(2\(^{m}\)). IEEE Trans. Comput. 41(8), 962–971 (May 1992)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Itoh, T., Tsujii, S.: Structure of parallel multipliers for a class of finite fields GF(2\(^{m}\)). Inform. and Comput. 83(8), 21–40 (1989)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Koç, Ç.K., Acar, T.: Montgomery multiplication in \(GF(2^{k})\). Des. Codes Cryptogr. 14(1), 57–69 (April 1998)CrossRefGoogle Scholar
  5. 5.
    Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, UK (1994)Google Scholar
  6. 6.
    Mastrovito, E.D.: VLSI designs for multiplication over finite fields over GF(2\(^{m}\)). In: Proc. Sixth Int’l Conf. Applied Algebra, Algebraic Algorithms, and Error Correcting Codes (AAECC-6), pp. 297–309, July 1988Google Scholar
  7. 7.
    Menezes, A.J.: Applications of Finite Fields. Kluwer, Massachusetts (1994)Google Scholar
  8. 8.
    Menezes, A.J.: Handbook of Applied Cryptography. CRC, Boca Raton, Florida (1997)Google Scholar
  9. 9.
    Reyhani-Masoleh, A., Hasan, M.A.: Low complexity bit-parallel architectures for polynomial basis multiplication over GF(\(2^{m}\)). IEEE Trans. Comput. 53(8), 945–959 (August 2004)CrossRefGoogle Scholar
  10. 10.
    Sunar, B., Koç, Ç.K.: Mastrovito multiplier for all trinomials. IEEE Trans. Comput. 48(5), 522–527 (May 1999)CrossRefGoogle Scholar
  11. 11.
    Wu, H., Hasan, M.A.: Low-complexity bit-parallel multipliers for a class of finite fields. IEEE Trans. Comput. 47(8), 883–887 (Aug. 1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wu, H.: Montgomery multiplier and squarer for a class of finite fields. IEEE Trans. Comput. 51(5), 521–529 (May 2002)CrossRefGoogle Scholar
  13. 13.
    Wu, H.: Bit-parallel finite field multiplier and squarer using polynomial basis. IEEE Trans. Comput. 51(7), 750–758 (July 2002)Google Scholar
  14. 14.
    Zhang, T., Parhi, K.K.: Systematic design of original and modified Mastrovito multipliers for general irreducible polynomials. IEEE Trans. Comput. 50(7), 734–749 (July 2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • Serdar S. Erdem
    • 1
  • Tuğrul Yanık
    • 2
  • Çetin K. Koç
    • 3
  1. 1.Electronics Engineering DepartmentGebze Institute of TechnologyGebzeTurkey
  2. 2.Computer Engineering DepartmentFatih UniversityIstanbulTurkey
  3. 3.Information Security Research CenterIstanbul Commerce UniversityIstanbulTurkey

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