Gauss Period, Sparse Polynomial, Redundant Basis, and Efficient Exponentiation for a Class of Finite Fields with Small Characteristic

  • Soonhak Kwon
  • Chang Hoon Kim
  • Chun Pyo Hong
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2906)

Abstract

We present an efficient exponentiation algorithm in a finite field GF(q n ) using a Gauss period of type (n,1). Though the Gauss period α of type (n,1) in GF(q n ) is never primitive, a computational evidence says that there always exists a sparse polynomial (especially, a trinomial) of α which is a primitive element in GF(q n ). Our idea is easily generalized to the field determined by a root of unity over GF(q) with redundant basis technique. Consequently, we find primitive elements which yield a fast exponentiation algorithm for many finite fields GF(q n ), where a Gauss period of type (n,k) exists only for larger values of k or the existing Gauss period is not primitive and has large index in the multiplicative group GF(q n )×.

Keywords

Finite field Gauss period exponentiation root of unity trinomial redundant basis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brickell, E.F., Gordon, D.M., McCurley, K.S., Wilson, D.B.: Fast exponentiation with precomputation. In: Rueppel, R.A. (ed.) EUROCRYPT 1992. LNCS, vol. 658, pp. 200–207. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  2. 2.
    Tenenbaum, G.: Introduction to analytic and probabilistic number theory. Cambridge Univ. Press, Cambridge (1995)Google Scholar
  3. 3.
    Gao, S., von zur Gathen, J., Panario, D.: Gauss periods and fast exponentiation in finite fields. In: Baeza-Yates, R., Poblete, P.V., Goles, E. (eds.) LATIN 1995. LNCS, vol. 911, pp. 311–322. Springer, Heidelberg (1995)Google Scholar
  4. 4.
    Gao, S., von zur Gathen, J., Panario, D.: Orders and cryptographical applications. Math. Comp. 67, 343–352 (1998)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gao, S., Vanstone, S.: On orders of optimal normal basis generators. Math. Comp. 64, 1227–1233 (1995)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Lim, C.H., Lee, P.J.: More flexible exponentiation with precomputation. In: Desmedt, Y.G. (ed.) CRYPTO 1994. LNCS, vol. 839, pp. 95–107. Springer, Heidelberg (1994)Google Scholar
  7. 7.
    de Rooij, P.: Efficient exponentiation using precomputation and vector addition chains. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 389–399. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  8. 8.
    Kwon, S., Kim, C.H., Hong, C.P.: Efficient exponentiation for a class of finite fields GF(2n) determined by Gauss periods. In: Walter, C.D., Koç, Ç.K., Paar, C. (eds.) CHES 2003. LNCS, vol. 2779, pp. 228–242. Springer, Heidelberg (2003) (to appear)CrossRefGoogle Scholar
  9. 9.
    Menezes, A.J., Blake, I.F., Gao, S., Mullin, R.C., Vanstone, S.A., Yaghoobian, T.: Applications of finite fields. Kluwer Academic Publisher, Dordrecht (1993)MATHGoogle Scholar
  10. 10.
    Feisel, S., von zur Gathen, J., Shokrollahi, M.: Normal bases via general Gauss periods. Math. Comp. 68, 271–290 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    von zur Gathen, J., Shparlinski, I.: Constructing elements of large order in finite fields. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC 1999. LNCS, vol. 1719, pp. 404–409. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  12. 12.
    von zur Gathen, J., Nöcker, M.J.: Exponentiation in finite fields: Theory and Practice. In: Mattson, H.F., Mora, T. (eds.) AAECC 1997. LNCS, vol. 1255, pp. 88–133. Springer, Heidelberg (1997)Google Scholar
  13. 13.
    von zur Gathen, J., Shparlinski, I.: Orders of Gauss periods in finite fields. In: Staples, J., Katoh, N., Eades, P., Moffat, A. (eds.) ISAAC 1995. LNCS, vol. 1004, pp. 208–215. Springer, Heidelberg (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Soonhak Kwon
    • 1
  • Chang Hoon Kim
    • 2
  • Chun Pyo Hong
    • 2
  1. 1.Inst. of Basic Science and Dept. of MathematicsSungkyunkwan UniversitySuwonKorea
  2. 2.Dept. of Computer and Information EngineeringDaegu UniversityKyungsanKorea

Personalised recommendations