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)


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 )×.


Finite field Gauss period exponentiation root of unity trinomial redundant basis 


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© 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

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