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International Conference on Finite Fields and Applications

Fq 2003: Finite Fields and Applications pp 197–214Cite as

Primitive Polynomials over Small Fields

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2948))

Abstract

Given a positive integer n, any root of a primitive polynomial x n + a 1 x n − 1 + ... + a n over the finite field \(\mathbb{F}_q\) of q elements (where q is a power of a prime p) is a primitive (generating) element of the extension \(\mathbb{F}_{q^n}\), by definition having multiplicative order q n – 1.

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References

  1. Cohen, S.D.: Primitive elements and polynomials with arbitrary trace. Discrete Math. 83, 1–7 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cohen, S.D.: Primitive elements and polynomials: existence results. In: Finite fields, coding theory, and advances in communications and computing. Lecture Notes in Pure and Appl. Math., vol. 141, pp. 43–55. Dekker, New York (1993)

    Google Scholar 

  3. Cohen, S.D., Mills, D.: Primitive polynomials with first and second coefficients prescribed. Finite Fields Appl. 9, 334–350 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fan, S.-Q., Han, W.-B.: p-adic formal series and primitive polynomials over finite fields. Proc. Amer. Math. Soc. 132, 15–31 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fan, S.-Q., Han, W.-B.: p-adic formal series and Cohen’s problem. Glasgow Math. J. (to appear)

    Google Scholar 

  6. Fan, S.-Q., Han, W.-B.: Character sums over Galois rings and primitive polynomials over finite fields. Finite Fields Appl. (to appear)

    Google Scholar 

  7. Han, W.-B.: The coefficients of primitive polynomials over finite fields. Math. Comp. 65, 331–340 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Han, W.-B.: On Cohen’s problem. In: Chinacrypt 1996, pp. 231–235. Academic Press, China (1996)

    Google Scholar 

  9. Han, W.-B.: On two exponential sums and their applications. Finite Fields Appl. 3, 115–130 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Han, W.-B.: The distribution of the coefficients of primitive polynomials over finite fields. In: Cryptography and computational number theory. Progr. Comput. Sci. Appl. Logic, vol. 20, pp. 43–57. Birkhuser, Basel (2001)

    Google Scholar 

  11. Jungnickel, D., Vanstone, S.A.: On primitive polynomials over finite fields. J. Algebra 124, 337–353 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Koblitz, N.: p-adic Numbers, p-adic Analysis, and Zeta-Functions. Springer, New York (1984)

    Google Scholar 

  13. Li, W.-C.W.: Character sums over p-adic fields. J. Number Theory 74, 181–229 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mills, D.: Existence of primitive polynomials with three coefficients prescribed. JP J. Algebra Number Theory Appl. (to appear)

    Google Scholar 

  15. Ren, D.-B.: On the coefficients of primitive polynomials over finite fields. Sichuan Daxue Xuebao 38, 33–36

    Google Scholar 

  16. Shparlinski, I.E.: On primitive polynomials. Prob. Peredachi Inform. 23, 100–103 (1988) (Russian)

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, Scotland

    Stephen D. Cohen

Authors
  1. Stephen D. Cohen
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Editor information

Editors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, 16802, PA, USA

    Gary L. Mullen

  2. University Paul Sabatier, AAECC/IRIT, 118 route de Narbonne, 31300, Toulouse cedex, France

    Alain Poli

  3. Sabancı University - FENS, Orhanli - Tuzla, 34956, Istanbul, Turkey

    Henning Stichtenoth

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© 2004 Springer-Verlag Berlin Heidelberg

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Cohen, S.D. (2004). Primitive Polynomials over Small Fields. In: Mullen, G.L., Poli, A., Stichtenoth, H. (eds) Finite Fields and Applications. Fq 2003. Lecture Notes in Computer Science, vol 2948. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24633-6_16

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  • DOI: https://doi.org/10.1007/978-3-540-24633-6_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21324-6

  • Online ISBN: 978-3-540-24633-6

  • eBook Packages: Springer Book Archive

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