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Gauß Periods in Finite Fields

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Finite Fields and Applications

Abstract

In this survey, we review two recent applications of a venerable tool: Gauß periods.

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References

  1. W. H. Bussey (1906).Galois field tables for p n ≤ 169. Bulletin of the American Mathematical Society 12, 22–38.

    Article  MathSciNet  Google Scholar 

  2. W. H. Bussey, Tables of Galois fields of order less than 1, 000. Bulletin of the American Mathematical Society 16 (1910), 188–206.

    Article  MATH  MathSciNet  Google Scholar 

  3. David G. Cantor, On arithmetical algorithms over finite fields. Journal of Combinatorial Theory Series A 50 (1989), 285–300.

    Article  MATH  MathSciNet  Google Scholar 

  4. August L. Grelle, Table des racines primitives etc. pour les nombres premiers depuis 3 jusqu’à 101, précédée d’une note sur le calcul de cette table. Journal für die Reine und Angewandte Mathematik 9 (1832), 27–53.

    Google Scholar 

  5. Sandra Feisel, Joachim von zur Gathen, and M. Amin Shokrollahi, Normal bases via general Gauβ periods. Mathematics of Computation 68 (225) (1999), 271–290.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Gao and H. W. Lenstra, Jr., Optimal normal bases. Designs, Codes, and Cryptography 2 (1992), 315–323.

    Article  MATH  MathSciNet  Google Scholar 

  7. SHuhong Gao, Joachim von zur Gathen, and Daniel Panario, Gauss periods and fast exponentiation in finite fields. In Proceedings of LATIN’ 95, Valparaíso, Chile, Lecture Notes in Computer Science 911. Springer-Verlag, 1995, 311–322.

    Google Scholar 

  8. Shuhong Gao, Joachim von zur Zgathen, Daniel Panario, and Victor Shoup, Algorithms for exponentiation in finite fields. Journal of Symbolic Computation 26(6) (2000), 879–889.

    Article  Google Scholar 

  9. Joachim von zur Gathen, Efficient and optimal exponentiation in finite fields, computational complexity 1 (1991), 360–394.

    Article  MATH  MathSciNet  Google Scholar 

  10. Joachim von zur Gathen and Michael Nöcker, Exponentiation in finite fields: Theory and practice. In Applied Algebra, Algebraic Algorithms and Error-CorrectingCodes: AAECC-12, Toulouse, Prance, ed. Teo Mora and Harold Mattson, Lecture Notes in Computer Science 1255. Springer-Verlag, 1997, 88–113.

    Google Scholar 

  11. Joachim von zur Gathen and Michael Nöcker, Normal bases, Gauss periods, and fast arithmetic. In Abstracts of the Fifth International Conference on Finite Fields and Applications, August 2–6, 1999, University of Augsburg, 1999, p. 70.

    Google Scholar 

  12. Joachim von zur Gathen & Francesco Pappalardi (2000). Density estimates for Gauβ periods. In Proc. Workshop on Cryptography and Computational Number Theory. Birkhäuser Verlag. To appear.

    Google Scholar 

  13. Joachim von zur Gathen and Victor Shoup, Computing Frobenius maps and factoring polynomials. computational complexity 2 (1992), 187–224.

    Article  MATH  MathSciNet  Google Scholar 

  14. Joachim von zur Gathen and Igor Shparlinski, Orders of Gauss periods in finite fields. Applicable Algebra in Engineering, Communication and Computing 9 (1998), 15–24.

    Article  MATH  MathSciNet  Google Scholar 

  15. Joachim Von zur Gathen and Igor Shparlinski, Constructing elements of large order in finite fields. In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: AAECC-13, Hawaii, Lecture Notes in Computer Science 1719, Springer-Verlag, 1999, 404–409.

    Google Scholar 

  16. Carl Friedrich Gauss, Disquisitiones Arithmeticae. Gerh. Fleischer Iun., Leipzig, 1801. English translation by Arthur A. Clarke, Springer-Verlag, New York, 1986.

    Google Scholar 

  17. D. R. Heath-Brown, Artin’s conjecture for primitive roots. Quarterly Journal of Mathematics 37 (1986), 27–38.

    Article  MATH  MathSciNet  Google Scholar 

  18. C. Hooley, On Artin’s conjecture. Journal für die Reine und Angewandte Mathematik 225 (1967), 209–220.

    MATH  MathSciNet  Google Scholar 

  19. C. G. J. Jacobi, Canon Arithmeticus sive tabulae quibus exhibentur pro singulis numeris primis vel primorum potestatibus infra 1000 numeri ad datos indices et indices ad datos numeros pertinentes. Typus Academicus, Berlin, 1839.

    Google Scholar 

  20. Nikolai M. Korobov, On the distribution of digits in periodic fractions. Math. USSR — Sbornik 18 (1972), 659–676. (In Russian).

    Article  Google Scholar 

  21. Nikolai M. Korobov, Exponential sums and their applications. Kluwer Acad. Publ., Dordrecht, 1992.

    MATH  Google Scholar 

  22. R. C. Mullin, I. M. Onyszchuk, S. A. Vanstone, and R. M. Wilson, Optimal normal bases in GF(p n). Discrete Applied Mathematics 22 (1989), 149–161.

    Article  MATH  MathSciNet  Google Scholar 

  23. K. Prachar, Primzahlverteilung. Springer-Verlag, Berlin, 1957.

    MATH  Google Scholar 

  24. A. Schönhage and V. Strassen, Schnelle Multiplikation groβer Zahlen. Computing 7 (1971), 281–292.

    Article  MATH  Google Scholar 

  25. Igor E. Shparlinski, On the dimension of BCH codes. Problemy Peredachi Inform. 25(1) (1988), 100–103. (In Russian).

    MathSciNet  Google Scholar 

  26. I. M. Vinogradov, Elements of number theory. Dover Publications, Inc., New York, 1954.

    MATH  Google Scholar 

  27. Alfred Wassermann, Konstruktion von Normalbasen. Bayreuther Math. Schriften 31 (1990), 155–164.

    MATH  MathSciNet  Google Scholar 

  28. Alfred Wassermann, Zur Arithmetik in endlichen Körpern. Bayreuther Math. Schriften 44 (1993), 147–251.

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Fachbereich 17 Mathematik-Informatik, Universität Paderborn, D-33095, Paderborn, Germany

    Joachim von zur Gathen

  2. Department of Computing, Macquarie University, Sydney, NSW, 2109, Australia

    Igor Shparlinski

Authors
  1. Joachim von zur Gathen
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  2. Igor Shparlinski
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Editor information

Editors and Affiliations

  1. Lehrstuhl für Diskrete Mathematik, Optimierung und Operations Research, Universität Augsburg, 86135, Augsburg, Germany

    Dieter Jungnickel

  2. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Harald Niederreiter

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© 2001 Springer-Verlag Berlin Heidelberg

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von zur Gathen, J., Shparlinski, I. (2001). Gauß Periods in Finite Fields. In: Jungnickel, D., Niederreiter, H. (eds) Finite Fields and Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56755-1_14

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  • DOI: https://doi.org/10.1007/978-3-642-56755-1_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62498-8

  • Online ISBN: 978-3-642-56755-1

  • eBook Packages: Springer Book Archive

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