Abstract
We show that Gauss periods of special type give an explicit polynomial-time construction of elements of exponentially large multiplicative order in some finite fields. It can be considered as a step towards solving the celebrated problem of finding primitive roots in finite fields in polynomial time.
Preview
Unable to display preview. Download preview PDF.
References
M. Clausen, A. Dress, J. Grabmeier and M. Karpinski, “On zero testing and interpolation of k-sparse multivariate polynomials over finite field”, Theor. Comp. Sci. 84 (1991), 151–164.
S. Gao, J. von zur Gathen and D. Panario, “Gauss periods and fast exponentiation in finite fields”, Proceedings LATIN '95, Springer Lecture Notes in Comp. Sci., 911 (1995), 311–322.
D. R. Heath-Brown, “Artin's conjecture for primitive roots”, Quart. J. Math., 37 (1986), 27–38.
C. Hooley, “On Artin's conjecture”, J. Reine Angew. Math., 225 (1967), 209–220.
N. M. Korobov, “Exponential sums with exponential functioms and the distribution of digits in periodic fractions”, Matem. Zametki, 8 (1970), 641–652 (in Russian).
N. M. Korobov, “On the distribution of digits in periodic fractions”, Matem. Sbornik, 89 (1972), 654–670 (in Russian).
H. W. Lenstra and C. Pomerance, “A rigorous time bound for factoring integers”, J. Amer. Math. Soc., 5 (1992), 483–516.
W. Narkiewicz, Classical problems in number theory, Polish Sci. Publ., Warszawa, 1986.
J.B. Rosser and L. Schoenfeld, “Approximate functions for some functions of prime numbers“, Illinois J. Math. 6 (1962) 64–94.
R. J. Schoof, “Quadratic fields and factorization”, Computational Methods in Number Theory, Amsterdam, 1984, 235–279.
D. Shanks, “Class number, a theory of factorization and genera”, Proc. Symp. in Pure Math., Amer. Math. Soc., Providence, 1971, 415–420.
I. Shparlinski Computational and algorithmic problems in finite fields, Kluwer Acad. Publ., Dordrecht, 1992.
I. Shparlinski, “On finding primitive roots in finite fields”, Theor. Comp. Sci. (to appear).
R. Tijdeman, “On the maximal distance between integers composed of small primes”, Compos. Math., 28 (1974), 159–162.
K. Werther, “The complexity of sparse polynomials interpolation over finite fields”, Appl. Algebra in Engin., Commun. and Comp., 5 (1994), 91–103.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
von zur Gathen, J., Shparlinski, I. (1995). Orders of Gauss periods in finite fields. In: Staples, J., Eades, P., Katoh, N., Moffat, A. (eds) Algorithms and Computations. ISAAC 1995. Lecture Notes in Computer Science, vol 1004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0015425
Download citation
DOI: https://doi.org/10.1007/BFb0015425
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60573-7
Online ISBN: 978-3-540-47766-2
eBook Packages: Springer Book Archive