Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-28T00:39:34.968Z Has data issue: false hasContentIssue false

SUM–PRODUCT ESTIMATES AND MULTIPLICATIVE ORDERS OF γ AND γ+γ−1 IN FINITE FIELDS

Published online by Cambridge University Press:  30 November 2011

IGOR SHPARLINSKI*
Affiliation:
Department of Computing, Macquarie University, NSW 2109, Australia (email: igor.shparlinski@mq.edu.au)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using a recent result on the sum–product problem, we estimate the number of elements γ in a prime finite field such that both γ and γ+γ−1 are of small order.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Ahmadi, O., Shparlinski, I. and Voloch, J. F., ‘Multiplicative order of Gauss periods’, Int. J. Number Theory 6 (2010), 877882.CrossRefGoogle Scholar
[2]Blake, I. F., Gao, S., Menezes, A. J., Mullin, R., Vanstone, S. and Yaghoobian, T., Applications of Finite Fields (Kluwer Academic Press, Dordrecht, 1993).Google Scholar
[3]Bourgain, J. and Garaev, M. Z., ‘On a variant of sum-product estimates and explicit exponential sum bounds in prime fields’, Math. Proc. Cambridge Philos Soc. 146 (2008), 121.Google Scholar
[4]Cohen, S. D., ‘The orders of related elements of a finite field’, Ramanujan J. 7 (2003), 169183.Google Scholar
[5]Garaev, M. Z., ‘Sums and products of sets and estimates for rational trigonometric sums in fields of prime order’, Russian Math. Surveys 65 (2010), 599658.CrossRefGoogle Scholar
[6]von zur Gathen, J. and Shparlinski, I., ‘Orders of Gauss periods in finite fields’, Appl. Algebra Engrg. Comm. Comput. 9 (1998), 1524.Google Scholar
[7]von zur Gathen, J. and Shparlinski, I., ‘Constructing elements of large order in finite fields and Gauss periods’, Proc. the 13th Symp. on Appl. Algebra, Algebraic Algorithms, and Error-Correcting Codes, Honolulu, HI, 1999, Lecture Notes in Computer Science, 1719 (Springer, Berlin, 1999), pp. 404–497.CrossRefGoogle Scholar
[8]von zur Gathen, J. and Shparlinski, I., ‘Gauss periods in finite fields’, Proc. 5th Conference of Finite Fields and their Applications, Augsburg, 1999 (Springer, Berlin, 2001), pp. 162–177.CrossRefGoogle Scholar
[9]Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (Oxford University Press, Oxford, 1979).Google Scholar
[10]Rudnev, M., ‘An improved sum-product inequality in fields of prime order’, Preprint, 2010. arXiv:1011.2738.Google Scholar
[11]Shparlinski, I., ‘On the multiplicative orders of γ and γ+γ −1 over finite fields’, Finite Fields Appl. 7 (2001), 327331.CrossRefGoogle Scholar
[12]Tao, T., ‘The sum-product phenomenon in arbitrary rings’, Contrib. Discrete Math. 4 (2009), 5982.Google Scholar
[13]Voloch, J. F., ‘On the order of points on curves over finite fields’, Integers 7 (2007), A49.Google Scholar
[14]Voloch, J. F., ‘Elements of high order on finite fields from elliptic curves’, Bull. Aust. Math. Soc. 81 (2010), 425429.Google Scholar