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On a New Exponential Sum

Published online by Cambridge University Press:  20 November 2018

Daniel Lieman  and
Igor Shparlinski
Daniel Lieman
Affiliation:
Department of Mathematics University of Missouri Columbia, Missouri 65211 USA, email: [email protected]
Igor Shparlinski
Affiliation:
Department of Computing Macquarie University NSW 2109 Australia, email: [email protected]
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Abstract

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Let $p$ be prime and let $\vartheta \,\in \,\mathbb{Z}_{p}^{*}$ be of multiplicative order $t$ modulo $p$. We consider exponential sums of the form

$$S\left( a \right)\,=\,\sum\limits_{x=1}^{t}{\exp \left( 2\pi ia{{\vartheta }^{{{x}^{2}}}}\,/\,p \right)}$$

and prove that for any $\varepsilon \,>\,0$

$$\underset{\gcd (a,\,p)\,=\,1}{\mathop{\max }}\,\,\left| S\left( a \right) \right|\,=\,O\left( {{t}^{5/6+\varepsilon }}\,{{p}^{1/8}} \right)$$

Keywords

11L0711T2311B5011K3111K38
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 1 , 01 March 2001 , pp. 87 - 92
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Canetti, R., Friedlander, J. B., Konyagin, S., Larsen, M., Lieman, D. and Shparlinski, I. E., On the statistical properties of the Diffie-Hellman distribution. Israel J. Math., (to appear).Google Scholar
[2] Friedlander, J. B., Lieman, D. and Shparlinski, I. E., On the distribution of the RSA generator. Proc. Intern. Conf. on Sequences and their Applications (SETA ‘98), Singapore, (eds. C. Ding, T. Helleseth and H. Niederreiter), Springer-Verlag, London, 1999, 205–212.Google Scholar
[3] Friedlander, J. B. and Shparlinski, I. E., On the distribution of the Power generator. Math. Comp., to appear.Google Scholar
[4] Konyagin, S. and Shparlinski, I. E., Character sums with exponential functions and their applications. Cambridge Univ. Press, Cambridge, 1999.Google Scholar
[5] Korobov, N. M., On the distribution of digits in periodic fractions. Math. USSR-Sb. 18 (1972), 659–676.Google Scholar
[6] Korobov, N. M., Exponential sums and their applications. Kluwer Acad. Publ., Dordrecht, 1992.Google Scholar
[7] Niederreiter, H., Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Amer.Math. Soc. 84 (1978), 957–1041.Google Scholar
[8] Niederreiter, H., Random number generation and Quasi-Monte Carlo methods. SIAM Press, 1992.Google Scholar
[9] Prachar, K., Primzahlverteilung. Springer-Verlag, Berlin, 1957.Google Scholar