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Character sums with exponential functions

Part of: Exponential sums and character sums

Published online by Cambridge University Press:  26 February 2010

John B. Friedlander ,
Jan Hansen  and
Igor E. Shparlinski
John B. Friedlander
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada. E-mail: frdlndr@math.toronto.edu
Jan Hansen
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia. E-mail: jhansen@ics.mq.edu.au
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia. E-mail: igor@ics.mq.edu.au
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Abstract

Let ϑ be an integer of multiplicative order t≥1 modulo a prime p. Sums of the form

are introduced and estimated, with a sequence such that kz1, …, kzT is a permutation of z1, …, zT, both sequences taken modulo t, for sufficiently many distinct modulo t values of k. Such sequences include

xn for x = 1 ,…,t with an integer n≥1;

xn for x = 1 ,…,t and gcd (x, t) = 1 with an integer n≥1;

ex for x = 1 ,…,T with an integer e, where T is the period of the sequence ex modulo t.

Some of the results can be extended to composite moduli and to sums of multiplicative characters as well. Character sums with the above sequences have some cryptographic motivation and applications and have been considered in several papers by J. B. Friedlander, D. Lieman and I. E. Shparlinski. In particular several previous bounds are generalized and improved.

MSC classification

Secondary: 11L07: Estimates on exponential sums
Type
Research Article
Information
Mathematika , Volume 47 , Issue 1-2 , December 2000 , pp. 75 - 85
Copyright
Copyright © University College London 2000

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References

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