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Certain Exponential Sums and Random Walks on Elliptic Curves

Published online by Cambridge University Press:  20 November 2018

Tanja Lange  and
Igor E. Shparlinski
Tanja Lange
Affiliation:
Institute for Information Security and Cryptology, Ruhr-University of Bochum, D-44780 Bochum, Germany, e-mail: lange@itsc.ruhr-uni-bochum.de
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia, e-mail: igor@comp.mq.edu.au
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Abstract

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For a given elliptic curve $\mathbf{E}$, we obtain an upper bound on the discrepancy of sets of multiples ${{z}_{s}}G$ where ${{z}_{s}}$ runs through a sequence $Z\,=\,\left( {{z}_{1}},\ldots ,{{z}_{T}} \right)$ such that $k{{z}_{1}},\ldots ,k{{z}_{T}}$ is a permutation of ${{z}_{1}},\ldots ,{{z}_{T}}$, both sequences taken modulo $t$, for sufficiently many distinct values of $k$ modulo $t$.

We apply this result to studying an analogue of the power generator over an elliptic curve. These results are elliptic curve analogues of those obtained for multiplicative groups of finite fields and residue rings.

Keywords

11L0711T2311T7114H5294A60
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 57 , Issue 2 , 01 April 2005 , pp. 338 - 350
Copyright
Copyright © Canadian Mathematical Society 2005

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