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# Certain Exponential Sums and Random Walks on Elliptic Curves

Published:2005-04-01
Printed: Apr 2005
• Tanja Lange
• Igor E. Shparlinski
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## Abstract

For a given elliptic curve $\E$, we obtain an upper bound on the discrepancy of sets of multiples $z_sG$ where $z_s$ runs through a sequence $\cZ=$z_1, \dots, z_T$$ such that $k z_1,\dots, kz_T$ is a permutation of $z_1, \dots, z_T$, 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.
 MSC Classifications: 11L07 - Estimates on exponential sums 11T23 - Exponential sums 11T71 - Algebraic coding theory; cryptography 14H52 - Elliptic curves [See also 11G05, 11G07, 14Kxx] 94A60 - Cryptography [See also 11T71, 14G50, 68P25, 81P94]