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Certain Exponential Sums and Random Walks on Elliptic Curves

Open Access article
 Printed: Apr 2005
  • Tanja Lange
  • Igor E. Shparlinski
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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, 11T23, 11T71, 14H52, 94A60 show english descriptions Estimates on exponential sums
Exponential sums
Algebraic coding theory; cryptography
Elliptic curves [See also 11G05, 11G07, 14Kxx]
Cryptography [See also 11T71, 14G50, 68P25, 81P94]
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]

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