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# Character Sums with Division Polynomials

Published:2011-06-27
Printed: Dec 2012
• Igor E. Shparlinski,
Department of Computing, Macquarie University, North Ryde, Sydney, NSW 2109, Australia
• Katherine E. Stange,
Department of Mathematics, Stanford University, Stanford, CA 94305, USA
 Format: LaTeX MathJax PDF

## Abstract

We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, \dots$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \varepsilon}$ for some fixed $\varepsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author.
 Keywords: division polynomial, character sum
 MSC Classifications: 11L40 - Estimates on character sums 14H52 - Elliptic curves [See also 11G05, 11G07, 14Kxx]

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