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Character Sums with Division Polynomials

Published online by Cambridge University Press:  20 November 2018

Igor E. Shparlinski  and
Katherine E. Stange
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, North Ryde, Sydney, NSW 2109, Australiae-mail: igor.shparlinski@mq.edu.au
Katherine E. Stange
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USAe-mail: stange@pims.math.ca
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Abstract

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We obtain nontrivial estimates of quadratic character sums of division polynomials ${{\text{ }\!\!\psi\!\!\text{ }}_{n}}\left( P \right)$, $n\,=\,1,\,2,\,\ldots $ , 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

11L4014H52division polynomial, character sum
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 4 , 01 December 2012 , pp. 850 - 857
Copyright
Copyright © Canadian Mathematical Society 2012

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