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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
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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 division polynomial, character sum
MSC Classifications: 11L40, 14H52 show english descriptions Estimates on character sums
Elliptic curves [See also 11G05, 11G07, 14Kxx]
11L40 - Estimates on character sums
14H52 - Elliptic curves [See also 11G05, 11G07, 14Kxx]
 

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