Character Sums with Division Polynomials

http://dx.doi.org/10.4153/CMB-2011-126-x
Canad. Math. Bull. 55(2012), 850-857

  Published:2011-06-27
 Printed: Dec 2012

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.