Character sums with division polynomials

TL;DR

This paper provides new bounds for quadratic character sums of division polynomials evaluated at points on elliptic curves over finite fields, with implications for cryptographic sequence analysis.

Contribution

It introduces nontrivial estimates for character sums of division polynomials on elliptic curves, addressing an open problem related to cryptographic sequence indistinguishability.

Findings

Bounds are nontrivial when the point's order exceeds q^{1/2 + ε}.

Results contribute to understanding cryptographic sequence properties.

Addresses an open question by Lauter and the second author.

Abstract

We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, ...$, 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 $q^{1/2 + \epsilon}$ for some fixed $\epsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences which has recently been brought up by K. Lauter and the second author.