Character sums with division polynomials
Igor E Shparlinski, Katherine E. Stange

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 , , evaluated at a given point on an elliptic curve over a finite field of elements. Our bounds are nontrivial if the order of is at least for some fixed . 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.
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