Exponential Sums Over Points of Elliptic Curves With Reciprocals of Primes

Abstract

Abstract. We consider exponential sums with x-coordinates of points qG and q −1 G where G is a point of order T on an elliptic curve modulo a prime p and q runs through all primes up to N (with gcd(q, T ) = 1 in the case of the points q −1 G). We obtain a new bound on exponential sums with q −1 G and correct an imprecision in the work of W. D. Banks, J. B. Friedlander, M. Z. Garaev and I. E. Shparlinski on exponential sums with qG. We also note that similar sums with g 1/q for an integer g with gcd(g, p) = 1 have been estimated by J. Bourgain and I. E. Shparlinski. §1. Introduction. Let p ≥ 5 be a prime and E be an elliptic curve defined over a finite field F p of p elements given by an affine Weierstraß equationwith some A, B ∈ F p , see [1,3,21].We recall that the set of all points on E forms an abelian group, with the "point at infinity" O as the neutral element, and we use ⊕ to denote the group operation. As usual, we write every point P = O on E as P = (x(P), y(P)).Let E(F p ) denote the set of F p -rational points on E. We recall that the celebrated result of Bombieri [4] implies in particular an estimate of order p 1/2 for exponential sums with functions from the function field of E taken over all points of E(F p ). More recently, various character sums over points of elliptic curves have been considered in a number of papers, see [2, 7, 9, 10, 14-16, 18, 20] and references therein; many of these estimates are motivated by applications to pseudorandom number generators on elliptic curves [19].Let G ∈ E(F p ) be a point of order T , in other words, T is the cardinality of the cyclic group G generated by G in E(F p ).We also denote e(z) = exp(2πiz) and e m (z) = e(z/m), and consider the sums S a (N ) = q≤N q prime gcd(q,T )=1

Exponential sums over points of elliptic curves

“…Let E(F q ) denote the set of F q -rational points on E. We recall that the celebrated result of Bombieri [6] implies, in particular, an estimate of order q 1/2 for exponential sums with functions from the function field of E taken over all points of E(F q ). More recently, various character sums over points of elliptic curves have been considered in a number of papers, see [1,3,8,12,13,17,18,19,25,26,28,30,32] and references therein. These estimates are motivated by various applications to such areas as • pseudorandom number generators from elliptic curves, see the most recent works [4,8,20,21,22,23] and also the survey [31];…”