Some special character sums over elliptic curves

Let E(C _q) be the set of C _q-rational points on an elliptic curve E over a finite field C _q of q elements given by an affine Weierstraß equation. We use x(P) to denote the x-component of a point P = (x(P), y(P)), ⋯ E and for an integer n consider character sums S _n(a, b) = ∑ Ψ (ax(P) + bx(nP)) , a, b ⋯ C _q, P⋯E(C _q) with an additive character Ψ of C _q. In the case when gcd (n, #E(C _q)) is sufficiently large, we obtain a new bound for such sums. In particular, we show that for any positive integer n ⌊ # E(F _q), we have S _n(a, b) = O(q ⁹/ ¹⁰uniformly over n, a ⋯ C* _q and b ⋯ C _q.