## Abstract

Let E be an elliptic curve over a finite field F_{q} of q elements given by an affine Weierstraß equation. Let ⊕ denote the group operation in the Abelian group of points on E. We also use x (P) to denote the x-component of a point P = (x (P), y (P)) ∈ E. We estimate character sumsW_{ρ, θ{symbol}} (ψ, U, V) = under(∑, U ∈ U) under(∑, V ∈ V) ρ (U) θ{symbol} (V) ψ (x (U ⊕ V)), where U and V are arbitrary sets of F_{q}-rational points on E, ψ is a nontrivial additive character of F_{q} and ρ (U) and θ{symbol} (V) are arbitrary bounded complex functions supported on U and V, respectively. Our bound of sums W_{ρ, θ{symbol}} (ψ, U, V) is nontrivial whenever# U > q^{1 / 2 + ε} and # V > q^{ε} for some fixed ε > 0. We also give various applications of this bound.

Original language | English |
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Pages (from-to) | 132-141 |

Number of pages | 10 |

Journal | Finite Fields and their Applications |

Volume | 14 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2008 |