Bilinear character sums over elliptic curves

Igor Shparlinski*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


Let E be an elliptic curve over a finite field Fq 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 sumsWρ, 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 Fq-rational points on E, ψ is a nontrivial additive character of Fq and ρ (U) and θ{symbol} (V) are arbitrary bounded complex functions supported on U and V, respectively. Our bound of sums Wρ, θ{symbol} (ψ, U, V) is nontrivial whenever# U > q1 / 2 + ε and # V > qε for some fixed ε > 0. We also give various applications of this bound.

Original languageEnglish
Pages (from-to)132-141
Number of pages10
JournalFinite Fields and their Applications
Issue number1
Publication statusPublished - Jan 2008


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