Pseudorandom bits from points on elliptic curves

Let be an elliptic curve over a finite field of elements, with gcd(q, 6)= 1, given by an affine Weierstraß equation. We use x(P) to denote the x-component of a point P = (x(P), y(P)) ε E. We estimate character sums of the form ^NΣ _n=1X(x(nP)x(nQ)) and ^NΣ _n1,...,nk=1Π ( ^kΣj=1cjx(( ^jΠ _i=1ni)R)) on average over all F _qrational points P,Q, and R on E, where X is a quadratic character, φ is a nontrivial additive character in Fq, and (c ₁,...,ck) ε F ^k _qis a nonzero vector. These bounds confirm several recent conjectures of Jao, Jetchev, and Venkatesan, related to extracting random bits from various sequences of points on the elliptic curves.