Pseudorandom Bits From Points on Elliptic Curves

Let E be an elliptic curve over a finite field F
_q
of q 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)) ∈ E. We estimate character sums of the form Σ
_n=1
^N
χ(x(nP)x(nQ)) and Σ
_n1,⋯,nk=1
^N
ψ(Σ
_j=1
^k
c
_j
x((Π
_i=1
^j
n
_i
)R)) on average over all F
_q
rational points P, Q, and R on E, where χ is a quadratic character, ψ is a nontrivial additive character in F
_q
, and (c
₁
,..., c
_k
) ∈ F
_q
^k
is 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.