Extractors for binary elliptic curves

We propose a simple and efficient deterministic extractor for an ordinary elliptic curve E, defined over double-struck F sign₂ⁿ, where n = 2ℓ and ℓ is a positive integer. Our extractor, for a given point P on E, outputs the first double-struck F sign₂^ℓ -coefficient of the abscissa of the point P. We also propose a deterministic extractor for the main subgroup G of E, where E has minimal 2-torsion. We show that if a point P is chosen uniformly at random in G, the bits extracted from the point P are indistinguishable from a uniformly random bit-string of length ℓ.