Let be a separable polynomial with coefficients in a (large) finite prime field We study the distribution properties of the sets obtained by projecting the elliptic curve onto the x-axis, proving that according to a quasi-randomness criterion of Chung and Graham, are quasi-random subsets of Every can be naturally represented as a ?elliptic? walk of length p. We use chi-squared based statistical tests to show that, from the point of view of the number of returns to the origin, a certain class of initial segments of elliptic walks cannot be distinguished from genuine random walks.

elliptic curves, finite fields, Legendre symbol, hybrid sums quasi-randomness, random walks.