Some Remarks on Arithmetical Properties of Recursive Sequences on Elliptic Curves over a Finite Field

In connection with problems of information theory, we study arithmetical progressions constructed at the points of elliptic curves over a finite field. For certain types of such curves, we establish the distribution of the quadratic residues at the $x$-coordinates of the sequence of points corresponding to progressions if the elliptic curves is defined over a simple field. A description of the set of all progressions on elliptic curves over a finite field is also given.