The number of points on an elliptic curve with square x-coordinates

TL;DR

This paper investigates the distribution of square x-coordinates on a specific class of elliptic curves over finite fields, demonstrating that half of the x-coordinates are squares, thus extending previous results.

Contribution

It generalizes earlier work by proving that exactly half of the x-coordinates on certain elliptic curves over finite fields are squares.

Findings

Half of the x-coordinates are squares on the considered elliptic curves.

The result extends previous findings from 2009.

Provides insight into the distribution of points on elliptic curves over finite fields.

Abstract

Let K be a finite field. We know that a half of elements of K* is a square. So it is natural to ask how many of them appear as x-coordinate of points on an elliptic curve over K. We consider a specific class of elliptic curves over finite fields and show that a half of x-coordinate on an elliptic curve is a square. This result generalizes my old paper posted 30 Dec 2009.